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Fluid FieldsA Wikibook

ContentsArticles

Navier–Stokes equations 1Incompressible flow 12Compressible flow 14Isochoric process 18Compressibility factor 20Solenoidal vector field 24Conservative vector field 25Laplacian vector field 28Stokes' law 29Projection method (fluid dynamics) 32Mach number 34Viscosity 38Rheology 56

ReferencesArticle Sources and Contributors 66Image Sources, Licenses and Contributors 68

Article LicensesLicense 69

Navier–Stokes equations 1

Navier–Stokes equationsIn physics the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe themotion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together withthe assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity),plus a pressure term.The equations are useful because they describe the physics of many things of academic and economic interest. Theymay be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. TheNavier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study ofblood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell'sequations they can be used to model and study magnetohydrodynamics.The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, giventheir wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions alwaysexist (existence), or that if they do exist, then they do not contain any singularity (smoothness). These are called theNavier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the sevenmost important open problems in mathematics and has offered a US$1,000,000 prize for a solution or acounter-example.[1]

The Navier–Stokes equations dictate not position but rather velocity. A solution of the Navier–Stokes equations iscalled a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space andtime. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found.This is different from what one normally sees in classical mechanics, where solutions are typically trajectories ofposition of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for afluid; however for visualization purposes one can compute various trajectories.

Properties

NonlinearityThe Navier–Stokes equations are nonlinear partial differential equations in almost every real situation. In somecases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linearequations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to theturbulence that the equations model.The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocityover position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example ofconvective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a smallconverging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.

Navier–Stokes equations 2

TurbulenceTurbulence is the time dependent chaotic behavior seen in many fluid flows. It is generally believed that it is due tothe inertia of the fluid as a whole: the culmination of time dependent and convective acceleration; hence flows whereinertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected byinertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulenceproperly.The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to thesignificantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requiressuch a fine mesh resolution that the computational time becomes significantly infeasible for calculation (see Directnumerical simulation). Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteadysolution, which fails to converge appropriately. To counter this, time-averaged equations such as theReynolds-averaged Navier-Stokes equations (RANS), supplemented with turbulence models (such as the k-ε model),are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Anothertechnique for solving numerically the Navier–Stokes equation is the Large eddy simulation (LES). This approach iscomputationally more expensive than the RANS method (in time and computer memory), but produces better resultssince the larger turbulent scales are explicitly resolved.

ApplicabilityTogether with supplemental equations (for example, conservation of mass) and well formulated boundary conditions,the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agreewith real world observations.The Navier–Stokes equations assume that the fluid being studied is a continuum not moving at relativistic velocities.At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce resultsdifferent from the continuous fluids modeled by the Navier–Stokes equations. Depending on the Knudsen number ofthe problem, statistical mechanics or possibly even molecular dynamics may be a more appropriate approach.Another limitation is very simply the complicated nature of the equations. Time tested formulations exist forcommon fluid families, but the application of the Navier–Stokes equations to less common families tends to result invery complicated formulations which are an area of current research. For this reason, these equations are usuallywritten for Newtonian fluids. Studying such fluids is "simple" because the viscosity model ends up being linear; trulygeneral models for the flow of other kinds of fluids (such as blood) do not, as of 2011, exist.

Derivation and descriptionThe derivation of the Navier–Stokes equations begins with an application of Newton's second law: conservation ofmomentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In aninertial frame of reference, the general form of the equations of fluid motion is:[2]

where is the flow velocity, is the fluid density, p is the pressure, is the (deviatoric) stress tensor, and represents body forces (per unit volume) acting on the fluid and is the del operator. This is a statement of theconservation of momentum in a fluid and it is an application of Newton's second law to a continuum; in fact thisequation is applicable to any non-relativistic continuum and is known as the Cauchy momentum equation.This equation is often written using the substantive derivative Dv/Dt, making it more apparent that this is a statementof Newton's second law:

Navier–Stokes equations 3

The left side of the equation describes acceleration, and may be composed of time dependent or convective effects(also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation ofbody forces (such as gravity) and divergence of stress (pressure and shear stress).

Convective acceleration

An example of convection. Though the flow maybe steady (time independent), the fluid

decelerates as it moves down the diverging duct(assuming incompressible flow), hence there is an

acceleration happening over position.

A very significant feature of the Navier–Stokes equations is thepresence of convective acceleration: the effect of time independentacceleration of a fluid with respect to space. While individual fluidparticles are indeed experiencing time dependent acceleration, theconvective acceleration of the flow field is a spatial effect, oneexample being fluid speeding up in a nozzle. Convective accelerationis represented by the nonlinear quantity:

which may be interpreted either as or as with the tensor derivative of the velocityvector Both interpretations give the same result, independent of the coordinate system — provided isinterpreted as the covariant derivative.[3]

Interpretation as (v·∇)v

The convection term is often written as

where the advection operator is used. Usually this representation is preferred because it is simpler than theone in terms of the tensor derivative [3]

Interpretation as v·(∇v)

Here is the tensor derivative of the velocity vector, equal in Cartesian coordinates to the component bycomponent gradient. The convection term may, by a vector calculus identity, be expressed without a tensorderivative:[4] [5]

The form has use in irrotational flow, where the curl of the velocity (called vorticity) is equal to zero.Regardless of what kind of fluid is being dealt with, convective acceleration is a nonlinear effect. Convectiveacceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamiceffect is disregarded in creeping flow (also called Stokes flow) .

Navier–Stokes equations 4

StressesThe effect of stress in the fluid is represented by the and terms; these are gradients of surface forces,analogous to stresses in a solid. is called the pressure gradient and arises from the isotropic part of the stresstensor. This part is given by normal stresses that turn up in almost all situations, dynamic or not. The anisotropic partof the stress tensor gives rise to , which conventionally describes viscous forces; for incompressible flow, this isonly a shear effect. Thus, is the deviatoric stress tensor, and the stress tensor is equal to:[6]

where is the 3×3 identity matrix. Interestingly, only the gradient of pressure matters, not the pressure itself. Theeffect of the pressure gradient is that fluid flows from high pressure to low pressure.The stress terms p and are yet unknown, so the general form of the equations of motion is not usable to solveproblems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses tothe fluid motion.[7] For this reason, assumptions on the specific behavior of a fluid are made (based on naturalobservations) and applied in order to specify the stresses in terms of the other flow variables, such as velocity anddensity.The Navier–Stokes equations result from the following assumptions on the deviatoric stress tensor :[8]

• the deviatoric stress vanishes for a fluid at rest, and – by Galilean invariance – also does not depend directly onthe flow velocity itself, but only on spatial derivatives of the flow velocity

• in the Navier–Stokes equations, the deviatoric stress is expressed as the product of the tensor gradient of theflow velocity with a viscosity tensor , i.e. :

• the fluid is assumed to be isotropic, as valid for gases and simple liquids, and consequently is an isotropictensor; furthermore, since the deviatoric stress tensor is symmetric, it turns out that it can be expressed in terms oftwo scalar dynamic viscosities μ and μ”: where is the

rate-of-strain tensor and is the rate of expansion of the flow• the deviatoric stress tensor has zero trace, so for a three-dimensional flow 2μ + 3μ” = 0As a result, in the Navier–Stokes equations the deviatoric stress tensor has the following form:[8]

with the quantity between brackets the non-isotropic part of the rate-of-strain tensor The dynamic viscosity μ doesnot need to be constant – in general it depends on conditions like temperature and pressure, and in turbulencemodelling the concept of eddy viscosity is used to approximate the average deviatoric stress.The pressure p is modelled by use of an equation of state.[9] For the special case of an incompressible flow, thepressure constrains the flow in such a way that the volume of fluid elements is constant: isochoric flow resulting in asolenoidal velocity field with [10]

Other forcesThe vector field represents body forces. Typically these consist of only gravity forces, but may include othertypes(such as electromagnetic forces). In a non-inertial coordinate system, other "forces" such as that associated withrotating coordinates may be inserted.Often, these forces may be represented as the gradient of some scalar quantity. Gravity in the z direction, forexample, is the gradient of . Since pressure shows up only as a gradient, this implies that solving a problemwithout any such body force can be mended to include the body force by modifying pressure.

Navier–Stokes equations 5

Other equationsThe Navier–Stokes equations are strictly a statement of the conservation of momentum. In order to fully describefluid flow, more information is needed (how much depends on the assumptions made), this may include boundarydata (no-slip, capillary surface, etc.), the conservation of mass, the conservation of energy, and/or an equation ofstate.Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achievedthrough the mass continuity equation, given in its most general form as:

or, using the substantive derivative:

Incompressible flow of Newtonian fluidsA simplification of the resulting flow equations is obtained when considering an incompressible flow of a Newtonianfluid. The assumption of incompressibility rules out the possibility of sound or shock waves to occur; so thissimplification is invalid if these phenomena are important. The incompressible flow assumption typically holds welleven when dealing with a "compressible" fluid — such as air at room temperature — at low Mach numbers (evenwhen flowing up to about Mach 0.3). Taking the incompressible flow assumption into account and assumingconstant viscosity, the Navier–Stokes equations will read, in vector form:[11]

Here f represents "other" body forces (forces per unit volume), such as gravity or centrifugal force. The shear stressterm becomes the useful quantity ( is the vector Laplacian) when the fluid is assumed incompressible,homogeneous and Newtonian, where is the (constant) dynamic viscosity.[12]

It's well worth observing the meaning of each term (compare to the Cauchy momentum equation):

Note that only the convective terms are nonlinear for incompressible Newtonian flow. The convective acceleration isan acceleration caused by a (possibly steady) change in velocity over position, for example the speeding up of fluidentering a converging nozzle. Though individual fluid particles are being accelerated and thus are under unsteadymotion, the flow field (a velocity distribution) will not necessarily be time dependent.Another important observation is that the viscosity is represented by the vector Laplacian of the velocity field(interpreted here as the difference between the velocity at a point and the mean velocity in a small volume around).This implies that Newtonian viscosity is diffusion of momentum, this works in much the same way as the diffusionof heat seen in the heat equation (which also involves the Laplacian).If temperature effects are also neglected, the only "other" equation (apart from initial/boundary conditions) needed isthe mass continuity equation. Under the incompressible assumption, density is a constant and it follows that theequation will simplify to:

This is more specifically a statement of the conservation of volume (see divergence).

Navier–Stokes equations 6

These equations are commonly used in 3 coordinates systems: Cartesian, cylindrical, and spherical. While theCartesian equations seem to follow directly from the vector equation above, the vector form of the Navier–Stokesequation involves some tensor calculus which means that writing it in other coordinate systems is not as simple asdoing so for scalar equations (such as the heat equation).

Cartesian coordinatesWriting the vector equation explicitly,

Note that gravity has been accounted for as a body force, and the values of will depend on the orientationof gravity with respect to the chosen set of coordinates.The continuity equation reads:

When the flow is at steady-state, does not change with respect to time. The continuity equation is reduced to:

When the flow is incompressible, is constant and does not change with respect to space. The continuity equationis reduced to:

The velocity components (the dependent variables to be solved for) are typically named u, v, w. This system of fourequations comprises the most commonly used and studied form. Though comparatively more compact than otherrepresentations, this is still a nonlinear system of partial differential equations for which solutions are difficult toobtain.

Cylindrical coordinates

A change of variables on the Cartesian equations will yield[11] the following momentum equations for r, , and z:

The gravity components will generally not be constants, however for most applications either the coordinates arechosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressurefield (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressuregradient). The continuity equation is:

Navier–Stokes equations 7

This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen(the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that avelocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangentialvelocity ( ), and the remaining quantities are independent of :

Spherical coordinates

In spherical coordinates, the r, , and momentum equations are[11] (note the convention used: iscolatitude[13] ):

Mass continuity will read:

These equations could be (slightly) compacted by, for example, factoring from the viscous terms. However,doing so would undesirably alter the structure of the Laplacian and other quantities.

Navier–Stokes equations 8

Stream function formulationTaking the curl of the Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if2D Cartesian flow is assumed ( and no dependence of anything on z), where the equations reduce to:

Differentiating the first with respect to y, the second with respect to x and subtracting the resulting equations willeliminate pressure and any conservative force. Defining the stream function through

results in mass continuity being unconditionally satisfied (given the stream function is continuous), and thenincompressible Newtonian 2D momentum and mass conservation degrade into one equation:

where is the (2D) biharmonic operator and is the kinematic viscosity, . We can also express this

compactly using the Jacobian determinant:

This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematicviscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero.In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describethe velocity components of an incompressible flow with one scalar function.

Compressible flow of Newtonian fluidsThere are some phenomena that are closely linked with fluid compressibility. One of the obvious examples is sound.Description of such phenomena requires more general presentation of the Navier–Stokes equation that takes intoaccount fluid compressibility. If viscosity is assumed a constant, one additional term appears, as shown here:[14] [15]

where is the volume viscosity coefficient, also known as second viscosity coefficient or bulk viscosity. Thisadditional term disappears for an incompressible fluid, when the divergence of the flow equals zero.

Navier–Stokes equations 9

Application to specific problemsThe Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and theirproper application to specific problems can be very diverse. This is partly because there is an enormous variety ofproblems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated asmultiphase flow driven by surface tension.Generally, application to specific problems begins with some flow assumptions and initial/boundary conditionformulation, this may be followed by scale analysis to further simplify the problem. For example, after assumingsteady, parallel, one dimensional, nonconvective pressure driven flow between parallel plates, the resulting scaled(dimensionless) boundary value problem is:

Visualization of a) parallel flow and b) radial flow.

The boundary condition is the no slip condition. This problem is easily solved for the flow field:

From this point onward more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on theparallel flow above would be the radial flow between parallel plates; this involves convection and thus nonlinearity.The velocity field may be represented by a function that must satisfy:

Navier–Stokes equations 10

This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flowassumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficultproblem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots ofcubic polynomials). Issues with the actual existence of solutions arise for R > 1.41 (approximately. This is not thesquare root of two), the parameter R being the Reynolds number with appropriately chosen scales. This is anexample of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds numberflows.

Exact solutions of the Navier–Stokes equationsSome exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases — with the non-linearterms in the Navier–Stokes equations equal to zero — are Poiseuille flow, Couette flow and the oscillatory Stokesboundary layer. But also more interesting examples, solutions to the full non-linear equations, exist; for example theTaylor–Green vortex.[16] [17] [18] Note that the existence of these exact solutions does not imply they are stable:turbulence may develop at higher Reynolds numbers.

Wyld diagramsWyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via a perturbationexpansion of the fundamental continuum mechanics. Similar to the Feynman diagrams in quantum field theory, thesediagrams are an extension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words,these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated andinteracting fluid particles to obey stochastic processes associated to pseudo-random functions in probabilitydistributions.[19]

See also• Boltzmann equation• Churchill–Bernstein equation• Coandă effect• Computational fluid dynamics• Fokker–Planck equation• Large eddy simulation• Mach number• Navier–Stokes existence and smoothness — one of the Millennium Prize Problems as stated by the Clay

Mathematics Institute• Multiphase flow• Reynolds transport theorem• Reynolds-averaged Navier–Stokes equations• Adhémar Jean Claude Barré de Saint-Venant• Vlasov equation

Navier–Stokes equations 11

Notes[1] Millennium Prize Problems (http:/ / www. claymath. org/ millennium/ ), Clay Mathematics Institute, , retrieved 2009-04-11[2] Batchelor (1967) pp. 137 & 142.[3] Emanuel, G. (2001), Analytical fluid dynamics (second ed.), CRC Press, ISBN 0849391148 pp. 6–7.[4] See Batchelor (1967), §3.5, p. 160.[5] Eric W. Weisstein, Convective Derivative (http:/ / mathworld. wolfram. com/ ConvectiveDerivative. html), MathWorld, , retrieved

2008-05-20[6] Batchelor (1967) p. 142.[7] Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963), The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley,

ISBN 0-201-02116-1, Vol. 1, §9–4 and §12–1.[8] Batchelor (1967) pp. 142–148.[9] Batchelor (1967) p. 165.[10] Batchelor (1967) p. 75.[11] See Acheson (1990).[12] Batchelor (1967) pp. 21 & 147.[13] Eric W. Weisstein (2005-10-26), Spherical Coordinates (http:/ / mathworld. wolfram. com/ SphericalCoordinates. html), MathWorld, ,

retrieved 2008-01-22[14] Landau & Lifshitz (1987) pp. 44–45.[15] Batchelor (1967) pp. 147 & 154.[16] Wang, C.Y. (1991), "Exact solutions of the steady-state Navier–Stokes equations", Annual Review of Fluid Mechanics 23: 159–177,

doi:10.1146/annurev.fl.23.010191.001111[17] Landau & Lifshitz (1987) pp. 75–88.[18] Ethier, C.R.; Steinman, D.A. (1994), "Exact fully 3D Navier–Stokes solutions for benchmarking", International Journal for Numerical

Methods in Fluids 19 (5): 369–375, doi:10.1002/fld.1650190502[19] McComb, W.D. (2008), Renormalization methods: A guide for beginners, Oxford University Press, ISBN 0199236526 pp. 121–128.

References• Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series,

Oxford University Press, ISBN 0198596790• Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0521663962• Landau, L. D.; Lifshitz, E. M. (1987), Fluid mechanics, Course of Theoretical Physics, 6 (2nd revised ed.),

Pergamon Press, ISBN 0 08 033932 8, OCLC 15017127• Rhyming, Inge L. (1991), Dynamique des fluides, Presses Polytechniques et Universitaires Romandes, Lausanne• Polyanin, A.D.; Kutepov, A.M.; Vyazmin, A.V.; Kazenin, D.A. (2002), Hydrodynamics, Mass and Heat Transfer

in Chemical Engineering, Taylor & Francis, London, ISBN 0-415-27237-8• Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 0070150001

External links• Simplified derivation of the Navier–Stokes equations (http:/ / www. allstar. fiu. edu/ aero/ Flow2. htm)• http:/ / www. claymath. org/ millennium/ Navier-Stokes_Equations/ navierstokes. pdf Millennium Prize problem

description.• CFD online software list (http:/ / www. cfd-online. com/ Wiki/ Codes) A compilation of codes, including

Navier–Stokes solvers.

Incompressible flow 12

Incompressible flowIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in whichthe divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplifyanalysis. In reality, all materials are compressible to some extent. Note that isochoric refers to flow, not the materialproperty. This means that under certain circumstances, a compressible material can undergo (nearly) incompressibleflow. However, by making the 'incompressible' assumption, one can greatly simplify the equations governing theflow of the material.The equation describing an incompressible (isochoric) flow,

,where is the velocity of the material.The continuity equation states that,

This can be expressed via the material derivative as

Since , we see that a flow is incompressible if and only if,

that is, the mass density is constant following the material element.

Relation to compressibility factorIn some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressurevariations. This is best expressed in terms of the compressibility factor

If the compressibility factor is acceptably small, the flow is considered to be incompressible.

Relation to solenoidal fieldAn incompressible flow is described by a velocity field which is solenoidal. But a solenoidal field, besides having azero divergence, also has the additional connotation of having non-zero curl (i.e., rotational component).Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the velocity field isactually Laplacian.

Difference between incompressible flow and materialAs defined earlier, an incompressible (isochoric) flow is the one in which

.This is equivalent to saying that

i.e. the material derivative of the density is zero. Thus if we follow a material element, its mass density will remain constant. Note that the material derivative consists of two terms. The first term describes how the density of the

Incompressible flow 13

material element changes with time. This term is also known as the unsteady term. The second term, describes thechanges in the density as the material element moves from one point to another. This is the convection or theadvection term. For a flow to be incompressible the sum of these terms should be zero.On the other hand, a homogeneous, incompressible material is defined as one which has constant densitythroughout. For such a material, . This implies that,

and independently.

From the continuity equation it follows that

Thus homogeneous materials always undergo flow that is incompressible, but the converse is not true.It is common to find references where the author mentions incompressible flow and assumes that density is constant.Even though this is technically incorrect, it is an accepted practice. One of the advantages of using theincompressible material assumption over the incompressible flow assumption is in the momentum equation wherethe kinematic viscosity ( ) can be assumed to be constant. The subtlety above is frequently a source ofconfusion. Therefore many people prefer to refer explicitly to incompressible materials or isochoric flow when beingdescriptive about the mechanics.

Related flow constraintsIn fluid dynamics, a flow is considered to be incompressible if the divergence of the velocity is zero. However,related formulations can sometimes be used, depending on the flow system to be modelled. Some versions aredescribed below:1. Incompressible flow: . This can assume either constant density (strict incompressible) or varying

density flow. The varying density set accepts solutions involving small perturbations in density, pressure and/ortemperature fields, and can allow for pressure stratification in the domain.

2. Anelastic flow: . Principally used in the field of atmospheric sciences, the anelastic constraintextend incompressible flow validity to stratified density and/or temperature as well as pressure. This allow thethermodynamic variables to relax to an 'atmospheric' base state seen in the lower atmosphere when used in thefield of meteorology, for example. This condition can also be used for various astrophysical systems.[1]

3. Low Mach-number flow / Pseudo-incompressibility: . The low Mach-number constraint can bederived from the compressible Euler equations using scale analysis of non-dimensional quantities. The restraint,like the previous in this section, allows for the removal of acoustic waves, but also allows for large perturbationsin density and/or temperature. The assumption is that the flow remains within a Mach number limit (normally lessthan 0.3) for any solution using such a constraint to be valid. Again, in accordance with all incompressible flowsthe pressure deviation must be small in comparison to the pressure base state.[2]

These methods make differing assumptions about the flow, but all take into account the general form of theconstraint for general flow dependent functions and .

Incompressible flow 14

Numerical approximations of incompressible flowThe stringent nature of the incompressible flow equations means that specific mathematical techniques have beendevised to solve them. Some of these methods include:1. The projection method (both approximate and exact)2. Artificial compressibility technique (approximate)3. Compressibility pre-conditioning

References[1] Durran, D.R. (1989). "Improving the Anelastic Approximation" (http:/ / ams. allenpress. com/ archive/ 1520-0469/ 46/ 11/ pdf/

i1520-0469-46-11-1453. pdf). Journal of the Atmospheric Sciences 46 (11): 1453–1461.doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2. .

[2] Almgren, A.S.; Bell, J.B.; Rendleman, C.A.; Zingale, M. (2006). "Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics"(http:/ / seesar. lbl. gov/ ccse/ Publications/ car/ LowMachSNIa. pdf). Astrophysical Journal 637: 922–936. doi:10.1086/498426. .

See also• Compressible flow• Navier-Stokes equations

Compressible flowCompressible fluid mechanics is a combination of the fields of traditional fluid mechanics and thermodynamics. Itis related to the more general study of compressibility. In fluid dynamics, a flow is considered to be a compressibleflow if the density of the fluid changes with respect to pressure. This is often the case where the Mach number (theratio of the flow speed to the local speed of sound) of the flow exceeds 0.3.

DefinitionCompressible flow theory is distinguished from incompressible flow theory in that the density can no longer beconsidered a constant. As such, where incompressible flow theory is governed mainly by the conservation of massand conservation of momentum equations, compressible flows require that the conservation of energy equation alsoto be considered. Compressible flow appears in many natural and many technological processes. Compressible flowdeals with more than air, including steam, natural gas, nitrogen and helium, etc. For instance, the flow of natural gasin a pipe system, a common method of heating in the u.s., should be considered a compressible flow. The aboveflows that were mentioned are called internal flows. Compressible flow also includes flow around bodies such as thewings of an airplane, and is considered an external flow.These processes include situations not expected to have a compressible flow, such as manufacturing process such asthe die casting, injection molding. The die casting process is a process in which liquid metal, mostly aluminum, isinjected into a mold to obtain a near final shape. The air is displaced by the liquid metal in a very rapid manner, in amatter of milliseconds, therefore the compressibility has to be taken into account. Maintaining assumption of acalorically perfect gas, these equations can be solved simultaneously to obtain temperature, pressure, and densityprofiles that vary with local Mach number.<1>When the Mach number of the flow is high enough so that the effects of compressibility can no longer be neglectedas the flow will even out density differences. Below Mach 0.3 fluid flows experience less than a 5% change indensity.

Compressible flow 15

Subsonic Compressible Flows

Compressible Flow Correction FactorsDue to the complexities of compressible flow theory, many times it is easier to first calculate the incompressibleflow characteristics, and then employ a correction factor to obtain the actual flow properties. Several correctionfactors exist towards this end with varying degrees of complexity and accuracy.

Prandtl–Glauert transformation

The Prandtl-Glauert transformation is found by linearizing the potential equations associated with compressible,inviscid flow. The Prandtl–Glauert transformation or Prandtl–Glauert rule (also Prandtl–Glauert–Ackeret rule) is anapproximation function which allows comparison of aerodynamical processes occurring at different Mach numbers.It was discovered that the linearized pressures in such a flow were equal to those found from incompressible flowtheory multiplied by a correction factor. The Prandtl-Glauert correction factor will always underestimate themagnitude of the pressure within the fluid. This correction factor is given below. [1] :

where• cp is the compressible pressure coefficient• cp0 is the incompressible pressure coefficient• M is the Mach number.This correction factor works well for all Mach numbers M<0.7 and M>1.3. It should be noted that since thiscorrection factor is derived from linearized equations, the pressures calculated is always less in magnitude than theactual pressures within the fluid.

Karman-Tsien Correction Factor

The Karman-Tsien transformation is a nonlinear correction factor to find the pressure coefficient of a compressible,inviscid flow. It is an empirically derived correction factor that tends to slightly overestimate the magnitude of thefluid's pressure. In order to employ this correction factor, the incompressible, inviscid fluid pressure must be knownfrom previous investigation.[2]

where• cp is the compressible pressure coefficient• cp0 is the incompressible pressure coefficient• M is the Mach number.This correction factor is valid for M<0.8.

Compressible flow 16

Supersonic FlowsA flow where the local Mach number reaches or exceeds 1 can contain shock waves. A shock is an abrupt change inthe properties of a flow; the typical thickness of a shock scales with the molecular mean free path in the fluid(typically a few micrometers).

Shock WavesShocks form because information about conditions downstream of a point of sonic or supersonic flow cannotpropagate back upstream past the sonic point. A shock can occur in at least two different mechanisms. The first iswhen a large difference (above a small minimum value) between the two sides of a membrane, and when themembrane bursts (see the discussion about the shock tube). Of course, the shock travels from the high pressure to thelow pressure side. The second is when many sound waves ``run into each other and accumulate (some refer to it as``coalescing) into a large difference, which is the shock wave. In fact, the sound wave can be viewed as an extremelyweak shock. In the speed of sound analysis, it was assumed the medium is continuous, without any abrupt changes.This assumption is no longer valid in the case of a shock.[1]

Transonic FlowsThe is the behaviour of a fluid changes radically as it starts to move above the speed of sound (in that fluid), ie. whenthe Mach number is greater than 1. For example, in subsonic flow, a stream tube in an accelerating flow contracts,but in a supersonicflow a stream tube in an accelerating flow expands. To interpret this in another way, considersteady flow in a tube that has a sudden expansion: the tube's cross section suddenly widens, so the cross-sectionalarea increases, see Whitcomb area rule.

Applications

AerodynamicsAerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them.Aerodynamics is often used synonymously with gas dynamics, with the difference being that gas dynamics applies toall gases. Understanding the motion of air (often called a flow field) around an object enables the calculation offorces and moments acting on the object. Typical properties calculated for a flow field include velocity, pressure,density and temperature as a function of position and time.

NozzlesIn subsonic flow, the fluid speed drops after the expansion (as expected). In supersonic flow, the fluid speedincreases. This sounds like a contradiction, but it isn't: the mass flux is conserved but because supersonic flow allowsthe density to change, the volume flux is not constant. This effect is utilized in De Laval nozzles.Nozzles can bedescribed as convergent (narrowing down from a wide diameter to a smaller diameter in the direction of the flow) ordivergent (expanding from a smaller diameter to a larger one). A de Laval nozzle has a convergent section followedby a divergent section and is often called a convergent-divergent nozzle ("con-di nozzle").Increasing the nozzle pressure ratio further will not increase the throat Mach number beyond unity. Downstream (i.e.external to the nozzle) the flow is free to expand to supersonic velocities. Note that the Mach 1 can be a very highspeed for a hot gas; since the speed of sound varies as the square root of absolute temperature. Thus the speedreached at a nozzle throat can be far higher than the speed of sound at sea level. This fact is used extensively inrocketry where hypersonic flows are required, and where propellant mixtures are deliberately chosen to furtherincrease the sonic speed.

Compressible flow 17

Shock TubesIn addition to measurements of rates of chemical kinetics shock tubes have been used to measure dissociationenergies and molecular relaxation rates they have been used in aerodynamic tests. The fluid flow in the driven gascan be used much as a wind tunnel, allowing higher temperatures and pressures therein replicating conditions in theturbine sections of jet engines. However, test times are limited to a few milliseconds, either by the arrival of thecontact surface or the reflected shock wave.They have been further developed into shock tunnels, with an added nozzle and dump tank. The resultant hightemperature hypersonic flow can be used to simulate atmospheric re-entry of spacecraft or hypersonic craft, againwith limited testing times.

See also• Gas dynamics• Transonic flow• Supersonic flow• Hypersonic flow• Fanno flow• Rayleigh flow• Isothermal flow• Mach number• Aerodynamics• Nozzles

References[1] Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179[2] The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1 , p.237

• Shapiro, Ascher H. (1953). The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1. RonaldPress. ISBN 978-0-471-06691-0.

• Anderson, John D. (2004). Modern Compressible Flow. McGraw-Hill. ISBN 0071241361.• Liepmann, H. W.; Roshko A. (2002). Elements of Gasdynamics. Dover Publications. ISBN 0486419630.• von Mises, Richard (2004). Mathematical Theory of Compressible Fluid Flow. Dover Publications.

ISBN 0486439410.• Meyer, Richard E. (2007). Introduction to Mathematical Fluid Dynamics. Dover Publications.

ISBN 0-486-45887-3.• Saad, Michael A. (1985). Compressible Fluid Flow. Prentice Hall. ISBN 0-13-163486-0.• Hodge, B. K.; Koenig K. (1995). Compressible Fluid Dynamics with Personal Computer Applications. Prentice

Hall. ISBN 013308552X.• Schreier, S. (1982). Compressible Flow. Wiley-Interscience. ISBN 0-471-05691-X.• Lakshminarayana, B. (1995). Fluid Dynamics and Heat Transfer of Turbomachinery. Wiley-Interscience.

ISBN 978-0-471-85546-0.

Compressible flow 18

External links• NASA Beginner's Guide to Compressible Aerodynamics (http:/ / www. grc. nasa. gov/ WWW/ K-12/ airplane/

bgc. html)• Purdue University Compressible Flow Calculators (https:/ / engineering. purdue. edu/ ~wassgren/ applet/ java/

comp_calculator/ Index. html)• (http:/ / www. potto. org/ gasDynamics/ node40. php)

Isochoric processAn isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process,is a thermodynamic process during which the volume of the closed system undergoing such a process remainsconstant. In nontechnical terms, an isochoric process is exemplified by the heating or the cooling of the contents of asealed non-deformable container: The thermodynamic process is the addition or removal of heat; the isolation of thecontents of the container establishes the closed system; and the inability of the container to deform imposes theconstant-volume condition.

FormalismAn isochoric thermodynamic process is characterized by constant volume, i.e. . The process does nopressure-volume work, since such work is defined by

,where P is pressure. The sign convention is such that positive work is performed by the system on the environment.For a reversible process, the first law of thermodynamics gives the change in the system's internal energy:

Replacing work with a change in volume gives

Since the process is isochoric, , the previous equation now gives

Using the definition of specific heat capacity at constant volume,

,

Integrating both sides yields

Where is the specific heat capacity at constant volume, is initial temperature and is final temperature. Weconclude with:

Isochoric process 19

Isochoric Process in the Pressure volume diagram. In this diagram,pressure increases, but volume remains constant.

On a pressure volume diagram, an isochoric processappears as a straight vertical line. Its thermodynamicconjugate, an isobaric process would appear as astraight horizontal line.

Ideal gas

If an ideal gas is used in an isochoric process, and thequantity of gas stays constant, then the increase inenergy is proportional to an increase in temperature andpressure. Take for example a gas heated in a rigidcontainer: the pressure and temperature of the gas willincrease, but the volume will remain the same.

Ideal Otto cycle

The ideal Otto cycle is an example of an isochoric process when it is assumed that the burning of the gasoline-airmixture in an internal combustion engine car is instantaneous. There is an increase in the temperature and thepressure of the gas inside the cylinder while the volume remains the same.

EtymologyThe noun isochor and the adjective isochoric are derived from the Greek words ἴσος (isos) meaning "equal", andχώρα (chora) meaning "space."

References

External links• http:/ / lorien. ncl. ac. uk/ ming/ webnotes/ Therm1/ revers/ isocho. htm

Compressibility factor 20

Compressibility factorThe compressibility factor (Z), also known as the compression factor, is a useful thermodynamic property formodifying the ideal gas law to account for the real gas behaviour.[1] In general, deviations from ideal behaviorbecome more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure.Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virialequation which take compound specific empirical constants as input. Alternatively, the compressibility factor forspecific gases can be read from generalized compressibility charts[1] that plot Z as a function of pressure at constanttemperature.

Definition and physical significanceThe compressibility factor is defined as:

where, is the pressure, is the molar volume of the gas, is the temperature, and is the gas constant.For an ideal gas the compressibility factor is per definition. In many real world applications requirements foraccuracy demand that deviations from ideal gas behaviour, i.e. real gas behaviour, is taken into account. The value of

generally increases with pressure and decreases with temperature. At high pressures molecules are colliding moreoften. This allows repulsive forces between molecules to have a noticeable effect, making the volume of the real gas( ) greater than the volume of an ideal gas ( ) which causes to increase above one.[2] Whenpressures are lower, the molecules are more free to move. In this case attractive forces dominate, making .The closer the gas is to its critical point or its boiling point, the more deviates from the ideal case.

Generalized compressibility factor graphs for pure gases

Generalized compressibility factor diagram.

The unique relationship between thecompressibility factor and the reducedtemperature, Tr, and the reducedpressure, Pr, was first recognized byJohannes Diderik van der Waals in1873 and is known as thetwo-parameter principle ofcorresponding states. The principle ofcorresponding states expresses thegeneralization that the properties of agas which are dependent onintermolecular forces are related to thecritical properties of the gas in auniversal way. That provides a mostimportant basis for developingcorrelations of molecular properties.

As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the samereduced temperature, Tr, and reduced pressure, Pr, should have the same compressibility factor.The reduced temperature and pressure are defined as:

Compressibility factor 21

and

Tc and Pc are known as the critical temperature and critical pressure of a gas. They are characteristics of eachspecific gas with Tc being the temperature above which it is not possible to liquify a given gas and Pc is theminimum pressure required to liquify a given gas at its critical temperature. Together they define the critical point ofa fluid above which distinct liquid and gas phases of a given fluid do not exist. The pressure-volume-temperature(PVT) data for real gases varies from one pure gas to another. However, when the compressibility factors of varioussingle-component gases are graphed versus pressure along with temperature isotherms many of the graphs exhibitsimilar isotherm shapes. In order to obtain a generalized graph that can be used for many different gases, the reducedpressure and temperature, Pr and Tr, are used to normalize the compressibility factor data. Figure 2 is an example ofa generalized compressibility factor graph derived from hundreds of experimental P-V-T data points of 10 puregases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide andsteam. There are more detailed generalized compressibility factor graphs based on as many as 25 or more differentpure gases, such as the Nelson-Obert graphs. Such graphs are said to have an accuracy within 1-2 percent for Zvalues greater than 0.6 and within 4-6 percent for Z values of 0.3-0.6. The generalized compressibility factor graphsmay be considerably in error for strongly polar gases which are gases for which the centers of positive and negativecharge do not coincide. In such cases the estimate for Z may be in error by as much as 15-20 percent. The quantumgases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure andtemperature for those three gases should be redefined in the following manner to improve the accuracy of predictingtheir compressibility factors when using the generalized graphs:

and

Theoretical modelsThe virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few gases aremono-atomic) as it is derived directly from statistical mechanics:

Where the coefficients in the numerator are known as virial coefficients and are functions of temperature.The virial coefficients account for interactions between successively larger groups of molecules. For example, Baccounts for interactions between pairs, C for interactions between three gas molecules, and so on. Becauseinteractions between large numbers of molecules are rare, the virial equation is usually truncated after the thirdterm.[3]

The Real gas article features more theoretical methods to compute compressibility factors

Experimental valuesIt is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomesimportant. As a rule of thumb, the ideal gas law is reasonably accurate up to a pressure of about 2 atm, and evenhigher for small non-associating molecules. For example methyl chloride, a highly polar molecule and therefore withsignificant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of10 atm and temperature of 100 °C[4] . For air (small non-polar molecules) at approximately the same conditions, thecompressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K).

Compressibility factor 22

Compressibility of airNormal air comprises in crude numbers 80 percent nitrogen N2 and 20 percent oxygen O2. Both molecules are smalland non-polar (and therefore non-associating). We can therefore expect that the behaviour of air within broadtemperature and pressure ranges can be approximated as an ideal gas with reasonable accuracy. Experimental valuesfor the compressibility factor confirm this.

Z for air as function of pressure 1-500 bar

75-200 K isotherms 250-1000 K isotherms

Compressibility factor for air (experimental values)

Pressure, bar (absolute)

Temp, K 1 5 10 20 40 60 80 100 150 200 250 300 400 500

75 0.0052 0.0260 0.0519 0.1036 0.2063 0.3082 0.4094 0.5099 0.7581 1.0125

80 0.0250 0.0499 0.0995 0.1981 0.2958 0.3927 0.4887 0.7258 0.9588 1.1931 1.4139

90 0.9764 0.0236 0.0453 0.0940 0.1866 0.2781 0.3686 0.4681 0.6779 0.8929 1.1098 1.3110 1.7161 2.1105

100 0.9797 0.8872 0.0453 0.0900 0.1782 0.2635 0.3498 0.4337 0.6386 0.8377 1.0395 1.2227 1.5937 1.9536

120 0.9880 0.9373 0.8860 0.6730 0.1778 0.2557 0.3371 0.4132 0.5964 0.7720 0.9530 1.1076 1.5091 1.7366

140 0.9927 0.9614 0.9205 0.8297 0.5856 0.3313 0.3737 0.4340 0.5909 0.7699 0.9114 1.0393 1.3202 1.5903

160 0.9951 0.9748 0.9489 0.8954 0.7803 0.6603 0.5696 0.5489 0.6340 0.7564 0.8840 1.0105 1.2585 1.4970

180 0.9967 0.9832 0.9660 0.9314 0.8625 0.7977 0.7432 0.7084 0.7180 0.7986 0.9000 1.0068 1.2232 1.4361

200 0.9978 0.9886 0.9767 0.9539 0.9100 0.8701 0.8374 0.8142 0.8061 0.8549 0.9311 1.0185 1.2054 1.3944

250 0.9992 0.9957 0.9911 0.9822 0.9671 0.9549 0.9463 0.9411 0.9450 0.9713 1.0152 1.0702 1.1990 1.3392

300 0.9999 0.9987 0.9974 0.9950 0.9917 0.9901 0.9903 0.9930 1.0074 1.0326 1.0669 1.1089 1.2073 1.3163

350 1.0000 1.0002 1.0004 1.0014 1.0038 1.0075 1.0121 1.0183 1.0377 1.0635 1.0947 1.1303 1.2116 1.3015

400 1.0002 1.0012 1.0025 1.0046 1.0100 1.0159 1.0229 1.0312 1.0533 1.0795 1.1087 1.1411 1.2117 1.2890

450 1.0003 1.0016 1.0034 1.0063 1.0133 1.0210 1.0287 1.0374 1.0614 1.0913 1.1183 1.1463 1.2090 1.2778

500 1.0003 1.0020 1.0034 1.0074 1.0151 1.0234 1.0323 1.0410 1.0650 1.0913 1.1183 1.1463 1.2051 1.2667

600 1.0004 1.0022 1.0039 1.0081 1.0164 1.0253 1.0340 1.0434 1.0678 1.0920 1.1172 1.1427 1.1947 1.2475

800 1.0004 1.0020 1.0038 1.0077 1.0157 1.0240 1.0321 1.0408 1.0621 1.0844 1.1061 1.1283 1.1720 1.2150

1000 1.0004 1.0018 1.0037 1.0068 1.0142 1.0215 1.0290 1.0365 1.0556 1.0744 1.0948 1.1131 1.1515 1.1889

Compressibility factor 23

Source: Perry's chemical engineers' handbook (6ed ed.). MCGraw-Hill. 1984. ISBN 0-07-049479-7. (table 3-162).Z-value are calculated from values of pressure, volume (or density), and temperature in Vassernan, Kazavchinskii,and Rabinovich, "Thermophysical Properties of Air and Air Components;' Moscow, Nauka, 1966, and NBS-NSFTrans. TT 70-50095, 1971: and Vassernan and Rabinovich, "Thermophysical Properties of Liquid Air and ItsComponent, "Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.

Compressibility of ammonia gasAmmonia is small but highly polar molecule with significant interactions. Values can be obtained from Perry 4th ed(awaits future library visit)

See also• Real gas• Theorem of corresponding states• Principle of corresponding states• Van der Waals equation

References[1] Properties of Natural Gases (http:/ / iptibm1. ipt. ntnu. no/ ~jsg/ undervisning/ naturgass/ parlaktuna/ Chap3. pdf). Includes a chart of

compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document)[2] McQuarrie, Donald A. and Simon, John D. (1999). Molecular Thermodynamics. University Science Books. ISBN 1-891389-05-X. page 55[3] Smith, J.M. et al. (2005). Introduction to Chemical Engineering Thermodynamics (Seventh Edition ed.). McGraw Hill. ISBN 0-07-310445-0.

page73[4] Perry's chemical engineers' handbook (6ed ed.). MCGraw-Hill. 1984. ISBN 0-07-049479-7. page 3-268

External links• Compressibility factor (gases) (http:/ / en. citizendium. org/ wiki/ Compressibility_factor_(gases)) A Citizendium

article.• Real Gases (http:/ / www. cbu. edu/ ~rprice/ lectures/ realgas. html) includes a discussion of compressibility

factors.• Compressibility factor calculator (http:/ / www. enggcyclopedia. com/ welcome-to-enggcyclopedia/

thermodynamics/ compressibility-factors-for-gases)

Solenoidal vector field 24

Solenoidal vector fieldIn vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vector field v withdivergence zero:

The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotationaland a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vectorpotential component, because the definition of the vector potential A as:

automatically results in the identity (as can be shown, for example, using Cartesian coordinates):

The converse also holds: for any solenoidal v there exists a vector potential A such that (Strictlyspeaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)The divergence theorem, gives the equivalent integral definition of a solenoidal field; namely that for any closedsurface , the net total flux through the surface must be zero:

,

where is the outward normal to each surface element.

EtymologySolenoidal has its origin in the Greek word for solenoid, which is σωληνοειδές (sōlēnoeidēs) and meaningpipe-shaped. This contains σωλην (sōlēn) or pipe. In the present context of solenoidal it means constrained like in apipe, so with a fixed volume.

Examples• the magnetic field B is solenoidal (see Maxwell's equations);• the velocity field of an incompressible fluid flow is solenoidal;• the vorticity field is solenoidal• the electric field D in regions where ρe = 0;

• the current density, J, if .

References• Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0486661105

Conservative vector field 25

Conservative vector fieldIn vector calculus a conservative vector field is a vector field which is the gradient of a function, known in thiscontext as a scalar potential. Conservative vector fields have the property that the line integral from one point toanother is independent of the choice of path connecting the two points: it is path independent. Conversely, pathindependence is equivalent to the vector field being conservative. Conservative vector fields are also irrotational,meaning that (in three-dimensions) they have vanishing curl. In fact, an irrotational vector field is necessarilyconservative provided that a certain condition on the geometry of the domain holds: it must be simply connected.An irrotational vector field which is also solenoidal is called a Laplacian vector field because it is the gradient of asolution of Laplace's equation.

DefinitionA vector field is said to be conservative if there exists a scalar field such that : Here denotesthe gradient of . When the above equation holds, is called a scalar potential for . The fundamental theoremof vector calculus states that any vector field can be expressed as the sum of a conservative vector field and asolenoidal field.

Path independenceA key property of a conservative vector field is that its integral along a path depends only on the endpoints of thatpath, not the particular route taken. Suppose that is a region of three-dimensional space, and that is arectifiable path in with start point and end point . If is a conservative vector field then

This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.An equivalent formulation of this is to say that

for every closed loop in S. The converse is also true: if the circulation of v around every closed loop in an open set Sis zero, then v is a conservative vector field.

Conservative vector field 26

Irrotational vector fields

The above field v(x,y,z) = (−y/(x2+y2), +x/(x2+y2), 0) includes a vortex at its center,so it is non-irrotational; it is neither conservative, nor does it have path

independence. However, any simply connected subset that excludes the vortex line(0,0,z) will have zero curl, ∇×v = 0. Such vortex-free regions are examples of

irrotational vector fields.

A vector field is said to be irrotational ifits curl is zero. That is, if

For this reason, such vector fields are sometimes referred to as curl-free vector fields.It is an identity of vector calculus that for any scalar field :

Therefore every conservative vector field is also an irrotational vector field.Provided that is a simply-connected region, the converse of this is true: every irrotational vector field is also aconservative vector field.The above statement is not true if is not simply-connected. Let be the usual 3-dimensional space, except withthe -axis removed; that is . Now define a vector field by

Then exists and has zero curl at every point in ; that is is irrotational. However the circulation of aroundthe unit circle in the -plane is equal to . Therefore does not have the path independence propertydiscussed above, and is not conservative.In a simply-connected region an irrotational vector field has the path independence property. This can be seen bynoting that in such a region an irrotational vector field is conservative, and conservative vector fields have the pathindependence property. The result can also be proved directly by using Stokes' theorem. In a connected region anyvector field which has the path independence property must also be irrotational.

Conservative vector field 27

More abstractly, a conservative vector field is an exact 1-form. That is, it is a 1-form equal to the exterior derivativeof some 0-form (scalar field) . An irrotational vector field is a closed 1-form. Since d2 = 0, any exact form isclosed, so any conservative vector field is irrotational. The domain is simply connected if and only if its firsthomology group is 0, which is equivalent to its first cohomology group being 0. The first de Rham cohomologygroup is 0 if and only if all closed 1-forms are exact.

Irrotational flowsThe flow velocity of a fluid is a vector field, and the vorticity of the flow can be defined by

A common alternative notation for vorticity is .[1]

If is irrotational, with , then the flow is said to be an irrotational flow. The vorticity of anirrotational flow is zero.[2]

Kelvin's circulation theorem states that a fluid that is irrotational in an inviscid flow will remain irrotational. Thisresult can be derived from the vorticity transport equation, obtained by taking the curl of the Navier-stokesequations.For a two-dimensional flow the vorticity acts as a measure of the local rotation of fluid elements. Note that thevorticity does not imply anything about the global behaviour of a fluid. It is possible for a fluid traveling in a straightline to have vorticity, and it is possible for a fluid which moves in a circle to be irrotational.

Conservative forcesIf the vector field associated to a force is conservative then the force is said to be a conservative force.The most prominent example of a conservative force is the force of gravity. According to Newton's law ofgravitation, the gravitational force, , acting on a mass , due to a mass which is a distance away,obeys the equation

where is the Gravitational Constant and is a unit vector pointing from towards . The force of gravity isconservative because , where

is the Gravitational potential energy.For conservative forces, path independence can be interpreted to mean that the work done in going from a point to a point is independent of the path chosen, and that the work W done in going around a closed loop is zero:

The total energy of a particle moving under the influence of conservative forces is conserved, in the sense that a lossof potential energy is converted to an equal quantity of kinetic energy or vice versa.

Conservative vector field 28

See also• Beltrami vector field• Complex lamellar vector field• Helmholtz decomposition• Solenoidal vector field• Longitudinal and transverse vector fields

References• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press

(2005)• D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press (2005)

Notes[1] Clancy, L.J., Aerodynamics, Section 7.11, Pitman Publishing Limited, London. ISBN 0 273 01120 0[2] Liepmann, H.W.; Roshko, A. (1993) [1957], Elements of gasdynamics, Courier Dover Publications, ISBN 0486419630, pp. 194–196.

Laplacian vector fieldIn vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If thefield is denoted as v, then it is described by the following differential equations:

A Laplacian vector field in the plane satisfies the Cauchy-Riemann equations: it is holomorphic.Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential (see irrotationalfield) φ :

.Then, since the divergence of v is also zero, it follows from equation (1) that

which is equivalent to

.Therefore, the potential of a Laplacian field satisfies Laplace's equation.

Stokes' law 29

Stokes' law

Creeping flow past a sphere: streamlines, drag force Fdand force by gravity Fg.

In 1851, George Gabriel Stokes derived an expression, now knownas Stokes' law, for the frictional force — also called drag force —exerted on spherical objects with very small Reynolds numbers(e.g., very small particles) in a continuous viscous fluid. Stokes'law is derived by solving the Stokes flow limit for small Reynoldsnumbers of the generally unsolvable Navier–Stokes equations:[1]

where:• Fd is the frictional force acting on the interface between the fluid and the particle (in N),• η is the fluid's viscosity (in [kg m-1 s-1]),• R is the radius of the spherical object (in m), and• v is the particle's velocity (in m/s).

If the particles are falling in the viscous fluid by their own weight due to gravity, then a terminal velocity, alsoknown as the settling velocity, is reached when this frictional force combined with the buoyant force exactly balancethe gravitational force. The resulting settling velocity (or terminal velocity) is given by:[2]

where:• vs is the particles' settling velocity (m/s) (vertically downwards if ρp > ρf, upwards if ρp < ρf ),• g is the gravitational acceleration (m/s2),• ρp is the mass density of the particles (kg/m3), and• ρf is the mass density of the fluid (kg/m3).

Note that for molecules Stokes' law is used to define their Stokes radius.The CGS unit of kinematic viscosity was named "stokes" after his work.

Stokes' law 30

ApplicationsStokes's law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. Asphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminalvelocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be usedfor opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid,Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameter isnormally used in the classic experiment to improve the accuracy of the calculation. The school experiment usesglycerine or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used inprocesses. It includes many different oils, and polymer liquids such as solutions.The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least 3Nobel Prizes.[3]

Stokes' law is important to understanding the swimming of microorganisms and sperm; also, the sedimentation,under the force of gravity, of small particles and organisms, in water.[4]

In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air(as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equationcan be made in the settlement of fine particles in water or other fluids.

Stokes flow around a sphere

Steady Stokes flowIn Stokes flow, at very low Reynolds number, the convective acceleration terms in the Navier–Stokes equations areneglected. Then the flow equations become, for an incompressible steady flow:[5]

where:• p is the fluid pressure (in Pa),• u is the flow velocity (in m/s), and• ω is the vorticity (in s-1), defined as  By using some vector calculus identities, these equations can be shown to result in Laplace's equations for thepressure and each of the components of the vorticity vector:[5]

  and   Additional forces like those by gravity and buoyancy have not been taken into account, but can easily be added sincethe above equations are linear, so linear superposition of solutions and associated forces can be applied.

Flow around a sphereFor the case of a sphere in a uniform far field flow, it is advantageous to use a cylindrical coordinate system( r , φ , z ). The z–axis is through the centre of the sphere and aligned with the mean flow direction, while r is theradius as measured perpendicular to the z–axis. The origin is at the sphere centre. Because the flow is axisymmetricaround the z–axis, it is independent of the azimuth φ.In this cylindrical coordinate system, the incompressible flow can be described with a Stokes stream function ψ,depending on r and z:[6] [7]

Stokes' law 31

with v and w the flow velocity components in the r and z direction, respectively. The azimuthal velocity componentin the φ–direction is equal to zero, in this axisymmetric case. The volume flux, through a tube bounded by a surfaceof some constant value ψ, is equal to 2π ψ and is constant.[6]

For this case of an axisymmetric flow, the only non-zero of the vorticity vector ω is the azimuthal φ–componentωφ

[8] [9]

The Laplace operator, applied to the vorticity ωφ, becomes in this cylindrical coordinate system with axisymmetry:[9]

From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocityV in the z–direction and a sphere of radius R, the solution is found to be[10]

The viscous force per unit area σ, exerted by the flow on the surface on the sphere, is in the z–direction everywhere.More strikingly, it has also the same value everywhere on the sphere:[1]

with ez the unit vector in the z–direction. For other shapes than spherical, σ is not constant along the body surface.Integration of the viscous force per unit area σ over the sphere surface gives the frictional force Fd according toStokes' law.

Terminal velocity and settling timeAt terminal velocity — or settling velocity — the frictional force Fd on the sphere is balanced by the excess force Fgdue to the difference of the weight of the sphere and its buoyancy, both caused by gravity:[2]

with ρp and ρf the mass density of the sphere and the fluid, respectively, and g the gravitational acceleration.Demanding force balance: Fd = Fg and solving for the velocity V gives the terminal velocity Vs. If terminal velocityis reached relatively quickly, an average settling time can be calculated by dividing the height the particle will fall byits terminal velocity.

Notes[1] Batchelor (1967), p. 233.[2] Lamb (1994), §337, p. 599.[3] Dusenbery, David B. (2009). Living at Micro Scale, p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.[4] Dusenbery, David B. (2009). Living at Micro Scale. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.[5] Batchelor (1967), section 4.9, p. 229.[6] Batchelor (1967), section 2.2, p. 78.[7] Lamb (1994), §94, p. 126.[8] Batchelor (1967), section 4.9, p. 230[9] Batchelor (1967), appendix 2, p. 602.[10] Lamb (1994), §337, p. 598.

Stokes' law 32

References• Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.• Lamb, H. (1994). Hydrodynamics (6th edition ed.). Cambridge University Press. ISBN 9780521458689.

Originally published in 1879, the 6th extended edition appeared first in 1932.

Projection method (fluid dynamics)The projection method is an effective means of numerically solving time-dependent incompressible fluid-flowproblems. It was originally introduced by Alexandre Chorin in 1967 and independently by Roger Temam[1] as anefficient means of solving the incompressible Navier-Stokes equations. The key advantage of the projection methodis that the computations of the velocity and the pressure fields are decoupled.

The algorithmThe algorithm of projection method is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodgedecomposition) of any vector field into a solenoidal part and an irrotational part. Typically, the algorithm consists oftwo stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint iscomputed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space ofdivergence-free velocity field to get the next update of velocity and pressure.

Helmholtz–Hodge decompositionThe theoretical background of projection type method is the decomposition theorem of Ladyzhenskaya sometimesreferred to as Helmholtz–Hodge Decomposition or simply as Hodge decomposition. It states that the vector field defined on a simply connected domain can be uniquely decomposed into a divergence-free (solenoidal) part and an irrotational part .[2] Thus,

since for some scalar function, . Taking the divergence of equation yields

This is a Poisson equation for the scalar function . If the vector field is known, the above equation can besolved for the scalar function and the divergence part of can be extracted using the relation

This is the essence of solenoidal projection method for solving incompressible Navier–Stokes equations.

Chorin's projection methodThe incompressible Navier-Stokes equation (differential form of momentum equation) may be written as

In Chorin's original version of the projection method [3] , the intermediate velocity, , is explicitly computed usingthe momentum equation ignoring the pressure gradient term:

where is the velocity at th time level. In the next (projection) step, we have

Projection method (fluid dynamics) 33

Re-writing the above equation for the velocity at level, we have

Computing the right-hand side of the above equation requires a knowledge of the pressure, , at level.This is obtained by taking the divergence and requiring that , which is thedivergence-free(continuity) condition, thereby deriving the following Poisson equation for ,

It is instructive to note that, the equation written in the following way

is the standard Hodge decomposition if boundary condition for on the domain boundary, is .For the explicit method, the boundary condition for in equation (1) is natural. If on , isprescribed, then the space of divergence-free vector field will be orthogonal to the space of irrotational vector fields,and from equation (2) one has

The explicit treatment of the boundary condition may be circumvented by using a staggered grid and requiring thatvanish at the pressure nodes that are adjacent to the boundaries.

A distinguishing feature of Chorin's projection method is that the velocity field is forced to satisfy a discretecontinuity constraint at the end of each time step.

General methodTypically the projection method operates as a two-stage fractional step scheme, a method which uses multiplecalculation steps for each numerical time-step. In many projection algorithms, the steps are split as follows:1. First the system is progressed in time to a mid-time-step position, solving the above transport equations for mass

and momentum using a suitable advection method. This is denoted the predictor step.2. At this point an initial projection may be implemented such that the mid-time-step velocity field is enforced as

divergence free.3. The corrector part of the algorithm is then progressed. These use the time-centred estimates of the velocity,

density, etc. to form final time-step state.4. A final projection is then applied to enforce the divergence restraint on the velocity field. The system has now

been fully updated to the new time.

References[1] Temam, R. (1968), "Une méthode d'approximation des solutions des équations Navier-Stokes,", Bull. Soc. Math. France 98: 115–152[2] Chorin, A. J.; J. E. Marsden (2000). A Mathematical Introduction to Fluid Mechanics (3rd ed.). Springer-Verlag. ISBN 0387979182.[3] Chorin, A. J. (1968), "Numerical Solution of the Navier-Stokes Equations", Math. Comp. 22: 745–762

Mach number 34

Mach number

An F/A-18 Hornet at transonic speed and displaying the Prandtl–Glauertsingularity just before reaching the speed of sound

Mach number ( or ) (generallypronounced /ˈmɑːk/, sometimes /ˈmɑːx/ or/ˈmæk/) is the speed of an object movingthrough air, or any other fluid substance,divided by the speed of sound as it is in thatsubstance for its particular physicalconditions, including those of temperatureand pressure. It is commonly used torepresent the speed of an object when it istravelling close to or above the speed ofsound.

whereis the Mach number

is the relative velocity of the source to the medium andis the speed of sound in the medium

The Mach number is named after Austrian physicist and philosopher Ernst Mach, a designation proposed byaeronautical engineer Jakob Ackeret. Because the Mach number is often viewed as a dimensionless quantity ratherthan a unit of measure, with Mach, the number comes after the unit; the second Mach number is "Mach 2" instead of"2 Mach" (or Machs). This is somewhat reminiscent of the early modern ocean sounding unit "mark" (a synonym forfathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade precedingfaster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never"Mach 1."[1]

In French, the Mach number is sometimes called the "nombre de Sarrau" ("Sarrau number") after Émile Sarrau,researching on explosions in the 1870s and 1880s.[2]

OverviewThe Mach number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluidflows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is adimensionless number. At Standard Sea Level conditions (corresponding to a temperature of 15 degrees Celsius), thespeed of sound is 340.3 m/s[3] (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. Thespeed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmosphericcomposition and largely independent of pressure. In the stratosphere, where the temperatures are constant, it does notvary with altitude even though the air pressure changes significantly with altitude.Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at

Mach number 35

the same Mach number. So, an aircraft traveling at Mach 1 at 20°C or 68°F will experience shock waves in much thesame manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at -50°C or -58F, even though it is traveling atonly 86% of its speed at higher temperature like 20°C or 68°F.

High-speed flow around objectsFlight can be roughly classified in six categories:

Regime Subsonic Transonic Sonic Supersonic Hypersonic High-hypersonic

Mach <0.75 0.75–1.2 1.0 1.2–5.0 5.0–10.0 >10.0

For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = Mach 25.4 in air at highaltitudes. The speed of light in a vacuum corresponds to a Mach number of approximately 881,000 (relative to air atsea level).At transonic speeds, the flow field around the object includes both sub- and supersonic parts. The transonic periodbegins when first zones of M>1 flow appear around the object. In case of an airfoil (such as an aircraft's wing), thistypically happens above the wing. Supersonic flow can decelerate back to subsonic only in a normal shock; thistypically happens before the trailing edge. (Fig.1a)As the speed increases, the zone of M>1 flow increases towards both leading and trailing edges. As M=1 is reachedand passed, the normal shock reaches the trailing edge and becomes a weak oblique shock: the flow decelerates overthe shock, but remains supersonic. A normal shock is created ahead of the object, and the only subsonic zone in theflow field is a small area around the object's leading edge. (Fig.1b)

(a) (b)

Fig. 1. Mach number in transonic airflow around an airfoil; M<1 (a) and M>1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier) a large pressure difference is created just in front of theaircraft. This abrupt pressure difference, called a shock wave, spreads backward and outward from the aircraft in acone shape (a so-called Mach cone). It is this shock wave that causes the sonic boom heard as a fast moving aircrafttravels overhead. A person inside the aircraft will not hear this. The higher the speed, the more narrow the cone; atjust over M=1 it is hardly a cone at all, but closer to a slightly concave plane.At fully supersonic speed, the shock wave starts to take its cone shape and flow is either completely supersonic, or(in case of a blunt object), only a very small subsonic flow area remains between the object's nose and the shockwave it creates ahead of itself. (In the case of a sharp object, there is no air between the nose and the shock wave: theshock wave starts from the nose.)As the Mach number increases, so does the strength of the shock wave and the Mach cone becomes increasinglynarrow. As the fluid flow crosses the shock wave, its speed is reduced and temperature, pressure, and densityincrease. The stronger the shock, the greater the changes. At high enough Mach numbers the temperature increasesso much over the shock that ionization and dissociation of gas molecules behind the shock wave begin. Such flowsare called hypersonic.It is clear that any object traveling at hypersonic speeds will likewise be exposed to the same extreme temperaturesas the gas behind the nose shock wave, and hence choice of heat-resistant materials becomes important.

Mach number 36

High-speed flow in a channelAs a flow in a channel crosses M=1 becomes supersonic, one significant change takes place. The conservation ofmass flow rate leads one to expect that contracting the flow channel would increase the flow speed (i.e. making thechannel narrower results in faster air flow) and at subsonic speeds this holds true. However, once the flow becomessupersonic, the relationship of flow area and speed is reversed: expanding the channel actually increases the speed.The obvious result is that in order to accelerate a flow to supersonic, one needs a convergent-divergent nozzle, wherethe converging section accelerates the flow to M=1, sonic speeds, and the diverging section continues theacceleration. Such nozzles are called de Laval nozzles and in extreme cases they are able to reach incredible,hypersonic speeds (Mach 13 at 20°C).An aircraft Machmeter or electronic flight information system (EFIS) can display Mach number derived fromstagnation pressure (pitot tube) and static pressure.

Calculating Mach NumberAssuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is derivedfrom Bernoulli's equation for M<1:[4]

where:is Mach number

is impact pressure andis static pressureis the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air).

The formula to compute Mach number in a supersonic compressible flow is derived from the Rayleigh SupersonicPitot equation:

or for air, a simplfied formula:

where:is now impact pressure measured behind a normal shock.

The Mach number at which an aircraft is flying at can be calculated by

where:is Mach number

is velocity of the moving aircraft andis the speed of sound at the given altitude

Note that the dynamic pressure can be found as:

Mach number 37

See also• Critical Mach number• Machmeter• Ramjet• Scramjet• Speed of sound• True airspeed

Notes[1] Bodie, Warren M., The Lockheed P-38 Lightning, Widewing Publications ISBN 0-9629359-0-5[2] Blackmore, John T. (1972). Ernst Mach: his live, work, and influence. University of California Press. p. 112. ISBN 9780520018495.[3] Clancy, L.J. (1975), Aerodynamics, Table 1, Pitman Publishing London, ISBN 0 273 01120 0[4] Olson, Wayne M. (2002). "AFFTC-TIH-99-02, Aircraft Performance Flight Testing." (PDF). Air Force Flight Test Center, Edwards AFB,

CA, United States Air Force.

External links• Gas Dynamics Toolbox (http:/ / web. ics. purdue. edu/ ~alexeenk/ GDT/ index. html) Calculate Mach number and

normal shock wave parameters for mixtures of perfect and imperfect gases.• NASA's page on Mach Number (http:/ / www. grc. nasa. gov/ WWW/ K-12/ airplane/ mach. html) Interactive

calculator for Mach number.• NewByte standard atmosphere calculator and speed converter (http:/ / www. newbyte. co. il/ calc. html)

Viscosity 38

Viscosity

Viscosity

Clear liquid above has lower viscosity than the substance below

SI symbol: μ, η

SI unit: Pa·s = kg/(s·m)

Derivations from other quantities: μ = G·t

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. Ineveryday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus, water is "thin", having alower viscosity, while honey is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greaterits ease of movement (fluidity).[1]

Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Forexample, high-viscosity felsic magma will create a tall, steep stratovolcano, because it cannot flow far before itcools, while low-viscosity mafic lava will create a wide, shallow-sloped shield volcano. All real fluids (exceptsuperfluids) have some resistance to stress and therefore are viscous, but a fluid which has no resistance to shearstress is known as an ideal fluid or inviscid fluid.The study of flowing matter is known as rheology, which includes viscosity and related concepts.

Viscosity 39

EtymologyThe word "viscosity" derives from the Latin word "viscum alba" for mistletoe. A viscous glue called birdlime wasmade from mistletoe berries and used for lime-twigs to catch birds.[2]

Properties and behavior

Laminar shear of fluid between two plates. Friction between the fluid and the movingboundaries causes the fluid to shear. The force required for this action is a measure of

the fluid's viscosity. This type of flow is known as a Couette flow.

Overview

In general, in any flow, layers move atdifferent velocities and the fluid'sviscosity arises from the shear stressbetween the layers that ultimatelyopposes any applied force.

The relationship between the shearstress and the velocity gradient can beobtained by considering two platesclosely spaced at a distance y, andseparated by a homogeneous substance.Assuming that the plates are very large,with a large area A, such that edgeeffects may be ignored, and that thelower plate is fixed, let a force F beapplied to the upper plate. If this forcecauses the substance between the platesto undergo shear flow at velocity u (as opposed to just shearing elastically until the shear stress in the substancebalances the applied force), the substance is called a fluid.

The applied force is proportional to the area and velocity of the plate and inversely proportional to the distancebetween the plates. Combining these three relations results in the equation:

Viscosity 40

Laminar shear, the non-constant gradient, is a result of the geometry the fluid is flowingthrough (e.g. a pipe).

,

where μ is the proportionality factor called viscosity.

This equation can be expressed in terms of shear stress . Thus as expressed in differential form by Isaac

Newton for straight, parallel and uniform flow, the shear stress between layers is proportional to the velocity gradientin the direction perpendicular to the layers:

Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.

Note that the rate of shear deformation is which can be also written as a shear velocity, .

James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposesshear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.

Viscosity 41

Types of viscosity

Viscosity, the slope of each line, varies among materials

Newton's law of viscosity, given above, is a constitutive equation (like Hooke's law, Fick's law, Ohm's law). It is nota fundamental law of nature but an approximation that holds in some materials and fails in others. Non-Newtonianfluids exhibit a more complicated relationship between shear stress and velocity gradient than simple linearity. Thusthere exist a number of forms of viscosity:• Newtonian: fluids, such as water and most gases which have a constant viscosity.• Shear thickening: viscosity increases with the rate of shear.• Shear thinning: viscosity decreases with the rate of shear.• Thixotropic: materials which become less viscous over time when shaken, agitated, or otherwise stressed.• Rheopectic: materials which become more viscous over time when shaken, agitated, or otherwise stressed.• A Bingham plastic is a material that behaves as a solid at low stresses but flows as a viscous fluid at high

stresses.• A magnetorheological fluid is a type of "smart fluid" which, when subjected to a magnetic field, greatly

increases its apparent viscosity, to the point of becoming a viscoelastic solid.

Viscosity 42

Viscosity coefficientsViscosity coefficients can be defined in two ways:• Dynamic viscosity, also absolute viscosity, the more usual one (typical units Pa·s, Poise, cP);• Kinematic viscosity is the dynamic viscosity divided by the density (typical units m2/s, Stokes, cSt).Viscosity is a tensorial quantity that can be decomposed in different ways into two independent components. Themost usual decomposition yields the following viscosity coefficients:• Shear viscosity, the most important one, often referred to as simply viscosity, describing the reaction to applied

shear stress; simply put, it is the ratio between the pressure exerted on the surface of a fluid, in the lateral orhorizontal direction, to the change in velocity of the fluid as you move down in the fluid (this is what is referredto as a velocity gradient).

• Volume viscosity (also called bulk viscosity or second viscosity) becomes important only for such effects wherefluid compressibility is essential. Examples would include shock waves and sound propagation. It appears in theStokes' law (sound attenuation) that describes propagation of sound in Newtonian liquid.

Alternatively,• Extensional viscosity, a linear combination of shear and bulk viscosity, describes the reaction to elongation,

widely used for characterizing polymers. For example, at room temperature, water has a dynamic shear viscosityof about 1.0 × 10−3 Pa·s and motor oil of about 250 × 10−3 Pa·s.[3]

Viscosity measurementViscosity is measured with various types of viscometers and rheometers. A rheometer is used for those fluids whichcannot be defined by a single value of viscosity and therefore require more parameters to be set and measured than isthe case for a viscometer. Close temperature control of the fluid is essential to accurate measurements, particularly inmaterials like lubricants, whose viscosity can double with a change of only 5 °C.For some fluids, viscosity is a constant over a wide range of shear rates (Newtonian fluids). The fluids without aconstant viscosity (non-Newtonian fluids) cannot be described by a single number. Non-Newtonian fluids exhibit avariety of different correlations between shear stress and shear rate.One of the most common instruments for measuring kinematic viscosity is the glass capillary viscometer.In paint industries, viscosity is commonly measured with a Zahn cup, in which the efflux time is determined andgiven to customers. The efflux time can also be converted to kinematic viscosities (centistokes, cSt) through theconversion equations.Also used in paint, a Stormer viscometer uses load-based rotation in order to determine viscosity. The viscosity isreported in Krebs units (KU), which are unique to Stormer viscometers.A Ford viscosity cup measures the rate of flow of a liquid. This, under ideal conditions, is proportional to thekinematic viscosity.Vibrating viscometers can also be used to measure viscosity. These models such as the Dynatrol use vibration ratherthan rotation to measure viscosity.Extensional viscosity can be measured with various rheometers that apply extensional stress.Volume viscosity can be measured with an acoustic rheometer.Apparent viscosity is a calculation derived from tests performed on drilling fluid used in oil or gas well development.These calculations and tests help engineers develop and maintain the properties of the drilling fluid to thespecifications required.

Viscosity 43

Units

Dynamic viscosityThe usual symbol for dynamic viscosity used by mechanical and chemical engineers — as well as fluid dynamicists— is the Greek letter mu (μ).[4] [5] [6] The symbol η is also used by chemists, physicists, and the IUPAC.[7]

The SI physical unit of dynamic viscosity is the pascal-second (Pa·s), (equivalent to N·s/m2, or kg/(m·s)). If a fluidwith a viscosity of one Pa·s is placed between two plates, and one plate is pushed sideways with a shear stress of onepascal, it moves a distance equal to the thickness of the layer between the plates in one second.The cgs physical unit for dynamic viscosity is the poise[8] (P), named after Jean Louis Marie Poiseuille. It is morecommonly expressed, particularly in ASTM standards, as centipoise (cP). Water at 20 °C has a viscosity of 1.0020cP or 0.001002 kg/(m·s).

1 P = 1 g·cm−1·s−1.1 Pa·s = 1 kg·m−1·s−1 = 10 P.

The relation to the SI unit is1 P = 0.1 Pa·s,1 cP = 1 mPa·s = 0.001 Pa·s.

Kinematic viscosityIn many situations, we are concerned with the ratio of the inertial force to the viscous force (i.e. the Reynoldsnumber, ) , the latter characterized by the fluid density ρ. This ratio is characterized by thekinematic viscosity (Greek letter nu, ν), defined as follows:

The SI unit of ν is m2/s. The SI unit of ρ is kg/m3.The cgs physical unit for kinematic viscosity is the stokes (St), named after George Gabriel Stokes. It is sometimesexpressed in terms of centistokes (cSt or ctsk). In U.S. usage, stoke is sometimes used as the singular form.

1 St = 1 cm2·s−1 = 10−4 m2·s−1.1 cSt = 1 mm2·s−1 = 10−6m2·s−1.

Water at 20 °C has a kinematic viscosity of about 1 cSt.The kinematic viscosity is sometimes referred to as diffusivity of momentum, because it has the same unit as and iscomparable to diffusivity of heat and diffusivity of mass. It is therefore used in dimensionless numbers whichcompare the ratio of the diffusivities.

FluidityThe reciprocal of viscosity is fluidity, usually symbolized by φ = 1 / μ or F = 1 / μ, depending on the conventionused, measured in reciprocal poise (cm·s·g−1), sometimes called the rhe. Fluidity is seldom used in engineeringpractice.The concept of fluidity can be used to determine the viscosity of an ideal solution. For two components and ,the fluidity when a and b are mixed is

which is only slightly simpler than the equivalent equation in terms of viscosity:

Viscosity 44

where χa and χb is the mole fraction of component a and b respectively, and μa and μb are the components pureviscosities.

Non-standard unitsThe Reyn is a British unit of dynamic viscosity.Viscosity index is a measure for the change of kinematic viscosity with temperature. It is used to characteriselubricating oil in the automotive industry.At one time the petroleum industry relied on measuring kinematic viscosity by means of the Saybolt viscometer, andexpressing kinematic viscosity in units of Saybolt Universal Seconds (SUS).[9] Other abbreviations such as SSU(Saybolt Seconds Universal) or SUV (Saybolt Universal Viscosity) are sometimes used. Kinematic viscosity incentistoke can be converted from SUS according to the arithmetic and the reference table provided in ASTM D2161.[10]

Molecular origins

Pitch has a viscosity approximately 230 billion(2.3×1011) times that of water.[11]

The viscosity of a system is determined by how molecules constitutingthe system interact. There are no simple but correct expressions for theviscosity of a fluid. The simplest exact expressions are theGreen–Kubo relations for the linear shear viscosity or the TransientTime Correlation Function expressions derived by Evans and Morrissin 1985. Although these expressions are each exact in order tocalculate the viscosity of a dense fluid, using these relations requiresthe use of molecular dynamics computer simulations.

Gases

Viscosity in gases arises principally from the molecular diffusion thattransports momentum between layers of flow. The kinetic theory ofgases allows accurate prediction of the behavior of gaseous viscosity.Within the regime where the theory is applicable:• Viscosity is independent of pressure and• Viscosity increases as temperature increases.[12]

James Clerk Maxwell published a famous paper in 1866 using thekinetic theory of gases to study gaseous viscosity.[13] To understandwhy the viscosity is independent of pressure consider two adjacent boundary layers (A and B) moving with respectto each other. The internal friction (the viscosity) of the gas is determined by the probability a particle of layer Aenters layer B with a corresponding transfer of momentum. Maxwell's calculations showed him that the viscositycoefficient is proportional to both the density, the mean free path and the mean velocity of the atoms. On the otherhand, the mean free path is inversely proportional to the density. So an increase of pressure doesn't result in anychange of the viscosity.

Viscosity 45

Relation to mean free path of diffusing particles

In relation to diffusion, the kinematic viscosity provides a better understanding of the behavior of mass transport of adilute species. Viscosity is related to shear stress and the rate of shear in a fluid, which illustrates its dependence onthe mean free path, λ, of the diffusing particles.From fluid mechanics, for a Newtonian fluid, the shear stress, τ, on a unit area moving parallel to itself, is found tobe proportional to the rate of change of velocity with distance perpendicular to the unit area:

for a unit area parallel to the x-z plane, moving along the x axis. We will derive this formula and show how μ isrelated to λ.Interpreting shear stress as the time rate of change of momentum, p, per unit area A (rate of momentum flux) of anarbitrary control surface gives

where is the average velocity along x of fluid molecules hitting the unit area, with respect to the unit area.Further manipulation will show[14]

, assuming that molecules hitting the unit area come from all distances between 0 and λ

(equally distributed), and that their average velocities change linearly with distance (always true for smallenough λ). From this follows:

whereis the rate of fluid mass hitting the surface,

ρ is the density of the fluid,

ū is the average molecular speed ( ),

μ is the dynamic viscosity.

Effect of temperature on the viscosity of a gas

Sutherland's formula can be used to derive the dynamic viscosity of an ideal gas as a function of the temperature:[15]

This in turn is equal to

where which is a constant.

in Sutherland's formula:• μ = dynamic viscosity in (Pa·s) at input temperature T,• μ0 = reference viscosity in (Pa·s) at reference temperature T0,• T = input temperature in kelvins,• T0 = reference temperature in kelvins,• C = Sutherland's constant for the gaseous material in question.

Viscosity 46

Valid for temperatures between 0 < T < 555 K with an error due to pressure less than 10% below 3.45 MPa.Sutherland's constant and reference temperature for some gases

Gas C [K] T0 [K] μ

0 [μPa s]

air 120 291.15 18.27

nitrogen 111 300.55 17.81

oxygen 127 292.25 20.18

carbon dioxide 240 293.15 14.8

carbon monoxide 118 288.15 17.2

hydrogen 72 293.85 8.76

ammonia 370 293.15 9.82

sulfur dioxide 416 293.65 12.54

helium 79.4 [16] 273 19 [17]

See also [18].

Viscosity of a dilute gas

The Chapman-Enskog equation[19] may be used to estimate viscosity for a dilute gas. This equation is based on asemi-theoretical assumption by Chapman and Enskog. The equation requires three empirically determinedparameters: the collision diameter (σ), the maximum energy of attraction divided by the Boltzmann constant (є/к)and the collision integral (ω(T*)).

with• T* = κT/ε — reduced temperature (dimensionless),• μ0 = viscosity for dilute gas (μPa.s),• M = molecular mass (g/mol),• T = temperature (K),• σ = the collision diameter (Å),• ε / κ = the maximum energy of attraction divided by the Boltzmann constant (K),• ωμ = the collision integral.

Viscosity 47

Liquids

Video showing three liquids with differentViscosities

In liquids, the additional forces between molecules become important.This leads to an additional contribution to the shear stress though theexact mechanics of this are still controversial. Thus, in liquids:• Viscosity is independent of pressure (except at very high pressure);

and• Viscosity tends to fall as temperature increases (for example, water

viscosity goes from 1.79 cP to 0.28 cP in the temperature rangefrom 0 °C to 100 °C); see temperature dependence of liquidviscosity for more details.

The dynamic viscosities of liquids are typically several orders ofmagnitude higher than dynamic viscosities of gases.

Viscosity of blends of liquids

The viscosity of the blend of two or more liquids can be estimatedusing the Refutas equation[20] . The calculation is carried out in threesteps.

The first step is to calculate the Viscosity Blending Number (VBN)(also called the Viscosity Blending Index) of each component of the blend:

(1)   where v is the kinematic viscosity in centistokes (cSt). It is important that the kinematic viscosity of each componentof the blend be obtained at the same temperature.The next step is to calculate the VBN of the blend, using this equation:

(2)   where xX is the mass fraction of each component of the blend.Once the viscosity blending number of a blend has been calculated using equation (2), the final step is to determinethe kinematic viscosity of the blend by solving equation (1) for v:

(3)  

where VBNBlend is the viscosity blending number of the blend.

Viscosity 48

Viscosity of selected substancesThe viscosity of air and water are by far the two most important materials for aviation aerodynamics and shippingfluid dynamics. Temperature plays the main role in determining viscosity.

Viscosity of air

Pressure dependence of the dynamicviscosity of dry air at the temperatures of

300, 400 and 500 K

The viscosity of air depends mostly on the temperature. At 15.0 °C, theviscosity of air is 1.78×10−5 kg/(m·s), 17.8 μPa.s or 1.78×10−5 Pa.s.. One canget the viscosity of air as a function of temperature from the Gas ViscosityCalculator [21]

Viscosity of water

Dynamic Viscosity of Water

The dynamic viscosity of water is 8.90 × 10−4 Pa·s or8.90 × 10−3 dyn·s/cm2 or 0.890 cP at about 25 °C.Water has a viscosity of 0.0091 poise at 25 °C, or 1centipoise at 20 °C.As a function of temperature T (K): μ(Pa·s) = A ×10B/(T−C)

where A=2.414 × 10−5 Pa·s ; B = 247.8 K ; and C = 140K .

Viscosity of liquid water at different temperatures up tothe normal boiling point is listed below.

Viscosity 49

Temperature [°C] Viscosity [mPa·s]

10 1.308

20 1.002

30 0.7978

40 0.6531

50 0.5471

60 0.4668

70 0.4044

80 0.3550

90 0.3150

100 0.2822

Viscosity of various materials

Example of the viscosity of milk and water.Liquids with higher viscosities will not make

such a splash when poured at the same velocity.

Some dynamic viscosities of Newtonian fluids are listed below:

Viscosity 50

Honey being drizzled.

Peanut butter is a semi-solid and can thereforehold peaks.

Viscosity of selected gases at 100 kPa, [μPa·s]

Gas at 0 °C (273 K) at 27 °C (300 K)[22]

air 17.4 18.6

hydrogen 8.4 9.0

helium 20.0

argon 22.9

xenon 21.2 23.2

carbon dioxide 15.0

methane 11.2

ethane 9.5

Viscosity 51

Fluid Viscosity [Pa·s] Viscosity [cP]

honey 2–10 2,000–10,000

molasses 5–10 5,000–10,000

molten glass 10–1,000 10,000–1,000,000

chocolate syrup 10–25 10,000–25,000

molten chocolate* 45–130 [23] 45,000–130,000

ketchup* 50–100 50,000–100,000

peanut butter* c. 250 c. 250,000

shortening* c. 250 250,000

|+Viscosity of fluids with variable compositions

Viscosity of liquids at 25 °C

Liquid (): Viscosity [Pa·s] Viscosity [cP=mPa.s]

acetone[24] 3.06×10−4 0.306

benzene[24] 6.04×10−4 0.604

blood (37 °C)[25] (3–4)×10−3 3–4

castor oil[24] 0.985 985

corn syrup[24] 1.3806 1380.6

ethanol[24] 1.074×10−3 1.074

ethylene glycol 1.61×10−2 16.1

glycerol[26] 69 (at 20 °C) 1490

HFO-380 2.022 2022

mercury[24] 1.526×10−3 1.526

methanol[24] 5.44×10−4 0.544

Motor oil SAE 10 (20 °C)[12] 0.065 65

Motor oil SAE 40 (20 °C)[12] 0.319 319

nitrobenzene[24] 1.863×10−3 1.863

liquid nitrogen @ 77K 1.58×10−4 0.158

propanol[24] 1.945×10−3 1.945

olive oil .081 81

pitch 2.3e8 2.3e11

quark–gluon plasma[27] 5e11 5e14

sulfuric acid[24] 2.42×10−2 24.2

water 8.94×10−4 0.894

Viscosity 52

* These materials are highly non-Newtonian.

Viscosity of solidsOn the basis that all solids such as granite[28] flow to a small extent in response to small shear stress, someresearchers[29] have contended that substances known as amorphous solids, such as glass and many polymers, maybe considered to have viscosity. This has led some to the view that solids are simply "liquids" with a very highviscosity, typically greater than 1012 Pa·s. This position is often adopted by supporters of the widely heldmisconception that glass flow can be observed in old buildings. This distortion is more likely the result of the glassmaking process rather than the viscosity of glass.[30]

However, others argue that solids are, in general, elastic for small stresses while fluids are not.[31] Even if solids flowat higher stresses, they are characterized by their low-stress behavior. This distinction is muddled if measurementsare continued over long time periods, such as the Pitch drop experiment. Viscosity may be an appropriatecharacteristic for solids in a plastic regime. The situation becomes somewhat confused as the term viscosity issometimes used for solid materials, for example Maxwell materials, to describe the relationship between stress andthe rate of change of strain, rather than rate of shear.These distinctions may be largely resolved by considering the constitutive equations of the material in question,which take into account both its viscous and elastic behaviors. Materials for which both their viscosity and theirelasticity are important in a particular range of deformation and deformation rate are called viscoelastic. In geology,earth materials that exhibit viscous deformation at least three times greater than their elastic deformation aresometimes called rheids.

Viscosity of amorphous materials

Common glass viscosity curves.[32]

Viscous flow in amorphous materials (e.g.in glasses and melts)[33] [34] [35] is athermally activated process:

where Q is activation energy, T is temperature, R is the molar gas constant and A is approximately a constant.The viscous flow in amorphous materials is characterized by a deviation from the Arrhenius-type behavior: Qchanges from a high value QH at low temperatures (in the glassy state) to a low value QL at high temperatures (in theliquid state). Depending on this change, amorphous materials are classified as either• strong when: QH − QL < QL or• fragile when: QH − QL ≥ QL.

Viscosity 53

The fragility of amorphous materials is numerically characterized by the Doremus’ fragility ratio:

and strong material have RD < 2 whereas fragile materials have RD ≥ 2.The viscosity of amorphous materials is quite exactly described by a two-exponential equation:

with constants A1, A2, B, C and D related to thermodynamic parameters of joining bonds of an amorphous material.Not very far from the glass transition temperature, Tg, this equation can be approximated by aVogel-Fulcher-Tammann (VFT) equation.If the temperature is significantly lower than the glass transition temperature, T < Tg, then the two-exponentialequation simplifies to an Arrhenius type equation:

with:

where Hd is the enthalpy of formation of broken bonds (termed configuron [36] s) and Hm is the enthalpy of theirmotion. When the temperature is less than the glass transition temperature, T < Tg, the activation energy of viscosityis high because the amorphous materials are in the glassy state and most of their joining bonds are intact.If the temperature is highly above the glass transition temperature, T > Tg, the two-exponential equation alsosimplifies to an Arrhenius type equation:

with:

When the temperature is higher than the glass transition temperature, T > Tg, the activation energy of viscosity is lowbecause amorphous materials are melt and have most of their joining bonds broken which facilitates flow.

Eddy viscosityIn the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices(or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transportand dissipation of energy in the smaller-scale flow (see large eddy simulation). Values of eddy viscosity used inmodeling ocean circulation may be from 5x104 to 106 Pa·s depending upon the resolution of the numerical grid.

The linear viscous stress tensorViscous forces in a fluid are a function of the rate at which the fluid velocity is changing over distance. The velocityat any point r is specified by the velocity field v(r). The velocity at a small distance dr from point r may be written asa Taylor series:

where dv / dr is shorthand for the dyadic product of the del operator and the velocity:

This is just the Jacobian of the velocity field.

Viscosity 54

Viscous forces are the result of relative motion between elements of the fluid, and so are expressible as a function ofthe velocity field. In other words, the forces at r are a function of v(r) and all derivatives of v(r) at that point. In thecase of linear viscosity, the viscous force will be a function of the Jacobian tensor alone. For almost all practicalsituations, the linear approximation is sufficient.If we represent x, y, and z by indices 1, 2, and 3 respectively, the i,j component of the Jacobian may be written as∂i vj where ∂i is shorthand for ∂/∂xi. Note that when the first and higher derivative terms are zero, the velocity of allfluid elements is parallel, and there are no viscous forces.Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix, and this decomposition isindependent of coordinate system, and so has physical significance. The velocity field may be approximated as:

where Einstein notation is now being used in which repeated indices in a product are implicitly summed. The secondterm from the right is the asymmetric part of the first derivative term, and it represents a rigid rotation of the fluidabout r with angular velocity ω where:

For such a rigid rotation, there is no change in the relative positions of the fluid elements, and so there is no viscousforce associated with this term. The remaining symmetric term is responsible for the viscous forces in the fluid.Assuming the fluid is isotropic (i.e. its properties are the same in all directions), then the most general way that thesymmetric term (the rate-of-strain tensor) can be broken down in a coordinate-independent (and therefore physicallyreal) way is as the sum of a constant tensor (the rate-of-expansion tensor) and a traceless symmetric tensor (therate-of-shear tensor):

where δij is the unit tensor. The most general linear relationship between the stress tensor σ and the rate-of-straintensor is then a linear combination of these two tensors:[37]

where ς is the coefficient of bulk viscosity (or "second viscosity") and μ is the coefficient of (shear) viscosity.The forces in the fluid are due to the velocities of the individual molecules. The velocity of a molecule may bethought of as the sum of the fluid velocity and the thermal velocity. The viscous stress tensor described above givesthe force due to the fluid velocity only. The force on an area element in the fluid due to the thermal velocities of themolecules is just the hydrostatic pressure. This pressure term (−p δij) must be added to the viscous stress tensor toobtain the total stress tensor for the fluid.

The infinitesimal force dFi on an infinitesimal area dAi is then given by the usual relationship:

Viscosity 55

References[1] Symon, Keith (1971). Mechanics (Third ed.). Addison-Wesley. ISBN 0-201-07392-7.[2] "The Online Etymology Dictionary" (http:/ / www. etymonline. com/ index. php?term=viscous). Etymonline.com. . Retrieved 2010-09-14.[3] Raymond A. Serway (1996). Physics for Scientists & Engineers (4th ed.). Saunders College Publishing. ISBN 0-03-005932-1.[4] Victor Lyle Streeter, E. Benjamin Wylie, Keith W. Bedford Fluid Mechanics, McGraw-Hill, 1998 ISBN 0070625379[5] J. P. Holman Heat Transfer, McGraw-Hill, 2002 ISBN 0071226214[6] Frank P. Incropera, David P. DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 2007 ISBN 0471457280[7] IUPAC Gold Book, Definition of (dynamic) viscosity (http:/ / goldbook. iupac. org/ D01877. html)[8] "IUPAC definition of the Poise" (http:/ / www. iupac. org/ goldbook/ P04705. pdf#search="poise iupac"). . Retrieved 2010-09-14.[9] ASTM D 2161, Page one,(2005)[10] "Quantities and Units of Viscosity" (http:/ / www. uniteasy. com/ en/ unitguide/ Viscosity. htm). Uniteasy.com. . Retrieved 2010-09-14.[11] Edgeworth,, R.; Dalton, B.J.; Parnell, T.. "The pitch drop experiment" (http:/ / www. physics. uq. edu. au/ physics_museum/ pitchdrop.

shtml). University of Queensland. . Retrieved 2009-03-31.. A copy of: European Journal of Physics (1984) pp. 198–200.[12] Glenn Elert. "The Physics Hypertextbook-Viscosity" (http:/ / physics. info/ viscosity/ ). Physics.info. . Retrieved 2010-09-14.[13] Maxwell, J. C. (1866). "On the viscosity or internal friction of air and other gases". Philosophical Transactions of the Royal Society of

London 156: 249–268. doi:10.1098/rstl.1866.0013.[14] Salmon, R.L. (1998). Lectures on geophysical fluid dynamics. Oxford University Press. ISBN 0195108086., pp. 23–26.[15] Alexander J. Smits, Jean-Paul Dussauge Turbulent shear layers in supersonic flow (http:/ / books. google. com/ books?id=oRx6U4T8zcIC&

pg=PA46), Birkhäuser, 2006, ISBN 0387261400 p. 46[16] data constants for sutherland's formula (http:/ / arxiv. org/ pdf/ physics/ 0410237. pdf)[17] Viscosity of liquids and gases (http:/ / hyperphysics. phy-astr. gsu. edu/ Hbase/ tables/ viscosity. html)[18] http:/ / www. epa. gov/ EPA-AIR/ 2005/ July/ Day-13/ a11534d. htm[19] J.O. Hirshfelder, C.F. Curtis and R.B. Bird (1964). Molecular theory of gases and liquids (First ed.). Wiley. ISBN 0-471-40065-3.[20] Robert E. Maples (2000). Petroleum Refinery Process Economics (2nd ed.). Pennwell Books. ISBN 0-87814-779-9.[21] http:/ / www. lmnoeng. com/ Flow/ GasViscosity. htm[22] "Handbook of Chemistry and Physics", 83rd edition, CRC Press, 2002.[23] "Chocolate Processing" (http:/ / www. brookfieldengineering. com/ education/ applications/ laboratory-chocolate-processing. asp).

Brookfield Engineering website. . Retrieved 2007-12-03.[24] CRC Handbook of Chemistry and Physics, 73rd edition, 1992–1993[25] Glenn Elert. "Viscosity. The Physics Hypertextbook. by Glenn Elert" (http:/ / hypertextbook. com/ physics/ matter/ viscosity/ ).

Hypertextbook.com. . Retrieved 2010-09-14.[26] viscosity table at hyperphysics.phy-astr.gsu.edu (http:/ / hyperphysics. phy-astr. gsu. edu/ HBASE/ Tables/ viscosity. html), contains

glycerin(=glycerol) viscosity[27] Thomas Schäfer (October 2009). "Nearly perfect fluidity". Physics 2: 88. doi:10.1103/Physics.2.88.[28] Kumagai, Naoichi; Sadao Sasajima, Hidebumi Ito (15 February 1978). "Long-term Creep of Rocks: Results with Large Specimens Obtained

in about 20 Years and Those with Small Specimens in about 3 Years" (http:/ / translate. google. com/ translate?hl=en& sl=ja& u=http:/ / ci.nii. ac. jp/ naid/ 110002299397/ & sa=X& oi=translate& resnum=4& ct=result& prev=/ search?q=Ito+ Hidebumi& hl=en). Journal of theSociety of Materials Science (Japan) (Japan Energy Society) 27 (293): 157–161. . Retrieved 2008-06-16.

[29] Elert, Glenn. "Viscosity" (http:/ / hypertextbook. com/ physics/ matter/ viscosity/ ). The Physics Hypertextbook. .[30] "Antique windowpanes and the flow of supercooled liquids", by Robert C. Plumb, (Worcester Polytech. Inst., Worcester, MA, 01609, USA),

J. Chem. Educ. (1989), 66 (12), 994–6[31] Gibbs, Philip. "Is Glass a Liquid or a Solid?" (http:/ / math. ucr. edu/ home/ baez/ physics/ General/ Glass/ glass. html). . Retrieved

2007-07-31.[32] Alexander Fluegel. "Viscosity calculation of glasses" (http:/ / www. glassproperties. com/ viscosity/ ). Glassproperties.com. . Retrieved

2010-09-14.[33] R.H.Doremus (2002). "Viscosity of silica". J. Appl. Phys. 92 (12): 7619–7629. doi:10.1063/1.1515132.[34] M.I. Ojovan and W.E. Lee (2004). "Viscosity of network liquids within Doremus approach". J. Appl. Phys. 95 (7): 3803–3810.

doi:10.1063/1.1647260.[35] M.I. Ojovan, K.P. Travis and R.J. Hand (2000). "Thermodynamic parameters of bonds in glassy materials from viscosity-temperature

relationships". J. Phys.: Condensed matter 19 (41): 415107. doi:10.1088/0953-8984/19/41/415107.[36] http:/ / www. wikidoc. org/ index. php/ Configuron[37] L.D. Landau and E.M. Lifshitz (translated from Russian by J.B. Sykes and W.H. Reid) (1997). Fluid Mechanics (2nd ed.). Butterworth

Heinemann. ISBN 0-7506-2767-0.

ASTM D 2161, Standard Practice for Conversion of Kinematic Viscosity to Saybolt Universal Viscosity or toSaybolt Furol Viscosity

Viscosity 56

Additional reading• Massey, B. S. (1983). Mechanics of Fluids (Fifth ed.). Van Nostrand Reinhold (UK). ISBN 0-442-30552-4.

External links• Fluid properties (http:/ / webbook. nist. gov/ chemistry/ fluid/ ) High accuracy calculation of viscosity and other

physical properties of frequent used pure liquids and gases.• Fluid Characteristics Chart (http:/ / www. engineersedge. com/ fluid_flow/ fluid_data. htm) A table of viscosities

and vapor pressures for various fluids• Gas Dynamics Toolbox (http:/ / web. ics. purdue. edu/ ~alexeenk/ GDT/ index. html) Calculate coefficient of

viscosity for mixtures of gases• Glass Viscosity Measurement (http:/ / glassproperties. com/ viscosity/ ViscosityMeasurement. htm) Viscosity

measurement, viscosity units and fixpoints, glass viscosity calculation• Kinematic Viscosity (http:/ / www. diracdelta. co. uk/ science/ source/ k/ i/ kinematic viscosity/ source. html)

conversion between kinematic and dynamic viscosity.• Physical Characteristics of Water (http:/ / www. thermexcel. com/ english/ tables/ eau_atm. htm) A table of water

viscosity as a function of temperature• Vogel–Tammann–Fulcher Equation Parameters (http:/ / www. iop. org/ EJ/ abstract/ 0953-8984/ 12/ 46/ 305)• Calculation of temperature-dependent dynamic viscosities for some common components (http:/ / ddbonline.

ddbst. de/ VogelCalculation/ VogelCalculationCGI. exe)

RheologyRheology (pronounced /riːˈɒlədʒi/) is the study of the flow of matter: primarily in the liquid state, but also as 'softsolids' or solids under conditions in which they respond with plastic flow rather than deforming elastically inresponse to an applied force.[1] It applies to substances which have a complex molecular structure, such as muds,sludges, suspensions, polymers and other glass formers (e.g. silicates), as well as many foods and additives, bodilyfluids (e.g. blood) and other biological materials.The flow of these substances cannot be characterized by a single value of viscosity (at a fixed temperature). Whilethe viscosity of liquids normally varies with temperature, it is variations with other factors which are studied inrheology. For example, ketchup can have its viscosity reduced by shaking (or other forms of mechanical agitation)but water cannot. Since Sir Isaac Newton originated the concept of viscosity, the study of variable viscosity liquids isalso often called Non-Newtonian fluid mechanics.[1]

The term rheology was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920, from a suggestionby a colleague, Markus Reiner.[2] The term was inspired by the aphorism of Simplicius (often misattributed toHeraclitus), panta rei, "everything flows"[3] Plato in his dialogue Cratylus recounts on Heraclitus' saying that "allthings move and nothing remains still"[4] ; he also compares the etymology of the name of the Greek goddess Rhea(Ρέα) to the Greek name for flow (ῥοή). He notes the etymological relationship of the names of "streams"[5] given toCronus (Chronos - time) and Rhea (ῥοή – flow or space) and he argues that this relationship is not accidental.[6]

Compare also words ending in -rhea, such as gonorrhea, galactorrhoea, steatorrhea, diarrhea or diarrhea and similarwords.The experimental characterization of a material's rheological behavior is known as rheometry, although the termrheology is frequently used synonymously with rheometry, particularly by experimentalists. Theoretical aspects ofrheology are the relation of the flow/deformation behavior of material and its internal structure (e.g., the orientationand elongation of polymer molecules), and the flow/deformation behavior of materials that cannot be described byclassical fluid mechanics or elasticity.

Rheology 57

ScopeIn practice, rheology is principally concerned with extending the "classical" disciplines of elasticity and (Newtonian)fluid mechanics to materials whose mechanical behavior cannot be described with the classical theories. It is alsoconcerned with establishing predictions for mechanical behavior (on the continuum mechanical scale) based on themicro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or theparticle size distribution in a solid suspension. Materials flow when subjected to a stress, that is a force per area.There are different sorts of stress (e.g. shear, torsional, etc.) and materials can respond in various ways. Thus muchof theoretical rheology is concerned with the forces associated and external applied loads and stresses, and theresulting internal strains.[1]

Continuum mechanicsThe study of the physicsof continuous materials

Solid mechanicsThe study of the physics of continuousmaterials with a defined rest shape.

ElasticityDescribes materials that return to their rest shape after an applied stress.

PlasticityDescribes materials that permanentlydeform after a sufficient applied stress.

RheologyThe study of materials with bothsolid and fluid characteristics.

Fluid mechanicsThe study of the physics of continuousmaterials which take the shape of theircontainer.

Non-Newtonian fluids

Newtonian fluids

Rheology unites the seemingly unrelated fields of plasticity and non-Newtonian fluids by recognizing that both thesetypes of materials are unable to support a shear stress in static equilibrium. In this sense, a plastic solid is a fluid.Granular rheology refers to the continuum mechanical description of granular materials.One of the tasks of rheology is to empirically establish the relationships between deformations and stresses,respectively their derivatives by adequate measurements. These experimental techniques are known as rheometryand are concerned with the determination with well-defined rheological material functions. Such relationships arethen amenable to mathematical treatment by the established methods of continuum mechanics.The characterisation of flow or deformation originating from a simple shear stress field is called shear rheometry (orshear rheology). The study of extensional flows is called extensional rheology. Shear flows are much easier to studyand thus much more experimental data are available for shear flows than for extensional flows.

RheologistA rheologist is an interdisciplinary scientist who studies the flow of complex liquids or the deformation of softsolids. It is not taken as a primary degree subject, and there is no general qualification. He or she will usually have aprimary qualification in one of several fields: mathematics, the physical sciences (e.g. chemistry, physics, biology),engineering (e.g. mechanical, chemical or civil engineering), medicine, or certain technologies, notably materials orfood. Typically, a small amount of rheology may be studied when obtaining a degree, but the professional willextend this knowledge during postgraduate research or by attending short courses and by joining one of theprofessional associations (see below).

ViscoelasticityThe classical theory of elasticity deals with the mechanical properties of elastic solids, for which, according to Hooke's Law, stress is always directly proportional to strain in small deformations — but independent of the rate of strain. The classical theory of hydrodynamics deals with the properties of viscous liquids, for which, according to Newton's Law, the stress is always directly proportional to the rate of strain, but independent of the strain itself. These principles are, of course, applicable only for ideal materials under ideal conditions, although the behavior of many solids approaches Hooke's law for infinitesimal strains, and that of many liquids approaches Newton's law for

Rheology 58

infinitesimal rates of strain. Two types of deviations from linearity may be considered here.1) When finite strains are applied to solid bodies, the stress-strain relationships are often much more complicated(i.e. Non-Hookean). Similarly, in steady flow with finite strain rates, many fluids exhibit marked deviations fromNewton's law of stress-strain proportionality.2) Even if both strain and rate of strain are infinitesimal, a system may exhibit both liquid-like and solid-likecharacteristics. A good example of this is when a body which is not quite an elastic solid (i.e. an inelastic solid) doesnot maintain a constant deformation under constant stress, but rather continues to deform with time – or "creeps".When such a body is constrained at constant deformation, the stress required to hold it gradually diminishes—or"relaxes" with time.Similarly, a body which is not quite liquid (i.e. some elements of elasticity) may, while flowing under constantstress, store some of the energy input instead of dissipating it all as heat and random thermal motion of its molecularconstituents. In addition, it may never recover all of its deformation upon removal of the initial applied stress. Whensuch bodies are subjected to a sinusoidally oscillating stress, the strain is neither exactly in phase with the stress (as itwould be for a perfectly elastic solid) nor 90 degrees out of phase (as it would be for a perfectly viscous liquid) butrather exhibits a strain value which lies somewhere in between the two extreme cases. I.E. Some of the energy isstored and recovered in each cycle, and some is dissipated as heat. These are viscoelastic materials.Thus, liquids are generally associated with viscous behavior (a thick oil is a viscous liquid) and solids with elasticbehavior (an elastic string is an elastic solid). A more general point of view is to consider the material behavior atshort times (relative to the duration of the experiment/application of interest) and at long times.Fluid and solid character are relevant at long timesWe consider the application of a constant stress (a so-called creep experiment):• if the material, after some deformation, eventually resists further deformation, it is considered a solid• if, by contrast, the material flows indefinitely, it is considered a fluidBy contrast, elastic and viscous (or intermediate, viscoelastic) behavior is relevant at short times (transient behavior)We again consider the application of a constant stress:• if the material deformation strain increases linearly with increasing applied stress, then the material is purely

elastic• if the material deformation rate increases linearly with increasing applied stress, then the material is purely

viscous• if neither the deformation strain, nor its derivative with time (rate) follows the applied stress, then the material is

viscoelasticPlasticity is equivalent to the existence of a yield stress

A material that behaves as a solid under low applied stresses may start to flow above a certain level of stress, calledthe yield stress of the material. The term plastic solid is often used when this plasticity threshold is rather high, whileyield stress fluid is used when the threshold stress is rather low. However, there is no fundamental differencebetween the two concepts.

ApplicationsRheology has applications in materials science engineering, geophysics, physiology, human biology and pharmaceutics. Materials science is utilized in the production of many industrially important substances such as concrete, paint and chocolate have complex flow characteristics. In addition, plasticity theory has been similarly important for the design of metal forming processes. The science of rheology and the characeterization of viscoelastic properties in the production and use of polymeric materials has been critical for the production of many products for use in both the industrial and military sectors. Study of flow properties of liquids is important for

Rheology 59

pharmacists working in the manufacture of several dosage forms, such as simple liquids, ointments, creams, pastesetc. The flow behavior of liquids under applied stress is of great relevance in the field of pharmacy. Flow propertiesare used as important quality control tools to maintain the superiority of the product and reduce batch to batchvariations.

PolymersThe viscoelastic properties of polymers are determined by the effects of the many variables, including temperature,pressure, and time. Other important variables include chemical composition, molecular weight and weightdistribution, degree of branching branching and crystallinity, types of functionality, component concentration,dilution with solvents or plasticizers, and mixture with other materials to form composite systems. With guidance bymolecular theory, the dependence of viscoelastic properties on these variables can be simplified by introducingadditional concepts such as the free volume, the monomeric friction coefficient, and the spacing betweenentanglement loci, to provide a qualitative understanding and in many cases a quantitative prediction of how toachieve desired physical and chemical properties and ultimate microstructure.

Structure evidenced in a typical linear (non-branched) polymer found in a fossil fuel likepetroleum.

Viscoelastic behavior reflects the combined viscous and elastic responses, under mechanical stress, of materialswhich are intermediate between liquids and solids in character. Fundamentally, the viscoelasticity can be related tothe motions of flexible polymer molecules and their entanglements and network junctions—the molecular basis ofviscoelasticity. Thus, rearrangements on a local scale (kinks) are relatively rapid, while on a long-range scale(convolutions) very slow. In addition, a new assortment of configurations is obtained under stress. The response tothe local aspects of the new distribution is rapid, while the response to the long-range aspects is slow. Thus there isvery wide and continuous range of timescales covering the response of such a system to externally applied stress.From measurements of the viscoelastic properties of polymers, information can be obtained about the nature and therates of change of the configurational rearrangements, and the nature of the (macro)molecular interactions over arange of time scales.Examples may be given to illustrate the potential applications of these principles to practical problems in theprocessing and use of rubbers, plastics, and fibers. Polymers constitute the basic materials of the rubber and plasticindustries and are of vital importance to the textile, petroleum, automobile, paper, and pharmaceutical industries.Their viscoelastic properties determine the mechanical performance of the final products of these industries, and alsothe success of processing methods at intermediate stages of production.In viscoelastic materials, such as most polymers and plastics, the presence of liquid-like behavior depends on theproperties of and so varies with rate of applied load, i.e., how quickly a force is applied. The silicone toy 'Silly Putty'behaves quite differently depending on the time rate of applying a force. Pull on it slowly and it exhibits continuousflow, similar to that evidenced in a highly viscous liquid. Alternatively, when hit hard and directly, it shatters like asilicate glass.In addition, conventional rubber undergoes a glass transition, (often called a rubber-glass transition). E.G. The Space Shuttle Challenger disaster was caused by rubber O-rings that were being used well below their glass transition temperature on an unusually cold Florida morning, and thus could not flex adequately to form proper seals

Rheology 60

between sections of the two solid-fuel rocket boosters.

Biopolymers

Linear structure of cellulose -- the most common component of all organic plantlife on Earth. * Note the evidence of hydrogen bonding which increases theviscosity at any temperature and pressure. This is an effect similar to that of

polymer crosslinking, but less pronounced.

A major but defining difference betweenpolymers and biopolymers can be found intheir structures. Polymers, includingbiopolymers, are made of repetitive unitscalled monomers. While polymers are oftenrandomly constructed with massiveentanglement, biopolymers often have awell defined structure. In the case ofproteins, the exact chemical compositionand the sequence in which these units arearranged is called the primary structure.Many proteins spontaneously fold intocharacteristic compact shapes—whichdetermine their biological functions anddepend in a complicated way on theirprimary structures. Structural biology is thestudy of the structural properties of thebiopolymers, much of which can bedetermined by their viscoelastic response toa wide range of loading conditions.

Rheology 61

Sol-gel

Polymerization process of tetraethylorthosilicate (TEOS) and water toform amorphous hydrated silica particles (Si-OH) can be monitored

rheologically by a number of different methods.

Sol-gel science (aka chemical solution deposition)is a wet-chemical technique widely used in thefields of materials science, glass production andceramic engineering. Such methods are usedprimarily for the fabrication of materials (typicallya metal oxide) starting from a chemical solutionwhich acts as the precursor for an integratednetwork (or gel) of either discrete nanoparticles ornetwork polymers. Typical precursors are metalalkoxides and metal chlorides, which undergovarious forms of hydrolysis and polycondensationreactions in order to form a viscoelastic network(or solid).

One of the largest application areas is thin filmsand coatings, which can be produced on a piece ofsubstrate by spin coating or dip coating. Othermethods include spraying, electrophoresis, inkjetprinting or roll coating. Optical coatings, protectiveand decorative coatings, and electro-opticcomponents can be applied to glass, metal andother types of substrates with these methods. Withthe viscosity of a sol adjusted into a proper range,both optical quality glass fiber and refractoryceramic fiber can be drawn which are used for fiberoptic sensors and thermal insulation, respectively.The mechanisms of hydrolysis and condensation,and the rheological factors that bias the structure toward linear or branched structures are the most critical issues ofsol-gel science and technology.

Geophysics

Geophysics includes the flow of molten lava and debris flows (fluid mudslides). Also included in this disciplinarybranch are solid Earth materials which only exhibit flow over extended time scales. Those that display viscousbehavior are known as rheids. E.G. Granite can do a plastic flow with a vanishingly small yield stress, (i.e. a viscousflow). Long term creep experiments (~ 10 years) indicate that the viscosity of granite under ambient conditions is onthe order of 1020 poises. [7]

Rheology 62

Piledriving for a bridge in Napa, California.

Deep foundations are used for structures or heavyloads when shallow foundations cannot providesufficient adequate capacity. They may also beused to transfer building loads past weak orcompressible soil layers. While shallowfoundations rely solely on the bearing capacity ofthe soil beneath them, deep foundations can relyon end bearing resistance, frictional resistancealong their length, or both in developing therequired capacity. Geotechnical engineers usespecialized tools, such as the cone penetrationtest, to estimate the amount of skin and endbearing resistance available in the subsurface.

In addiiton, pile driving is often used to check forstability in varying soil types such as clay, sand,gravels, fractured shale, etc. Geotechnicalengineering (or 'soil engineering') often utilizessoil logs or bore logs to show what may beevidenced while driving piles through givenstratum and soil lenses. Wave equations mustoften be employed when using vibratory ormechanical impact hammers. The harmonics setup by vibratory or impact hammers drasticallychange the ability of given soils to create wallfriction on a given pile type, as well as the elastic alteration or resistance to penetration in a normal state.

Dynamic testing of soils may involve the attachment of transducers to pilings while they are being driven. Inaddition, theoretical bearing calculations using a nuclear densometer may be carried out in the field. In the end, afairly simple linear equation may suffice to give a good approximation of the bearing capacity of the soil. Oneexample of such an equation is given as follows:

( Wt. of pile hammer )( Drop Height ) / ( 0.85 )( Pile set )

= End Bearing Value

Rheology 63

Physiology

Heme

Physiology includes the study of many bodily fluids that have complexstructure and composition, and thus exhibit a wide range of viscoelasticflow characteristics. In particular there is a specialist study of bloodflow called hemorheology. This is the study of flow properties of bloodand its elements (plasma and formed elements, including red bloodcells, white blood cells and platelets). Blood viscosity is determined byplasma viscosity, hematocrit (volume fraction of red blood cell, whichconstitute 99.9% of the cellular elements) and mechanical behavior ofred blood cells. Therefore, red blood cell mechanics is the majordeterminant of flow properties of blood. [8]

Food rheology

Food rheology is important in the manufacture and processing of foodproducts. Food rheology is the study of the rheological properties of food, that is, the consistency and flow of foodunder tightly specified conditions. The consistency, degree of fluidity, and other mechanical properties are importantin understanding how long food can be stored, how stable it will remain, and in determining food texture. Theacceptability of food products to the consumer is often determined by food texture, such as how spreadable andcreamy a food product is. Food rheology is important in quality control during food manufacture and processing.

Thickening agents, or thickeners, are substances which, when added to an aqueous mixture, increase its viscositywithout substantially modifying its other properties, such as taste. They provide body, increase stability, and improvesuspension of added ingredients. Thickening agents are often used as food additives and in cosmetics and personalhygiene products. Some thickening agents are gelling agents, forming a gel. The agents are materials used to thickenand stabilize liquid solutions, emulsions, and suspensions. They dissolve in the liquid phase as a colloid mixture thatforms a weakly cohesive internal structure. Food thickeners frequently are based on either polysaccharides (starches,vegetable gums, and pectin), or proteins. [9]

Concrete RheologyConcrete's and mortar's workability is related to the rheological properties of the fresh cement paste. The mechanicalproperties of hardened concrete are better if less water is used in the preparation of concrete paste, however reducingthe water-to-cement ratio may decrease the ease of mixing and application. To avoid these undesired effects,superplasticizers are typically added to decrease the apparent yield stress and the viscosity of the fresh paste. Theiraddition highly improves concrete and mortar properties.

MeasurementRheometers are instruments used to characterize the rheological properties of materials, typically fluids and melts.These instruments impose a specific stress field or deformation to the fluid, and monitor the resultant deformation orstress. Instruments can be run in steady flow or oscillatory flow, in both shear and extension.

Deborah numberWhen the rheological behavior of a material includes a transition from elastic to viscous as the time scale increases (or, more generally, a transition from a more resistant to a less resistant behavior), one may define the relevant time scale as a relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number. Small Deborah numbers correspond to situations where the

Rheology 64

material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situationswhere the material behaves rather elastically.[10]

Note that the Deborah number is relevant for materials that flow on long time scales (like a Maxwell fluid) but notfor the reverse kind of materials (like the Voigt or Kelvin model) that are viscous on short time scales but solid onthe long term.

Reynolds numberIn fluid mechanics, the Reynolds number is a measure of the ratio of inertial forces (vsρ) to viscous forces (μ/L) andconsequently it quantifies the relative importance of these two types of effect for given flow conditions. Under lowReynolds numbers viscous effects dominate and the flow is laminar, whereas at high Reynolds numbers inertiapredominates and the flow may be turbulent. However, since rheology is concerned with fluids which do not have afixed viscosity, but one which can vary with flow and time, calculation of the Reynolds number can be complicated.It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with otherdimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similarflow patterns, in perhaps different fluids with possibly different flow rates, have the same values for the relevantdimensionless numbers, they are said to be dynamically similar.Typically it is given as follows:

where:• vs - mean fluid velocity, [m s−1]• L - characteristic length, [m]• μ - (absolute) dynamic fluid viscosity, [N s m−2] or [Pa s]• ν - kinematic fluid viscosity: ν = μ / ρ, [m² s−1]• ρ - fluid density, [kg m−3].

References[1] W. R. Schowalter (1978) Mechanics of Non-Newtonian Fluids Pergamon ISBN 0-08-021778-8[2] J. F. Steffe (1996) Rheological Methods in Food Process Engineering 2nd ed ISBN 0-9632036-1-4 page 1[3] Barnes, Jonathan (1982). The presocratic philosophers. ISBN 978-0415050791.[4] Plato, Cratylus [402a (http:/ / www. perseus. tufts. edu/ hopper/ text?doc=Perseus:text:1999. 01. 0172:text=Crat. :section=402a)][5] By the term "streams", Plato implies physical quantities that change with respect to time[6] Plato, Cratylus 402b (http:/ / www. perseus. tufts. edu/ hopper/ text?doc=Perseus:text:1999. 01. 0172:text=Crat. :section=402b)[7] Kumagai, N., Sasajima, S., Ito, H., Long-term Creep of Rocks, J. Soc. Mat. Sci. (Japan), Vol. 27, p. 157 (1978) Online (http:/ / translate.

google. com/ translate?hl=en& sl=ja& u=http:/ / ci. nii. ac. jp/ naid/ 110002299397/ & sa=X& oi=translate& resnum=4& ct=result& prev=/search?q=Ito+ Hidebumi& hl=en)

[8] Baskurt OK, Meiselman HJ (2003). "Blood rheology and hemodynamics". Seminars in Thrombosis and Haemostasis 29 (5): 435–450.doi:10.1055/s-2003-44551. PMID 14631543.

[9] B.M. McKenna, and J.G. Lyng (2003). Texture in food > Introduction to food rheology and its measurement (http:/ / books. google. com/?id=wM1asp1LL8EC& pg=PA130& dq=Food+ Rheology& q=Food Rheology). books.google.com. ISBN 9781855736733. . Retrieved2009-09-18.

[10] M. Reiner (1964) Physics Today volume 17 no 1 page 62 The Deborah Number

Rheology 65

Further reading• The Origins of Rheology: A short historical excursion (http:/ / www. ae. su. oz. au/ rheology/

Origin_of_Rheology. pdf) by Deepak Doraiswamy, University of Sidney

External linksJournals covering rheology• Applied Rheology (http:/ / www. appliedrheology. org/ )• Biorheology (http:/ / www. iospress. nl/ loadtop/ load. php?isbn=0006355x)• Journal of Rheology (http:/ / www. rheology. org/ sor/ publications/ j_rheology/ default. htm)• Journal of the Society of Rheology, JAPAN (http:/ / www. jstage. jst. go. jp/ browse/ rheology)• Journal of Non-Newtonian Fluid Mechanics (http:/ / www. sciencedirect. com/ science/ journal/ 03770257)• Korea-Australia Rheology Journal (http:/ / www. rheology. or. kr/ karj/ )• Rheologica Acta (http:/ / link. springer. de/ link/ service/ journals/ 00397/ )• Rheology Bulletin (http:/ / www. rheology. org/ SoR/ publications/ rheology_b/ default. htm)Organizations concerned with the study of rheology• The Romanian Society of Rheology (http:/ / reologie. ro)• The Society of Rheology (http:/ / www. rheology. org)• The Society of Rheology, JAPAN (http:/ / wwwsoc. nii. ac. jp/ srj/ )• The European Society of Rheology (http:/ / www. rheology-esr. org)• The British Society of Rheology (http:/ / www. bsr. org. uk)• Deutsche Rheologische Gesellschaft (http:/ / www. drg. bam. de)• Groupe Français de Rhéologie (http:/ / www. univ-lemans. fr/ sciences/ wgfr/ )• Belgian Group of Rheology (http:/ / cit. kuleuven. be/ ltrk/ bgr/ bgr. html)• Swiss Group of Rheology (http:/ / www. ar. ethz. ch/ FR/ )• Nederlandse Reologische Vereniging (http:/ / www. mate. tue. nl/ nrv/ index. html)• Società Italiana di Reologia (http:/ / www. sir-reologia. com)• Nordic Rheology Society (http:/ / www. sik. se/ nrs/ )• Australian Society of Rheology (http:/ / www. rheology. org. au)Rheology Conferences• Malvern Rheology Workshop (http:/ / reologie. ro/ 2009/ 07/ 29/ rheology-workshop/ )• Conferences on Rheology & Soft Matter Materials (http:/ / www. ar. ethz. ch/ conf. html)• Summer School of Rheology 2010 (http:/ / reologie. ro/ 2009/ 12/ 24/ school-of-rheology/ )• VII Annual European Conference on Rheology (AERC 2011) (http:/ / www. rheology-esr. org/ AERC/ 2011/ )

Article Sources and Contributors 66

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Incompressible flow  Source: http://en.wikipedia.org/w/index.php?oldid=392605415  Contributors: Arthena, AugPi, BenFrantzDale, Bluap, COGDEN, Charles Matthews, Crowsnest, Davehi1,Dudecon, Fropuff, J-Wiki, John, Linas, MFNickster, MarSch, Mashford, Mogaman, Moink, Mythealias, Oleg Alexandrov, Peter Harriman, Salih, Suspekt, Tobyeharris, 老陳, 37 anonymous edits

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Compressibility factor  Source: http://en.wikipedia.org/w/index.php?oldid=399016749  Contributors: 14 June, Aushulz, Deklund, Derryfraser, Dhollm, Ecorahul, Fraserlawson, Gmkung,Ilnyckyj, Jdpipe, JimVC3, Mbeychok, Mion, Mjpotter, Mythealias, Power.corrupts, Pushkar.gaikwad, RexxS, Rikki1989, Sachith v, Su-no-G, THEN WHO WAS PHONE?, Tim R, Turbojet,WilfriedC, 18 anonymous edits

Solenoidal vector field  Source: http://en.wikipedia.org/w/index.php?oldid=400649672  Contributors: Abar, AeroSpace, AugPi, Brews ohare, CBM, Catslash, Charles Matthews, Complexica,Crowsnest, Fredrik, Gareth McCaughan, Hellisp, Hpesoj00, JJ Harrison, KasugaHuang, MarSch, Michael Hardy, Oleg Alexandrov, Owlbuster, Plober, RDBury, Silverfish, Smack, The Anome,13 anonymous edits

Conservative vector field  Source: http://en.wikipedia.org/w/index.php?oldid=401196355  Contributors: Acyso, Allen McC., AugPi, Billlion, Brews ohare, CBM, Charles Matthews,Commentor, Complexica, Crowsnest, Dolphin51, Encephalon, Ferengi, Geometry guy, Hao2lian, Hephaestos, Hess88, Infovarius, Kar.ma, Keenan Pepper, Kotjze, MFNickster, MarSch, Mauxb,NOrbeck, Nistra, Oleg Alexandrov, PV=nRT, Paga19141, Patrick, Rabbanis, Siddhant, Silverfish, Spoon!, Sverdrup, Sławomir Biały, Terry Bollinger, That Guy, From That Show!, Trammel,Zvika, 27 anonymous edits

Laplacian vector field  Source: http://en.wikipedia.org/w/index.php?oldid=328503607  Contributors: AugPi, Boggie, CBM, Charles Matthews, MFNickster, MarSch, Oleg Alexandrov, RDBury,Silverfish, 3 anonymous edits

Stokes' law  Source: http://en.wikipedia.org/w/index.php?oldid=406253243  Contributors: Access Denied, Adiel lo, Awickert, Ben pcc, Biglovinb, Charles Matthews, Crowsnest, Cwmonroe,Donfbreed, Dougalc, Giftlite, Gogo Dodo, Jeodesic, JohnCD, JohnOwens, Kevmitch, Leszek Jańczuk, Looxix, Mbgopackgo160, Michael Hardy, Mindtrik, Mintleaf, Nickpowerz, Peterlewis,Rahnle, Retaggio, Robinh, Robthecar, Salih, Siddhant, Stan J Klimas, Steg55, Suffusion of Yellow, Syrthiss, Tagishsimon, The wub, Uvainio, Vishahu, Vossman, WaitingWang, Willgold,WingkeeLEE, Wolfkeeper, Xaven, Yevgeny Kats, 95 anonymous edits

Projection method (fluid dynamics)  Source: http://en.wikipedia.org/w/index.php?oldid=386344047  Contributors: Jitse Niesen, Koavf, Mdd, Michael Hardy, Salih, T-1000, Tobyeharris,Yeokaiwei, 6 anonymous edits

Mach number  Source: http://en.wikipedia.org/w/index.php?oldid=406730029  Contributors: Abdull, Abqwildcat, Abyab, Acroterion, Adamrush, Ahoerstemeier, Airboyd, Alexbrewer, AndreEngels, Anonymous Dissident, Arpingstone, Asambhav.x, Austin Hair, AxelBoldt, Axemanstan, B4hand, BQZip01, BW95, BadWolf42, Barnaby dawson, Bart133, Bdesham, BenFrantzDale,Bennylin, BesigedB, BilCat, Binksternet, Birddog165, Bjh21, Boothy443, Bryanholmberg, Can't sleep, clown will eat me, Carbuncle, Catgut, Charles Matthews, Chovain, Chrislk02, ChuckCarroll, Contratrombone64, Contrawiki, Conversion script, Crowsnest, Cutler, Danish Ali Khan, Davewho2, Dhaluza, DirkvdM, Dolphin51, Donkrodgers, DouglasHeld, Dougweller, EMBaero,Ellywa, Eranb, Ericg, Erik9, Fabometric, FocalPoint, Fredrik, Frencheigh, Fresheneesz, Freso, Garik, Garrisonroo, Giftlite, GregorB, Gugganij, Gups, Hede2000, Helwer7, HexaChord, Icairns,J.delanoy, Jaganath, Jason237, Jaydec, Jdpipe, Jimmy Slade, Jimp, Joseph Solis in Australia, Jusdafax, Jyril, Karada, Keta, Kwamikagami, Latics, LiamE, MS86, Magister Mathematicae,Malone23kid, Marc Venot, Marek69, Marsian, Matt Crypto, Mayooranathan, Mbeychok, Merovingian, MimirZero, MisterSheik, Mostafaardeshiri, Mpeisenbr, Mulad, Mythealias, Norandav,Nuno Tavares, Ocknock, Officially Mr X, Palapala, Patrick, Peterlin, Pibwl, Planetscared, Prkl75, Profoss, Ps ttf, Pureme, Qmark42, R'n'B, Ragged sail, Rchandra, ReallyNiceGuy, RenniePet,RexNL, Roadrunner, Roscoe x, ScriptedGhost, SeanMack, Shadowjams, Shawn81, SlaveToTheWage, Smyth, Soap, Someone42, Spiffy sperry, Synook, Tangotango, Tdadamemd, The Epopt,The1exile, TheRaytracer, Theon, Thunderbird2, Tide rolls, TigerShark, Tobias Hoevekamp, Topjetpilot1, Treesmill, Twinsday, Whitehatnetizen, Wilsonkd, Wimt, WojPob, Wolfkeeper,Wraithdart, XJamRastafire, Xyb, Zanimum, 266 anonymous edits

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Rheology  Source: http://en.wikipedia.org/w/index.php?oldid=407340805  Contributors: 195.93.33.xxx, Alex Bakharev, AndreiDukhin, AnnaFrance, Barticus88, Bcartolo, BiT, BillC, Cfp,Charles Matthews, Chemical Engineer, Colloid07, Colonies Chris, Conversion script, CorbinSimpson, Cutler, Cypgay, Danh, Daveh4h, Dhollm, DonQ1906, Fadesga, Fmorriso9, GCW50,GeoGreg, Giftlite, Glennwells, Gobonobo, Graham87, Greensburger, Heron, Hike395, Hydroli, Igodard, Isnow, Ivan Štambuk, Jarlaxl, José Gnudista, Kaytles, Kosebamse, Kwamikagami, Lightcurrent, LilHelpa, Logger9, Looxix, Marie Poise, Mark Christensen, Martarius, Mattisse, MaxPower, Mejor Los Indios, Michael Hardy, Nbarth, Nihiltres, Odysses, Pcg19, PhilKnight, RHSydnor,Retaggio, Rmhermen, Robinh, Rogermw, Rokmonkey, Roque345, Rune.welsh, Sivar, Solsikche, Spiegelberg88, Stay cool, Suicidalhamster, Superiorsarcaser, Taeedxy, Talkstosocks, Telente,Tiddly Tom, Tobias Hoevekamp, Um123abc, Uncle Milty, Vsmith, Wdanwatts, Woohookitty, Xis, Zencowboy27, 89 anonymous edits

Image Sources, Licenses and Contributors 68

Image Sources, Licenses and ContributorsImage:ConvectiveAcceleration.png  Source: http://en.wikipedia.org/w/index.php?title=File:ConvectiveAcceleration.png  License: Public Domain  Contributors: Ben pccImage:NSConvection.png  Source: http://en.wikipedia.org/w/index.php?title=File:NSConvection.png  License: Public Domain  Contributors: Ben pcc, JoelholdsworthFile:Isochore Zustandsänderung.png  Source: http://en.wikipedia.org/w/index.php?title=File:Isochore_Zustandsänderung.png  License: GNU Free Documentation License  Contributors:Original uploader was Pikarl at de.wikipediaImage:Diagramma generalizzato fattore di compressibilità.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Diagramma_generalizzato_fattore_di_compressibilità.jpg  License:Creative Commons Attribution-Sharealike 3.0  Contributors: User:AushulzImage:Compressibility Factor of Air 75-200 K.png  Source: http://en.wikipedia.org/w/index.php?title=File:Compressibility_Factor_of_Air_75-200_K.png  License: Public Domain Contributors: User:Power.corruptsImage:Compressibility Factor of Air 250 - 1000 K.png  Source: http://en.wikipedia.org/w/index.php?title=File:Compressibility_Factor_of_Air_250_-_1000_K.png  License: Public Domain Contributors: User:Power.corruptsImage:Irrotationalfield.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Irrotationalfield.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:AllenMcC.Image:Stokes sphere.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Stokes_sphere.svg  License: GNU Free Documentation License  Contributors: User:KraaiennestImage:FA-18 Hornet breaking sound barrier (7 July 1999).jpg  Source: http://en.wikipedia.org/w/index.php?title=File:FA-18_Hornet_breaking_sound_barrier_(7_July_1999).jpg  License:Public Domain  Contributors: Alison, Apalsola, Arnomane, Denniss, Didactohedron, Diego pmc, Dual Freq, Duesentrieb, Duffman, Hailey C. Shannon, Makthorpe, Paul Richter, RaySys,Raymond, Schekinov Alexey Victorovich, Svetovid, Werneuchen, 2 anonymous editsImage:Transsonic flow over airfoil 1.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Transsonic_flow_over_airfoil_1.gif  License: Public Domain  Contributors: Original uploaderwas Prkl75 at en.wikipediaImage:Transsonic flow over airfoil 2.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Transsonic_flow_over_airfoil_2.gif  License: Public Domain  Contributors: Original uploaderwas Prkl75 at en.wikipediaImage:Viscosity.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Viscosity.gif  License: Creative Commons Attribution-Sharealike 3.0  Contributors: User:AnynobodyImage:Laminar shear.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Laminar_shear.svg  License: unknown  Contributors: User:StanneredImage:Laminar shear flow.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Laminar_shear_flow.svg  License: Public Domain  Contributors: Mdd, StanneredFile:Viscous regimes chart.png  Source: http://en.wikipedia.org/w/index.php?title=File:Viscous_regimes_chart.png  License: Public Domain  Contributors: Dhollm, 2 anonymous editsImage:University of Queensland Pitch drop experiment-6-2.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:University_of_Queensland_Pitch_drop_experiment-6-2.jpg  License:GNU Free Documentation License  Contributors: John MainstoneFile:Viscosity video science museum.ogv  Source: http://en.wikipedia.org/w/index.php?title=File:Viscosity_video_science_museum.ogv  License: GNU Free Documentation License Contributors: geniFile:Air dry dynamic visocity on pressure temperature.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Air_dry_dynamic_visocity_on_pressure_temperature.svg  License: CreativeCommons Attribution-Sharealike 3.0  Contributors: User:Stan J KlimasFile:Dynamic Viscosity of Water.png  Source: http://en.wikipedia.org/w/index.php?title=File:Dynamic_Viscosity_of_Water.png  License: Creative Commons Attribution 3.0  Contributors:User:WilfriedCImage:Drop 0.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Drop_0.jpg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: Henningklevjer, Nagy,Rgoodermote, Roomba, 2 anonymous editsImage:Runny hunny.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Runny_hunny.jpg  License: Public Domain  Contributors: Anna reg, Bdk, Cookie, EvaK, Gveret Tered, Hiart,Jonik, MattesImage:PeanutButter.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:PeanutButter.jpg  License: GNU Free Documentation License  Contributors: User:PiccoloNamekImage:Glassviscosityexamples.png  Source: http://en.wikipedia.org/w/index.php?title=File:Glassviscosityexamples.png  License: Public Domain  Contributors: AfluegelImage:Poly Pet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Poly_Pet.jpg  License: Public Domain  Contributors: User:Logger9Image:Cellulose strand.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Cellulose_strand.jpg  License: GNU Free Documentation License  Contributors: User:Laghi.lImage:Sol-gel silicate bonds.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sol-gel_silicate_bonds.svg  License: Public Domain  Contributors: User:SquidoniusImage:PileDriving.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:PileDriving.jpg  License: Creative Commons Attribution 2.5  Contributors: Original uploader was Argyriou aten.wikipediaFile:Heme.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Heme.svg  License: Public Domain  Contributors: User:Lennert B

License 69

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/