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Continuum Fundamentals

James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009

Conserved Quantities Let a conseved quantity have amount F per unit . Examples are as follows: For conservation of , we take F = ρ , where ρ is mass . For conservation of linear , F = ρu, where u is the vector (relative to some Newtonian reference frame). For conservation of , F = ρx × u, where x is the vector (relative to a fixed point in that frame).

1 2 For conservation of (first law of ), F = ρ (e + 2 |u| ) where e is the 1 2 per unit mass and 2 |u| is the per unit mass. There is also the non- of (second law of thermodynamics) for which the same framework here will be used, replacing an “ = ” with a “ ≥ ”, and in which F = ρs where s is entropy per unit mass. We assume that at the continuum scales of interest here, the mass density ρ as well as the the momentum per unit mass ρu can be interpreted as well-defined averages of microscopic and momenta. That is, they are averages over minute spatial regions and intervals, which can nevertheless be chosen large enough that there are negligible statistical fluctuations about those averages at a given x,t in different macroscopic realizations of the same or flow process. Accordingly, u must then be interpreted as the mass averaged velocity. When studying the of a surface S in the continuum which is said to move with the material, we will interpret that as moving with the local velocity u. Correspondingly, then, in writing the law of conservation of, e.g., linear momentum for a region of material momentarily occupying volume V , the surface flux rate of momentum T (where T is the vector along the bounding surface S of V ), is understood to include contributions not just from acting on S but also from transfers of momentum across S associated with microscopic scale relative to the mass-averaged motion u. We consider a single-component continuum. More complicated expressions than given here would be required to describe multi-component continua in which one component moves macroscopically relative to the other(s), like for fluids entraining particles or droplets of other fluids, or for deformable porous with fluid infiltration, etc.

Representation of Accumulation Rates in Conservation Laws; Reynolds Trans- port Theorem

1 We use Eulerian coordinates (fixed in ) so, focusing attention on an arbitrarily selected volume V of space, with bounding surface S. Conservation laws have the form:

Rate of accumulation of quantity F within the which, at time t, occupies region V

= Rate of generation of that quantity within V and and along its surface S

Because of the Eulerian coordinates, to properly judge the rate of accumulation of F , we must account for the change within V as well as for the fact that some of the matter within V at time t will, at an infinitesimally later time t + δt, have exited V across S at locations where n · u > 0 (here n is the outer unit normal to S), and that other matter will have entered V through S at places where n · u < 0. (Had we used Lagrangian coordinates (i.e., fixed along material points), the terms crossing S would not appear and, rather, we’d say that the boundary of V moves with velocity u.) Hence, with the Eulerian description,

Accumulation of quantity in V within time δt Z Z = [F (x, t + δt) − F (x, t)]dV + F (x, t)(n · uδt) dS V S to first order in δt. Dividing by δt and letting δt → 0, this shows that

Z ∂F Z Z ∂F  Rate of accumulation of quantity in V = dV + n · uF dS = + ∇·(uF ) dV V ∂t S V ∂t where the Theorem has been used in the last step. So we simply choose F = ρ, 1 2 or ρu, or ρx×u, or ρ(e+ 2 |u| ) to get the left side of the conservation law for the respective cases of mass, linear momentum, angular momentum, and energy. (The result of the above calculation, e.g., when rewritten as

d Z Z ∂F  F dV = + ∇·(uF ) dV , dt V (t) V ∂t

is sometimes referred to as the Reynolds transport theorem.)

Representation of Generation Rates in Conservation Laws The form taken by the generation rate will, if non-zero, be of the form Z Z Generation rate = H dV + hn dS V S Thus, in form, each conservation law will read

Z ∂F  Z Z + ∇·(uF ) dV = H dV + hn dS . V ∂t V S

2 In thise we note that H must be of the same physical , and same tensorial rank, as is F , whereas hn must have physical dimensions with length elevated by a of one relative to F and H (a surface density rather than a volume density) but must share the same tensorial rank as F and H.

Tensors Associated with Surface An important conclusion may now be drawn, from the conservation law itself, about the allowable form of any surface generation term hn: Divide both sides of the last equation by the total surface S of the region, and then let S → 0 and V → 0 in such a way that the greatest linear of V also → 0 (i.e., V and S shrink onto a point). In that limit the volume intergrals scale with V and do not contribute in the statement of the conservation law because V/S → 0. Thus 1 Z hn dS → 0 as S → 0 . S S

Now, in this limit, as V and S are shrunk onto a point, the quantity hn can only vary with the orientation n of the surface element at that point. The Cauchy tetrahedron argument then requires that for general n (= n1, n2, n3), the value hn on a surface element of that orientation has the form

(1) (2) (3) (k) hn = n1h + n2h + n3h ≡ nkh (1) (2) (3) Here h , h and h are the values of hn for surface elements with normals n pointing, respectively, in the positive 1, 2, and 3 directions. This result suffices for us to conclude that the quantity h defined by

(1) (2) (3) (k) h = e1h + e2h + e3h ≡ ekh (with hn = n · h) is a dyad (hence has components following transformation laws) having a tensorial rank which is higher by unity than the rank of F , H, and hn.

[For example, in the case of conservation of linear momentum, F , H, and hn are of rank 1, i.e., are vectors, so that h (the stress tensor) is of rank 2. In the case of , F , H, and hn are of rank 0, i.e., they are scalars, so that h (one term of which is the flux vector) is of rank 1.]

Local Form of a Conservation Law R Thus in the above integral form of a conservation law, the term hndS may now be rewritten R R S as S n · h dS = V ∇· h dV by the divergence theorem. Therefore, if the conservation law applies for any choice of V and if the terms within it are assumed to be locally continuous, then the local (pde) form of the conservation law is [with notation (...), t ≡ ∂(...)/∂t]

F, t + ∇·(uF ) = ∇· h + H

3 The conservation laws now follow. Mass:

In the case of mass, F = ρ and there will be no generation rate, H = hn = 0. Thus

ρ, t + ∇ · (ρu) = 0

This may be rearranged to Dρ ρ + u · ∇ρ + ρ ∇ · u = 0 or + ρ ∇ · u = 0 , t Dt

where D(...)/Dt ≡ (...), t + u · ∇(...) is the short notation for a rate of change following a material particle moving with velocity u. Linear Momentum: Linear momentum, for which F = ρu, is generated by , which may be divided, ac- cording to the Cauchy stress hypothesis, into a force ρg per unit volume within V and by a force per unit area acting along S. The sum of that force per unit area on S plus the contribution already noted from momentum flux per unit area associated with microscopic motions relative to the mass-averaged velocity u defines the surface stress vector T. Thus H = ρg (here g is per unit mass) and hn = T. Accordingly, the conservation law requires the existence of a second rank tensor, the stress tensor σ (= σijeiej), satisfying

T = n · σ (Tj = niσij)

Evidently, σijej is the stress vector T associated with a surface element having the unit outward normal n = ei at the point of interest.

Thus, with F = ρu, H = ρg and hn = n · σ, the local form is

(ρu), t + ∇ · (ρuu) = ∇ · σ + ρ g and simplifying with use of , this becomes Du ρ (u + u · ∇u) = ∇ · σ + ρ g or ρ = ∇ · σ + ρ g , t Dt

Angular Momentum:

In this case, F = ρ x × u, H = ρ x × g, and hn = x × T = x × (n · σ) so that the associated 2nd rank tensor h, associated with the vector hn by hn = n · h, is h = −σ × x. The local expression is then

(ρ x × u), t + ∇·(u(ρ x × u)) = ∇· (−σ × x) + ρ x × g

4 That can be greatly simplified by use of conservation of mass and of linear momentum so that there results an expression which is given most simply in component form as

σij ei × ej = 0 (implying that σij = σji) That is, σ is symmetric; σ = σT . Energy:

1 2 Now F = ρ (e + 2 |u| ), and the energy generation terms are H = ρ g · u + r in the volume, where r is the rate of radiant per unit volume, and hn = T · u − qn on the surface, where qn is the rate of heat outflow per unit area across S. As before, hn can be represented as n·h where in this case, for which hn is a scalar, h will be a vector. In particular, h = σ · u − q where the heat flow vector q, the existence of which is a consequence of the conservation law, satisfies qn = n · q on S. The resulting local form is 1 1 [ρ (e + |u|2)] + ∇·[ρu (e + |u|2)] = ∇ · (σ · u) + ρ g · u − ∇ · q + r 2 , t 2 and simplifying with the use of conservation of mass and linear and angular momentum, the energy equation is De ρ (e + u · ∇e) = σ : D − ∇ · q + r or ρ = σ : D − ∇ · q + r , t Dt where 1 D = sym(∇u) ≡ [(∇u) + (∇u)T ] 2 is the rate of deformation tensor.

It may be observed that σ : D (= σijDij) is the rate of stress working per unit volume, and therfore that w, the rate of stress working per unit mass, is given by σ : D w ≡ ρ In terms of w, the last form of the energy equation reads De/Dt = w + (r − ∇ · q)/ρ, which is the form expected from elementary considerations when it is recognized that (r −∇ · q)/ρ is the rate of heat absorption per unit mass. For example, in the case that the stress state is just a pure hydrostatic p, i.e. when σij = −p δij, like for inviscid fluid models, it is seen that σ : D ≡ σijDij = −p Djj = −p ∇ · u = p (Dρ/Dt)/ρ, where the last form given above for conservation of mass has been used. Thus w = p (Dρ/Dt)/ρ2 = −p D(1/ρ)/Dt, which is the expected result in this case. [Aside on (not required for ES 220): In the case of elastic materials, energy e and other thermodynamic functions of state would depend on strain E from a reference

5 configuration as well as on another state variable such as temperatuure or entropy s per unit mass. Several appropriate definitions of E can be made. For example, letting ρo and X denote ρ and x in the reference configuration, one commonly adopted definition of E is as the Green strain E, which is based on the change in under deformation. It is defined by requiring that dx · dx = dX · dX + 2 dX · E · dX for arbitrary choices of dX. Here dx denotes the material line element which was described by dX when in the reference configuration. Thinking of x as a of X and t, and writing dxi = (∂xi/∂Xk)dXk this then defines the components of E as     1 ∂xi ∂xi 1 ∂ξk ∂ξl ∂ξi ∂ξi Ekl = − δkl = + + 2 ∂Xk ∂Xl 2 ∂Xl ∂Xk ∂Xk ∂Xk where ξ = x − X is the of the considered material particle from its reference position. A -conjugate symmetric stress tensor S, called the second Piola-Kirchhoff stress, may then be defined with the property σ : D S : DE/Dt w = = which implies that S = det[F] F−1 · σ · FT −1 ρ ρo Here F (not to be confused with use of F above as a general symbol for density of a con- T served quantity) is defined by dx = F·dX, so that it has components Fij = ∂xi/∂Xj, F T −1 −1 is its (Fij = Fji), F is its inverse (dX = F ·dx), and det[F] (= ρo/ρ) is its determinant.] Entropy: Entropy is not a and, rather, can only be constrained to be non-decreasing in time (second law of thermodynamics). In this case the general integral and local forms of a conservation law above are replaced by the inequalities Z ∂F  Z Z + ∇·(uF ) dV ≥ H dV + hn dS (with hn = n·h) V ∂t V S

and F, t + ∇·(uF ) ≥ ∇· h + H In these expressions we take F = ρs, H = r/T , and h = −q/T to form what is called the Clausius-Duhem inequality, of which the local form is q r (ρs) + ∇·(ρus) ≥ −∇· ( ) + , t T T and which becomes, after simplification with conservation of mass, q r Ds q · ∇T ∇ · q r ρ (s + u · ∇s) ≥ −∇· ( ) + or ρ ≥ − + , t T T Dt T 2 T T

The last expression can be combined with the energy equation to rewrite the Clausius-Duhem inequality as De q · ∇T σ : D Ds + ≤ + T Dt ρT ρ Dt

6 Thermodynamic Constraints on Constitutive Laws, Simple Viscous For simple viscous fluids, which we think of as compressible in general, we assume that the stress tensor σ is a function of ρ, T and the instantaneous rate of deformation tensor D (but not of its history, like would be the case for complex fluids).

eq eq When D = 0, σ must reduce to its equilibrium value σ where σij = −p(ρ, T )δij Here p(ρ, T ) is the equilibrium pressure given by the thermal . Then when D 6= 0, we write

eq σij = τij + σij ≡ τij − p(ρ, T )δij

where p(ρ, T ) continues to be given by the thermal equation of state and the tensor τ may be called the viscous part of the stress tensor σ. Other thermodynamic properties such as e and s relate to ρ, T and p just like at equilibium, i.e., in a manner constraind by continuing validity of the perfect differential form

−p d(1/ρ) + T ds = de ,

that is, of D(1/ρ) Ds De −p + T = . Dt Dt Dt

We now observe that because Dkk = (D(1/ρ)/Dt) /(1/ρ), D(1/ρ) pD σeq : D (σ − τ ): D −p = − kk ≡ ≡ Dt ρ ρ ρ

and the equation before this last one then becomes an expression for De/Dt in the form

De Ds (σ − τ ): D = T + Dt Dt ρ

Substituting this last expression into the thermodynamic inequality given just before the start of this section, and concelling like terms on each side, we get (after multiplying through by −ρ) τ : D − q · (∇T )/T ≥ 0 . This must hold for all possible deformation rates and , and for all ρ and T . Now, by our assumptions about the viscous stress, τ is a function only of ρ, T and D, but not of ∇T . Thus, the inequality still gives a valid general constraint at points in space-time

7 for which ∇T = 0, and such points could in principle be encountered for any values of ρ, T and D. Hence we must require that τ : D ≥ 0 for all ρ, T and D. In a related manner, when D = 0 it is clear that the inequality

−q · ∇T ≥ 0

It must hold for all D if we assume that q is a function only of ρ, T and ∇T , but not of D. Linear viscous resistance and heat conduction relations: We here make the conven- tional additional assumptions that τ is linear in D, and that q is linear in ∇T , in ways that recognize the isotropy of fluid properties. The most general* laws of that type are

τ 0 = 2µD0 and tr(τ )/3 = κ tr(D) (Newtonian )

0 0 [where τ ≡ τ − tr(τ )I/3 and tr(τ ) ≡ τkk; similarly D ≡ D − tr(D)I/3 and tr(D) ≡ Dkk; here “tr” stands for “trace”, or sum of diagonal elements] and

q = −K∇T (Fourier law for heat conduction) ,

The factors in those equations are the shear viscosity µ, the bulk viscosity κ, and the thermal conductivity K. In general, all are functions of ρ and T . The “primed” quantities τ 0, D0, and σ0 are called the deviatoric parts of τ , D and σ, respectively, and we note that tr(τ 0) = 0, tr(D0) = 0 and tr(σ0) = 0. Also, the relation σ = τ − shows that the stress tensor satisfies

σ0 ≡ τ 0 = 2µD0 and tr(σ)/3 ≡ tr(τ )/3 − p = κ tr(D) − p .

That may be rewritten as an equivalent single expression for σ (rather than separately for its deviatoric part and its trace) as

σ = τ − pI = 2µD + (κ − 2µ/3)tr(D)I − pI ,

and the expressions may be inverted to represent D as 1  1 1  1  1 1  1 D = τ − − tr(τ )I = σ − − tr(σ)I + pI . 2µ 6µ 9κ 2µ 6µ 9κ 3κ

In these equations, we recall that p is the function p = p(ρ, T ) characterizing the thermal equation of state, and is not to be equated to −tr(σ)/3 (unless κ = 0 and/or tr(D) = 0). For example, if an “isotropic” stress state σ11 = σ22 = σ33 = −C (the notation C is used to denote a compressive stress), with off-diagonal σij = 0, is applied to a fluid element, the corresponding deformation rates will satisfy D11 = D22 = D33, and off-diagonal Dij = 0, where 3κD11 = p(ρ, T ) − C. Recognizing that 3D11 = −(1/ρ)Dρ/Dt in this circumstance,

8 it is seen that ρ would evolve according to the first order differential equation κDρ/Dt = ρ [C − p(ρ, T )]. To characterize the solution to that differential equation, we would have to say something about the thermal conditions on the fluid element. In the simplest case, when we assume T is held fixed by appropriate heat extraction from the element, it is easy to see that for fixed C, ρ will evolve monotonically, at a finite rate, towards the value which makes p(ρ, T ) = C, provided that κ > 0 (see next sub-section) and that the isothermal compressibility is positive (assured by ∂p(ρ, T )/∂ρ > 0). [* At the start of this sub-section expressions were given for τ 0, tr(τ ) and q (or, equivaletly, for τ and q) which were claimed to provide the most general possible linear relations to D and ∇T for an isotropic material. It is easy to understand that the result stated for q satisfies that claim, but the relation given between τ and D may merit more discussion. Such is provided here, making use of the of τ (which holds because of symmetry of σ and I). That symmetry means that at any given x and t, there exist three mutually orthogonal principal directions so that τ is diagonal relative to those axes. Let those directions be chosen as the 1, 2 and 3 directions, so that the τij vanish if i 6= j. Now we make use of linearity and isotropy. By linearity the instantaneous respose D to τ is the sum of the separate responses to τ11, τ22 and τ33. By isotropy the response to τ11 alone must be of the form D11 = A τ11,D22 = D33 = −B τ11,Dij = 0 for i 6= j

where A and B are scalar fluid properties. The response to τ22 alone, and to τ33 alone, may be written similarly, changing 1,2,3 to 2,3,1, and to 3,1,2, respectively, keeping the same factors A and B. Thus, summing these separate linear responses, we conclude that, e.g.,

D11 = A τ11 − B τ22 − B τ33 = (A + B) τ11 − B (τ11 + τ22 + τ33)

with similar expressions for D22 and D33, and with the off-diagonal Dij = 0. Thus, in this special, local principal axes ,

Dij = (A + B) τij − B δij τkk

holds for all i, j, i.e., for all deformation and stress components. However, Dij, τij and δij are second rank , whereas τkk is under coordinate transformation. Thus if we choose any other orthogonal coordinate directions which are rotated from the principal axes, the same relation between components of D and τ must hold in that new system. Renaming A and B in terms of µ and κ, that justifies the form asserted for relations between D and τ towards the start of this sub-section.] Requirement of non-negative and conductivity: It is easy to show that the viscous rate τ : D satisfies

τ : D = τ 0 : D0 + tr(τ ) tr(D)/3 = 2µ (D0 : D0) + κ [tr(D)]2 .

Thus the thermodynamic inequality τ : D − q · (∇T )/T ≥ 0 becomes 2µ (D0 : D0) + κ [tr(D)]2 + K |∇T |2/T ≥ 0 .

9 That can be satisfied for all possible D [meaning all possible D0 and tr(D)] and ∇T as may arise in flow processes only if all of µ, κ and K are non-negative.

Energy Equation We now recall the equation ρDe/Dt = σ : D − ∇ · q + r as well as the above transformation of the thermodynamic identity −p d(1/ρ) + T ds = de into ρDe/Dt = ρT Ds/Dt + (σ − τ ): D. Eliminating De/Dt from the expressions, and using q = −K∇T , allows the energy equation to be rewritten as Ds ∇ · (K ∇T ) + τ : D + r = ρ T Dt In this expression, we recall that s is related to p, ρ and T (themselves related by p = p(ρ, T )) just as for equilibrium states. Thus we may write s = s(T, ρ) or s = s(T, p), according to preference, and write

Ds ∂s(T, ρ) DT ∂s(T, ρ) Dρ Ds ∂s(T, p) DT ∂s(T, p) Dp = + or = + , Dt ∂T Dt ∂ρ Dt Dt ∂T Dt ∂p Dt

Two of the above partial derivatives of s are given in terms of specific per unit mass, cv or cp, at constant ρ or p, respectively, as ∂s(T, ρ)/∂T = cv/T and ∂s(T, p)/∂T = cp/T . Maxwell reciprocal relations associated with recognition that −p d(1/ρ) − s dT and dp/ρ − s dT are perfect differentials enable the other partial derivatives to be computed from the thermal equation of state. The result may be written as DT p Dρ ∇ · (K ∇T ) + τ : D + r = ρc − βˆ , v Dt ρ Dt DT Dp or as ∇ · (K ∇T ) + τ : D + r = ρc − αˆ , p Dt Dt

T ∂p(ρ, T ) T ∂ρ(p, T ) where βˆ = andα ˆ = − . p ∂T ρ ∂T

It may be noted thatα ˆ > 0 when the under fixed pressure is positive, and then βˆ > 0 too, so long as p > 0 and the isothermal compressibility of the fluid is postive (as we would always assume on grounds of thermodynamic stability). Also, for an ideal , βˆ =α ˆ = 1. Simplified energy equation: In simplified analyses of some flow and prob- lems, distinctions between cv abd cp are ignored (with notation c used instead), and the

10 Dρ/Dt and Dp/Dt terms are ignored (they are often small compared to other terms), so that the energy equation is written approximately as DT ∇ · (K ∇T ) + τ : D + r = ρc . Dt

11