Week 8 lectures. ρ t +u ρ+ρ u = 0. where µ and λ are viscosity and second viscosity coefficients, respectively and S is the strain tensor:

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1 Week 8 lectures. Equations for motion of fluid without incomressible assumtions Recall from week notes, the equations for conservation of mass and momentum, derived generally without any incomressibility assumtion leads to (. t + (u = t +u + u = 0 ( u (.2 t +(u u = + T+b, where T is the second order tress Tensor. For Newtonian fluids, (.3 T = 2µ ( 3 ( ui + [λ+ 23 ] µ ( ui, where µ and λ are viscosity and second viscosity coefficients, resectively and is the strain tensor: (.4 ij = 2 (u i,j +u j,i 2 ( xj u i + xi u j Using (.3 and (.4, (.2 becomes ( u (.5 t +(u u = +µ u+(λ+µ ( u+b In general, (. and (.5 are not enough to determine the flow since these constitute four scalar equations for five scalar unknowns u,,. Additional equations are needed, which come from energy conservation and thermo-dynamics as in the ensuing. 2. Energetic Considerations Consider an arbitary but fixed volume W Ω R 3 with smooth boundary having outwards normal n. Consider rate of change of energy contained in W: (2.6 d dt W ( ( 2 u 2 +E dx = t W 2 u 2 +E,dx where E(x,t is the internal energy erunit mass at a location x at time t. The inwards flux of energy through W, which includes heat

2 2 flux as well (, is given by ( (2.7 W 2 u 2 +E u ndx+ n (k Tdx W { ( [ ] } = u 2 u 2 +E (k T dx, W where T(x,t is the temerature at osition x at time t and k is the thermal conductivity. Now consider the work done by forces in the system. It is convenient to define the total tress tensor Σ so that (2.8 Σ ij = δ i,j +T ij The work done on the fluid inside W in unit time, including body and surface forces, is given by (2.9 u bdx+ u i Σ ij n j dx = (u i b i +u i,j Σ ij +u i Σ ij,j dx W W From conservation of energy, the sum of rate of change of energy given by (2.6 equals the sum of inwards energy flux (2.7 and the work done by forces er unit time given by (2.9. This is true for arbitrary W and so (2.0 ( [ ( ] t 2 u 2 +E + u 2 u 2 +E (k T = u i b i +u i,j Σ ij +u i Σ ij,j Using (. and (.2, noting that Σ ij = n j +T ij, we obtain (2. W DE = u i,jσ ij + (k T, where we recall oerator D = t +u. 3. Thermodynamic consideration We now recall some facts from equilibrium Thermodynamics (2. The intensive quantities used include internal energy er unit mass E, temerature T, ressure, secific volume v =, i.e. volume of fluid er unit mass, entroy er unit mass, which is a characteristic of disorder in the system (3 The fundamental ansatz in thermodynamics, which is suorted by latter theoretical develoments in statistical mechanics as ( This is because according to first law of thermodynamics, heat inutted into the system is transferred to work done by the system and change of internal energy. (2 For most uroses it is good enough to ignore non-equilibrium effects since the thermodynamic time-scale is far shorter than the time scale in which fluid moves (3 Using statistical mechanics, entroy can be quantified to be roortional to the log of the number of states in the system consistent with a given thermodynamic state.

3 well as exeriment, is that any two of these variables comletely characterize a system. For instance, if T and v are considered as indeendent variables, the rest of the variables,, E, etc., are each functions of T and v. Choice of different indeendent variables is suitable for different uroses. The first law of thermodynamics states that at each location occuied by a gas or fluid, (3.2 de = Td dv, where dq = Td denotes the infinitesimal heat generated, dw = dv is the infinitesimal work done on the system, er unit time. What is imortant about the relation (3.2 is that E is a state variable, which mathematically means that de is an exact differential, where in (3.2 we hae used and v as the two indeendent variables. Therefore, it follows from (3.2 that E E = T and =. From equality of mixed v second artials, where we assume E to be smooth enough function of and v, it follows that (3.3 ( ( = v v E is not the only state variable. I = E + v called enthaly er unit mass is another state variable. If we consider instead and to be two indeendent variables, it follows from (3.2 that (3.4 di = d(e +v = Td +vd Equality of two second mixed artials of I gives ( ( v (3.5 = Yet another state variable is the Gibbs free energy F = E T er unit mass. If we use v and T as two indeendent variables, then (3.2 imlies (3.6 df = dv dt The equality of mixed second artials of F gives ( ( (3.7 = v v Again, F +v is also a state variable. If we use and T as indeendent variable, it follows from (3.6 that (3.8 d(f +v = vd dt, T 3

4 4 equality of mixed artials immediately imlies ( ( v (3.9 = It is also convenient to define (3.20 c = T (3.2 c v = T (, (, which is the secific heat er unit mass for constant ressure and constant volume resectively, since dq = Td is the heat content (4 er unit mass er unit temerature. We also define volumetric exansion rate with temerature for fixed ressure ( v (3.22 β = 4. Full equation of fluid dynamics We return to (2.. We note from (3.2 that ( ( DT (4.23 T +T D = TD = DE + D Using (3.9, (3.20 and (4.23 and (2., we obtain (4.24 c DT βtd = TD = u i,jσ ij + (k T+ u Using (.3 and (2.8, it follows that (4.25 T D = c DT βtd = Φ+ λ+ 2µ 3 ( u 2 + (k T, T where the ositive definite quantity (4.26 Φ = 2µ ( ij ( 3 ( uδ i,j ij 3 ( uδ i,j = 2µ ( i,j i,j 3 ( u2 has the hysical interretation of dissiation of mechanical energy erunit mass As stated earlier, we also have an equation of state that follows from equilibrium thermodynamics (4.27 f(,,t = 0 v T (4 Note that in our formulation the unit of heat and unit of energy are chosen the same

5 Equations (., (.5, (4.25 and (4.27 constitute a comlete set of six scalar equations for the unknowns,, T and u, since the viscosity coefficients µ = µ(,t, λ = λ(,t and thermal conductivity k = k(,t are known functions. 4.. Bernoulli law for steady flow of a frictionless non-conducting fluid. Assume that the force erunit mass b = Ψ, is time-indeendent. Then, from (2.0, using (., (4.28 D (E + 2 u 2 = u Ψ+ x j (u i Σ ij + (k T Using (.3, (.4 and (., it follows that (4.29 x (uiσij = j u u+2µ x j = D ( + t + 2µ x j {u i [ ij 3 ]} ( uδi,j + (λ+ 23 µ ( u 2 {u i [ ij 3 ( uδi,j ]} + (λ+ 23 µ xj (u j u Therefore, for a frictionless non-conducting fluid limit, i.e. when viscosity coefficients µ, λ, as well as conductivity coefficient k is negligible, and the flow is steady, i.e. t = 0, we have from (4.28 and (4.29, (4.30 D ( 2 u 2 +E + +Ψ This means that along a streamline: (4.3 H = 2 u 2 +E + +Ψ = 0 is a constant; this is generalization of Bernoulli s rincile for a general comressible fluid that is frictionless, non-conducting and steady. 5. Isentroic and Homentroic flows ( We note from (4.25 if the ositive definite dissiation terms Φ, λ+ 2 µ ( u 2 are small, along with temerature diffusion term 3 (k T, each of which is the result of molecular diffusion, then we can aroximate D = 0, i.e. entroy is constant along a stream line, though it need not be the same constant on different straight lines. When the latter is true, the flow is said to be homentroic. For isentroic flow, it is useful to take equation of state in the form (5.32 = (, and define (5.33 ( = c 2 5

6 6 Then ignoring molecular diffusion terms in (.5, which is consistent with isentroic aroximation rocess, (. and (.5 simlify under isentroic aroximation to the follow relation (5.34 (5.35 D c 2 + u = 0 Du = b Equations (5.34, (5.35 along with D = 0 andequation of state (4.27 determine u,, and for an isentroic flow. For a homentroic flow, since is a constant everywhere, = ( and c 2 = c 2 ((, in which case the set of equations (5.34 and (5.35 comletely determine the unknowns and u; or equivalently and u if we choose to rewrite (5.34 and (5.35 in the form (5.36 D + u = 0 (5.37 Du = b w,where w = c 2 ( d As shall be seen later, these form a air of nonlinear hyerbolic PDEs; the solutions are tyically characterized by shocks, i.e. solutions need to be weak solutions with jums, and classical solutions have to be matched across the two sides of the shocks by considering conserved quantities on stream lines. Note from (4.23, ( D (5.38 E + = T D + D dt Therefore, it follows that for Isentroic flows, the Bernoulli rincile for steady flow (4.30 reduces to (5.39 H = c 2 (, 2 u 2 + d +Ψ = constant on astreamline Note that if a flow has a shock region, molecular effects like viscous dissiation and heat conduction become imortant; so tyically H will jum across the shock by an amount deending on how much dissiation and heat conduction takes lace in a thin shock region.

7 5.. ound roagation in homentroic flows. Consider small dearture from an equilibrium flow where u = 0. In equilibrium (5.37 imlies = 0 (5.40 b = 0 0 = w 0 uch an equilibrum is when b is time-indeendent and curl-free. Then, if we decomose (5.4 = 0 +, = 0 + and linearize for small, and u, then (5.34 and (5.35 imly ( c 2 t + u = 0 0 ( u t = b Dividing (5.43 by 0 and taking divergence of the resulting equation, while taking the time derivative of (5.42, we obtain on elimination of u ( ( 2 ( c 2 0 t = 2 b 0 0 Thisistheequationforroagationofsound. If 0 issatiallyuniform, and effect of force b is negligible, we obtain the usual wave equation (5.45 c t 2 =, where c 0 will be the seed of roagation of sound, as we know from elementary PDE class. We note that sound roagation is ossible in a medium with comressibility taken into account Vorticity in homentroic flow and otential flow aroximation. We recall the identity (5.46 u u = 2 u 2 u ω, where ω = u is the vorticity, it follows from taking curl of (5.37, recalling Du = tu+u u, (5.47 ω t + (ω u = b If force b is conservative, we note that if ω = 0 initially, then ω = 0 for all t (5, i.e. a otential flow aroximation u = φ is valid under those conditions. In this case, for steady flow, Bernoulli law (show 7 (5 This assumes uniqueness of solutions, which is true, but is yet to be roved.

8 8 as an exercise (5.39 is globally valid, i..e H is the same constant everywhere Validity of incomressible u = 0 aroximation. We return to the full set of equations (.,(.5, (4.25 and (4.27 for,, T and u and examine in a formal sense the conditions under which incomressibility u = 0 assumtions would remain valid. We denote atyicalvelocityscalebyscalaru andatyicallengthscaleofvariation of u to be L. o, from (., incomressibility assumtion is valid if (5.48 D << U L If we choose and entroy er unit mass as indeendent variables, then ( ( D (5.49 = D + D = ( D c 2 + D o condition (5.48 translates to (5.50 D c 2 ( D << U c 2 L This will be satisfied if each of the conditions D c 2 << U and L ( D c 2 << U D. are satisfied. Normally, in estimating, we L may assume flow to be isentroic since of molecular diffusion effects like viscosities and thermal conductivity only effect distribution of ressure gradient rather than its order of magnitude. Then using (5.35, it follows that the condition on ressure becomes (5.5 c 2 t D 2c 2 u 2 + c 2u b U << L Noting that time scale associated with choice of velocity scales U and L is L, the second term term D U 2c 2 u 2 << U, when L U2 /c 2 <<. Now conider the magnitude of the first term on the left of (5.5, which deends on the unsteadiness of the flow. If the flow-field is oscillatory with frequency n, noting that / scales as U 2, then the condition << U will be satisfied when n2 L 2 <<. If n = O( U, c 2 t L c 2 L then the above condition is satisfied when U 2 /c 2 <<. If n >> U, L then the condition is more exacting. Consider the last term in (5.5. It is clear that u b << U if c 2 L b L <<. If b is gravity, this condition is satisfied in most laboratory c 2 conditions, given the relatively large value of seed of sound.

9 We now consider the second term on the left of (5.50 with isentroic assumtion. We first note that from thermodynamic relations derived earlier (5.52 c 2 ( = c 2 ( ( = ( ( = βt c Using (4.25, it follows from (5.52 that the second term on the left of (5.50 << U if L (5.53 β { Φ+ ( k } U << c x j x j L, which is hysically the statement that variation of density of a material element in the fluid due to internal dissiation and conduction of heat is small. 9