energy eqn

8/10/2019 energy eqn

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Chapter 6 - Equations of Motion and Energy in Cartesian Coord inates

Equations of motion of a Newtonian fluidThe Reynolds numberDissipation of Energy by Viscous Forces

The energy equationThe effect of compressibilityResume of the development of the equationsSpecial cases of the equations

Restrictions on types of motionIsochoric motionIrrotational motionPlane flow

Axisymmetric flowParallel flow perpendicular to velocity gradient

Specialization on the equations of motion

HydrostaticsSteady flowCreeping flowInertial flowBoundary layer flowLubrication and film flow

Specialization of the constitutive equationIncompressible flowPerfect (inviscid, nonconducting) fluidIdeal gasPiezotropic fluid and barotropic flowNewtonian fluids

Boundary conditionsSurfaces of symmetryPeriodic boundarySolid surfacesFluid surfacesBoundary conditions for the potentials and vorticity

Scaling, dimensional analysis, and similarityDimensionless groups based on geometryDimensionless groups based on equations of motion and energyFriction factor and drag coefficients

Bernoulli theoremsSteady, barotropic flow of an inviscid, nonconducting fluid withconservative body forcesCoriolis forceIrrotational flowIdeal gas

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Reading assignmentChapter 2&3 in BSLChapter 6 in Aris

Equations of motion of a Newtonian fluid

We will now substitute the constitutive equation for a Newtonian fluid intoCauchys equation of motion to derive the Navier-Stokes equation.Cauchys equation of motion is

,

or

ii i

Dva f

Dt

D

Dt

= = +

= = + v

a f

ij jT

T

e

.

The constitutive equation for a Newtonian fluid is

( ) 2

or

( ) 2

ij ij ijT p

p

= + +

= + +T I e

.

The divergence of the rate of deformation tensor needs to be restated with amore meaningful expression.

,

2

2

2

1

2

1 1

2 2

1 1( )

2 2

or

1 1( )

2 2

jiij j

j j i

ji

j j i j

i

i

vve

x x x

vv

x x x

vx

= +

= +

= +

= +

v

e v v

x

.

Thus

2

,

2

( ) ( )

or

( )

ij j i

i i

pT v

x x

p

= + + +

= + + +

v

T v

.

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derive the Reynolds number NRe from the Navier-Stokes equation at this point.The Reynolds number is the characteristic ratio of the inertial and viscous forces.When it is very large the inertial terms dominate the viscous terms and vice versawhen it is very small. Its value gives the justification for assumptions of thelimiting cases of inviscid flow and creeping flow.

We will consider the case of single-phase flow with conservative bodyforces (e.g., gravitational) and density a single valued function of pressure. Thepressure and potential from the body force can be combined into a singlepotential.

1

where

p

p

dpgz

=

=

f

If the change in density is small enough, the potential can be approximated bypotential that has the units of pressure.

, small change in density

where

P

P p g z

=

Suppose that the flow is characterized by a certain linear dimension, L, a

velocity U, and a density . For example, if we consider the steady flow past anobstacle, Lmay be its diameter and Uand the velocity and density far from theobstacle. We can make the variables dimensionless with the following

2

2 2 2

, , ,

,

U Pt t P

U L L

L L

= = = =

= =

v xv x

U .

The conservative body force, Navier-Stokes equation is made dimensionless withthese variables.

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2

2 22

2 2

2

2

Re

2

Re

( )

( / 1)

( / 1)

( / 1)

where

/

DP

Dt

U D U U U P

L Dt L L L

U L D PDt

DN P

Dt

U L UN

U L

= + + +

= + + +

+ = + +

+ = + +

= =

vv

vv

v v

vv

The Reynolds number partitions the Navier Stokes equation into two

parts. The left side or inertial and potential terms, which dominates for large NReand the right side or viscous terms, which dominates for small NRe. The potentialgradient term could have been on the right side if the dimensionless pressure

was defined differently, i.e., normalized with respect to (U)/L, the shear stressrather than kinetic energy. Note that the left side has only first derivatives of thespatial variables while the right side has second derivatives. We will see laterthat the left side may dominate for flow far from solid objects but the right sidebecomes important in the vicinity of solid surfaces.

The nature of the flow field can also be seen form the definition of theReynolds number. The second expression is the ratio of the characteristic kineticenergy and the shear stress.

The alternate form of the dimensionless Navier-Stokes equation with theother definition of dimensionless pressure is as follows.

2

Re ( / 1)

/

DN P

Dt

PP

U L

= + + +

=

vv

Dissipation of Energy by Viscous ForcesIf there was no dissipation of mechanical energy during fluid motion then

kinetic energy and potential energy can be exchanged but the change in the sumof kinetic and potential energy would be equal to the work done to the system.However, viscous effects result in irreversible conversion of mechanical energyto internal energy or heat. This is known as viscous dissipation of energy. Wewill identify the components of mechanical energy in a flowing system beforeembarking on a total energy balance.

The rate that work Wis done on fluid in a material volume Vwith a surfaceSis the integral of the product of velocity and the force at the surface.

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( )

( )

n

S

S

V

dWdS

dt

dS

dV

=

=

=

v t

v T n

v T

The last integrand is rather complicated and is better treated with index notation.

, , ,

,

2

,

( )

1

2

i ij j ij i j i ij j

iij i j i i

ij i j i i

v T T v v T

DvT v v f

Dt

DvT v f v

Dt

= +

= +

= +

( )21

2

Dv

Dt = + v T T : v f v

We made use of Cauchys equation of motion to substitute for the divergence ofthe stress tensor. The integrals can be rearranged as follows.

( )

22

2

,

1 1

2 2

( )

1

2

where

V V

V V V

ij i j

d Dvv dV dV

dt Dt

dV dV dV

Dv

Dt

T v

=

= +

= +

=

f v v T T : v

f v v T T : v

T : v

The left-hand term can be identified to be the rate of change of kinetic energy.The first term on the right-hand side is the rate of change of potential energy dueto body forces. The second term is the rate at which surface stresses do work onthe material volume. We will now focus attention on the last term.

The last term is the double contracted product of the stress tensor with thevelocity gradient tensor. Recall that the stress tensor is symmetric for a nonpolarfluid and the velocity gradient tensor can be split into symmetric andantisymmetric parts. The double contract product of a symmetric tensor with anantisymmetric tensor is zero. Thus the last term can be expressed as a doublecontracted product of the stress tensor with the rate of deformation tensor. Wewill use the expression for the stress of a Newtonian fluid.

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[ ]

,

2 2

( ) 2

2

2 ( 2 )

ij i j ij ij

ij ij ij

ii ij ij

T v T e

p e e

p e e e

p

=

= + +

=

=

= T : v

where is the second invariant of the rate of deformation tensor. Thus the rateat which kinetic energy per unit volume changes due to the internal stresses isdivided into two parts:(i) a reversible interchange with strain energy, ( / )( / )p p D Dt = ,(ii) a dissipation by viscous forces,

2( 2 ) 4 +

Since is always positive, this last term is always dissipative. If Stokes

relation is used this term is

2 2

24 43

for incompressible flow it is

4 .

(The above equation is sometimes written -4, where is called the dissipationfunction. We have reserved the symbol for the second invariant of the rate ofdeformation tensor, which however is proportional to the dissipation function for

incompressible flow. is the symbol used later for the negative of thedissipation by viscous forces.)

The energy equationWe need the formulation of the energy equation since up to this point we

have more unknowns than equations. In fact we have one continuity equation(involving the density and three velocity components), three equations of motion(involving in addition the pressure and another thermodynamic variable, say thetemperature) giving four equations in six unknowns. We also have an equation

of state, which in incompressible flow asserts that is a constant reducing thenumber of unknowns to five. In the compressible case it is a relation

( , )f p T=

which increases the number of equations to five. In either case, there remains agap of one equation, which is filled by the energy equation.

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The equations of continuity and motion were derived respectively from theprinciples of conservation of mass and momentum. We now assert the first lawof thermodynamics in the form that the increase in total energy (we shall consideronly kinetic and internal energies) in a material volume is the sum of the heattransferred and work done on the volume. Let q denote the heat flux vector,

then, since n is the outward normal to the surface, -qn is the heat flux into thevolume. Let Udenote the specific internal energy, then the balance expressedby the first law of thermodynamics is

2

( )2

V V S S

d vU dV dV dS dS

dt + = + f v v T n q n

This may be simplified by subtracting from it the expression we already have forthe rate of change of kinetic energy, using Reynolds transport theorem, andGreens theorem.

( ) 0V

DUdV

Dt

+ = q T : v

Since this is valid for any arbitrary material volume, we have assume continuity ofthe integrand

( )DU

Dt = + q T : v .

We assume Fouriers law for the conduction of heat.

k T= qWe assume a Newtonian fluid for the dissipation of energy.

.2

( ) ( )

( 2 ) 4

p

= +

= +

T : v v

Substituting this back into the energy balance we have

( ) ( )DU k T pDt

= + v

Physically we see that the internal energy increases with the influx of heat, thecompression and the viscous dissipation.

If we write the equation in the form

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( )DU p D

k TDt Dt

= +

the left-hand side can be transformed by one of the fundamental thermodynamicidentities. For if Sis the specific entropy,

2

(1/ )

( / )

T dS dU p d

dU p d

= +

=

Substituting this into the last equation for internal energy gives

( )DS

T k TDt

= + .

Giving an equation for the rate of change of entropy. Dividing by T and

integrating over a material volume gives

2

2

2

2

1( )

( ) ( )

( )

V V

V

V

S V

DS ddV S dV

Dt dt

k T dV T T

k kT T

T T T

kdS T dV

T T T

=

= +

= + +

dV

= + +

q n

The second law of thermodynamics requires that the rate of increase ofentropy should be no less than the flux of heat divided by temperature. Theabove equation is consistent with this requirement because the volume integral

on the right-hand side cannot be negative. It is zero only if kor Tand arezero. This equation also shows that entropy is conserved during flow if thethermal conductivity and viscosity are zero.

0, when 0, and 0DS

k

Dt

= = =

Assignment 6.2 Do exercises 6.3.1 and 6.3.2 in Aris.

The Effect of Compressibi lity(Batcehlor, 1967)Isentropic flow. The condition of zero viscosity and thermal conductivity

results in conservation of entropy during flow or isentropic flow. This idealcondition is useful for illustration the effect of compressibility on fluid dynamics.

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The conservation of entropy during flow implies that density, pressure, andtemperature is changing in a reversible manner during flow. The relationbetween entropy, density, temperature, and pressure is given bythermodynamics.

1/

where

1=-

p T

p

p

p

p

S SdS dT dpT p

CdT dp

T T

CdT dp

T

T

= +

=

=

These relations may be combined with the condition that the material derivativeof entropy is zero to obtain a relation between temperature and pressure duringflow.

, when 0, and 0pDT T Dp

C kDt Dt

= = =

The equation of state expresses the density as a function of temperature andpressure. During isentropic flow the pressure and temperature are not

independent but are constrained by constant entropy or adiabatic compressionand expansion. The density in this case is given as

( , )p S =

We now have as many equations as unknowns and the system can bedetermined. The simplifying feature of isentropic flow is that exchanges betweenthe internal energy and other forms of energy are reversible, and internal energyand temperature play passive roles, merely changing in response to thecompression of a material element.

The continuity equation and equation of motion governing isotropic flow

may now be expressed as follows.

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2

2

10

where

S

Dp

c Dt

Dp

Dt

pc

+ =

=

=

v

vf

The physical significance of the parameter c, which has the dimensions ofvelocity, may be seen in the following way. Suppose that a mass of fluid of

uniform density o is initially at rest, in equilibrium, so that the pressure po isgiven by

o op = f .

The fluid is then disturbed slightly (all changes being isentropic), by somematerial being compressed and their density changed by small amounts, and issubsequently allowed to return freely to equilibrium and to oscillate about it. (Thefluid is elastic, and so no energy is dissipated, so oscillations about the

equilibrium are to be expected.) The perturbation quantities 1(= - o) and p1(= p - po) and vare all small in magnitude and a consistent approximation to thecontinuity equation and equations of motion is

1

2

1 1

10o

o

o

p

tc

pt

+ =

=

v

vf

where cois the value of cat =o. On eliminating vwe have

221 1

1 12 22

1

o o

p pp

tc c

=

f

f

The body forces commonly arise from the earths gravitational field, in which

case the divergence is zero and the last term is negligible except in the unlikelyevent of a length scale of the pressure variation not being small compared with

co2/g (which is about 1.2 104 m for air under normal conditions and is even

larger for water). Thus under these conditions the above equation reduces to the

wave equation for p1 and 1 satisfies the same equation. The solutions of thisequation represents plane compression waves, which propagate with velocity coand in which the fluid velocity v is parallel to the direction of propagation. In

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another words, co is the speed of propagation of sound waves in a fluid whose

undisturbed density is o.

Conditions for the velocity distribution to be approximately solenoidal. Theassumption of solenoidal or incompressible fluid flow is often made without a

rigorous justification for the assumption. We will now reexamine this assumptionand make use of the results of the previous section to express the conditions forsolenoidal flow in terms of identifiable dimensionless groups.

The condition of solenoidal flow corresponds to the divergence of thevelocity field vanishing everywhere. We need to characterize the flow field by acharacteristic value of the change in velocity Uwith respect to both position andtime and a characteristic length scale over which the velocity changes L. Thespatial derivatives of the velocity then is of the order of U/L. The velocitydistribution can be said to be approximately solenoidal if

i.e., if

1

U

L

D U

Dt L


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is satisfied, the changes in density of a material element due to pressurevariations are negligible, i.e., the fluid is behaving as if it were incompressible.This is by far the more practically important of the two requirements for vto be a

solenoidal vector field. In estimating /Dp Dt we shall lose little generality by

assuming the flow to be isentropic, because the effects of viscosity and thermal

conductivity are normally to modify the distribution of pressure rather than tocontrol the magnitude of pressure variation. We may then rewrite the lastequation with the aid of equations of motion of an isentropic fluid derived in thelast section.

2

2

Dp pp

Dt t

p d

t dt

p dv

t d

= +

= +

= +

v

vv f

v f t

Thus

2

2 2 2

1 1

2

p Dv U

c t c c Dt L

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gradient is the order of the rate of change of momentum, the spatial pressure

variation over a region of length Lis LU. Since the pressure is also oscillating,

the magnitude of p/t is then LU2. Thus the condition that the first term besmall compared to U/Lis

2 2

2 1L

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The foundation of the study of fluid motion lies in kinematics, the analysisof motion and deformations without reference to the forces that are brought intoplay. To this we added the concept of mass and the principle of the conservationof mass, which leads to the equation of continuity,

( )( )DDt t0 + = + =

v v

An analysis of the nature of stress allows us to set up a stress tensor, whichtogether with the principle of conservation of linear momentum gives theequations of motion

D

Dt = +

vf T .

If the conservation of moment of momentum is assumed, it follows that the stress

tensor is symmetric, but it is equally permissible to hypothesize the symmetry ofthe stress tensor and deduce the conservation moment of momentum. For acertain class of fluids however (hereafter called polar fluids) the stress tensor isnot symmetric and there may be an internal angular momentum as well as theexternal moment of momentum.

As yet nothing has been said as to the constitution of the fluid and certainassumptions have to be made as to its behavior. In particular we have noticedthat the hypothesis of Stokes that leads to the constitutive equation of aStokesian fluid (not-elastic) and the linear Stokesian fluid which is the Newtonianfluid.

( ) , Stokesian fluid

( ) 2 , Newtonian fluid

ij ij ij ik kj

ij ij ij

T p e e e

T p e

= + + += + +

.

The coefficients in these equations are functions only of the invariants of the rateof deformation tensor and of the thermodynamic state. The latter may bespecified by two thermodynamic variables and the nature of the fluid is involvedin the equation of state, of which one form is

( , )f p T= .

If we substitute the constitutive equation of a Newtonian fluid into theequations of motion, we have the Navier-Stokes equation.

2( ) ( )D

pDt

= + + + v

f v v

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Finally, the principle of the conservation of energy is used to give anenergy equation. In this, certain assumptions have to be made as to the energytransfer and we have only considered the conduction of heat, giving

( ) ( )DU

k T p

Dt

= + v

These equations are both too general and too special. They are toogeneral in the sense that they have to be simplified still further before any largebody of results can emerge. They are too special in the sense that we havemade some rather restrictive assumptions on the way, excluding for exampleelastic and electromagnetic effects.

Special cases of the equationsThe full equations may be specialized is several ways, of which we shall

consider the following:

(i) restrictions on the type of motions,(ii) specializations on the equations of motion,(iii) specializations of the constitutive equation or equation of state.This classification is not the only one and the classes will be seen to overlap. Weshall give a selection of examples and of the resulting equations, but the list is byno means exhaustive.

Under the first heading we have any of the specializations of the velocityas a vector field. These are essentially kinematic restrictions.

(ia) Isochoric motion. (i.e., constant density) The velocity field is solenoidal

1 0DDt

= = v =

The equation of continuity now gives

0D

Dt

= ,

that is, the density does not change following the motion. This does not meanthat it is uniform, though, if the fluid is incompressible, the motion is isochoric.

The other equations simplify in this case for we have , , and of theconstitutive equations functions of only and of the invariants of the rate ofdeformation tensor. In particular for a Newtonian fluid

2

2ij ij ijT p e

Dp

Dt

= +

= + v

f v

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The energy equation is

( )DU

k TDt

= + ,

and for a Newtonian fluid

4 = .

Because the velocity field is solenoidal, the velocity can be expressed asthe curl of a vector potential.

.= v AThe Laplacian of the vector potential can be expressed in terms of the vorticity.

2

2

( )

( )

, if

= =

=

= =

w vA

A A

A A 0

If the body force is conservative, i.e., gradient of a scalar, the body force andpressure can be eliminated from the Navier-Stokes equation by taking the curl ofthe equation.

2D

Dt = +

w

w v w

where is the kinematic viscosity.Isochoric motion is a restriction that has to be justified. Because it is

justified in so many cases, it is easier to identify the cases when it does notapply. We showed during the discussion of the effects of compressibility thatcompressibility or non-isochoric is important in the cases of significant Machnumber, high frequency oscillations such as in acoustics, large dimensions suchas in meteorology, and motions with significant viscous or compressive heating.

(ib) Irrotational motion. The velocity field is irrotational

0= =w v

It follows that there exists a velocity potential (x,t) from which the velocity canbe derived as

= v

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and in place of the three components of velocity we seek only one scalarfunction. (Note that some authors express the velocity as the gradient of a scalarand others as the negative of a gradient of a scalar. We will use either toconform to the book from which it was extracted.) The continuity equation

becomes

2 0D

Dt

+ =

so that for an isochoric (or incompressible), irrotational motion, is a potentialfunction satisfying

.2 0 =The Navier-Stokes equations become

2 21 1( ) ( 2 ) (

2p

t

)

+ = + + f .

In the case of an irrotational body force = f and when pis a function only of, this has an immediate first integral since every term is a gradient. Thus if

( ) /P dp = ,

2 21( ) ( ) ( 2 ) (

2

P gt

)t

+ + + + =

is a function of time only.Irrotational motions with finite viscosity are only very special motions

because the no-slip boundary conditions on solid surfaces usually will causegeneration of rotation. Usually irrotational motion is associated with inviscidfluids because the no-slip boundary condition then will not apply and initiallyirrotational motion will remain irrotational.

(ic) Complex lamellar motions, Betrami motions, ect. These names can beapplied when the velocity field is of this type. Various simplifications are possibleby expressing the velocity in terms of scalar fields. We shall not discuss themfurther here.

(id) Plane flow. Here the motion is restricted to two dimensions which may betaken to be the 012plane. Then v3= 0and x3does not occur in the equations.

Also, the vector potential and the vorticity have only one nonzero component.

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Incompressible plane flow. Since the flow is solenoidal, the velocity canbe expressed as the curl of the vector potential. The nonzero component of thevector potential is the stream function.

3 3, ,

1 2 3

2 1

( , , ) , , 0

i ij j ij jv A

v v vx x

=

= =

=

v A

The vorticity has only a single component, that in the 03direction, which we willwrite without suffix

,

2 1

1 2

2

, , 1,2i ijk k jw v j k

v vwx x

=

= =

=

=

w v

.

If the body force is conservative, i.e., gradient of a scalar, then the body forceand pressure disappear from the Navier-Stokes equation upon taking the curl ofthe equations. In plane flow

2Dw wDt

=

Thus for incompressible, plane flow with conservative body forces, the continuityequation and equations of motion reduce to two scalar equations.

Incompressible, irrotational plane motion. A vector field that is irrotationalcan be expressed as the gradient of a scalar. Since the flow is incompressible,the velocity vector field is solenoidal and the Laplacian of the scalar is zero, i.e., itis harmonic or an analytical function.

2

0

=

=

=

v

v

Since the flow is incompressible, it also can be expressed as the curl of thevector potential, or in plane flow as derivatives of the stream function as above.Since the flow is irrotational, the vorticity is zero and the stream function is also

an analytical function, i.e., ( )2 2, 0 , 0= = = = + =v A w v A A .Thus

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1

1 2

2

2 1

vx x

vx x

= =

= =

These relations are the Cauchy-Riemann relations show that the complex

function f i = + is an analytical function of 1z x i x2= + . The whole resourcesof the theory of functions of a complex variable are thus available and manysolutions are known.

Steady, plane flow. If the fluid is compressible but the flow is steady (i.e.,no quantity depends on t) the equation of continuity becomes

1 2

1 2

( ) ( )0

v v

x x

+ =

A stream function can again be introduced, this time in the form

1 2

2 1

1 1,v v

x x

= =

.

The vorticity is now given by

2w

= +

(ie) Axisymmetric flows. Here the flow has an axis of symmetry such that theflow field can be expressed as a function of only two coordinates by usingcurvilinear coordinates. The curl of the vector potential and velocity has only onenon-zero component and a stream function can be found.

(if) Parallel flow perpendicular to velocity gradient. If the flow is parallel, i.e., thestreamlines are parallel and are perpendicular to the velocity gradient, then theequations of motion become linear in velocity if the fluid is Newtonian.

If 0, then , and D

Dt t

t

= =

= +

v vv v

vf T

The second type of specializations are limiting cases of the equations ofmotion.

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(iia) Hydrostatics. When there is no flow, the only non-zero terms in theequations of motion are the body forces and pressure gradient.

p = f .

If the body force is conservative, i.e., the gradient of a scalar, then the hydrostaticpressure can be determined from this scalar.

( ) 0

constant

where

p

p

dp

=

=

=

= +

=

f

If the body force is due to gravity then

g z =where zis the elevation above a datum such as the mean sea level.

(iib) Steady flow. Examples of this have already been given and indeed it mighthave been considered as a restriction of the first class. All partial derivatives withrespect to time vanish and the material derivative reduce to the following

D

Dt= v .

In particular, the continuity equation is

( ) 0 =v

so that the mass flux field is solenoidal.

(iic) Creeping flow. It is sometimes justifiable to assume that the velocity is so

small that the square of velocity is negligible by comparison with the velocityitself. This linearizes the equations and allows them to be solved more readily.For example, the Navier-Stokes equation becomes

2( ) ( )p

t

= + + +

v

f v v .

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In particular, for steady, incompressible, creeping flow with conservative bodyforces

.

2

where

P

P p g z

=

=

v

However, since the continuity equation is v= 0, we have

2 0P =or P is a harmonic potential function. This is the starting point for Stokessolution of the creeping flow about a sphere and for its various improvements.

One may ask, How small must the velocity be in order to neglect thenonlinear terms? To answer this question, we need to examine the value of the

Reynolds number. However, this time the pressure and body forces will be madedimensionless with respect to the shear forces rather than the kinetic energy.

**

2 2 2

, , ,/

,

U Pt t P

U L L U

L L

= = = =

= =

v xv x

L

The dimensionless Navier-Stokes equation for incompressible flow is now asfollows

** 2Re

2

Re

where

/

DN PDt

U L UN

U L

= +

= =

v v

.

Creeping flow is justified if the Reynolds number is small enough to neglect theleft-hand side of the above equation. If the dimensionless variables and theirderivatives are the order of unity, then creeping flow is justified if the Reynoldsnumber is small compared to unity.

Another specialization of the equations of motion where the equations aremade linear arises in stability theory when the basic flow is known but perturbedby a small amount. Here it is the squares and products of small perturbationsthat are regarded as negligible.

(iid) Inertial flow. The flow is said to be inviscid when the inertial terms aredominant and the terms with viscosity in the equations of motion can beneglected. We can examine the conditions when this may be justified from the

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dimensionless equations of motion with the pressure and body forces normalizedwith respect to the kinetic energy.

2

Re

2

Re

2

( / 1)

where

/

DN P

Dt

U L UN

U L

PP

U

+ = + +

= =

=

vv

The limit of inviscid flow may occur when the Reynolds number is becomes verylarge such that the right-hand side of the above equation is negligible. Noticethat if the right-hand side of the equation vanishes, the equation goes from being

second order in spatial derivatives to first order. The differential equation goesfrom being parabolic to being hyperbolic and the number of possible boundarycondition decreases. The no-slip boundary condition at solid surfaces can nolonger apply for inviscid flow. These types of problems are known as singularperturbation problems where the differential equations are first order except nearboundaries where they become second order. Physically, the form dragdominates the skin friction in inertial flow. The macroscopic momentum balanceis described by Bernoulli theorems.

Notice that inviscid flow is necessary for irrotational flow past solid objectsbut inviscid flow may be rotational. If fact much of the classical fluid dynamics ofvorticity is based on inviscid flow.

(iie) Boundary-layer flows. Ideal fluid or inviscid flow may be assumed far froman object but real fluids have no-slip boundary conditions on solid surfaces. Theresult is a boundary-layerof viscous flow with a large vorticity in a thin layer nearsolid surfaces that merges into the ideal fluid flow at some distance from the solidsurface. It is possible to neglect certain terms of the equations of motioncompared to others. The basic case of steady incompressible flow in twodimensions will be outlined. If a rigid barrier extends along the positive 01axisthe velocity components v1and v2are both zero there. In the region distant from

the axis the flow is v1= U(x1), v2 0, and may be expected to be of the formshown in Fig. 6.1, in which v1 differs from U and v2 from zero only within a

comparatively short distance from the plate. To express this we suppose L is atypical dimension along the plate and a typical dimension of this boundary layerand that V1and V2are typical velocities of the order of magnitude of v1and v2.We then introduce dimensionless variables

x x

Ly

xu

v

Vv

v

V

* * * *, , ,= = = =1 2 11

2

2

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which will be of the order of unity. This in effect is a stretching upward of thecoordinates so that we can compare orders of magnitude of the various terms inthe equations of motion, for now all dimensionless quantities will be of order of

unity. It is assumed that


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The circumstances under which these simplified equations are valid are given bythe term above that we assumed to be of order of unity, which we rewrite as

1

OL V L

=

.

Since it was assumed that


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33 3 0

30

12 123

12 120

* * *

12

* *

12

( )( / )

( )( / )

hh

h

o

o

o

vdx v

x

v h vdx O h L

x x

h v L hO h L

x U t t

=

= +

+ =

The characteristic time can be chosen as to make the dimensionless group equalto unity.

o

Lt

U=

The dimensionless variables can now be substituted into the equations ofmotion for zero Reynolds number with gravitational body force and Newtonianfluid. For the components in the plane of the film,

2 ** * *2 * 1212 12 12 2 *

3

0o o o

U U LP v

P L P h x

= + + 2

v

The dimensionless group in the last term can be made equal to unity because ofthe two characteristic quantities, Po and U, only one is specified and the othercan be determined from the first.

2

2

2

1

and

, or

o o

o oo

o

U L

P h

P h L U U P

L h

=

= =

The equations of motion in the plane of the film is now

2 *

* * 21212 *2

3

0 ( ov

P O hx

= + + / )L

The dimensionless variables can now be substituted into the equation of motionperpendicular to the plane of the film.

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24 2 ***2 * 312 3* 2 2

3 3

*2

*

3

0

0 ( / )

o o

o o o o

o

h hP U L U Lv

*2

v

x P h L P h L x

PO h L

x

= + +

= +

This last equation shows that the pressure is approximately uniform over the

thickness of the film. Thus the velocity profile of can be determined by

integrating the equation of motion in the plane of the film twice and applying theappropriate boundary conditions. The average velocity in the film can bedetermined by integrating the velocity profile across the film.

*

12v

Certain of the specializations based on the constitutive equation or

equation of state have turned up already in the previous cases. We mentionhere a few important cases.

(iiia) Incompressible fluid. An incompressible fluid is always isochoric and theconsiderations of (ia) apply. It should be remembered that for an incompressiblefluid the pressure is not defined thermodynamically, but is an variable of themotion.

(iiib) Perfect fluid. A perfect fluid has no viscosity so that

and

p

Dp

Dt

=

=

T I

vf

.

If, in addition, the fluid has zero conductivity the energy equation becomes

0DS

Dt=

and the flow is isentropic.

(iiic) Ideal gas. An ideal gas is a fluid with the equation of state

p R T= .

The entropy of an ideal gas is given by

lnvdT

S c RT

=

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which for constant specific heats gives

/ vS cp e =

(iiid) Piezotropic fluid and barotropic flow. When the pressure and density are

directly related, the fluid is said to be piezotropic. A simple relation between pand allows us to write

1( )

pdp

p P

= = .

(iiie) Newtonian fluids. Here the assumption of a linear relation between stressand strain leads to the constitutive equation

( ) 2p = + +T I e .

The equations of motion becomes the Navier-Stokes equations.

Assignment 6.3 Carry out the steps in specializing the continuity and equationsof motion for boundary layer and lubrication or film flows.

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Boundary conditionsThe flow field is often desired for a finite region of space that is bounded

by a surface. Boundary conditions are needed on these surfaces and at internalinterfaces for the flow field to be determined. The boundary conditions fortemperature and heat flux are continuity of both across internal interfaces that

are not sources or sinks and either a specified temperature, heat flux, or acombination of both at external boundaries.Surfaces of symmetry. Surfaces of symmetry corresponds to reflection

boundary conditions where the normal component of the gradient of thedependent variables are zero. Thus surfaces of symmetry have zero momentumflux, zero heat flux, and zero mass flux. Because the momentum flux is zero, theshear stress is zero across a surface of symmetry.

Periodic boundary. Periodic boundaries are boundaries where thedependent variables and its derivatives repeat themselves on oppositeboundaries. The boundaries may or may not be symmetry boundary conditions.

An example of when periodic boundaries are not symmetry boundaries are the

boundaries of = 0 and = 2of a non-symmetric system with cylindrical polarcoordinate system.Solid surfaces. A solid surface is a material surface and kinematics

require that the mass flux across the surface to be zero. This requires thenormal component of the fluid velocity to be that of the solid. The tangentialcomponent of velocity depends on the assumption made about the fluid viscosity.If the fluid is assumed to have zero viscosity the order of the equations of motionreduce to first order and the tangential components of velocity can not bespecified. Viscous fluids stick to solid surfaces and the tangential components ofvelocity is equal to that of the solid. Exception to the no-slip boundaryconditions is when the mean free path of as gas is similar to the dimensions of

the solid. An example is the flow of gas through a fine pore porous media.Porous surface. A porous surface may not be a no-flow boundary. Fluxthrough a porous material is generally described by Darcys law.

Fluid surfaces. If there is no mass transfer across a fluid-fluid interface,the interface is a material surface and the normal component of velocity on eitherside of the interface is equal to the normal component of the velocity of theinterface. The tangential component of velocity at a fluid interface is not knownapriori unless the interface is assumed to be immobile as a result of adsorbedmaterials. The boundary condition at fluid interfaces is usually jump conditionson the normal and tangential components of the stress tensor. Aris give athorough discussion on the dynamical connection between the surface and itssurroundings. If we assume that the interfacial tension is constant and that it ispossible to neglect the surface density and the coefficients of dilational and shearsurface viscosity then the jump condition across a fluid-fluid interface is

[ ]

2

2

ij j iT n H n

H

= = T n n

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where the bracket denotes the jump condition across the interface, His the mean

curvature of the interface, and is the interfacial or surface tension. The jumpcondition on the normal component of the stress is the jump in pressure across acurved interface given by the Laplace-Young equation. The tangentialcomponents of stress are continuous if there are no surface tension gradients

and surface viscosity. Thus the tangential stress at the clean interface with aninviscid fluid is zero.For boundary conditions at a fluid interface with adsorbed materials and

thus having interfacial tension gradients and surface viscosity, see Chapter 10 ofAris and the thesis of Singh (1996).

Boundary conditions for the potentials and vorticity. Some fluid flowproblems are more conveniently calculated through the scalar and vectorpotentials and the vorticity.

2

2

= +

=

=

v A

v

A w

The boundary condition on the scalar potential is that the normal derivative isequal to the normal component of velocity.

n

=

n

n v

The boundary condition on the vector potential is that the tangential components

vanish and the normal derivative of the normal component vanish (Hirasaki andHellums, 1970). Wong and Reizes (1984) introduced a method where the needfor calculation of the scaler potential is replaced by the use of an irrotationalcomponent of velocity.

( )

( )

0

0

t

n

n

=

=

A

A

In two dimensional or axisymmetric incompressible flow, it is notnecessary to have a scalar potential and the single nonzero component of thevector potential is the stream function. The boundary condition on the streamfunction for flow in the x1, x2plane of Cartesian coordinates is

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1 2

2 1

3

where

n nx x

A

=

=

=

n v n A

The boundary condition on the normal component of the vorticity of a fluidwith a finite viscosity on a solid surface is determined from the tangentialcomponents of the velocity of the solid. It is zero if the solid is not rotating. If theboundary is an interface between two viscous fluids then the normal componentof vorticity is continuous across the interface (C. Truesdell, 1960). If the interfaceis with an inviscid fluid, the tangential components of vorticity vanishes and thenormal derivative of the normal component vanishes for a plane interface(Hirasaki, 1967).

If the boundary is at along a region of space for which the velocity field is

known, the vorticity can be calculated from the derivatives of the velocity field.If the boundary is as surface of symmetry, the tangential components of

vorticity must vanish because the normal component of velocity and the normalderivative of velocity vanish. The normal derivative of the normal component ofvorticity vanishes from the solenoidal property of vorticity.

Scaling, Dimensional Analysis, and SimilarityWe have already seen some examples of scaling and dimensional

analysis when we determined when the continuity equations and equations ofmotion could be simplified. The concept of similarity states that the solution oftransport problems do not need to be determined separately for each value of the

parameters. Rather the variables and parameters can be grouped intodimensionless variables and dimensionless numbers and the solution will havefewer degrees of freedom. Also, in some cases the partial differential equationscan have the independent variables combined to fewer independent variablesand be expressed as ordinary differential equations. The concept of similaritydoes not apply only to mathematical solutions but is also used to design physicalanalogs of systems on a smaller scale or with different transport mechanism. Forexample, before numerical simulation the streamlines and pressure gradients forflow in petroleum reservoirs were studied by electrical conduction on a laboratoryscale model that is geometrically similar.

Dimensionless groups based on geometry. The aspect ratiois the ratio of

the characteristic lengths of the system. The symbol is normally used todenote the aspect ratio. We saw how the aspect ratio simplified the equations ofmotion for boundary layer flow and lubrication or film flow. The table below listssome examples of aspect ratio expressions for different problems.

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Examples of aspect ratio, /L boundary layer flow

ho/L lubrication or film flow

Ly/Lx width/length

D/L entrance effects in pipe flowDmax/Dmin eccentricity ratio

It may be difficult to define the characteristic length of an irregularlyshaped conduit or object. The characteristic dimension for an irregular conduit orobject can be determined by the hydraulic diameter.

4, conduit

4, 2 D object

6, 3 D object

xh

w

w

h

w

AD

P

V

AD

V

A

=

=

whereAxand Pware the cross-sectional area and wetted perimeter of the conduitand V and Aw are the volume and wetted area of an object. The definitionsreduce to the diameter of a cylinder or sphere for regular objects. (The readershould be aware that some definitions such as the hydraulic radius in BSL maynot reduce to the dimension of the regular object.) The hydraulic diameter may

provide length scales but exact similarity is not satisfied unless the conduits orobjects are geometrically similar.

Dimensionless groups based on equations of motion and energy. Wederived earlier the Reynolds number from the equations of motion anddimensionless groups from the energy equation when compressibility isimportant. We will discuss these further and additional dimensionless groups.

Interpretation of the Reynolds number

U L

basic definition

2

/U

U L

kinetic energy

shear stress

2

( )

v v

v

inertial force

viscous force

The Reynolds number can be interpreted as a ratio of kinetic energy toshear stress. We will see later that some of the dimensionless numbers differ by

6 - 32


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whether it is normalized with respect to the kinetic energy or the shear stressterm, i.e., it is a product of the Reynolds number and another dimensionlessnumber.

Suppose the characteristic value of the body force is gL and the

characteristic value of pressure is /L , i.e., due to capillary forces. The

dimensionless NavierStokes equation can be expressed as follows.

2

2 2

Re

* 1* * * *

*

Dv g Lp

Dt U g U L N

= +

fv

We can now define dimensionless groups that include gravity or buoyancy forcesand gravity forces.

Dimensionless groups based on gravity and capillarity2 2

Fr U UNg L g L

= = Froude number

2 2

/We

U L UN

L

= =

Weber number

Re

/

/Ca We

U U LN

LN N

= = =

capillary number

2

Re

/ Fr

Ng L g L

U U L N

= =

gravity number

2

/

WeBo

Fr

Ng L g LN

L N

= = = Bond number

We derived scale factors that have to be small in order to neglect theeffects of compressibility. They are summarized here.

Dimensionless groups necessary for incompressible flow

Ma

UN

c=

Mach number

L

c

(frequency length)/sonic velocity

2g L g Lc p

= density change due to body forces

Friction factor and drag coefficients. Friction factor and drag coefficientsare the force on the wall of a conduit or on an object normalized with respect tokinetic energy. There are some ambiguities in the literature that one should beaware of.

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Friction factors

2

wSP

m

fU

=

Stanton-Pannel

2

2 wF

mf U

= Fanning

2

8 wDW

m

fU

=

Darcy-Weisbach

2

2 /4Moody F DW

m

D P Lf f f

U=

= =

Moody

21/ 2( )

riction drag

drag

p

F Ff

A U

+=

drag coefficient

2

riction drag

t

p

F FC

A U

+=

drag coefficient

Bernoulli TheoremsWhen the viscous effects are negligible compared with the inertial forces

(i.e., large Reynolds number) there are a number of generalizations that can bemade about the flow. These are described by the Bernoulli theorems. The fluidis assumed to be inviscid and have zero thermal conduction so that the flow isalso barotropic (density a single-valued function of pressure). The first of theBernoulli theorems is derived for flow that may be rotational. A special case isfor motions relative to a rotating coordinate system where Coriolis forces arise.For irrotational flow, the Bernoulli theorem is a statement of the conservation

kinetic energy, potential energy, and the expansion energy. A macroscopicenergy balance can be made that includes the effects of viscous dissipation andthe work done by the system.

Steady, barotropic flow of an inviscid, nonconducting fluid withconservative body forces. The equations of motion for a Newtonian fluid is

2( ) ( )D

pDt

= + + + v

f v v .

The assumptions of steady, inviscid flow simplify the equations to

( ) p = v v f .

The assumptions of barotropic flow with conservative body forces allow,

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, ( )

( ) ( ( ))

pdp

p

p

= =

= +

f

v v

.

and by virtue of the identity

212

( ) ( )v = + v v w v

the equations of motion can be written

212

212

( ( ) )

( )

p v

H

H p v

+ + =

=

+ +

v w

v w .

If the body force is gravitational then = g z.Let H denote the function of which the gradient occurs on the left-hand

side of this equation. H is a vector normal to the surfaces of constant H.However, v

wis a vector perpendicular to both vand wso that these vectors aretangent to the surface. However, v and w are tangent to the streamlines andvortex lines respectively so that these must lie in a surface of constant H. Itfollows that His constant along the streamlines and vortex lines. The surfaces ofconstant H which are crossed with this network of stream and vortex lines areknown as Lamb surfaces and are illustrated in Fig. 6.2.

Coriolis force. Suppose that the motion is steady relative to a steadily

rotating axis with an angular velocity, . Batchelor (1967) derives the equationsof motion in this rotating frame and shows that we must now include the potentialfrom the centrifugal force and the addition of a Coriolis force term.

221

2

( 2 )

( )( )

2

H

H p v

= +

+ +

v w

x

Irrotational flow. If the flow is also irrotational, then w= 0 and hence theenergy function

212( )H p + + v

is constant everywhere.Ideal gas. For an ideal gas we have

( )p vp c c T=

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37/37

, if is constant

. . .

p

p T

p

p p

dpc dT

dpc dT

c T c

Q E D

=

=

=

energy eqn

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