A STUDY ON APPLICATION OF MATHEMATICAL FLUID MECHANICS · A STUDY ON APPLICATION OF MATHEMATICAL FLUID MECHANICS Daruri Venugopal Research Scholar, OPJS University, Churu, Rajasthan

Airo International Research Journal

Volume XV, ISSN: 2320-3714 January, 2018 Impact Factor 0.75 to 3.19 UGC Approval Number 63012

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A STUDY ON APPLICATION OF MATHEMATICAL FLUID

MECHANICS

Daruri Venugopal

Research Scholar, OPJS University, Churu, Rajasthan

Dr. Ashwani Kumar Nagpal

Assistant Professor, OPJS University, Churu, Rajasthan

ABSTRACT

Fluid Mechanics is the train inside the wide field of connected mechanics worried about the conduct of

fluids and gasses in motion or at rest. All things considered it incorporates a huge range of issues that may

fluctuate from substantial scale geophysical flows to the little scale investigation of blood flow in vessels.

Inside the geophysical flows, the modeling of sea waves is of incredible interest for a few fields of

science and engineering, i.e., oceanography, submerged acoustics, waterfront and sea engineering or

navigation.

KEYWORDS: Fluid mechanics, Equations

INTRODUCTION

Numerous industries have a solid requirement

for enhanced fluids that can exchange warm

more proficiently. The innately poor warm

conductivity of ordinary fluids puts a central

point of confinement on warm exchange.

Subsequently, for over a century researchers and

architects have attempted extraordinary

endeavors to break this principal restrict. The

proposed traditional approach to improve warm

move in warm frameworks is to expand the

warmth exchange surface region of cooling

gadgets and the flow speed or to scatter strong

particles in warm exchange fluids. The idea of

utilizing suspensions of strong particles to

upgrade warm conductivity of convectional

warm exchange fluids was started by Maxwell.

By scattering millimeter or micrometer-sized

particles in liquids, Maxwell could upgrade the

thermos physical properties of base fluids.

We as of now have a presence of mind thought

of when we are working with a fluid, instead of

a strong: Fluids tend to flow when we

collaborate with them (e.g., when you mix your

morning espresso); solids have a tendency to

misshape or twist (e.g., when you write on a

console, the springs under the keys pack).

Designers require a more formal and exact

definition of a fluid: A fluid is a substance that

deforms continuously under the application of a

shear (distracting) stretch regardless of how little

the shear stress might be.

In this manner fluids involve the liquid and gas

(or vapor) periods of the physical forms in

which matter exists. The qualification between a

fluid and the strong condition of issue is clear in

the event that you look at fluid and strong

conduct. A strong deforms when a shear push is

connected, yet its distortion does not keep on

increasing with time.



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In Fig the distortions of a strong (Fig. 5a) and a

fluid (Fig. Life) under the activity of a consistent

shear compel are differentiated. In Fig. 5a the

shear constrain is connected to the strong

through the upper of two plates to which the

strong has been fortified.

At the point when the shear constrain is

connected to the plate, the square is distorted as

appeared. From our past work in mechanics, we

realize that, gave the versatile furthest reaches of

the strong material is not surpassed, the

disfigurement is relative to the connected shear

push, T = FIA, where a is the region of the

surface in contact with the plate.

To rehash the explore different avenues

regarding a fluid between the plates, utilize a

color marker to layout a fluid component as

appeared by the strong lines (Fig. 5b). At the

point when the shear constrain, F, is connected

to the upper plate, the disfigurement of the fluid

component keeps on expanding as long as the

power is connected. The fluid in coordinate

contact with the strong boundary has an

indistinguishable speed from the boundary itself;

there is no slip at the boundary. This is an

experimental truth in light of various perceptions

of fluid conduct.

1 The shape of the fluid element, at successive

instants of time t2> tx> t0, isshown (Fig.) by the

dashed lines, which represent the positions of

the dyemarkers at successive times. Because the

fluid motion continues under the applicationof a

shear stress, we can also define a fluid as a

substance that cannot sustain ashear stress when

at rest.

Fig. Behavior of a solid and a fluid, under the action of a constantshear force.

Fluid mechanics manages the conduct of fluids

at rest and in motion. We may pose the inquiry:

"Why ponder fluid mechanics?" Knowledge and

comprehension of the fundamental standards and

ideas of fluid mechanics are basic to dissect any

framework in which a fluid is the working

medium. We can give numerous cases. The

design of for all intents and purposes all

methods for transportation requires application

of the standards of fluid mechanics. Included are

subsonic and supersonic air ship, surface ships,

submarines, and autos. As of late vehicle makers

have given more thought to aerodynamic design.

This has been valid for quite a while for the

designers of both hustling autos and pontoons.

The design of drive frameworks for space flight

and also for toy rockets depends on the

standards of fluid mechanics. The fall of the

Tacoma Narrows Bridge in 1940 is proof of the

conceivable outcomes of ignoring the essential



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standards of fluid mechanics.2 It is ordinary

today to perform model investigations to decide

the aerodynamic powers on, and flow fields

around, structures and structures. These

incorporate investigations of high rises, baseball

stadiums, smokestacks, and malls.

The design of a wide range of fluid apparatus

including pumps, fans, blowers, compressors,

and turbines obviously requires information of

the essential standards of fluid mechanics.

Oil is an application of significant significance

in fluid mechanics. Warming and ventilating

frameworks for private homes and substantial

office structures and the design of pipeline

frameworks are further cases of specialized

problem zones requiring information of fluid

mechanics. The circulatory arrangement of the

body is basically a fluid framework. It is not

shocking that the design of blood substitutes,

simulated hearts, heart-lung machines, breathing

guides, and other such gadgets must depend on

the essential standards of fluid mechanics.

Indeed, even some of our recreational attempts

are specifically identified with fluid mechanics.

The cutting and snaring of golf balls can be

clarified by the standards of fluid mechanics

(despite the fact that they can be adjusted just by

a golf master!). This rundown of real-world

applications of fluid mechanics could go on

uncertainly. Our fundamental point here is that

fluid mechanics is not a subject concentrated for

simply scholarly interest; rather, it is a subject

with across the board significance both in our

regular encounters and in present day

innovation.

Unmistakably, we can't plan to consider in detail

even a little level of these and other particular

problems of fluid mechanics. Rather, the

motivation behind this content is to exhibit the

fundamental laws and related physical ideas that

give the premise or beginning stage in the

examination of any problem in fluid mechanics.

Examination of any problem in fluid mechanics

fundamentally incorporates articulation of the

essential laws administering the fluid motion.

The basic laws, which are applicable to anyfluid,

are:

1. The conservation of mass.

2. Newton's second law of motion.

3. The principle of angular momentum.

4. The first law of thermodynamics,

5. The second law of thermodynamics.

Not all basic laws are always required to solve

any one problem. On the other hand,in many

problems it is necessary to bring into the

analysis additional relations thatdescribe the

behavior of physical properties of fluids under

given conditions. +For example, you probably

recall studying properties of gases in basic

physicsor thermodynamics. The ideal gas

equation of state

is a model that relates density to pressure and

temperature for many gases under

normalconditions. In Eq. 1.1, /? is the gas

constant. Values of R for several common gases;

and T in Eq. 1.1 are the absolute pressure and

absolutetemperature, respectively; is density

(mass per unit volume). Example Problem

illustrates use of the ideal gas equation of state.

Clearly the essential laws with which we might

bargain are the same as those utilized as a part of

mechanics and thermodynamics. Our errand will

be to define these laws in appropriate forms to



779

take care of fluid flow problems and to apply

them to a wide assortment of circumstances.

We should accentuate that there are, as we might

see, numerous obviously straightforward

problems in fluid mechanics that can't be tackled

scientifically. In such cases we should turn to

more entangled numerical arrangements or

potentially results of experimental tests, the first

step in tackling a problem is to characterize the

framework that you are endeavoring to

investigate. In essential mechanics, we

influenced broad utilization of the free-body to

chart. We will utilize a framework or a control

volume, contingent upon the problem being

examined. These ideas are indistinguishable to

the ones you utilized as a part of

thermodynamics (with the exception of you may

have called them shut framework and open

framework, individually). We can utilize it is

possible that one to gel mathematical

articulations for each of the essential laws. In

thermodynamics they were for the most part

used to get articulations for protection of mass

and the first and second laws of

thermodynamics; in our investigation of fluid

mechanics, we will be most interested in

preservation of mass and Newton's second law

of motion. In thermodynamics our concentration

was vitality; in fluid mechanics it will

principally be powers and motion.

APPLICATIONS OF MATHEMATICS TO

FLUID MECHANICS

What is fluid? How does a fluid (liquid or gas)

contrast from a strong? We can answer these

questions either in terms of tiny properties or in

terms of plainly visible properties.

Solids:

• frequently have minute long-extend

order; the iotas or atoms shape a

customary grid (elastic and plastic

are remarkable exemptions);

• tend to frame faceted precious stones if

become under the correct conditions;

• hurt when you kick them; they have a

non-zero "shear modulus".

Liquids:

• have infinitesimal short-extend order,

however no long-go order;

• flow affected by gravity;

• have zero shear modulus, so they flow

aside when you kick them (not very

hard);

• have a settled volume at low weight and

are typically difficult to pack.

Gases:

• have next to no short-go order (perfect

gasses have none);

• have zero shear modulus and you can

without much of a stretch travel

through them;

• grow to possess the accessible volume

and are very compressible.

In this way, fluid is a material that is

interminably deformable or flexible. A fluid may

oppose moving starting with one shape then onto

the next however opposes a similar sum every

which way and in all shapes.

The fundamental normal for the fluid is that it

can flow.

Fluids are isolated in two classes.

Incompressible (fluids that move at far subsonic



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speeds and don't change their thickness) and

compressible fluids.

Fluid motions are by and large ordered into three

gatherings: Laminar flows, Laminar-Turbulent

change flows and Turbulent flows. Laminar flow

is the stream-lined motion of the fluid, while the

turbulent flow is irregular in space and time,

while the laminar-turbulent change concerns

precarious flows.

In order to demonstrate the way along which the

fluid is flowing we utilize the streamlines.

Thus, streamlines are those lines that the

digression at one point on it gives the heading of

the fluid speed by then.

In segment two, we examine some fundamental

structure for working out problems of fluid

mechanics, from mathematical perspective.

SYMMETRY, SIMILARITY PRINCIPLE

AND GROUP-THEORETIC CRITERIA IN

FLUID MECHANICS

First, we describe a simple incompressible fluid

flow and its characteristics depending on a

control parameter, namely Reynolds number.

Let us consider a flow of uniform velocity, say

incident on an infinitecylinder of

circular cross-section, from left to right and

parallel to x-axis (Frisch, 1999)1(Fig.).

1Frisch U., Turbulence, Cambridge Univ. Press, South

Asia edition, 1999.



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Fig. Flow around a circular cylinder

V = a characteristic fluid velocity;L =a

characteristic length scale (diameter of the

cylinder); , kinematic viscosity; is

the viscosity of the fluid and is the densityof

the fluid.

Since the fluid is assumed incompressible, is

constant.The Reynolds number of the fluid flow

is defined by

(1)

The similarity principle for incompressible flow

is taken here as (Frisch, 1999).

Proposition 1 For a given geometrical shape of

the boundaries, the Reynolds number (Re) is the

only control parameter of the fluid.2

This implies that the analysis of incompressible

flow around cylinders of different diameters, of

course of infinitely long size, can be made in the

2Munson B. & Young D., Fundamentals of Fluid

Mechanics, John Wiley & Sons Inc., 3rd edition.

same manner depending on the Reynolds

number.

Now, the analysis of the flow around circular

cylinder by flow visualization technique reveals

how the flow changes from laminar state and

tends towards turbulent state. This is done by

increasing Reynolds number gradually and

taking the pictures of the flow situations around

the cylinder. The flow around the cylinder is

governed by the Navier-Stokes equation, namely

(2)

and the mass conservation equation

(3)

U. Frisch (1999) noted some apparent

symmetries in the flow around cylinder at low

Reynolds number. These symmetries are:

Left-right (x-reversal);

Up-down (y-reversal);



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Time translation (t-invariance);

Space-translation parallel to the axis of cylinder

(z-invariance).

In Fig. schematic diagram of symmetries in

flows around circular cylinder are shown.

Fig. Symmetries (1) to (4)

If , and are the components of velocity,

the left-right symmetry is:

(4)

and up-down symmetry is:

(5)

From Fig.7, the picture at Re = 0.16, it appears

that left-right symmetry holds good. But through

examination of the figure indicates that the left-

right symmetry is approximately correct. The

left-right symmetry is broken slightly because of

the fact that interactions among eddies does not

occur exactly in the same manner in the front

and rear sides of the cylinder.

So, this symmetry is not consistent with the full

Navier-Stokes equation. If the non-linear term is

dropped the symmetry is then consistent with the

Stokes equation (slightly broken symmetry).

From mathematical point of view the Group

theoretic approach is considered useful to solve

many problems of fluid flows. G. Birkoff has

discussed such group theoretic approach to

problems of Fluid Mechanics in his well-known

book on Hydrodynamics. We now discuss

symmetries in fluid flows from the concepts of

discrete or continuous invariance groups of

dynamical theory. Here the term symmetry is

used for the invariance group. A group of

transformations acting on space-time functions

,

which are spatially periodic and divergence less,

is denoted by G.3

Proposition 2 G is said to be a symmetry group

of Navier-Stokes equation, if for all solutions

of the Navier-Stokes equation, and all the

function is alsoa solution.Frisch

(1999), noted the following symmetries of the

Navier-Stokes equation:

3Birkoff G., Hydrodynamics.



783

Space-translation:

Time-translations:

Galilean Transformations:

Parity P:

Rotations:

Scaling:

For the Galilean transformations, when

is substituted for ,there is a

cancellation of terms between and

For the last case we have the following:

When t is changed into into and into

all terms of N-S equationsare multiplied by

except viscous term which is multiplied

by . Thus, forviscosity only is

permitted.

Such scaling transformations allow the Reynolds

number to be unchanged and accordingly the

symmetry is equivalent to the

well-known SimilarityPrincipleof Fluid

Dynamics.

Remark 1 It appears that when the Reynolds

number is sufficiently high to neglect the

viscous term, many scaling groups with proper

scaling exponent h may be employed to the

corresponding Fluid Mechanics problems. From

mathematical point of view such approaches are

surely to be appreciated.4

THE EQUATIONS OF MOTION

In this study we develop the basic equations of

fluid mechanics. Theseequations are derived

from the conservation laws of mass, momentum,

andenergy. We begin with the simplest

assumptions, leading to Euler’s equationsfor a

perfect fluid. These assumptions are relaxed in

this studyto allow for viscous effects that arise

from the molecular transport ofmomentum.5

Euler’s Equations -

Let D be a region in two- or three-dimensional

space filled with a fluid. Our object is to

4Tritton, Physical Fluid Dynamics, 2008.

5Munson, B.R., Rothmayer, A.P., Okiishi, T.H., Huebsch,W.W.: Fundamentals of Fluid Mechanics. Wiley, New York (2012) (Introductory Text on Fluid Mechanics, directed to Engineering Students, with Emphasis on the Fundamental Aspects of Fluid Flows, Including Eulerian and Lagrangian Descriptions)



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describe the motion of such a fluid. Let

be a point

in D and consider the particle of fluid moving

through x at time t. Relative to standard

Euclidean coordinates in space, we write

. Imaginea particle (think

of a particle of dust suspended) in the fluid; this

particle traverses a well-defined trajectory. Let

denote the velocity of theparticle of

fluid that is moving through x at time t. Thus,

for each fixed time, u is a vector field on D. as in

Figure. We call u the (spatial) velocity field of

the fluid.

For each time t, assume that the fluid has a well-

defined mass density . Thus, if W is any

subregion of D, the mass of fluid in W at time t

Figure. Fluid particles flowing in a region D.

is given by

where dV is the volume element in the plane or

in space.

In what follows we shall assume that the

functions 11 and (and others to be introduced

later) are smooth enough so that the standard

operations of calculus may be performed on

them. This assumption is open to criticism and

indeed we shall come back and analyze it in

detail later.6

The assumption that exists is a continuum

assumption. Clearly, it does not hold if the

molecular structure of matter is taken into

account. For most macroscopic phenomena

occurring in nature, it is believed that this

assumption is extremely accurate.Our derivation

of the equations is based on three basic

principles:

Mass is neither created nor destroyed;

The rate of change of momentum of a portion of

the fluid, equals the force applied to it (Newton's

second law);

Energy is neither created nor destroyed.

Let us treat these three principles in turn.

i Conservation of Mass

Let W be a fixed subregion of D (W does not

change with time). The rate of changeof mass in

W is

Let denote the boundary of W, assumed to

be smooth; let 11 denote the unit outward

6Choudhuri, A. R.: The Physics of Fluids and Plasmas.

Cambridge University Press, Cambridge (1998) Referred to in Chapter 1. Includes a good discussion of the Euler equation, especially in MHD



785

normal defined at points of ; and let dA

denote the area element on . The volume

flow rate across per unit area is and the

mass flow rate per unit area is (see

Figure 9).

Figure . The mass crossing the boundary per

unit time equals the surface integral of

over

The principle of conservation of mass can be

more precisely stated as follows: The rate of

increase of mass in W equals the rate at which

mass is crossing in the inward direction; i.e.,

This is the integral form of the law of

conservation of mass. By the divergence

theorem, this statement is equivalent to

Because this is to hold for all W, it is equivalent

to

The last equation is the differential form of the

law of conservation of mass, also known as the

continuity equation.

If and u are not smooth enough to justify the

steps that lead to the differential form of the law

of conservation of mass, then the integral form is

the one to use.

ii Balance of Momentum

Let be the path

followed by a fluid particle, so thatthe

velocity field is given by

that is,

This and the calculation following explicitly use

standard Euclidean coordinates in space (delete

for plane flow).1

The acceleration of a fluid particle is given by

By the chain rule, this becomes



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Using the notation

And

we obtain

which we also write as

where

We call

the material derivative; it takes into account the

fact that the fluid is moving and that the

positions of fluid particles change with time.

Indeed, if is any function of

position and time (scalar or vector), thenby the

chain rule,

For any continuum, forces acting on a piece of

material are of two types. First, there are forces

of stress, whereby the piece of material is acted

on by forces across its surface by the rest of the

continuum. Second, there are external, or body,

forces such as gravity or a magnetic field, which

exert a force per unit volume on the continuum.

The clear isolation of surface forces of stress in a

continuum is usually attributed to Cauchy.7

Later, we shall examine stresses more generally,

but for now let us define an ideal fluid, as one

with the following property: For any motion of

the fluid there is a function called

thepressure such that if is asurface in

the fluid with a chosen unit, normal n, the force

of stress exerted across the surface per unit

area at at time is ;

i.e.,force across S per unit area =

n.

Note that the force is in the direction 11 and that

the force acts orthogonally to the surface ; that

is, there are no tangential forces (see Figure).

Figure. Pressure forces across a surface .

7Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical

Methods for Physicists. Academic, New York (2012) (New Edition of a Classic Book on Mathematical Methods of Physics)



787

Of course, the concept of an ideal fluid as a

mathematical definition is not subject to dispute.

However, the physical relevance of the notion

(or mathematical theorems we deduce from it)

must be checked by experiment. As we shall see

later, ideal fluids exclude many interesting real

physicalphenomena, but nevertheless form a

crucial component of a more complete theory.

Intuitively, the absence of tangential forces

implies that there is no way for rotation to start

in a fluid, nor, if it is there at the beginning, to

stop. This idea will be amplified in the next

section. However, even here we can detect

physical trouble for ideal fluids because of the

abundance of rotation in real fluids (near the

oars of a rowboat, in tornadoes, etc.).

If W is a region in the fluid at a particular instant

of time £, the total force exerted on the fluid

inside W by means of stress on its boundary is

(negative because 11 points outward). If e is any

fixed vector in space, the divergence theorem

gives

Thus,

If denotes the given body forceper unit

mass, the total bodyforce is

Thus, on any piece of fluid material,

force per unit volume =

By Newton’s second law (force = mass x

acceleration) we are led to the differential form

of the law of balance of momentum:

(BM1)

Next we shall derive an integral form of balance

of momentum in two ways. We derive it first as

a deduction from the differential form and

second from basic principles.

From balance of momentum in differential form,

we have

and so, using the equation of continuity,



788

If e is any fixed vector in space, one checks that

Therefore, if W is a fixed volume in space, the rate of change of momentum in direction e in W is

by the divergence theorem. Thus, the integral form of balance of momentum becomes:

(BM2)

The quantity is the momentum

flux per unit area crossing , where 11 is the

unit outward normal to This derivation of

the integral balance law for momentum

proceeded via the differential law. With an eye

to assuming as little differentiability as possible,

it is useful to proceed to the integral law directly

and, as with conservation of mass, derive the

differential form from it. To do this carefully

requires us to introduce some useful notions.8

As earlier, let D denote the region in which the

fluid is moving. Let and let us write

for the trajectory followed by the

particle that is atpoint x at time . We will

assume is smooth enough so the following

8James A. Fay (June 1994). Introduction to Fluid Mechanics. MIT Press. ISBN 0-262-06165-1. see "4.5 Euler's Equation in Streamline Coordinates" pp. 150–152 <https://books.google.com/books?id=XGVpue4954wC&pg=150>

manipulations are legitimate and for fixed

is an invertible mapping. Let denote the map

; that is, with fixed , this map

advanceseach fluid particle from its position at

time to its position at time . Here, of

course, the subscript does not denote

differentiation. We call the fluid flow map. If

W is a region in D, then is the

volumeW moving with the fluid. See Figure.

The “primitive" integral form of balance of

momentum states that

https://en.wikipedia.org/wiki/International_Standard_Book_Number

https://en.wikipedia.org/wiki/Special:BookSources/0-262-06165-1

https://books.google.com/books?id=XGVpue4954wC&pg=150





789

(BM3)

that is, the rate of change of momentum of a

moving piece of fluid equals the total force

(surface stresses plus body forces) acting on it.

These two forms of balance of momentum

(BMl) and (BM3) are equivalent. To prove this,

we use the change of variables theorem to write

Figure. is the image of W as particles of fluid in W flow for time t.

where is the Jacobian determinant, of the map Because the Volume is fixed at its initial

position, we may differentiate under the integral sign. Note that

is the material derivative, as was shown earlier (If you prefer, this equality says that isdifferentiation

following the fluid) Next, we learn how to differentiate .

CONCLUSION

The fundamental fluid properties, similar to

weight, thickness, consistency, mass and

volumetric flow rates, height, weight and speed

heads, head misfortunes, and how they influence



790

each other. We are additionally learned of

essential fluid mechanics processes in different

straightforward plant segments, similar to

elbows, openings, venturis, valves and pumps.

We comprehend and can perceive some

operating problems related with fluid mechanics,

similar to flow instigated vibrations, cavitation,

water sledge and steam pound. We have checked

on some great operating practices to maintain a

strategic distance from these problems.

REFERENCES

[1]. R. W. Braun, R. Meise, B.A. Taylor,

Surjectivity of constant coefficient

partial differential operators on

and Whitney’s C4 cone, Bull. Soc. Roy.

Liege 70 (2001) 195–206.

[2]. R. W. Braun, The subjectivity of a

constant coefficient homogeneous

differential operator on the real analytic

functions and the geometry of its

symbol, Ann. Inst. Fourier 45 (1995)

223–249.

[3]. T. Kawai, On the global existence of

real analytic solutions of linear

differential equations I, J. Math. Soc.

Japan 24 (1972) 481–517.

[4]. Frisch U., Turbulence, Cambridge Univ.

Press, South Asia edition, 1999.

[5]. Munson B. & Young D., Fundamentals

of Fluid Mechanics, John Wiley &

SonsInc., 3rd edition.Birkoff G.,

Hydrodynamics.

[6]. Tritton, Physical Fluid Dynamics, 2008.

[7]. Munson, B.R., Rothmayer, A.P.,

Okiishi, T.H., Huebsch,W.W.

Fundamentals of Fluid Mechanics.

Wiley, New York (2012) (Introductory

Text on Fluid Mechanics, Directed to

EngineeringStudents, with Emphasis on

the Fundamental Aspects of Fluid

Flows, Including Eulerian

andLagrangian Descriptions)

[8]. Choudhuri, A. R.: The Physics of Fluids

and Plasmas. Cambridge University

Press, Cambridge(1998) Referred to in

Chapter 1. Includes a good discussion of

the Euler equation, especially inMHD

[9]. Arfken, G.B., Weber, H.J., Harris, F.E.:

Mathematical Methods for Physicists.

Academic,New York (2012) (New

Edition of a Classic Book on

Mathematical Methods of Physics)

[10]. James A. Fay (June

1994). Introduction to Fluid Mechanics.

MIT Press. ISBN 0-262-06165-1. see

"4.5 Euler's Equation in Streamline

Coordinates" pp. 150–152

<https://books.google.com/books?id=X

GVpue4954wC&pg=150>

Documents

A STUDY ON APPLICATION OF MATHEMATICAL FLUID MECHANICS · A STUDY ON APPLICATION OF MATHEMATICAL FLUID MECHANICS Daruri Venugopal Research Scholar, OPJS University, Churu, Rajasthan