Chapter 6 - Equations of Motion and Energy in Cartesian Coordinates

Transcription

1 Chater 6 - Equations of Motion and Energy in Cartesian Coordinates Equations of motion of a Newtonian fluid The Reynolds number Dissiation of Energy by Viscous Forces The energy equation The effect of comressibility Resume of the develoment of the equations Secial cases of the equations Restrictions on tyes of motion Isochoric motion Irrotational motion Plane flow Axisymmetric flow Parallel flow erendicular to velocity gradient Secialization on the equations of motion Hydrostatics Steady flow Creeing flow Inertial flow Boundary layer flow Lubrication and film flow Secialization of the constitutive equation Incomressible flow Perfect (inviscid, nonconducting) fluid Ideal gas Piezotroic fluid and barotroic flow Newtonian fluids Boundary conditions Surfaces of symmetry Periodic boundary Solid surfaces Fluid surfaces Boundary conditions for the otentials and vorticity Scaling, dimensional analysis, and similarity Dimensionless grous based on geometry Dimensionless grous based on equations of motion and energy Friction factor and drag coefficients Bernoulli theorems Steady, barotroic flow of an inviscid, nonconducting fluid with conservative body forces Coriolis force Irrotational flow Ideal gas 6-1

2 Reading assignment Chater &3 in BSL Chater 6 in Aris Equations of motion of a Newtonian fluid We will now substitute the constitutive equation for a Newtonian fluid into Cauchy s equation of motion to derive the Navier-Stokes equation Cauchy s equation of motion is i ai fi or Dv = = + T ij, j Dv a= = f + T The constitutive equation for a Newtonian fluid is T or = ( + λ Θ ) δ + µ e ij ij ij T= ( + λ Θ ) I+ µ e The divergence of the rate of deformation tensor needs to be restated with a more meaningful exression e ij, j 1 v v i j = + x j xj x i 1 vi 1 v = + x x x x 1 1 ( v) = vi + xi or j j i j j Thus 1 1 ( ) e= v+ v T ij, j = + ( λ + µ ) ( v) + µ vi x x i or T= + ( λ + µ ) Θ+ µ v i 6 -

3 Substituting this exression into Cauchy s equation gives the Navier-Stokes equation Dvi = fi + ( λ + µ ) ( v) + µ vi x x or i i Dv = f + ( λ + µ ) Θ+ µ v The Navier-Stokes equation is sometimes exressed in terms of the acceleration by dividing the equation by the density Dv v a= = + ( v ) v t 1 = f + ( λ + ν) Θ+ ν v where ν = µ/ and λ = λ/ ν is known as the kinematic viscosity and if Stokes relation is assumed λ + ν = ν/3 Using the identities v ( v) ( w v the last equation can be modified to give Dv 1 a= = f + ( λ + ν) Θ ν w v) If the body force f can be exressed as the gradient of a otential (conservative body force) and density is a single valued function of ressure (iezotroic), the Navier-Stokes equation can be exressed as follows Dv a= = [ Ω+ P( ) ( λ + ν) Θ] ν w where f = Ω and P( ) = d Assignment 61 Do exercises 6111, 6113, and 6114 in Aris The Reynolds number Later we will discuss the dimensionless grous resulting from the differential equations and boundary conditions However, it is instructive to 6-3

4 derive the Reynolds number N Re from the Navier-Stokes equation at this oint 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 versa when it is very small Its value gives the justification for assumtions of the limiting cases of inviscid flow and creeing flow We will consider the case of single-hase flow with conservative body forces (eg, gravitational) and density a single valued function of ressure The ressure and otential from the body force can be combined into a single otential 1 f = Ω where d Ω= gz If the change in density is small enough, the otential can be aroximated by otential that has the units of ressure P Ω, small change in density where P= gz Suose that the flow is characterized by a certain linear dimension, L, a velocity U, and a density For examle, if we consider the steady flow ast an obstacle, L may be it s diameter and U and the velocity and density far from the obstacle We can make the variables dimensionless with the following v v x U P =, x =, t = t, P = U L L = L, = L U The conservative body force, Navier-Stokes equation is made dimensionless with these variables 6-4

5 Dv = P + ( λ + µ ) Θ+ µ v U Dv U µ U µ U = P + ( λ / µ + 1) Θ + L L L L UL Dv + P = ( λ / µ + 1) Θ + v µ N Re Dv + P = ( λ / µ + 1) Θ + where N Re UL U = = µ µ U / L v v The Reynolds number artitions the Navier Stokes equation into two arts The left side or inertial and otential terms, which dominates for large N Re and the right side or viscous terms, which dominates for small N Re The otential gradient term could have been on the right side if the dimensionless ressure was defined differently, ie, normalized with resect to (µu)/l, the shear stress rather than kinetic energy Note that the left side has only first derivatives of the satial variables while the right side has second derivatives We will see later that the left side may dominate for flow far from solid objects but the right side becomes imortant in the vicinity of solid surfaces The nature of the flow field can also be seen form the definition of the Reynolds number The second exression is the ratio of the characteristic kinetic energy and the shear stress The alternate form of the dimensionless Navier-Stokes equation with the other definition of dimensionless ressure is as follows Dv NRe = P + ( λ / µ + 1) Θ + P P = µ U / L v Dissiation of Energy by Viscous Forces If there was no dissiation of mechanical energy during fluid motion then kinetic energy and otential energy can be exchanged but the change in the sum of kinetic and otential energy would be equal to the work done to the system However, viscous effects result in irreversible conversion of mechanical energy to internal energy or heat This is known as viscous dissiation of energy We will identify the comonents of mechanical energy in a flowing system before embarking on a total energy balance The rate that work W is done on fluid in a material volume V with a surface S is the integral of the roduct of velocity and the force at the surface 6-5

6 dw dt = v t S S ( n) = v T n = ( v T) V ds ds dv The last integrand is rather comlicated and is better treated with index notation ( vt) = T v + vt i ij, j ij i, j i ij, j Dvi = Tij vi, j + vi fi 1 Dv = Tij vi, j + fi vi 1 Dv ( v T) = T: v+ f v We made use of Cauchy s equation of motion to substitute for the divergence of the stress tensor The integrals can be rearranged as follows d dt V 1 1 Dv vdv= dv 1 Dv V = f vdv + ( v T) dv T: vdv V V V where T: v= T v ij i, j ( ) = f v+ v T 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 otential energy due to body forces The second term is the rate at which surface stresses do work on the material volume We will now focus attention on the last term The last term is the double contracted roduct of the stress tensor with the velocity gradient tensor Recall that the stress tensor is symmetric for a nonolar fluid and the velocity gradient tensor can be slit into symmetric and antisymmetric arts The double contract roduct of a symmetric tensor with an antisymmetric tensor is zero Thus the last term can be exressed as a double contracted roduct of the stress tensor with the rate of deformation tensor We will use the exression for the stress of a Newtonian fluid 6-6

7 T v = T e ij i, j ij ij = ( + λθ ) δ + µ e e = λ Θ e µ e e [ ] ij ij ij ii ij ij λ µ ( ) = Θ Θ Θ Φ = T: v where Φ is the second invariant of the rate of deformation tensor Thus the rate at which kinetic energy er unit volume changes due to the internal stresses is divided into two arts: (i) a reversible interchange with strain energy, Θ = ( / )( D / ), (ii) a dissiation by viscous forces, ( λ + µ ) Θ 4µ Φ Since Θ Φ is always ositive, this last term is always dissiative If Stokes relation is used this term is 4 Θ 4Φ 3 µ for incomressible flow it is 4µ Φ (The above equation is sometimes written -4µΦ, where Φ is called the dissiation function We have reserved the symbol Φ for the second invariant of the rate of deformation tensor, which however is roortional to the dissiation function for incomressible flow Υ is the symbol used later for the negative of the dissiation by viscous forces) The energy equation We need the formulation of the energy equation since u to this oint we have more unknowns than equations In fact we have one continuity equation (involving the density and three velocity comonents), three equations of motion (involving in addition the ressure and another thermodynamic variable, say the temerature) giving four equations in six unknowns We also have an equation of state, which in incomressible flow asserts that is a constant reducing the number of unknowns to five In the comressible case it is a relation = f ( T, ) which increases the number of equations to five In either case, there remains a ga of one equation, which is filled by the energy equation 6-7

8 The equations of continuity and motion were derived resectively from the rinciles of conservation of mass and momentum We now assert the first law of thermodynamics in the form that the increase in total energy (we shall consider only kinetic and internal energies) in a material volume is the sum of the heat transferred and work done on the volume Let q denote the heat flux vector, then, since n is the outward normal to the surface, -q n is the heat flux into the volume Let U denote the secific internal energy, then the balance exressed by the first law of thermodynamics is d dt v ( + U) dv = f vdv + v T nds q nds V V S S This may be simlified by subtracting from it the exression we already have for the rate of change of kinetic energy, using Reynolds transort theorem, and Green s theorem DU + q T: ( v ) dv = 0 V Since this is valid for any arbitrary material volume, we have assume continuity of the integrand DU = q+ T: ( v ) We assume Fourier s law for the conduction of heat q = k T We assume a Newtonian fluid for the dissiation of energy T: ( v) = ( v) +Υ Υ= ( λ + µ ) Θ 4µ Φ Substituting this back into the energy balance we have DU = ( k T) ( v ) +Υ Physically we see that the internal energy increases with the influx of heat, the comression and the viscous dissiation If we write the equation in the form 6-8

9 DU D = ( k T) +Υ the left-hand side can be transformed by one of the fundamental thermodynamic identities For if S is the secific entroy, TdS= du+ d(1/ ) = du ( / ) d Substituting this into the last equation for internal energy gives DS T = ( k T) +Υ Giving an equation for the rate of change of entroy Dividing by T and integrating over a material volume gives V DS dv d = dt V 1 Υ = ( k T) + dv T T V k k Υ ( T) ( T) T T T dv = + + V q n k Υ = + ( ) + T T T S S dv ds T dv V The second law of thermodynamics requires that the rate of increase of entroy should be no less than the flux of heat divided by temerature The above equation is consistent with this requirement because the volume integral on the right-hand side cannot be negative It is zero only if k or T and Υ are zero This equation also shows that entroy is conserved during flow if the thermal conductivity and viscosity are zero DS = 0, when µ = 0, and k = 0 Assignment 6 Do exercises 631 and 63 in Aris The Effect of Comressibility (Batcehlor, 1967) Isentroic flow The condition of zero viscosity and thermal conductivity results in conservation of entroy during flow or isentroic flow This ideal condition is useful for illustration the effect of comressibility on fluid dynamics 6-9

10 The conservation of entroy during flow imlies that density, ressure, and temerature is changing in a reversible manner during flow The relation between entroy, density, temerature, and ressure is given by thermodynamics S S ds = dt + d T T C 1/ = dt T T C dt β = d T where 1 β =- T d These relations may be combined with the condition that the material derivative of entroy is zero to obtain a relation between temerature and ressure during flow C DT βt D =, when µ = 0, and k = 0 The equation of state exresses the density as a function of temerature and ressure During isentroic flow the ressure and temerature are not indeendent but are constrained by constant entroy or adiabatic comression and exansion The density in this case is given as = (, S) We now have as many equations as unknowns and the system can be determined The simlifying feature of isentroic flow is that exchanges between the internal energy and other forms of energy are reversible, and internal energy and temerature lay assive roles, merely changing in resonse to the comression of a material element The continuity equation and equation of motion governing isotroic flow may now be exressed as follows 6-10

11 1 D + v = 0 c Dv = f where c = S The hysical significance of the arameter c, which has the dimensions of velocity, may be seen in the following way Suose that a mass of fluid of uniform density o is initially at rest, in equilibrium, so that the ressure o is given by = f o o The fluid is then disturbed slightly (all changes being isentroic), by some material being comressed and their density changed by small amounts, and is subsequently allowed to return freely to equilibrium and to oscillate about it (The fluid is elastic, and so no energy is dissiated, so oscillations about the equilibrium are to be exected) The erturbation quantities 1 (= - o ) and 1 (= - o ) and v are all small in magnitude and a consistent aroximation to the continuity equation and equations of motion is c 1 o 1 + o v = 0 t v o = f t 1 1 where c o is the value of c at = o On eliminating v we have c 1 f 1 1 = 1 1 f t o co The body forces commonly arise from the earth s gravitational field, in which case the divergence is zero and the last term is negligible excet in the unlikely event of a length scale of the ressure variation not being small comared with c o /g (which is about 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 1 and 1 satisfies the same equation The solutions of this equation reresents lane comression waves, which roagate with velocity c o and in which the fluid velocity v is arallel to the direction of roagation In 6-11

12 another words, c o is the seed of roagation of sound waves in a fluid whose undisturbed density is o Conditions for the velocity distribution to be aroximately solenoidal The assumtion of solenoidal or incomressible fluid flow is often made without a rigorous justification for the assumtion We will now reexamine this assumtion and make use of the results of the revious section to exress the conditions for solenoidal flow in terms of identifiable dimensionless grous The condition of solenoidal flow corresonds to the divergence of the velocity field vanishing everywhere We need to characterize the flow field by a characteristic value of the change in velocity U with resect to both osition and time and a characteristic length scale over which the velocity changes L The satial derivatives of the velocity then is of the order of U/L The velocity distribution can be said to be aroximately solenoidal if v << ie, if U L 1 D U << L For a homogeneous fluid we may choose and the entroy er unit mass S as the two indeendent arameters of state, in which case the rate of change of ressure exerienced by a material element can be exressed as (, ) = S D D DS S = c + The condition that the velocity field should be aroximately solenoidal is 1 D 1 DS U < < c c S L This condition will normally be satisfied only if each of the two terms on the left-hand side has a magnitude small comared with U/L We will now examine each of these terms I When the condition 1 D U << c L 6-1

13 is satisfied, the changes in density of a material element due to ressure variations are negligible, ie, the fluid is behaving as if it were incomressible This is by far the more ractically imortant of the two requirements for v to be a solenoidal vector field In estimating D / we shall lose little generality by assuming the flow to be isentroic, because the effects of viscosity and thermal conductivity are normally to modify the distribution of ressure rather than to control the magnitude of ressure variation We may then rewrite the last equation with the aid of equations of motion of an isentroic fluid derived in the last section D = + v t dv = + v t f dt dv = + v f t dt Thus 1 v f 1 Dv U + << c t c c L Showing that in general three searate conditions, viz that each term on the lefthand side should have a magnitude small comared with U/L if the flow field is to be incomressible I (i) Consider first the last term on the left-hand side of the above equation The order of magnitude of Dv / will be the same as that of v /t or v v (ie, U 3 /L), which ever is greater Thus the condition arising from this term can be exressed as U << 1 c or N Ma where N Ma << 1 U = c I (ii) The magnitude of the artial derivative of ressure with resect to time deends directly on the unsteadiness of the flow Let us suose that the flow field is oscillatory and that ν is a measure of the dominant frequency The rate of change of momentum is then the order of Uν Since the ressure 6-13

14 gradient is the order of the rate of change of momentum, the satial ressure variation over a region of length L is LUν Since the ressure is also oscillating, the magnitude of / t is then LUν Thus the condition that the first term be small comared to U/L is ν L c << 1 This condition is equivalent to the condition that the length of the system should be small enough that a ressure transient due to comression is felt instantaneously throughout the system I(iii) If we regard the body forces arising from gravity, the term from the body forces, v f/c, has a magnitude of order gu/c, so the condition that it be small comared to U/L is gl << 1 c In the case of air, we can use the isoentroic equation of state to find gl gl = c γ This shows that the condition is satisfied rovided the difference between the static-fluid ressure at two oints at vertical distance L aart is a small fraction of the absolute ressure, ie, rovided the length scale L characteristic of the velocity distribution is small comared to /g, the scale height of the atmoshere, which is about 84 km for air under normal conditions The fluid will thus behave as if it were incomressible when the three conditions I(i), I(ii), and I(iii) are satisfied The first is not satisfied in near sonic or hyersonic gas dynamics, the second is not satisfied in acoustics, and the third is not satisfied dynamical meteorology II The second condition necessary for incomressible flow is that arising from entroy This condition requires that variation of density of a material element due to internal dissiative heating or due to molecular conduction of heat into the element be small We will show later how the small variation of density leading to natural convection can be allowed by yet assume incomressible flow Resume of the develoment of the equations We have now obtained a sufficient number of equations to match the number of unknown quantities in the flow of a fluid This does not mean that we can solve them nor even that the solution will exists, but it certainly a necessary beginning It will be well to review the rinciles that have been used and the assumtions that have been made 6-14

15 The foundation of the study of fluid motion lies in kinematics, the analysis of motion and deformations without reference to the forces that are brought into lay To this we added the concet of mass and the rincile of the conservation of mass, which leads to the equation of continuity, D + ( v) = + ( v ) = 0 t An analysis of the nature of stress allows us to set u a stress tensor, which together with the rincile of conservation of linear momentum gives the equations of motion Dv = f + T If the conservation of moment of momentum is assumed, it follows that the stress tensor is symmetric, but it is equally ermissible to hyothesize the symmetry of the stress tensor and deduce the conservation moment of momentum For a certain class of fluids however (hereafter called olar fluids) the stress tensor is not symmetric and there may be an internal angular momentum as well as the external moment of momentum As yet nothing has been said as to the constitution of the fluid and certain assumtions have to be made as to its behavior In articular we have noticed that the hyothesis of Stokes that leads to the constitutive equation of a Stokesian fluid (not-elastic) and the linear Stokesian fluid which is the Newtonian fluid Tij = ( + α) δij + β eij + γ eik ekj, Stokesian fluid T = ( + λθ ) δ + µ e, Newtonian fluid ij ij ij The coefficients in these equations are functions only of the invariants of the rate of deformation tensor and of the thermodynamic state The latter may be secified by two thermodynamic variables and the nature of the fluid is involved in the equation of state, of which one form is = f ( T, ) If we substitute the constitutive equation of a Newtonian fluid into the equations of motion, we have the Navier-Stokes equation Dv = + λ + µ + µ f ( ) ( v) v 6-15

16 Finally, the rincile of the conservation of energy is used to give an energy equation In this, certain assumtions have to be made as to the energy transfer and we have only considered the conduction of heat, giving DU = ( k T) ( v ) +Υ These equations are both too general and too secial They are too general in the sense that they have to be simlified still further before any large body of results can emerge They are too secial in the sense that we have made some rather restrictive assumtions on the way, excluding for examle elastic and electromagnetic effects Secial cases of the equations The full equations may be secialized is several ways, of which we shall consider the following: (i) restrictions on the tye of motions, (ii) secializations on the equations of motion, (iii) secializations of the constitutive equation or equation of state This classification is not the only one and the classes will be seen to overla We shall give a selection of examles and of the resulting equations, but the list is by no means exhaustive Under the first heading we have any of the secializations of the velocity as a vector field These are essentially kinematic restrictions (ia) Isochoric motion (ie, constant density) The velocity field is solenoidal 1 D = v =Θ= 0 The equation of continuity now gives D = 0, that is, the density does not change following the motion This does not mean that it is uniform, though, if the fluid is incomressible, the motion is isochoric The other equations simlify in this case for we have α, β, and γ of the constitutive equations functions of only Φ and Ψ of the invariants of the rate of deformation tensor In articular for a Newtonian fluid T = δ + µ e ij ij ij Dv = f + µ v 6-16

17 The energy equation is DU = ( k T) +Υ, and for a Newtonian fluid Υ= 4µ Φ Because the velocity field is solenoidal, the velocity can be exressed as the curl of a vector otential v = A The Lalacian of the vector otential can be exressed in terms of the vorticity w = v = ( A) = ( A) A =, if = A A 0 If the body force is conservative, ie, gradient of a scalar, the body force and ressure can be eliminated from the Navier-Stokes equation by taking the curl of the equation w = w v + ν w D where ν 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 not aly We showed during the discussion of the effects of comressibility that comressibility or non-isochoric is imortant in the cases of significant Mach number, high frequency oscillations such as in acoustics, large dimensions such as in meteorology, and motions with significant viscous or comressive heating (ib) Irrotational motion The velocity field is irrotational w = v=0 It follows that there exists a velocity otential ϕ(x,t) from which the velocity can be derived as v = ϕ 6-17

18 and in lace of the three comonents of velocity we seek only one scalar function (Note that some authors exress the velocity as the gradient of a scalar and others as the negative of a gradient of a scalar We will use either to conform to the book from which it was extracted) The continuity equation becomes D + ϕ = 0 so that for an isochoric (or incomressible), irrotational motion, ϕ is a otential function satisfying ϕ = 0 The Navier-Stokes equations become ϕ ( ϕ) = + ( λ + ν) ( ϕ) t f In the case of an irrotational body force f = Ω and when is a function only of, this has an immediate first integral since every term is a gradient Thus if P( ) = d/, ϕ 1 ( ) ( ) ( ) + ϕ +Ω+ P λ + ν ϕ = g ( t) t is a function of time only Irrotational motions with finite viscosity are only very secial motions because the no-sli boundary conditions on solid surfaces usually will cause generation of rotation Usually irrotational motion is associated with inviscid fluids because the no-sli boundary condition then will not aly and initially irrotational motion will remain irrotational (ic) Comlex lamellar motions, Betrami motions, ect These names can be alied when the velocity field is of this tye Various simlifications are ossible by exressing the velocity in terms of scalar fields We shall not discuss them further here (id) Plane flow Here the motion is restricted to two dimensions which may be taken to be the 01 lane Then v 3 = 0 and x 3 does not occur in the equations Also, the vector otential and the vorticity have only one nonzero comonent 6-18

19 Incomressible lane flow Since the flow is solenoidal, the velocity can be exressed as the curl of the vector otential The nonzero comonent of the vector otential is the stream function v= A v = ε A = ε ψ i ij3 3, j ij, j ψ ψ ( v1, v, v3) =,,0 x x1 The vorticity has only a single comonent, that in the 03 direction, which we will write without suffix w = v w = ε v, j, k = 1, i ijk k, j v v w = x x 1 1 = ψ If the body force is conservative, ie, gradient of a scalar, then the body force and ressure disaear from the Navier-Stokes equation uon taking the curl of the equations In lane flow Dw = ν w Thus for incomressible, lane flow with conservative body forces, the continuity equation and equations of motion reduce to two scalar equations Incomressible, irrotational lane motion A vector field that is irrotational can be exressed as the gradient of a scalar Since the flow is incomressible, the velocity vector field is solenoidal and the Lalacian of the scalar is zero, ie, it is harmonic or an analytical function v = ϕ v = 0 = ϕ Since the flow is incomressible, it also can be exressed as the curl of the vector otential, or in lane flow as derivatives of the stream function as above Since the flow is irrotational, the vorticity is zero and the stream function is also v = A, 0 = w = v= A+ A, ψ = 0 an analytical function, ie, ( ) Thus 6-19

20 v 1 v ϕ ψ = = x x 1 ϕ ψ = = x x 1 These relations are the Cauchy-Riemann relations show that the comlex function f = ϕ + iψ is an analytical function of z = x1+ ix The whole resources of the theory of functions of a comlex variable are thus available and many solutions are known Steady, lane flow If the fluid is comressible but the flow is steady (ie, no quantity deends on t) the equation of continuity becomes ( v1) ( v) + = 0 x x 1 A stream function can again be introduced, this time in the form 1 ψ 1 ψ v =, v = 1 x x1 The vorticity is now given by w = ψ + ψ (ie) Axisymmetric flows Here the flow has an axis of symmetry such that the flow field can be exressed as a function of only two coordinates by using curvilinear coordinates The curl of the vector otential and velocity has only one non-zero comonent and a stream function can be found (if) Parallel flow erendicular to velocity gradient If the flow is arallel, ie, the streamlines are arallel and are erendicular to the velocity gradient, then the equations of motion become linear in velocity if the fluid is Newtonian Dv v If v v= 0, then =, and t v = f + T t The second tye of secializations are limiting cases of the equations of motion 6-0

21 (iia) Hydrostatics When there is no flow, the only non-zero terms in the equations of motion are the body forces and ressure gradient f = If the body force is conservative, ie, the gradient of a scalar, then the hydrostatic ressure can be determined from this scalar f = Ω = Ω Φ Ω ( ) = 0 Φ=Ω+ constant where d Φ= If the body force is due to gravity then Ω= gz where z is the elevation above a datum such as the mean sea level (iib) Steady flow Examles of this have already been given and indeed it might have been considered as a restriction of the first class All artial derivatives with resect to time vanish and the material derivative reduce to the following D = v In articular, the continuity equation is ( v ) = 0 so that the mass flux field is solenoidal (iic) Creeing flow It is sometimes justifiable to assume that the velocity is so small that the square of velocity is negligible by comarison with the velocity itself This linearizes the equations and allows them to be solved more readily For examle, the Navier-Stokes equation becomes v = + ( λ + µ ) ( ) + t f v µ v 6-1

22 In articular, for steady, incomressible, creeing flow with conservative body forces P = µ v where P= gz However, since the continuity equation is v = 0, we have P = 0 or P is a harmonic otential function This is the starting oint for Stokes solution of the creeing flow about a shere and for its various imrovements One may ask, How small must the velocity be in order to neglect the nonlinear terms? To answer this question, we need to examine the value of the Reynolds number However, this time the ressure and body forces will be made dimensionless with resect to the shear forces rather than the kinetic energy v v x U P x t t P U L L µ U / L ** =, =, =, = = L, = L The dimensionless Navier-Stokes equation for incomressible flow is now as follows N Re Dv N Re = P + ** v where UL U = = µ µ U / L Creeing flow is justified if the Reynolds number is small enough to neglect the left-hand side of the above equation If the dimensionless variables and their derivatives are the order of unity, then creeing flow is justified if the Reynolds number is small comared to unity Another secialization of the equations of motion where the equations are made linear arises in stability theory when the basic flow is known but erturbed by a small amount Here it is the squares and roducts of small erturbations that are regarded as negligible (iid) Inertial flow The flow is said to be inviscid when the inertial terms are dominant and the terms with viscosity in the equations of motion can be neglected We can examine the conditions when this may be justified from the 6 -

23 dimensionless equations of motion with the ressure and body forces normalized with resect to the kinetic energy N Re Dv + P = ( λ / µ + 1) Θ + where N Re P UL U = = µ µ U / L P = U v The limit of inviscid flow may occur when the Reynolds number is becomes very large such that the right-hand side of the above equation is negligible Notice that if the right-hand side of the equation vanishes, the equation goes from being second order in satial derivatives to first order The differential equation goes from being arabolic to being hyerbolic and the number of ossible boundary condition decreases The no-sli boundary condition at solid surfaces can no longer aly for inviscid flow These tyes of roblems are known as singular erturbation roblems where the differential equations are first order excet near boundaries where they become second order Physically, the form drag dominates the skin friction in inertial flow The macroscoic momentum balance is described by Bernoulli theorems Notice that inviscid flow is necessary for irrotational flow ast solid objects but inviscid flow may be rotational If fact much of the classical fluid dynamics of vorticity is based on inviscid flow (iie) Boundary-layer flows Ideal fluid or inviscid flow may be assumed far from an object but real fluids have no-sli boundary conditions on solid surfaces The result is a boundary-layer of viscous flow with a large vorticity in a thin layer near solid surfaces that merges into the ideal fluid flow at some distance from the solid surface It is ossible to neglect certain terms of the equations of motion comared to others The basic case of steady incomressible flow in two dimensions will be outlined If a rigid barrier extends along the ositive 01 axis the velocity comonents v 1 and v are both zero there In the region distant from the axis the flow is v 1 = U(x 1 ), v 0, and may be exected to be of the form shown in Fig 61, in which v 1 differs from U and v from zero only within a comaratively short distance from the late To exress this we suose L is a tyical dimension along the late and δ a tyical dimension of this boundary layer and that V 1 and V are tyical velocities of the order of magnitude of v 1 and v We then introduce dimensionless variables x x x v v = 1, y =, u = 1, v = L δ V V * * * * 1 6-3

24 which will be of the order of unity This in effect is a stretching uward of the coordinates so that we can comare orders of magnitude of the various terms in the equations of motion, for now all dimensionless quantities will be of order of unity It is assumed that δ << L, but the circumstances under which this is valid will become aarent later It is also assumed that the functions are reasonably smooth and no vast variations of gradient occur The equation of continuity becomes V u V v L x δ y * * 1 + = 0 * * which would lose its meaning if one of these terms were comletely negligible in comarison with the other It follows that V δ = O V L, 1 where the symbol O means "is of the order of" The Navier-Stokes equations become * * * * V1 * u V1V * u o V 1 u L u u + v = + ν * * * + * L x δ y L x L x δ y and VV v V v V δ v v L x δ y δ y δ L x y * * * * * 1 * * o u + v = + ν * * * + * * In the first of these equations the last term on the right-hand side dominates the Lalacian and u*/ x* can be neglected Dividing through by V 1 /L we see that the other terms will be of the same order of magnitude rovided Lν o = V1 and = O(1) δ V 1 * * Inserting these orders of magnitude in the second equation and the condition that δ << L results in the ressure gradient / y being the dominant term Thus is a function of x only Returning to the original variables, we have the equations v1 v + = 0 x1 x v v 1 d v1 v v = + x1 x dx ν x 6-4

25 The circumstances under which these simlified equations are valid are given by the term above that we assumed to be of order of unity, which we rewrite as δ ν = O L V1 L Since it was assumed that δ << L this equation shows that this will be the case if ν << VL 1 The assumtion that the ressure is O( V 1 ) is consistent with the assumtion that the outer flow is inertial denominated flow (iif) Lubrication and film flow Lubrication and film flow is another case where one dimension is small comared to the other dimensions Lubrication flow is usually between two solid surfaces with the relative velocity between the two surfaces secified Film flow is usually between a solid and fluid surfaces or between two fluid interfaces and the driving force for flow is usually buoyancy or the Lalace ressure between curved interfaces Because of the one dimension being small, lubrication and film flow is about always at low Reynolds numbers The equations of motion are secialized by normalizing the variables with characteristic quantities of the flow and droing the terms that are negligible Incomressible flow with low Reynolds number of a Newtonian fluid is assumed Let the direction normal to the film be the 03 direction The double subscrit, 1, will be used to denote the two coordinate directions in the lane of the film Let h o be a characteristic film thickness and L be a characteristic dimension in the lane of the film and h o /L << 1 The variables are normalized to be order of unity x x h t x =, x =, h =, t = * 1 * 3 * * 1 3 L ho ho to v vt gz v =, v =, P = U h P * 1 * 3 o * 1 3 o where σ µ U L Po = gl or Po = or Po = f R L ho o The film boundaries are material surfaces and the velocity erendicular to the lane of the film is the rate of change of film thickness v 3 h h = t ho h = t t o * * Substitution into the integral of the equation of continuity result in the following 6-5

26 h 0 h 0 v3 h dx 3 = v 3 0 x3 v x ( hv ) dx = + O( h / L) x1 * * * ( hv1) L h + Oh ( / ) * = * o L x1 Uto t o The characteristic time can be chosen as to make the dimensionless grou equal to unity t o L = U The dimensionless variables can now be substituted into the equations of motion for zero Reynolds number with gravitational body force and Newtonian fluid For the comonents in the lane of the film, U U L 0 µ µ v = + + * * * * * 1 1P 1v1 * PL o Ph o o x3 The dimensionless grou in the last term can be made equal to unity because of the two characteristic quantities, P o and U, only one is secified and the other can be determined from the first µ UL = 1 Ph o and o Ph o o Lµ U U =, or Po = Lµ h o The equations of motion in the lane of the film is now v 0 = + + ( o * * * 1 1 P O h / L ) * x3 The dimensionless variables can now be substituted into the equation of motion erendicular to the lane of the film 6-6

27 * 4 * µ h * * o µ h o v3 * 1v 3 * 3 o o o o 3 P U L U L 0 = + + x Ph L Ph L x * P 0 = + Oh ( o / L) x * 3 This last equation shows that the ressure is aroximately uniform over the * thickness of the film Thus the velocity rofile of v 1 can be determined by integrating the equation of motion in the lane of the film twice and alying the aroriate boundary conditions The average velocity in the film can be determined by integrating the velocity rofile across the film Certain of the secializations based on the constitutive equation or equation of state have turned u already in the revious cases We mention here a few imortant cases (iiia) Incomressible fluid An incomressible fluid is always isochoric and the considerations of (ia) aly It should be remembered that for an incomressible fluid the ressure is not defined thermodynamically, but is an variable of the motion (iiib) Perfect fluid A erfect fluid has no viscosity so that T= I and Dv = f If, in addition, the fluid has zero conductivity the energy equation becomes DS = and the flow is isentroic 0 (iiic) Ideal gas An ideal gas is a fluid with the equation of state = RT The entroy of an ideal gas is given by dt S = cv Rln T 6-7

28 which for constant secific heats gives = e S c / v γ (iiid) Piezotroic fluid and barotroic flow When the ressure and density are directly related, the fluid is said to be iezotroic A simle relation between and allows us to write 1 d = P( ) = (iiie) Newtonian fluids Here the assumtion of a linear relation between stress and strain leads to the constitutive equation T= ( + λ Θ ) I+ µ e The equations of motion becomes the Navier-Stokes equations Assignment 63 Carry out the stes in secializing the continuity and equations of motion for boundary layer and lubrication or film flows 6-8

29 Boundary conditions The flow field is often desired for a finite region of sace that is bounded by a surface Boundary conditions are needed on these surfaces and at internal interfaces for the flow field to be determined The boundary conditions for temerature and heat flux are continuity of both across internal interfaces that are not sources or sinks and either a secified temerature, heat flux, or a combination of both at external boundaries Surfaces of symmetry Surfaces of symmetry corresonds to reflection boundary conditions where the normal comonent of the gradient of the deendent variables are zero Thus surfaces of symmetry have zero momentum flux, zero heat flux, and zero mass flux Because the momentum flux is zero, the shear stress is zero across a surface of symmetry Periodic boundary Periodic boundaries are boundaries where the deendent variables and its derivatives reeat themselves on oosite boundaries The boundaries may or may not be symmetry boundary conditions An examle of when eriodic boundaries are not symmetry boundaries are the boundaries of θ = 0 and θ = π of a non-symmetric system with cylindrical olar coordinate 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 the normal comonent of the fluid velocity to be that of the solid The tangential comonent of velocity deends on the assumtion made about the fluid viscosity If the fluid is assumed to have zero viscosity the order of the equations of motion reduce to first order and the tangential comonents of velocity can not be secified Viscous fluids stick to solid surfaces and the tangential comonents of velocity is equal to that of the solid Excetion to the no-sli boundary conditions is when the mean free ath of as gas is similar to the dimensions of the solid An examle is the flow of gas through a fine ore orous media Porous surface A orous surface may not be a no-flow boundary Flux through a orous material is generally described by Darcy s law Fluid surfaces If there is no mass transfer across a fluid-fluid interface, the interface is a material surface and the normal comonent of velocity on either side of the interface is equal to the normal comonent of the velocity of the interface The tangential comonent of velocity at a fluid interface is not known ariori unless the interface is assumed to be immobile as a result of adsorbed materials The boundary condition at fluid interfaces is usually jum conditions on the normal and tangential comonents of the stress tensor Aris give a thorough discussion on the dynamical connection between the surface and its surroundings If we assume that the interfacial tension is constant and that it is ossible to neglect the surface density and the coefficients of dilational and shear surface viscosity then the jum condition across a fluid-fluid interface is T ij nj = Hσ ni T n= H σ n [ ] 6-9

30 where the bracket denotes the jum condition across the interface, H is the mean curvature of the interface, and σ is the interfacial or surface tension The jum condition on the normal comonent of the stress is the jum in ressure across a curved interface given by the Lalace-Young equation The tangential comonents of stress are continuous if there are no surface tension gradients and surface viscosity Thus the tangential stress at the clean interface with an inviscid fluid is zero For boundary conditions at a fluid interface with adsorbed materials and thus having interfacial tension gradients and surface viscosity, see Chater 10 of Aris and the thesis of Singh (1996) Boundary conditions for the otentials and vorticity Some fluid flow roblems are more conveniently calculated through the scalar and vector otentials and the vorticity v= ϕ + A ϕ = v A= w The boundary condition on the scalar otential is that the normal derivative is equal to the normal comonent of velocity n ϕ ϕ n = n v The boundary condition on the vector otential is that the tangential comonents vanish and the normal derivative of the normal comonent vanish (Hirasaki and Hellums, 1970) Wong and Reizes (1984) introduced a method where the need for calculation of the scaler otential is relaced by the use of an irrotational comonent of velocity A () t A = 0 ( n) n = 0 In two dimensional or axisymmetric incomressible flow, it is not necessary to have a scalar otential and the single nonzero comonent of the vector otential is the stream function The boundary condition on the stream function for flow in the x 1, x lane of Cartesian coordinates is 6-30

31 n v= n A ψ = n n 1 x x1 where ψ = A 3 ψ The boundary condition on the normal comonent of the vorticity of a fluid with a finite viscosity on a solid surface is determined from the tangential comonents of the velocity of the solid It is zero if the solid is not rotating If the boundary is an interface between two viscous fluids then the normal comonent of vorticity is continuous across the interface (C Truesdell, 1960) If the interface is with an inviscid fluid, the tangential comonents of vorticity vanishes and the normal derivative of the normal comonent vanishes for a lane interface (Hirasaki, 1967) If the boundary is at along a region of sace 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 comonents of vorticity must vanish because the normal comonent of velocity and the normal derivative of velocity vanish The normal derivative of the normal comonent of vorticity vanishes from the solenoidal roerty of vorticity Scaling, Dimensional Analysis, and Similarity We have already seen some examles of scaling and dimensional analysis when we determined when the continuity equations and equations of motion could be simlified The concet of similarity states that the solution of transort roblems do not need to be determined searately for each value of the arameters Rather the variables and arameters can be groued into dimensionless variables and dimensionless numbers and the solution will have fewer degrees of freedom Also, in some cases the artial differential equations can have the indeendent variables combined to fewer indeendent variables and be exressed as ordinary differential equations The concet of similarity does not aly only to mathematical solutions but is also used to design hysical analogs of systems on a smaller scale or with different transort mechanism For examle, before numerical simulation the streamlines and ressure gradients for flow in etroleum reservoirs were studied by electrical conduction on a laboratory scale model that is geometrically similar Dimensionless grous based on geometry The asect ratio is the ratio of the characteristic lengths of the system The symbol α is normally used to denote the asect ratio We saw how the asect ratio simlified the equations of motion for boundary layer flow and lubrication or film flow The table below lists some examles of asect ratio exressions for different roblems 6-31

32 Examles of asect ratio, α δ/l boundary layer flow h o /L lubrication or film flow L y /L x width/length D/L entrance effects in ie flow D max /D min eccentricity ratio It may be difficult to define the characteristic length of an irregularly shaed conduit or object The characteristic dimension for an irregular conduit or object can be determined by the hydraulic diameter 4Ax Dh =, conduit P w 4 V, D object Aw Dh = 6 V, 3 D object Aw where A x and P w are the cross-sectional area and wetted erimeter of the conduit and V and A w are the volume and wetted area of an object The definitions reduce to the diameter of a cylinder or shere for regular objects (The reader should be aware that some definitions such as the hydraulic radius in BSL may not reduce to the dimension of the regular object) The hydraulic diameter may rovide length scales but exact similarity is not satisfied unless the conduits or objects are geometrically similar Dimensionless grous based on equations of motion and energy We derived earlier the Reynolds number from the equations of motion and dimensionless grous from the energy equation when comressibility is imortant We will discuss these further and additional dimensionless grous Interretation of the Reynolds number UL basic definition µ U kinetic energy µ U / L shear stress v ( v) µ v inertial force viscous force The Reynolds number can be interreted as a ratio of kinetic energy to shear stress We will see later that some of the dimensionless numbers differ by 6-3

33 whether it is normalized with resect to the kinetic energy or the shear stress term, ie, it is a roduct of the Reynolds number and another dimensionless number Suose the characteristic value of the body force is gl and the characteristic value of ressure is σ/l, ie, due to caillary forces The dimensionless Navier Stokes equation can be exressed as follows Dv* g L f σ 1 = + * * * * * U v g U L NRe We can now define dimensionless grous that include gravity or buoyancy forces and gravity forces Dimensionless grous based on gravity and caillarity U U Froude number NFr = = gl gl U L U Weber number NWe = = σ σ / L U U / L N = µ µ Ca NWe NRe σ = σ / L = caillary number gl gl N gravity number Re = = µ U µ U / L NFr gl gl NWe N = Bo σ = σ / L = Bond number N Fr We derived scale factors that have to be small in order to neglect the effects of comressibility They are summarized here Dimensionless grous necessary for incomressible flow U Mach number NMa = c ν L (frequency length)/sonic velocity c gl gl density change due to body forces = c γ Friction factor and drag coefficients Friction factor and drag coefficients are the force on the wall of a conduit or on an object normalized with resect to kinetic energy There are some ambiguities in the literature that one should be aware of 6-33

34 Friction factors τ w fsp = U f f F DW τ = = m w Um 8τ w Um D P/ L f = 4 f = f f Moody= F DW Um drag F = A F Ct = A + F friction drag ( 1/U ) friction + F U drag Stanton-Pannel Fanning Darcy-Weisbach Moody drag coefficient drag coefficient Bernoulli Theorems When the viscous effects are negligible comared with the inertial forces (ie, large Reynolds number) there are a number of generalizations that can be made about the flow These are described by the Bernoulli theorems The fluid is assumed to be inviscid and have zero thermal conduction so that the flow is also barotroic (density a single-valued function of ressure) The first of the Bernoulli theorems is derived for flow that may be rotational A secial case is for 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, otential energy, and the exansion energy A macroscoic energy balance can be made that includes the effects of viscous dissiation and the work done by the system Steady, barotroic flow of an inviscid, nonconducting fluid with conservative body forces The equations of motion for a Newtonian fluid is Dv = + λ + µ + µ f ( ) ( v) v The assumtions of steady, inviscid flow simlify the equations to ( v ) v= f The assumtions of barotroic flow with conservative body forces allow, 6-34