A Course in Fluid Mechanicswith Vector Field TheorybyDennis C. PrieveDepartment of Chemical EngineeringCarnegie Mellon UniversityPittsburgh, PA 15213An electronic version of this book in Adobe PDF format was made available tostudents of 06-703, Department of Chemical Engineering,Carnegie Mellon University, Fall, 2000.Copyright 2000 by Dennis C. Prieve06-703 1 Fall, 2000Copyright 2000 by Dennis C. PrieveTable of ContentsALGEBRA OF VECTORS AND TENSORS ..............................................................................................................................1VECTOR MULTIPLICATION...................................................................................................................................................... 1Definition of Dyadic Product..............................................................................................................................................2DECOMPOSITION INTO SCALAR COMPONENTS.................................................................................................................... 3SCALAR FIELDS........................................................................................................................................................................... 3GRADIENT OF A SCALAR........................................................................................................................................................... 4Geometric Meaning of the Gradient..................................................................................................................................6Applications of Gradient .....................................................................................................................................................7CURVILINEAR COORDINATES................................................................................................................................................... 7Cylindrical Coordinates .....................................................................................................................................................7Spherical Coordinates.........................................................................................................................................................8DIFFERENTIATION OF VECTORS W.R.T. SCALARS................................................................................................................ 9VECTOR FIELDS......................................................................................................................................................................... 11Fluid Velocity as a Vector Field......................................................................................................................................11PARTIAL & MATERIAL DERIVATIVES.................................................................................................................................. 12CALCULUS OF VECTOR FIELDS............................................................................................................................................14GRADIENT OF A SCALAR (EXPLICIT).................................................................................................................................... 14DIVERGENCE, CURL, AND GRADIENT.................................................................................................................................... 16Physical Interpretation of Divergence............................................................................................................................16Calculation of .v in R.C.C.S.........................................................................................................................................16Evaluation of v and v in R.C.C.S. ...........................................................................................................................18Evaluation of .v, v and v in Curvilinear Coordinates ...................................................................................19Physical Interpretation of Curl ........................................................................................................................................20VECTOR FIELD THEORY...........................................................................................................................................................22DIVERGENCE THEOREM........................................................................................................................................................... 23Corollaries of the Divergence Theorem..........................................................................................................................24The Continuity Equation...................................................................................................................................................24Reynolds Transport Theorem............................................................................................................................................26STOKES THEOREM.................................................................................................................................................................... 27Velocity Circulation: Physical Meaning .......................................................................................................................28DERIVABLE FROM A SCALAR POTENTIAL........................................................................................................................... 29THEOREM III.............................................................................................................................................................................. 31TRANSPOSE OF A TENSOR, IDENTITY TENSOR.................................................................................................................... 31DIVERGENCE OF A TENSOR..................................................................................................................................................... 32INTRODUCTION TO CONTINUUM MECHANICS*.............................................................................................................34CONTINUUM HYPOTHESIS...................................................................................................................................................... 34CLASSIFICATION OF FORCES................................................................................................................................................... 36HYDROSTATIC EQUILIBRIUM................................................................................................................................................37FLOW OF IDEAL FLUIDS ..........................................................................................................................................................37EULER'S EQUATION.................................................................................................................................................................. 38KELVIN'S THEOREM.................................................................................................................................................................. 41IRROTATIONAL FLOW OF AN INCOMPRESSIBLE FLUID..................................................................................................... 42Potential Flow Around a Sphere.....................................................................................................................................45d'Alembert's Paradox.........................................................................................................................................................5006-703 2 Fall, 2000Copyright 2000 by Dennis C. PrieveSTREAM FUNCTION...................................................................................................................................................................53TWO-D FLOWS.......................................................................................................................................................................... 54AXISYMMETRIC FLOW (CYLINDRICAL)............................................................................................................................... 55AXISYMMETRIC FLOW (SPHERICAL).................................................................................................................................... 56ORTHOGONALITY OF =CONST AND =CONST................................................................................................................... 57STREAMLINES, PATHLINES AND STREAKLINES................................................................................................................. 57PHYSICAL MEANING OF STREAMFUNCTION....................................................................................................................... 58INCOMPRESSIBLE FLUIDS........................................................................................................................................................ 60VISCOUS FLUIDS ........................................................................................................................................................................62TENSORIAL NATURE OF SURFACE FORCES.......................................................................................................................... 62GENERALIZATION OF EULER'S EQUATION........................................................................................................................... 66MOMENTUM FLUX................................................................................................................................................................... 68RESPONSE OF ELASTIC SOLIDS TO UNIAXIAL STRESS....................................................................................................... 70RESPONSE OF ELASTIC SOLIDS TO PURE SHEAR................................................................................................................. 72GENERALIZED HOOKE'S LAW................................................................................................................................................. 73RESPONSE OF A VISCOUS FLUID TO PURE SHEAR............................................................................................................... 75GENERALIZED NEWTON'S LAW OF VISCOSITY.................................................................................................................... 76NAVIER-STOKES EQUATION................................................................................................................................................... 77BOUNDARY CONDITIONS........................................................................................................................................................ 78EXACT SOLUTIONS OF N-S EQUATIONS ...........................................................................................................................80PROBLEMS WITH ZERO INERTIA........................................................................................................................................... 80Flow in Long Straight Conduit of Uniform Cross Section..........................................................................................81Flow of Thin Film Down Inclined Plane ........................................................................................................................84PROBLEMS WITH NON-ZERO INERTIA.................................................................................................................................. 89Rotating Disk*....................................................................................................................................................................89CREEPING FLOW APPROXIMATION...................................................................................................................................91CONE-AND-PLATE VISCOMETER........................................................................................................................................... 91CREEPING FLOW AROUND A SPHERE (Re0).................................................................................................................... 96Scaling..................................................................................................................................................................................97Velocity Profile....................................................................................................................................................................99Displacement of Distant Streamlines ........................................................................................................................... 101Pressure Profile................................................................................................................................................................ 103CORRECTING FOR INERTIAL TERMS.................................................................................................................................... 106FLOW AROUND CYLINDER AS RE0................................................................................................................................. 109BOUNDARY-LAYER APPROXIMATION............................................................................................................................ 110FLOW AROUND CYLINDER AS Re ................................................................................................................................ 110MATHEMATICAL NATURE OF BOUNDARY LAYERS........................................................................................................ 111MATCHED-ASYMPTOTIC EXPANSIONS.............................................................................................................................. 115MAES APPLIED TO 2-D FLOW AROUND CYLINDER...................................................................................................... 120Outer Expansion .............................................................................................................................................................. 120Inner Expansion............................................................................................................................................................... 120Boundary Layer Thickness............................................................................................................................................. 120PRANDTLS B.L. EQUATIONS FOR 2-D FLOWS................................................................................................................... 120ALTERNATE METHOD: PRANDTLS SCALING THEORY.................................................................................................. 120SOLUTION FOR A FLAT PLATE............................................................................................................................................ 120Time Out: Flow Next to Suddenly Accelerated Plate................................................................................................ 120Time In: Boundary Layer on Flat Plate....................................................................................................................... 120Boundary-Layer Thickness ............................................................................................................................................ 120Drag on Plate ................................................................................................................................................................... 12006-703 3 Fall, 2000Copyright 2000 by Dennis C. PrieveSOLUTION FOR A SYMMETRIC CYLINDER......................................................................................................................... 120Boundary-Layer Separation.......................................................................................................................................... 120Drag Coefficient and Behavior in the Wake of the Cylinder................................................................................... 120THE LUBRICATION APPROXIMATION............................................................................................................................. 157TRANSLATION OF A CYLINDER ALONG A PLATE............................................................................................................ 163CAVITATION............................................................................................................................................................................ 166SQUEEZING FLOW.................................................................................................................................................................. 167REYNOLDS EQUATION........................................................................................................................................................... 171TURBULENCE............................................................................................................................................................................ 176GENERAL NATURE OF TURBULENCE.................................................................................................................................. 176TURBULENT FLOW IN PIPES................................................................................................................................................. 177TIME-SMOOTHING.................................................................................................................................................................. 179TIME-SMOOTHING OF CONTINUITY EQUATION.............................................................................................................. 180TIME-SMOOTHING OF THE NAVIER-STOKES EQUATION................................................................................................ 180ANALYSIS OF TURBULENT FLOW IN PIPES........................................................................................................................ 182PRANDTLS MIXING LENGTH THEORY............................................................................................................................... 184PRANDTLS UNIVERSAL VELOCITY PROFILE................................................................................................................. 187PRANDTLS UNIVERSAL LAW OF FRICTION....................................................................................................................... 189ELECTROHYDRODYNAMICS............................................................................................................................................... 120ORIGIN OF CHARGE................................................................................................................................................................. 120GOUY-CHAPMAN MODEL OF DOUBLE LAYER.................................................................................................................. 120ELECTROSTATIC BODY FORCES........................................................................................................................................... 120ELECTROKINETIC PHENOMENA.......................................................................................................................................... 120SMOLUCHOWSKI'S ANALYSIS (CA. 1918)............................................................................................................................. 120ELECTRO-OSMOSIS IN CYLINDRICAL PORES...................................................................................................................... 120ELECTROPHORESIS................................................................................................................................................................. 120STREAMING POTENTIAL....................................................................................................................................................... 120SURFACE TENSION................................................................................................................................................................. 120MOLECULAR ORIGIN.............................................................................................................................................................. 120BOUNDARY CONDITIONS FOR FLUID FLOW...................................................................................................................... 120INDEX........................................................................................................................................................................................... 21106-703 1 Fall, 2000Copyright 2000 by Dennis C. PrieveAlgebra of Vectors and TensorsWhereasheatandmassarescalars,fluidmechanicsconcernstransportofmomentum,whichisavector.Heat and mass fluxes are vectors, momentum flux is a tensor.Consequently, the mathematicaldescriptionoffluidflowtendstobemoreabstractandsubtlethanforheatandmasstransfer.Inanefforttomakethestudentmorecomfortablewiththemathematics,wewillstartwithareviewofthealgebraofvectorsandanintroductiontotensorsanddyads.Abriefreviewofvectoradditionandmultiplication can be found in Greenberg, pages 132-139.Scalar - a quantity having magnitude but no direction (e.g. temperature, density)Vector - (a.k.a. 1st rank tensor) a quantity having magnitude and direction (e.g. velocity, force,momentum)(2nd rank)Tensor-aquantityhavingmagnitudeandtwodirections(e.g.momentumflux,stress)VECTOR MULTIPLICATIONGiven two arbitrary vectorsa andb, there are three types of vector productsare defined:Notation Result DefinitionDot Product a.b scalar ab cosCross Product ab vector ab|sin|n where is an interior angle (0 ) andn is a unit vector which is normal to botha andb.Thesense of n is determined from the "right-hand-rule"Dyadic Product ab tensor

Greenberg, M.D., Foundations Of Applied Mathematics, Prentice-Hall, 1978. The right-hand rule:withthefingersoftherighthandinitiallypointinginthedirectionofthefirstvector,rotatethefingerstopointinthedirectionofthesecondvector;thethumbthenpointsinthedirection with the correct sense.Of course, the thumb should have been normal to the plane containingboth vectors during the rotation.In the figure above showing a and b, ab is a vector pointinginto thepage, while ba points out of the page.06-703 2 Fall, 2000Copyright 2000 by Dennis C. PrieveIn the above definitions, we denote the magnitude (or length) of vector a by the scalara.Boldface willbeusedtodenotevectorsanditalicswillbeusedtodenotescalars.Second-ranktensorswillbedenoted with double-underlined boldface; e.g. tensor T.Definition of Dyadic ProductReference:AppendixBfromHappel&Brenner.TheworddyadcomesfromGreek:dymeans two while ad means adjacent.Thus the name dyad refers to the way in which this product isdenoted:thetwovectorsarewrittenadjacenttooneanotherwithnospaceorotheroperatorinbetween.There is no geometrical picture that I can draw which will explain what a dyadic product is.It's bestto think of the dyadic product as a purely mathematical abstraction having some very useful properties:Dyadic Product ab - that mathematical entity which satisfies the following properties (where a,b, v, and w are any four vectors):1. ab.v = a(b.v) [which has the direction ofa; note thatba.v =b(a.v) which has the direction ofb.Thus ab ba since they dont produce the same result on post-dotting with v.]2. v.ab = (v.a)b [thus v.ab ab.v]3. abv = a(bv) which is another dyad4. vab = (va)b5. ab:vw = (a.w)(b.v) which is sometimes known as theinner-outer product or the double-dotproduct.*6. a(v+w) = av+aw (distributive for addition)7. (v+w)a = va+wa8. (s+t)ab=sab+tab(distributiveforscalarmultiplication--alsodistributivefordotandcrossproduct)9. sab = (sa)b = a(sb)

Happel, J., & H. Brenner, Low Reynolds Number Hydrodynamics, Noordhoff, 1973.* Brenner defines this as (a.v)(b.w).Althoughthetwodefinitionsarenotequivalent,eithercanbeused--aslongasyouareconsistent.Inthesenotes,wewilladoptthedefinitionaboveandignoreBrenner's definition.06-703 3 Fall, 2000Copyright 2000 by Dennis C. PrieveDECOMPOSITION INTO SCALAR COMPONENTSThree vectors (say e1, e2, and e3) are said to belinearly independent if none can be expressedas a linear combination of the other two (e.g.i, j, andk).GivensuchasetofthreeLIvectors,anyvector (belonging to E3) can be expressed as a linear combination of this basis:v = v1e1 + v2e2 + v3e3wheretheviarecalledthescalarcomponentsofv.Usually,forconvenience,wechooseorthonormal vectors as the basis:ei.ej = ij = 10 if if i ji jRSTalthough this is not necessary. ijiscalledtheKroneckerdelta.Justasthefamiliardotandcrossproducts can written in terms of the scalar components, so can the dyadic product:vw = (v1e1+v2e2+v3e3)(w1e1+w2e2+w3e3)= (v1e1)(w1e1)+(v1e1)(w2e2)+ ...= v1w1e1e1+v1w2e1e2+ ... (nine terms)where theeiejareninedistinctunitdyads.Wehaveappliedthedefinitionofdyadicproducttoperform these two steps: in particular items 6, 7 and 9 in the list above.More generally any nth rank tensor (in E3) can be expressed as a linear combination of the 3n unit n-ads.For example, ifn=2, 3n=9 and ann-ad is a dyad.Thus a generalsecond-rank tensorcanbedecomposed as a linear combination of the 9 unit dyads:T = T11e1e1+T12e1e2+ ... = i=1,3j=1,3TijeiejAlthough a dyad (e.g.vw)isanexampleofasecond-ranktensor,notall2nd rank tensorsTcanbeexpressedasadyadicproductoftwovectors.Toseewhy,notethatageneralsecond-ranktensorhasninescalarcomponentswhichneednotberelatedtooneanotherinanyway.Bycontrast, the 9 scalar components of dyadic product above involve onlysixdistinct scalars (the 3 components of v plus the 3 components of w).After a while you get tired of writing the summation signs and limits.So anabbreviation was adopted whereby repeated appearance of an index implies summation over the threeallowable values of that index:T = Tijeiej06-703 4 Fall, 2000Copyright 2000 by Dennis C. PrieveThis is sometimes called the Cartesian (implied) summation convention.SCALAR FIELDSSuppose I have some scalar function of position (x,y,z) which is continuously differentiable, thatisf = f(x,y,z)andf/x,f/y,andf/zexistandarecontinuousthroughoutsome3-Dregioninspace.Thisfunction is called ascalar field.Now considerf at a second point which is differentially close to thefirst.Thedifferenceinfbetweenthesetwopointsiscalled the total differential of f:f(x+dx,y+dy,z+dz) - f(x,y,z) dfFor any continuous function f(x,y,z), df is linearly relatedtothepositiondisplacements,dx,dyanddz.ThatlinearrelationisgivenbytheChainRuleofdifferentiation:dffx dxfy dyfz dz + +Instead of defining position using a particular coordinatesystem, we could also define position using a position vector r:r i j k + + x y zThe scalar field can be expressed as a function of a vector argument, representing position, instead of aset of three scalars:f = f(r)Consider an arbitrary displacement away from the point r, which we denote as dr to emphasize that themagnitude|dr|ofthisdisplacementissufficientlysmallthatf(r)canbelinearizedasafunctionofposition around r.Then the total differential can be written as06-703 5 Fall, 2000Copyright 2000 by Dennis C. Prievedf f d f + ( ) ( ) r r rGRADIENT OF A SCALARWe are now is a position to define an important vector associatedwiththisscalarfield.Thegradient(denotedasf)isdefinedsuchthatthedotproductofitandadifferentialdisplacementvector gives the total differential:df d f r.EXAMPLE: Obtain an explicit formula for calculating the gradient in Cartesian* coordinates.Solution: r = xi + yj + zkr+dr = (x+dx)i + (y+dy)j + (z+dz)ksubtracting: dr = (dx)i + (dy)j + (dz)kf = (f)xi + (f)yj + (f)zkdr.f = [(dx)i + ...].[(f)xi + ...]df = (f)xdx + (f)ydy + (f)zdz (1)Using the Chain rule: df = (f/x)dx + (f/y)dy + (f/z)dz (2)According to the definition of the gradient, (1) and (2) are identical.Equating them and collecting terms:[(f)x-(f/x)]dx + [(f)y-(f/y)]dy + [(f)z-(f/z)]dz = 0Think ofdx, dy, anddz as three independent variables which can assume an infinite number of values,even though they must remain small.The equality above must hold for all values of dx, dy, anddz.Theonly way this can be true is if each individual term separately vanishes:**

*Named after French philosopher and mathematician Ren Descartes (1596-1650), pronounced "day-cart", who first suggested plotting f(x) on rectangular coordinates** For any particular choice ofdx,dy, anddz,wemightobtainzerobycancellationofpositiveandnegative terms.However a small change in one of the three without changing the other two would causethesumtobenonzero.Toensureazero-sumforallchoices,wemustmakeeachtermvanishindependently.06-703 6 Fall, 2000Copyright 2000 by Dennis C. PrieveSo (f)x = f/x, (f)y = f/y, and (f)z = f/z,leaving + + ffxfyfzi j kOther ways to denote the gradient include:f = gradf = f/rGeometric Meaning of the Gradient1) direction: f(r)isnormaltothef=constsurfacepassingthroughthepointrinthedirectionofincreasing f.falso points in the direction of steepest ascent of f.2) magnitude:|f|istherateofchangeoffwithdistance along this directionWhat do we mean by an "f=const surface"?Consider anexample.Example:Supposethesteadystatetemperatureprofilein some heat conduction problem is given by:T(x,y,z) = x2 + y2 + z2PerhapsweareinterestedinTatthepoint(3,3,3)where T=27.T is normal to the T=const surface:x2 + y2 + z2 = 27which is a sphere of radius27 .Proof of 1).Let's use the definition to show that these geometric meanings are correct.df = dr.f

A vertical bar in the left margin denotes material which (in the interest of time) will be omitted from thelecture.06-703 7 Fall, 2000Copyright 2000 by Dennis C. PrieveConsideranarbitraryf.Aportionofthef=constsurfacecontaining the point r is shown in the figure at right.Choose adr which lies entirely on f=const.In other words, the surfacecontains both r and r+dr, sof(r) = f(r+dr)anddf = f(r+dr)-f(r) = 0Substituting this into the definition of gradient:df = 0 = dr.f = 1dr11f1cosSince 1dr1 and 1f1 are in general not zero, we are forcedto the conclusion that cos=0 or =90.This means that f is normal to dr which lies in the surface.2) can be proved in a similar manner: choose dr to be parallel tof.Doesf point toward higher orlower values of f?Applications of Gradientfind a vector pointing in the direction of steepest ascent of some scalar fielddetermine a normal to some surface (needed to apply b.c.s liken.v=0foraboundarywhichisimpermeable)determine the rate of change along some arbitrary direction:ifn is a unit vector pointing along somepath, thenn. ffsis the rate of change off with distance (s) along this path given byn. f sis called thedirectedderivative of f.CURVILINEAR COORDINATESInprinciple,allproblemsinfluidmechanicsandtransportcouldbesolvedusingCartesiancoordinates.Often,however,wecantakeadvantageofsymmetryinaproblembyusinganothercoordinate system.This advantage takes the form of a reduction in the number of independent variables(e.g. PDE becomes ODE).A familiar example of a non-Cartesian coordinate system is:06-703 8 Fall, 2000Copyright 2000 by Dennis C. PrieveCylindrical Coordinatesr = (x2+y2)1/2x = rcos = tan-1(y/x) y = rsinz = z z = zVectors are decomposed differently.Instead ofin R.C.C.S.: v = vxi + vyj + vzkin cylindrical coordinates, we writein cyl. coords.: v = vrer + ve + vzezwhere er, e, andez are new unit vectors pointing ther, andz directions.We also have a differentset of nine unit dyads for decomposing tensors:erer, ere, erez, eer, etc.LiketheCartesianunitvectors,theunitvectorsincylindricalcoordinatesformanorthonormalsetofbasis vectors for E3.Unlike Cartesian unit vectors, the orientation of er ande depend on position.Inother words:er = er()e = e()06-703 9 Fall, 2000Copyright 2000 by Dennis C. PrieveSpherical CoordinatesSphericalcoordinates(r,,)aredefinedrelativetoCartesiancoordinatesassuggestedinthefigures above (two views of the same thing).The green surface is the xy-plane, the red surface is thexz-plane, while the blue surface (at least in the left image) is the yz-plane.These three planes intersect attheorigin(0,0,0),whichliesdeeperintothepagethan(1,1,0).Thestraightredline,drawnfromtheorigin to the point (r,,) has length r, The angle is the angle the red line makes with thez-axis (thered circular arc labelled has radius r and is subtended by the angle).The angle (measured in thexy-plane)istheanglethesecondblueplane(actuallyitsonequadrantofadisk)makeswiththexy-plane (red).This plane which is a quadrant of a disk is a=const surface: all points on this plane havethe same coordinate.The second red (circular) arc labelled is also subtended by the angle .

This particular figure was drawn using r = 1, = /4 and = /3.06-703 10 Fall, 2000Copyright 2000 by Dennis C. PrieveAnumberofother=constplanesareshowninthefigureatright,alongwithasphereofradiusr=1.Alltheseplanesintersect along the z-axis, which also passesthrough the center of the sphere.( )2 2 21 2 21sin cossin sin tancos tanx r r x y zy r x y zz r y x + + +| ` + . , The position vector in spherical coordinatesis given byr = xi+yj+zk = r er(,)In this case all three unit vectors depend onposition:er = er(,), e = e(,), and e = e()where er is the unit vector pointing the direction of increasingr, holding and fixed;eistheunitvector pointing the direction of increasing , holding r and fixed; ande is the unit vector pointing thedirection of increasing , holding r and fixed.Theseunitvectorsareshowninthefigureatright.Notice that the surface=const is a plane containing thepoint r itself, the projection of the point onto the xy-planeand the origin.The unit vectors er and e lie in this planeaswellastheCartesianunitvectork(sometimesdenoted ez).If we tilt this =const planeinto the plane of the page (as in the sketch at left), we can more easily seethe relationship between these three unit vectors:( ) ( ) cos sinz r e e eThisisobtainedbydeterminedfromthegeometryoftherighttriangleinthe figure at left.When any of the unit vectors is position dependent, wesay the coordinates are:kunit circle on = constsurface06-703 11 Fall, 2000Copyright 2000 by Dennis C. Prievecurvilinear - at least one of the basis vectors is position dependentThiswillhavesomeprofoundconsequenceswhichwewillgettoshortly.Butfirst,weneedtotaketime-out to define:DIFFERENTIATION OF VECTORS W.R.T. SCALARSSuppose we have a vector v which depends on the scalar parameter t:v = v(t)Forexample,thevelocityofasatellitedependsontime.Whatdowemeanbythederivativeofavector with respect to a scalar.As in the Fundamental Theorem of Calculus, we define the derivativeas: ddtt t tt tv v v=( + ) - ( )lim RSTUVW0Note that dv/dt is also a vector.EXAMPLE: Compute der/d in cylindrical coordinates.Solution: From the definition of the derivative:ddr r r re e e e + RSTUVW RSTUVW lim( ) ( )lim 0 0Since the location of the tail of a vector is not partofthedefinitionofavector,let'smovebothvectorstotheorigin(keepingtheorientationfixed).Using the parallelogram law, we obtain thedifference vector.Its magnitude is:e er r( ) ( ) sin + 22Its direction is parallel to e(+/2), so:e e er r( ) ( ) sin + + 22 2e jRecalling that sinx tends to x as x0, we have06-703 12 Fall, 2000Copyright 2000 by Dennis C. Prievelim ( ) ( ) + 0e e er r l q chDividing this by , we obtain the derivative:der/d = eSimilarly, de/d = -erOne important application of differentiation with respect to ascalar is the calculation of velocity, given position as a function oftime.In general, if the position vector is known, then the velocitycan be calculated as the rate of change in position:r = r(t)v = dr/dtSimilarly,theaccelerationvectoracanbecalculatedasthederivative of the velocity vector v:a = dv/dtEXAMPLE:Giventhetrajectoryofanobjectincylindrical coordinatesr = r(t), = (t), and z = z(t)Find the velocity of the object.Solution:First,weneedtoexpressrinintermsoftheunitvectorsincylindricalcoordinates.Usingthefigureatright, we note by inspection that*r(r,,z) = rer() + zezNow we can apply the Chain Rule:

*Recalling that r = xi + yj + zk in Cartesian coordinates, you might be tempted to write r = rer +e +zez in cylindrical coordinates. Of course, this temptation gives the wrong result (in particular, the unitsof length in the second term are missing).06-703 13 Fall, 2000Copyright 2000 by Dennis C. Prievedrdr dzdz dr r d dzz rzr ddrr zr r zrr r re e ee e ee FHGIKJ +FHGIKJ +FHGIKJ + +, , ,b gDividing by dt, we obtain the velocity:vre e e + +ddtdrtdtr d tdtdz tdtvrv vzr zb g b g b g VECTOR FIELDSAvectorfieldisdefinedjustlikeascalarfield,exceptthatit'savector.Namely,avectorfieldisaposition-dependent vector:v = v(r)Commonexamplesofvectorfieldsincludeforcefields,likethegravitationalforceoranelectrostaticforce field.Of course, in this course, the vector field of greatest interest is:Fluid Velocity as a Vector FieldConsider steady flow around a submerged object.What do we mean by fluid velocity?Thereare two ways to measure fluid velocity.First, we could add tracer particles to the flow and measure thepositionofthetracerparticlesasafunctionoftime;differentiatingpositionwithrespecttotime,wewould obtain the velocity.A mathematical tracer particle is called a material point:Material point - fluid element - a given set of fluid molecules whose location may change withtime.

Actually, this only works for steady flows.In unsteady flows, pathlines, streaklines and streamlinesdiffer (see Streamlines, Pathlines and Streaklines on page 65).Inamolecular-leveldescriptionofgasesorliquids,evennearbymoleculeshavewidelydifferentvelocitieswhichfluctuatewithtimeasthemoleculesundergocollisions.Wewillreconcilethemolecular-leveldescriptionwiththemorecommoncontinuumdescriptioninChapter4.Fornow,wejuststatethatbylocationofamaterialpointwemeanthelocationofthecenterofmassofthemolecules.The point needs to contain a statistically large number of molecules so thatr(t) convergesto a smooth continuous function.06-703 14 Fall, 2000Copyright 2000 by Dennis C. PrieveSuppose the trajectory of a material point is given by:r = r(t)Then the fluid velocity at any time is vr ddt(3)A second way to measure fluid velocity is similar to the bucket-and-stopwatch method.We measurethe volume of fluid crossing a surface per unit time:n v.RSTUVW lim aqa 0whereaistheareaofasurfaceelementhavingaunitnormal n andq is the volumetric flowrate of fluid crossinga in the direction of n.When a is small enough so that this quotient has convergedin a mathematical sense and a is small enough so that the surface is locally planar so we can denote itsorientation by a unit normal n, we can replace a byda and q by dq and rewrite this definition as:dq = n.v da (4)Thisisparticularlyconvenienttocomputethevolumetricflowrateacrossanarbitrarycurvedsurface, given the velocity profile.We just have tosum up the contribution from each surface element:Q daA zn v.PARTIAL & MATERIAL DERIVATIVESLet f = f(r,t)represent some unsteady scalar field (e.g. the unsteady temperature profile inside a moving fluid).Thereare two types of time derivatives of unsteady scalar fields which we will find convenient to define.In theexample in which f represents temperature, these two time derivatives correspond to the rate of change(denotedgenericallyasdf/dt)measuredwithathermometerwhicheitherisheldstationaryinthemoving fluid or drifts along with the local fluid.partial derivative - rate of change at a fixed spatial point:06-703 15 Fall, 2000Copyright 2000 by Dennis C. Prievedf dft dt | `

. ,r 0where the subscript dr=0 denotes that we are evaluating the derivative along a path* onwhichthespatialpointrisheldfixed.Inotherwords,thereisnodisplacementinposition during the time intervaldt.As time proceeds, different material points occupythe spatial point r.material derivative (a.k.a. substantial derivative) - rate of change within a particularmaterialpoint (whose spatial coordinates vary with time):d dtDf dfDt dt| `

. ,r vwhere the subscript dr = v dt denotes that a displacement in position (corresponding tothemotionofthevelocity)occurs:herevdenotesthelocalfluidvelocity.Astimeproceeds,themovingmaterialoccupiesdifferentspatialpoints,sorisnotfixed.Inother words, we are following along with the fluid as we measure the rate of change off.A relation between these two derivatives can be derived using a generalized vectorial form of the ChainRule.Firstrecallthatforsteady(independentoft)scalarfields,theChainRulegivesthetotaldifferential (in invariant form) asdf f d f d f + r r r r b g b g.When t is a variable, we just add another contribution to the total differential which arises from changesin t, namely dt.The Chain Rule becomesdf f d t dt f tftdt d f + + + r r r r , , b g b g.The first term has the usual Chain-Rule form for changes due to a scalar variable; the second term giveschanges due to a displacement in vectorial position r.Dividing bydt holdingR fixed yields the materialderivative:

*BypathImeanaconstraintamongtheindependentvariables,whichinthiscasearetimeandposition (e.g.x,y,z and t).Forexample,Imightvaryoneoftheindependentvariables(e.g.x)whileholdingtheothersfixed.Alternatively,Imightvaryoneoftheindependentvariables(e.g.t)whileprescribingsomerelatedchangesintheothers(e.g.x(t),y(t)andz(t)).Inthelattercase,Iamprescribing (in parametric form) a trajectory through space, hence the name path.06-703 16 Fall, 2000Copyright 2000 by Dennis C. Prieve1d dt d dt d dtDf df f dt dfDt dt t dt dt | ` | ` | ` + . , . , . ,r v r v r vvr.But (dr/dt) is just v, leaving:DfDtftf + v.Thisrelationshipholdsforatensorofanyrank.Forexample,thematerialderivativeofthevelocityvector is the acceleration a of the fluid, and it can be calculated from the velocity profile according toav vv v + DDt t.We will define v in the next section.Calculus of Vector FieldsJust like there were three kinds of vector multiplication which can be defined, there are three kindsof differentiation with respect to position.Shortly,wewillprovideexplicitdefinitionsofthesequantitiesintermsofsurfaceintegrals.Letmeintroducethistypeofdefinitionusingamorefamiliarquantity:GRADIENT OF A SCALAR (EXPLICIT)Recall the previous definition for gradient:f = f (r): df = dr.f (implicit defn of f )Such an implicit definition is like definingf (x) as that function associated with f(x) which yields:f = f (x): df = (dx) f (implicit defn of f ' )Anequivalent,butexplicit,definitionofderivativeisprovidedbytheFundamentalTheoremoftheCalculus:f xxf x x f xxdfdx + RSTUVW ( )lim( ) ( ) 0(explicit defn of f ' )We can provide an analogous definition of fNotation ResultDivergence .v scalarCurl v vectorGradient v tensor06-703 17 Fall, 2000Copyright 2000 by Dennis C. Prieve RS|T|UV|W|zfV VfdaAlim01n (explicit defn of f )where f= any scalar fieldA=asetofpointswhichconstitutesanyclosedsurfaceenclosingthepointrat which f is to be evaluatedV = volume of region enclosed by Ada = area of a differential element (subset) ofAn = unit normal toda, pointingout of regionenclosed by Alim (V0) = limit asall dimensions ofA shrink to zero (in other words,A collapses about thepoint at which f is to be defined.)What is meant by this surface integral?Imagine A to be the skin of a potato.To compute the integral: 1) Carve the skin into a number of elements.Each element must be sufficiently small so that element can be considered planar (i.e. n is practically constant over the element) f is practically constant over the element2) For each element of skin, compute nf da3) Sum yields integralThis same type of definition can be used for each of the three spatial derivatives of a vector field:DIVERGENCE, CURL, AND GRADIENTDivergence RS|T|UV|W|z. .v n vlimV VdaA01Curl RS|T|UV|W|zv n vlimV VdaA0106-703 18 Fall, 2000Copyright 2000 by Dennis C. PrieveGradient RS|T|UV|W|zv nvlimV VdaA01Physical Interpretation of DivergenceLet the vector field v = v(r) represent the steady-state velocity profile in some 3-D region of space.What is the physical meaning of .v?n.vda =dq=volumetricflowrateoutthroughda(cm3/s).Thisquantityispositiveforoutflow and negative for inflow.A n.v da = net volumetric flowrate out of enclosed volume (cm3/s).This is also positive fora net outflow and negative for a net inflow. (1/V) A n.v da = flowrate out per unit volume (s-1).v = > A B