The Methods of Chapman-Enskog and Grad and Applications

Transcription

1 The Methods of Chapman-Enskog and Grad and Applications Gilberto Medeiros Kremer Departamento de Física Universidade Federal do Paraná Curitiba, BRAZIL Contents 1 Introduction 3 Boltzmann and Balance Equations 4 3 Chapman-Enskog Method 7 4 Application of Chapman-Enskog Method to Granular Gases 14 5 Grad s Method of Moments 19 6 Chapman-Enskog-Grad Combined Method 3 7 Fourteen Moment Theory for Granular Gases 5 RTO-EN-AVT

2 9 - RTO-EN-AVT-194

3 1 INTRODUCTION 1 Introduction We may say that one of the main objectives of the kinetic theory is to describe the macroscopic properties of gases like pressure, temperature, thermal conductivity, viscosity, diffusion, etc. from microscopic quantities that are associated with the molecules which compose the gases like mass, velocity, kinetic energy, internal degrees of freedom and interaction forces between the molecules. The foundations of the modern kinetic theory of gases were established in 1867 by James Clerk Maxwell ( ) who proposed a general transport equation for arbitrary macroscopic quantities associated with mean values of microscopic quantities. This equation of transport relates the time evolution of a macroscopic quantity with the motion of the molecules, collision between the molecules and action of external forces. Although the theory was valid for any molecular interaction potential, Maxwell could only determine the transport coefficients of shear viscosity, thermal conductivity and diffusion by assuming that the interaction potential was derived from a repulsive force which was inversely proportional to the fifth power of the relative distance between the molecules. Nowadays this type of potential is known as Maxwellian potential. The kinetic theory of gases gained a new impulse in 187 with the work by Ludwig Eduard Boltzmann ( ), who proposed an integro-differential equation the Boltzmann equation which represents the evolution of the velocity distribution function in the phase space spanned by the coordinates and velocities of the molecules. In the Boltzmann equation the temporal change of the distribution function has two terms, one of them is a drift term due to the motion of the molecules while the other one is a collision term related to encounters of the molecules. Based on this equation, Boltzmann proposed the so-called H function which decreases with time or remains constant. The identification of this function as the negative of the gas entropy gave a molecular interpretation of the increase of the entropy for irreversible processes. Furthermore, Boltzmann in the same work presented a rigorous deduction of the Maxwellian distribution function. From the Boltzmann equation one could determine the velocity distribution function hence the transport coefficients of rarefied gases, however this task was not so easy. Indeed, it took almost forty years after the proposition of Boltzmann equation, for David Hilbert ( ) to show how one could get an approximate solution of the integro-differential equation from a power series expansion of a parameter which is proportional to the mean free path. Further advances were due to Sydney Chapman ( ) and David Enskog ( ) who in the years 1916 and 1917 calculated independently and by different methods the transport coefficients for gases whose molecules interact according to any kind of spherically symmetric potential function. Another method derived from the Boltzmann equation was proposed in 1949 by Harold Grad ( ) who expanded the distribution function in terms of tensorial Hermite polynomials and introduced balance equations corresponding to higher order moments of the distribution function. The aim of these notes is to discuss the methods of Chapman-Enskog and Grad with an applications to granular gases. (a) Cartesian notation for tensors is used with Latin indexes i, j, k,... which may range from 1 to 3 denoting the three-dimensional system of spatial coordinates x, y, z; (b) Einstein s summation convention over repeated indexes is used, for example, T ij v j 3 j=1 T ijv j ; RTO-EN-AVT

4 BOLTZMANN AND BALANCE EQUATIONS (c) Parentheses around the indexes denote the symmetric part of a tensor, brackets its antisymmetric part while angular parentheses refer to its traceless symmetric part, for example, T (ij) = 1 (T ij + T ji ), T [ij] = 1 (T ij T ji ), T ij = T (ij) 1 3 T rrδ ij. Above, δ ij is Kronecker s symbol and the traceless tensor T ij is called tensor deviator. Boltzmann and Balance Equations We consider a monatomic rarefied gas with N molecules enclosed in a recipient of volume V, where a molecule is specified as a point in a six-dimensional phase space spanned by its coordinates x = (x 1, x, x 3 ) and velocity components c = (c 1, c, c 3 ). The state of a gas is characterized by the one-particle distribution function f(x, c, t) such that f(x, c, t)dx dc gives at time t, the number of molecules in the volume element with position vectors within the range x and x + dx and with velocity vectors within the range c and c + dc. An elastic binary collision is characterized by the conservation laws of momentum and kinetic energy, namely, mc + mc 1 = mc + mc 1, 1 mc + 1 mc 1 = 1 mc + 1 mc 1, (1) where (c, c 1 ) refer to pre-collisional velocities and (c, c 1) to post-collisional velocities of two molecules. The subindex 1 is used in order to distinguish two identical molecules that participate in the collision. The relative velocity is denoted by g = c 1 c and the energy conservation law implies that g = g. The space-time evolution of the one-particle distribution function in the phase space is governed by the Boltzmann equation, which in the absence of external forces, reads f t + c f i = (f x 1f f 1 f)g b db dε dc 1. () i We note that it is a non-linear integro-differential equation for f(x, c, t). The right-hand side of the Boltzmann equation is related to the collisions of the molecules through the product of two distribution functions. The relative motion of the molecules is characterized by the impact parameter 0 b < and by the azimuthal angle 0 ε π (see Figure 1). Furthermore, the following abbreviations were introduced f f(x, c, t), f 1 f(x, c 1, t), f f(x, c, t), f 1 f(x, c 1, t). The multiplication of the Boltzmann equation () by an arbitrary function ψ ψ(x, c, t) and the integration of the resulting equation over all values of the velocity components c leads to t ψf dc + [ ] ψ ψc i f dc t + c ψ i f dc = (ψ ψ)f 1 f g b db dε dc 1 dc = 1 (ψ 1 + ψ ψ 1 ψ )(f 4 1f f 1 f)g b db dε dc 1 dc, (3) where the two equalities in the right-hand side of the above equation where obtained from the symmetry properties of the collision operator. 9-4 RTO-EN-AVT-194

5 BOLTZMANN AND BALANCE EQUATIONS Direct collision g = c c 1 Restitution cylinder Collision cylinder g db b π O ε dε g Collision plane g Figure 1: Geometry of a binary collision. From the analysis of the last equality in the right-hand side of (3) we may infer that it vanishes for any kind of distribution function f when ψ 1 + ψ = ψ 1 + ψ. A function ψ which fulfills such condition is called a summational invariant. A summational invariant is given by a linear combination of the conservative quantities: mass, linear momentum and energy, and is expressed by ψ(c) = A + B c + Dc where A and D are two scalar functions and B a vectorial function, all of them being independent of c. In kinetic theory a macroscopic state of a gas is characterized by quantities that are defined in terms of the distribution function f(x, c, t). Firstly, based on the microscopic quantities of the gas molecules as mass m, linear momentum mc i and energy mc / we define the mass density ϱ, the momentum density ϱv i and the total energy density ϱu of the gas, namely, mc ϱ = mf(x, c, t) dc, ϱv i = mc i f(x, c, t) dc, ϱu = f(x, c, t) dc. (4) If we substitute the molecular velocity in (4) 3 by the peculiar velocity C i = c i v i such that mc i f dc = 0 we obtain ϱu = 1 ϱv + ϱε, where ϱε = 1 mc f(x, c, t) dc. (5) Hence, the total energy density of the gas is given by a sum of its kinetic energy density ϱv / and its internal energy density ϱε. We define the moment of the distribution function of order N by p i1 i...i N = mc i1 C i... C in f(x, c, t) dc, (6) which represents a symmetric tensor of order N with (N +1)(N +)/ distinct components. The zeroth moment represents the mass density, the first moment vanishes and the second moment, known as the pressure tensor, reads p ij = mc i C j f(x, c, t)dc. (7) RTO-EN-AVT

6 BOLTZMANN AND BALANCE EQUATIONS We denote by p ij the pressure deviator, which represents the traceless part of the pressure tensor. It is given by p ij = p ij 1 3 p rrδ ij, with p = 1 3 p rr = 1 mc f(x, c, t) dc, (8) 3 representing the hydrostatic pressure of the gas. The internal energy density ϱε, given by (5) is related to the hydrostatic pressure by p = ϱε/3. The equation of state of an ideal gas is given by p = nkt where k, n = ϱ/m and T denote the Boltzmann constant, the particle number density and the temperature of the gas, respectively. Hence, we can obtain the following expression for the temperature of a monatomic gas written in terms of the distribution function: T = p nk = m 3 k ε = 1 3nk mc f(x, c, t) dc. (9) The heat flux vector is defined as the contracted third order moment q i = 1 p jji = 1 mc C i f(x, c, t) dc, (10) and the moments of higher order do not have specific proper names. We obtain the balance equations for the moments of the distribution function from the transfer equation (3) by choosing ψ(x, c, t) equal to: (i) Balance of mass density (ψ = m): (ii) Balance of momentum density (ψ = mc i ): ϱv i t ϱ t + ϱv i = 0, (11) + x j (ϱv i v j + p ij ) = 0, (1) (iii) Balance of total energy density (ψ = mc /): [ ϱ (ε + 1 )] t v + [ ϱ (ε + 1 ] )v v i + q i + p ij v j = 0, (13) (iv) Balance of Nth order moment (ψ = mc i1 C i... C in ): p i1 i...i N t + x k (p i1 i...i N k + p i1 i...i N v k ) N ϱ p (i 1 i...i N 1 p in )k x k + Np k(i1 i...i N 1 v in ) x k = P i1 i...i N. (14) The above equation was obtained by eliminating the time derivative of the hydrodynamic velocity v i by the use of (1). The parenthesis around the indexes indicate a sum over all N! permutations of these indexes divided by N!. Furthermore, the production term due to the molecular collisions P i1 i...i N reads P i1 i...i N = m(c i 1 C i... C i N C i1 C i... C in )f f 1 g b db dε dc dc 1. (15) 9-6 RTO-EN-AVT-194

7 3 CHAPMAN-ENSKOG METHOD If we subtract from the balance equation of the total energy density (13) the momentum balance equation (1) multiplied by the bulk velocity v i, we obtain the balance equation for the internal energy density, namely, 3 Chapman-Enskog Method First let us introduce the dimensionless variables ϱε t + (ϱεv i + q i ) + p ij v i x j = 0. (16) t = t/t c, x = x/l c, c = c/ c, g = g/ḡ, b = b/d, ε = ε/π, f = f/f c, (17) where d denote the molecular diameter, c = 8kT/mπ the mean thermal velocity, and ḡ = c the mean relative velocity. Furthermore, the characteristic value for the distribution function f c is equal to n/ c 3 where n represents the particle number density of the gas and t c and L c are a representative time and length that characterize the fluid flow. By taking into account the dimensionless variables (17), the Boltzmann equation () can be written as Sr f t + f c i x i = 1 Kn (f 1 f f 1 f )g b db dε dc 1. (18) In the above equation we have introduced the Strouhal number Sr and the Knudsen number Kn which are defined by Sr = L c c t c, Kn = l L c, (19) where l = 1/ πd n is the mean free path. The Knudsen number is related to the degree of rarefaction of a gas. When Kn 1 the molecular collisions are very important, the distribution function is determined by the collision term of the Boltzmann equation and the gas is described by a continuum regime. Otherwise, when Kn 1 the molecular collisions become negligible, the distribution function is determined via a collisionless Boltzmann equation and the regime of the gas is known as free molecular flow. In the method of Chapman-Enskog the distribution function f is expanded as f = f (0) + Λf (1) + Λ f () + = Λ r f (r), (0) r=0 i.e., in power series of a parameter Λ which is of order of the Knudsen number. The distribution functions f (0), f (1) and f () represent the first, second and third approximation to the distribution function, and so on. The parameter Λ can be set later equal to unity, so that the proper dimensions of the Boltzmann equation are restored. Moreover, the approximations must satisfy the constraints ψf (r) dc = 0, r 1, (1) RTO-EN-AVT

8 3 CHAPMAN-ENSKOG METHOD where ψ represents the summational invariants m, mc i (or mc i ) and mc / (or mc /). If we insert the series expansion of the distribution function (0) into the definitions of the pressure tensor and of the heat flux vector we obtain 1 p ij = δ ij mc f (0) dc + mc i C j 3 m q i = C C i Λ r f (r) dc = r=1 r=1 r=1 Λ r f (r) dc = pδ ij + r=1 Λ r p (r) ij, () Λ r q (r) i. (3) The parameter Λ is introduced into the collision term of the Boltzmann equation (), yielding f Df + C i = 1 Q(f, f), (4) Λ where D = / t + v i / denotes the material time derivative. In (4) the collision term was expressed in terms of an integral for a bilinear quantity Q(F, G) = 1 (F 1G + F G 1 F 1 G F G 1 )g b db dε dc 1. (5) The material time derivative, like the distribution function, is also expanded in power series as D = D 0 + ΛD 1 + Λ D + = Λ r D r, (6) and the insertion of the expansions (), (3) and (6) into (11), (1) and (16) lead to the following decomposition of the balance equations D 0 ϱ + ϱ v i = 0, D r ϱ = 0 ( r 1), (7) (r) ij ϱd 0 v i + p = 0, ϱd r v i + p = 0 ( r 1), (8) x j 3 nkd 0T + p v i = 0, r=1 3 nkd rt + q(r) i + p (r) v i ij = 0 ( r 1). (9) x j We can now obtain the integral equations for the approximations of the distribution function by the inserting the expansions (0) and (6) into the Boltzmann equation (4) and equating equal powers of Λ. Hence, it follows Q ( f (0), f (0)) = 0, (30) Q ( f (0), f (1)) = D 0 f (0) + C i f (0) Q ( f (0), f ()) + Q ( f (1), f (1)) = D 0 f (1) + D 1 f (0) + C i f (1), (31), (3) and so on. The above equations represent the three first integral equations for f (0), f (1) and f (), respectively. 9-8 RTO-EN-AVT-194

9 3 CHAPMAN-ENSKOG METHOD For the determination of the first approximation to the distribution function f (0) from the integral equation (30) we note that its solution is given by f (0) f (0) 1 = f (0) f (0) 1 or ln f (0) + ln f (0) 1 = ln f (0) + ln f (0) 1. Hence, ln f (0) is a summation invariant so that it must be represented by ln f (0) = A + B C + DC. If we insert this representation for f (0) into the definitions of the mass density (4) 1, hydrodynamic velocity (4) and temperature (9) we obtain the Maxwellian distribution function, namely, ( ) 3 β f (0) = n e βc where β = m π kt. (33) For the determination of the second approximation f (1) we write f (1) = f (0) ϕ and the integral equation (31) reduces to Q ( f (0), f (0) ϕ ) { ( 1 = f (0) βc 5 ) } T v i C i + βc i C j T x j = I[ϕ] = f (0) f (0) 1 (ϕ 1 + ϕ ϕ 1 ϕ) g b db dε dc 1, (34) by eliminating the material time derivatives through the use of (7) 1, (8) 1 and (9) 1. The general solution of the integral equation (34) is given by a sum of the solution of the homogeneous integral equation plus a particular solution of it, i.e., ϕ = ϕ h + ϕ p. The solution of the homogeneous integral equation I[ϕ h ] = 0 is a summational invariant ϕ h = α 1 +α rc r +α 3 C, where α 1 and α 3 are scalar functions, while α r is a vector function, all of them do not depend on the peculiar velocity C. The particular solution of the integral equation is expressed as a linear combination of the temperature gradient and of the velocity gradient deviator, namely, ϕ p = A i T T βb ij v i x j. (35) where A i and B ij denote a vector function and a tensor function of (C, ϱ, T ), respectively. It is easy to show that the representations of the vector and tensor functions in terms of the peculiar velocity C are given by A i = A C i and B ij = BC i C j, where the scalar functions A and B depend only on (C, ϱ, T ). Hence, the solution of the non-homogeneous integral equation (34) is given by ϕ = A C i T T βbc i C j v i x j + α 1 + α rc r + α 3 C. (36) The second approximation f (1) = f (0) ϕ must satisfy the constraints (1) which imply the following restrictions α 1 + 3kT m α 3 = 0, α 1 + 5kT [ ] m α 3 = 0, mc A T + αj f (0) dc = 0. (37) T x j From the two first equations we may infer that α 1 = α 3 = 0, whereas the last equation implies that α j must be proportional to T/ x j. By writing the proportionality factor of α j as α/t, the constraint (37) 3 reduces to C Af (0) dc = 0, (38) RTO-EN-AVT

10 3 CHAPMAN-ENSKOG METHOD with A = A α being a new scalar function of (C, ϱ, T ). Hence, from the above results we may write the distribution function (0) up to the second approximation and with Λ = 1 as f = f (0) (1 + ϕ) = f {1 (0) A T C T v i i βbc i C j x j }. (39) Now the scalar coefficients A and B in the distribution function (39) can be obtained as solutions of the integral equations that follows from (34), namely, [ f (0) βc 5 ] C i = I[AC i ], f (0) C i C j = I[BC i C j ]. (40) In order to solve the integral equations (40) the scalar coefficients A(C, ϱ, T ) and B(C, ϱ, T ) are expanded in series of Sonine polynomials. The Sonine polynomials of order n and index (l + 1/) in the variable βc = mc /kt are defined by S (n) l+1/ (βc ) = n k=0 and obey the orthogonality conditions 0 Γ(n + l + 3/) k!(n k)!γ(k + l + 3/) ( βc ) k, (41) e βc C l+ S (n) l+1/ (βc )S (p) l+1/ (βc )β l+3/ dc = 1 Γ(n + l + 3/) δ np, (4) n! where Γ(n) denotes the gamma function. The first three Sonine polynomials read S (0) l+1/ (βc ) = 1, S (1) S () l+1/ (βc ) = 1 ( l + 5 )( l + 3 l+1/ (βc ) = l + 3 βc, (43) ) ( l + 5 ) βc + 1 β C 4. (44) The integral equations (40) in terms of the Sonine polynomials become f (0) S (1) 3/ (βc )C i = I [AC i ], f (0) S (0) 5/ (βc )C i C j = I [ BC i C j ]. (45) Moreover, the expansions of the scalar coefficients A(C, ϱ, T ) and B(C, ϱ, T ) in series of Sonine polynomials are written as A(C, ϱ, T ) = r=0 a (r) S (r) 3/ (βc ), B(C, ϱ, T ) = r=0 b (r) S (r) 5/ (βc ). (46) In the above equations, the scalar coefficients a (r) and b (r) are only functions of (ϱ, T ), and from this point on, the dependence of the Sonine polynomials in the variable βc is omitted. The coefficient A must satisfy the constraint (38), hence by using its representation (46) 1, we get 0 = C f (0) r=0 ( ) 3 a (r) S (r) β 3/ dc = 4πn π a (r) r=0 0 S (0) 3/ S(r) 3/ C4 e βc dc = 3 n β a(0), (47) 9-10 RTO-EN-AVT-194

11 3 CHAPMAN-ENSKOG METHOD due to (4) and (43) 1. From (47) it follows that a (0) = 0 and the scalar coefficient A reduces to A(C, ϱ, T ) = a (r) S (r) 3/. (48) Now the insertion of the expressions (48) and (46) into their respective integral equations (45), yields f (0) S (1) 3/ C i = a (r) I r=1 r=1 [ ] S (r) 3/ C i, f (0) S (0) 5/ C ic j = b (r) I r=0 [ S (r) 5/ C ic j ]. (49) In order to determine the coefficients a (r) e b (r) from (49) we proceeds as follows. First we multiply the integral equation (49) 1 by βs (s) 3/ C i and integrate the resulting equation over all values of C. By using (4) we obtain 15 4 n β δ(1,r) = α (r,s) a (s), where α (r,s) = 1 β s=1 S (r) 3/ C ii[s (s) 3/ C i]dc. (50) In the above equation δ (1,r) represents Kronecker s symbol. Following the same methodology, we multiply the integral equation (49) by β S (s) 5/ C ic j, integrate the resulting equation and get 5 n β δ(0,r) = s=0 β (r,s) b (s), where β (r,s) = S (r) 5/ C ic j I[S (s) 5/ C ic j ]dc. (51) The integrals in (50) and (51) depend on the molecular interaction potential and on the fields (ϱ, T ). Moreover, from the condition φi[ϕ]dc = ϕi[φ]dc which is valid for all arbitrary functions ϕ(x, c, t) and φ(x, c, t) we may infer that α (r,s) = α (s,r), β (r,s) = β (s,r) and α (r,0) = α (0,r) = 0. (5) We conclude that (50) 1 and (51) 1 represent an infinite system of algebraic equations for the coefficients a (n) and b (n), respectively. To obtain the constitutive equations for the pressure tensor and for the heat flux vector we proceed as follows. First, we insert the non-equilibrium distribution function (39) together with the representations of A and B given by (48) and (46), respectively into the definitions of the pressure tensor and heat flux vector and get p ij = q i = mc i C j fdc = p δ ij m C C i fdc = mβc i C j S (0) 5/ f (0) In (54) we have used the relationship βc = r=0 [ ] m 5 T β S(0) 3/ S(1) 3/ C i f (0) [ ] 5 S(0) 3/ S(1) 3/ b (r) S (r) 5/ C pc q v p x q dc, (53) r=1 a (r) S (r) 3/ C p T dc. (54) x p which follows from (43). Next, the substitution of the Maxwellian distribution function (33) into (53) and (54) and RTO-EN-AVT

12 3 CHAPMAN-ENSKOG METHOD the subsequent integration of the resulting equations over all values of C, yield p ij = p δ ij µ v i x j, where µ = ϱ β b(0), (55) q i = λ T, where λ = 5k 4m ϱ β a(1), (56) thanks to the orthogonality condition of the Sonine polynomials (4). Equations (55) 1 and (56) 1 are the mathematical expressions for the laws of Navier-Stokes and Fourier, with µ and λ representing the coefficients of shear viscosity and thermal conductivity, respectively. We note from (55) and (56) that the transport coefficients of shear viscosity and thermal conductivity are functions only of the coefficients a (1) and b (0), respectively. We can obtain these coefficients from the system of equations (50) 1 and (51) 1, which can be written as where A nn = det B nn = det 15 4 a (1) = lim n A nn A nn, b (0) = lim n B nn B nn, (57) n α (1,)... α (1,n) β 0 α (,)... α (,n) α (1,1)... α (1,n)....., A nn = det......, 0 α (n,)... α (n,n) α (n,1)... α (n,n) (58) n β (0,1)... β (0,n) β 0 β (1,1)... β (1,n) β (0,0)... β (0,n)....., B nn = det β (n,1)... β (n,n) β (n,0)... β (n,n) (59) 5 The coefficients a (1) and b (0) follow from (58) and (59) through a method of successive approximations, where the approximation of order p for the coefficients a (1) and b (0) denoted by [a (1) ] p and [b (0) ] p is given by the sub-determinant of order p. The first two approximations for the coefficients a (1) e b (0) are [a (1) ] = 15 4 [a (1) ] 1 = 15 4 n 1 β α, (1,1) [b(0) ] 1 = 5 n α (,) β α (1,1) α (,) α, (1,) [b(0) ] = 5 n 1, (60) β β (0,0) n β (1,1) β β (0,0) β (1,1) β The integrals α (r,s) and β (r,s) are given in terms of the collision integrals Ω (l,r) = 0 0 (0,1). (61) e γ γ r+3 ( 1 cos l χ ) b db dγ, (6) where χ = arccos(g g/g ) is the scattering angle and γ = β/ g represents a dimensionless relative velocity. The determination of the integrals α (r,s) and β (r,s) is not an easy task. As an example, let us compute the integral β (0,0). For its calculation a change of the integration variables 9-1 RTO-EN-AVT-194

13 3 CHAPMAN-ENSKOG METHOD is performed by replacing the velocities C and C 1 by the relative velocity g = C 1 C and by the center of mass velocity G = (C 1 + C)/. The linear momentum conservation law implies that G = G and the Jacobian of the transformation of the volume elements has modulus equal to one so that dcdc 1 = dgdg. In terms of the new variables the integral reads ( ) 3 β (0,0) = C i C j I[C i C j ]dc = n β e βg e β g π ( G i G j G i g j + 1 ) (g 4 g ) ig j i g j g i g j g b db dεdgdg. (63) The integration of (63) with respect to G and ε leads to β (0,0) = πn 4 ( ) 3 β ( e β g cos χ 1 ) g 5 b db dg, (64) π If we introduce the dimensionless relative velocity γ = β/ g, the integration of the above equation with respect to the angles of g leads to β (0,0) = πn ( β ) 5 Ω (,), (65) where Ω (,) denotes the collision integral defined by (6). By introducing the integrals α (r,s) = β 5 n π α(r,s), and β (r,s) = β 5 n π β(r,s), we can obtain in the same way the following expressions in terms of the collision integrals Ω (l,r) α (1,1) = 4Ω (,), α (1,) = 7Ω (,) Ω (,3), α (,) = 77 4 Ω(,) 7Ω (,3) + Ω (,4), (66) β (0,0) = α (1,1), β (0,1) = α (1,), β (1,1) = Ω(,) 7Ω (,3) +Ω (,4) = α (,) α(1,1). (67) As was pointed out the coefficients b (0) and a (1) are determined through a method of successive approximations, so are the transport coefficients µ and λ. Hence, it follows from (60), (61), (55) and (56) that the two first approximations to the coefficients of shear viscosity and thermal conductivity read [µ] = 5 4 mkt π [µ] 1 = 5 mkt 1, [λ] 4 π β (0,0) 1 = 75k mkt 1, (68) 16m π α (1,1) β (1,1) β (0,0) β (1,1) β (1,), [λ] = 75k mkt α (,) 16m π α (1,1) α (,) α (1,). (69) Now from (68) and (69) we can build the ratios [λ] 1 [µ] 1 = 5 c v, [λ] [λ] 1 = ( 1 α(1,) α (1,1) α (,) ) 1 > ( 1 β(0,1) β (0,0) β (1,1) ) 1 = [µ] [µ] 1. (70) RTO-EN-AVT

14 4 APPLICATION OF CHAPMAN-ENSKOG METHOD TO GRANULAR GASES We may infer from (70) 1 that the ratio of the first approximations to the coefficients of thermal conductivity and shear viscosity of monatomic ideal gases is equal to 5c v / for all types of spherically symmetrical molecular interaction potentials, where c v = 3k/m is the specific heat at constant volume. Furthermore, from the relationship (70) it follows that [λ] /[µ] > [λ] 1 /[µ] 1 = 5c v /, i.e., the ratio of the second approximations is larger than the ratio of the first ones. For a hard-sphere potential the impact parameter is given by b = d cos(χ/) with 0 χ π, so that the collision integral Ω (,r) becomes Ω (,r) = Hence, it follows from (66) and (67) that 0 e γ γ r d dγ = d 6 Γ ( r + 4 ). (71) α (1,1) = β (0,0) = 4d, α (1,) = β (0,1) = d, α (,) = 45 4 d, β (1,1) = 05 1 d, (7) and the two first approximations (68) and (69) to the coefficients of shear viscosity and thermal conductivity for the hard-sphere potential read [µ] 1 = 5 mkt 16d π, [λ] 1 = 5 [µ] 1 c v, [µ] = 05 [µ] , [λ] = 45 [λ] (73) We note that the second approximations to the transport coefficients is a small correction to the first ones, so is the ratio [λ] /[µ] [λ] 1 /[µ] 1. The fifth approximations to the transport coefficients for hard-sphere potential are [µ] 5 [µ] , [λ] 5 [λ] Application of Chapman-Enskog Method to Granular Gases The mechanical energy of a gas is conserved when its molecules undergo elastic collisions and, in this case, the gas relaxes towards an equilibrium state characterized by a Maxwellian distribution function. However, the inelastic collisions of the gas molecules transform the translational kinetic energy into heat and the mechanical energy lost implies a temperature decay of the gas. In the literature gases whose molecules undergo inelastic collisions are known as granular gases. The main premisses of the kinetic theory of granular gases are: (a) only binary collisions of hard spherical molecules are taken into account and (b) the energy lost from inelastic collisions is small. Let us consider the encounter of two identical molecules of mass m diameter d precollisional velocities (c, c 1 ) and post-collisional velocities (c, c 1). The momentum conservation law mc+mc 1 = mc +mc 1 holds for inelastic collisions, but the inelastic encounters are characterized by the relationship (g k) = α(g k) which relates the pre- and postcollisional relative velocities at collision. The parameter 0 α 1 here considered as a constant refers to the normal restitution coefficient and k is the unit vector directed 9-14 RTO-EN-AVT-194

15 4 APPLICATION OF CHAPMAN-ENSKOG METHOD TO GRANULAR GASES along the line which joins the molecules centers and pointing from center of the molecule labeled by the index 1 to the center of the molecule without label. The component of the velocity perpendicular to the collision vector k does not change in inelastic collisions, so that k g = k g. From the momentum conservation law it follows the relationships which give the post-collisional velocities in terms of the pre-collisional ones, as well as relationships which connect the relative velocities and their modulus: c = c α (g k)k, c 1 = c α (g k)k, (74) g = g (1 + α)(g k)k, g = g (1 α )(g k). (75) In the case of elastic collisions, α = 1 and it follows the conservation of the kinetic energy g = g. A restitution collision with pre-collisional velocities (c, c 1) that corresponds to the post-collisional velocities (c, c 1 ) are characterized by the equations c = c α (g k)k, α c 1 = c α (g k)k, (76) α where k = k and (g k) = α(g k). For the determination of the Boltzmann equation we have to know the transformation of the volume elements dc 1 dc = J dc 1 dc, where the modulus of the Jacobian is given by J = 1/α. Hence, it follows the relationship (g k ) dc dc 1 = 1 α (g k) dc dc 1, (77) and the Boltzmann equation for granular gases reads ( ) f t + c f 1 i = α f 1 f f 1 f d (g k) dk dc 1 Q I (f, f). (78) Above we have introduced the bilinear form Q I (F, G) = 1 ( 1 α F 1 G + 1 ) α F G 1 F 1 G F G 1 d (g k) dk dc 1. (79) The balance equations for the mass density ϱ, for the momentum density ϱv i and for the specific internal energy ε = 3kT/m are obtained from the Boltzmann equation (78) by multiplying it successively by m, mc i and mc / and integrating the resulting equations. Hence, it follows Dϱ + ϱ v i = 0, Dv i + p ij x j = 0, DT + 3nk ( ) qi v i + p ij + T ζ = 0. (80) x j We observe that the balance equation for the temperature (80) 3 has the additional term T ζ which represents the energy loss due to inelastic collisions. The coefficient ζ, known as cooling rate of the granular gas, is given by ζ = d m(1 α ) 1nkT f 1 f(g k) 3 dk dc 1 dc. (81) RTO-EN-AVT

16 4 APPLICATION OF CHAPMAN-ENSKOG METHOD TO GRANULAR GASES It is easy to verify from the above equation that the cooling rate vanishes for elastic collisions where α = 1. The methodology used to determine the distribution function by using the Chapman- Enskog method for granular gases is different from the one applied to a monatomic gas of elastic spherical molecules, since we have to take into account that there exists no equilibrium state characterized by a Maxwellian distribution function. The first difference, refers to the decomposition of the balance equations, since they are written as D 0 n = 0, D 1 n = n v i, (8) D 0 v i = 0, D 1 v i = 1 p. ϱ (83) D 0 T = T ζ (0), D 1 T = T ζ (1) T v i. 3 (84) The first and the second approximation to the cooling rate in (84) read as ζ (0) = d m(1 α ) 1nkT ζ (1) = d m(1 α ) 1nkT f (0) 1 f (0) (g k) 3 dk dc 1 dc, (85) ( f (1) 1 f (0) + f (0) 1 f (1) ) (g k) 3 dk dc 1 dc. (86) In terms of the parameter Λ we write the Boltzmann equation for granular gases (78) 1 Λ Df + C f i = 1 Λ Q I(f, f), (87) instead of the representation (4), indicating that the material time derivative and the collision term are of same order, while the spatial gradients are of higher order. The two first integral equations for the determination of f (0) and f (1) are obtained by inserting the expansions (0) and (6) into the Boltzmann equation (87) and equating the equal powers of Λ, yielding D 0 f (0) = Q I (f (0), f (0) ), D 1 f (0) + D 0 f (1) + C i f (0) = Q I (f (1), f (0) ). (88) In order to determine the first approximation of the distribution function the integral equation (88) 1 is written as (0) (0) f T ζ T = Q I(f (0), f (0) ), (89) through the elimination of the material time derivatives D 0 by using (8) 1, (83) 1 and (84) 1. We note that the solution of the integral equation (89) for the distribution function f (0) is not a Maxwellian, but we may write f (0) as a Maxwellian plus an expansion in Sonine polynomials of the peculiar velocity, i.e., f (0) = f M [1 + ] a n S (n) (βc ), where f M = n n=1 1 ( ) 3 β e βc, (90) π 9-16 RTO-EN-AVT-194

17 4 APPLICATION OF CHAPMAN-ENSKOG METHOD TO GRANULAR GASES denotes the Maxwellian distribution function. The scalar coefficients a n do not depend on the peculiar velocity C and we suppose that this series expansion converges for large values of the peculiar velocity. We may approximate the representation of the distribution function (90) as [ ] f (0) = f M 1 + a 1 S (1) 1 (βc ) + a S () 1 (βc ), where (91) S (1) 1 (βc ) = 3 βc, S () 1 (βc ) = βc + 1 β C 4. (9) If we insert the representation (91) into the definition of the temperature (9) and integrate the resulting equation we get that a 1 = 0. For the determination of the coefficient a we proceed as follows. First the product of the distribution functions is written as ( ) ( 3 ){ [ f (0) f (0) β 1 = n e β G + g a π 4 5β ) (G + g + β (G 4 + G g + g4 8 + (G g) )] }, (93) by neglecting all product of the coefficient a and changing the velocity variables (C, C 1 ) by the relative and the center of mass velocities (g, G). Next, we multiply the integral equation (89) by an arbitrary function of the peculiar velocity ψ(c ) and integrate the resulting equation over all values of c, yielding ψ(c (0) (0) f )T ζ T dc = 1 [ψ(c 1 ) + ψ(c ) ψ(c1) ψ(c ) ] f (0) 1 f (0) d (g k) dk dc 1 dc. (94) If we choose ψ(c ) = 1 and ψ(c ) = C in (94) the integration of the resulting equations lead to identities. However, if we choose ψ(c ) = C 4 in (94) and use the relationship C C 4 C1 4 C 4 = (α + 1) (g k) (G k) + (α 1) (g k) 4 8 +(α 1)(g k) G + (α 1) (g k) g 4(α + 1)(g k)(g k)(g g), (95) 4 we obtain through the integration of the resulting equation that the coefficient a has the form a = 16(1 α)(1 α ) 81 17α + 30α (1 α). (96) We may infer from the above equation that the coefficient a vanishes for elastic collisions where α = 1. From the knowledge of the coefficient a, the distribution function (91) is written as [ f (0) = f M (1 α)(1 ] α ) 81 17α + 30α (1 α) S() 1 (βc ). (97) RTO-EN-AVT

18 4 APPLICATION OF CHAPMAN-ENSKOG METHOD TO GRANULAR GASES α = 1.0 α = f (0) x Figure : Left frame: dimensionless distribution function π 3/ f (0) /nβ 3/ as a function of x = βc for α = 1 and 0.8. Right frame: cooling rate ζ (0) /nd versus the temperature and the normal restitution coefficient. If we substitute the distribution function (97) into the expression for the cooling rate (85) and integrate the resulting equation we get that ζ (0) = 4 3 nd (1 α ) πkt m [ ] 3(1 α)(1 α ) 1 +. (98) 81 17α + 30α (1 α) In the left frame of Figure it is represented the dimensionless distribution function π 3/ f (0) /nβ 3/ as a function of x = βc and α. We may infer from this figure that the peak of the curve decreases by decreasing the normal restitution coefficient. In the right frame of the Figure it is plotted the cooling rate ζ (0) /nd versus the temperature and the normal restitution coefficient. We note that the cooling rate increases by decreasing the normal restitution coefficient and by increasing the temperature. Furthermore, the cooling rate is zero when α = 1, i.e., for elastic collisions of the molecules. The determination of the second approximation to the distribution function f (1) from the integral equation (88) is more involved and the details of such kind of calculation will not be given here. From this integral equation we may conclude that f (1) is a function of the thermodynamic forces: deviator of the velocity gradient v i / x j, gradient of the particle number density n/ and gradient of temperature T/. Hence, we may write the second approximation f (1) as [ ] f (1) = f M γ 1 S (1) 3 (C ln T )C i + γ S (0) 5 (C v i )C i C j + γ 3 S (1) 3 (C ln n )C i, (99) x j where γ 1, γ and γ 3 are scalar coefficients that depend on n, T and α. If we solve the integral equation (88) we get that the coefficients of the distribution function f (1) are 9-18 RTO-EN-AVT-194

19 5 GRAD S METHOD OF MOMENTS given by γ 1 = γ 3 = 15 m d n(9 + 7α)(α + 1) 300(1 α) d n(9 + 7α)(α + 1)(19 3α) 15 1 ( m ) 3,(100) d n(13 α)(α + 1) π kt πkt, γ = m πkt. (101) We obtain the constitutive equations for the pressure tensor and heat flux vector through the substitution of the distribution function (99) into their definitions and by integrating the resulting equations. Hence, it follows the laws of Navier-Stokes and Fourier p ij = mc i C j fdc = p δ ij µ v i, (10) x j m q i = C C i fdc = λ T ϑt ln n. (103) Above, the coefficients of shear viscosity µ, thermal conductivity λ and the one associated with the gradient of particle number density ϑ read µ = ϑ = 15 mkt d (13 α)(α + 1) 750(1 α) k d n(9 + 7α)(α + 1)(19 3α) m π, λ = 75 k d (9 + 7α)(α + 1) m mkt π, (104) mkt π. (105) We note that the coefficients are proportional to T so that they increase by increasing of the temperature. With respect to the normal restitution coefficient α, we may infer that the coefficients also increase by decreasing the values of α. Furthermore, by considering elastic collisions where α = 1 the coefficients of shear viscosity and thermal conductivity reduce to the first approximations (73) 1,, while the coefficient ϑ vanishes. 5 Grad s Method of Moments Let us consider a thirteen scalar field description of a rarefied gas within Grad s moment method. In this case the fields are the mass density ϱ(x, t), the hydrodynamic velocity v i (x, t), the pressure tensor p ij (x, t) and the heat flux vector q i (x, t) and their balance equations read ϱ t ϱv i t p ij t q i t + ϱv i = 0, (106) + (ϱv iv j + p ij ) x j = 0, (107) + (p ijk + p ij v k ) x k + (q ij + q i v j ) x j + p ki v j x k + p kj v i x k = P ij, (108) v j v i + p ijk + q j p ki x k x j ϱ p kj x j 1 p rr ϱ p ij x j = Q i. (109) RTO-EN-AVT

20 5 GRAD S METHOD OF MOMENTS We note that the two first equations above represent the balance equations of mass density (11) and moment density (1), while the balance equation for the pressure tensor (108) was obtained from (14) by taking N =. The balance equation for the heat flux vector (109) follows from (14) by taking N = 3, contracting two indices and dividing the resulting equation by. Moreover, we have introduced the following definitions in (108) and (109): p ijk = mc i C j C k fdc, P ij = m ( ) C ic j C i C j ff1 dγ, (110) m m ( ) q ij = C C i C j fdc, Q i = C C i C C i ff1 dγ. (111) The tensors p ijk and q ij represent non-convective fluxes for the pressure tensor and heat flux vector, respectively, while P ij and Q i refer to their production terms. In the above equations we have introduced the abbreviation dγ = g b db dε dc 1 dc. If we analyze the system of partial differential equations (106) (109) we may infer that it cannot be considered a system of field equations for the basic fields ϱ, v i, p ij and q i. In fact, to obtain a closed system of differential equations, the non-convective fluxes p ijk, q ij and the production terms P ij, Q i must be expressed in terms of the basic fields, and for this end we must express the distribution function in terms of the thirteen scalar fields ϱ, v i, p ij and q i. In Grad s moment method the distribution function is expanded in series of tensorial Hermite polynomials H i1 i...i N (N = 0, 1,,... ) as follows f = f (ah (0) + a i H i + 1! a ijh ij ) N! a i 1 i...i N H i1 i...i N +..., (11) where a i1 i...i N (N = 0, 1,,... ) are tensorial coefficients that depend on x and t. Furthermore, the Maxwellian distribution function f (0) is written as f (0) = n where ω(ξ) is the weight function: ( ) 3 β ( e βc m ) 3 = n ω(ξ), (113) π kt ω(ξ) = 1 e ξ /, with ξ (π) 3 i = m kt C i. (114) On the basis of the weight function, the tensorial Hermite polynomials are written as H i1 i...i N (ξ) = ( 1)N ω N ω ξ i1 ξ i... ξ in, (115) being orthogonal with respect to ω(ξ), i.e., ω(ξ)h i1 i...i N (ξ)h j1 j...j M (ξ)dξ = δ MN i1 j 1 i j...i N j N. (116) In the above equation, i1 j 1 i j...i N j N represents the sum i1 j 1 i j...i N j N = δ i1 j 1 δ i j... δ in j N + (all permutations of the j s indices) and δ MN is Kronecker s symbol. 9-0 RTO-EN-AVT-194

21 5 GRAD S METHOD OF MOMENTS From (115) we can obtain the first four tensorial Hermite polynomials, namely, H(ξ) = 1, H i (ξ) = ξ i, H ij (ξ) = ξ i ξ j δ ij, (117) H ijk (ξ) = ξ i ξ j ξ k (ξ i δ jk + ξ j δ ik + ξ k δ ij ). (118) In terms of the tensorial Hermite polynomials we can write the thirteen moments of the distribution function as ( ) 3 ( ) kt kt ϱ = m fhdξ, 0 = m fh i dξ, (119) m m T = m ( ) 5 kt m f(hii + 3H)dξ, (10) 3kϱ m p ij = ( ) 5 ( kt m f H ij 1 ) m 3 H rrδ ij dξ, q i = m ( ) 3 kt f (H ijj + 5H i ) dξ.(11) m We recall that the temperature is given by T = mp rr /3kϱ. The coefficients a i1 i...i N that appear in the distribution function (11) are determined from the definition of the moments of the distribution function. Furthermore, for a thirteen moment theory the distribution function (11) is written as ( f = f (0) ah + a i H i + 1 a ijh ij + 1 ) 10 a rrih ssi, (1) where a, a i, a ij and a rri represent thirteen scalar coefficients to be determined. We note that the term 1 a 3! ijkh ijk was decomposed according to [ 1 a ijk + 1 ] 6 5 (a rriδ jk + a rrj δ ik + a rrk δ ij ) H ijk = 1 6 a ijk H ijk a rrih ssi, (13) and the part associated with the third order tensor 1a 6 ijk H ijk was not taken into account. If we insert now the distribution function (1) into the definitions of the moments (119) (11), integrate the resulting equations and use the orthogonality properties of the tensorial Hermite polynomials (116), we obtain that a = 1, a i = 0, a rr = 0, a ij = p ij p, a rri = q i m p kt. (14) Hence, the distribution function for the thirteen moments (with β = m/kt ) becomes { f = f (0) 1 + [p β ij C i C j + 45 ( ϱ q ic i βc 5 )]}, (15) which is the so-called Grad s distribution function. In order to determine the non-convective fluxes p ijk and q ij we insert Grad s distribution function (15) into the definitions (110) 1 and (111) 1 and obtain by integrating the resulting equations: p ijk = 5 (q iδ jk + q j δ ik + q k δ ij ), q ij = 5p ϱ δ ij + 7p ϱ p ij. (16) RTO-EN-AVT

22 5 GRAD S METHOD OF MOMENTS For the calculation of the production terms P ij and Q i we change the integration variables (c, c 1 ) (g, G) so that in these new variables (110) and (111) become m [ 1 ( ) ( ) P ij = g i g j g i g j Gi g j g j Gj (g i g i )] ff 1 dγ, (17) [ m 1 Q i = G r (g rg i g r g i ) 1 (G + 14 ) ] g (g i g i ) G i G r (g r g r ) ff 1 dγ, (18) where the product of Grad s distribution functions in a linearized theory reads ( ) { 3 β ff 1 = n e (βg + β g ) 1 + 4β π ϱ p kl [G k G l + 14 ] g kg l (19) ( + 8β 5ϱ q j [G j βg + β 4 g 5 ) ] } + βg j G k g k. (130) The integration of (17) and (18), yields P ij = p ij τ r, Q i = q i 3τ r, where τ r = 5 16n m πkt 1, (131) Ω (,) represents the relaxation time of the pressure deviator. Once we know the constitutive equations for the non-convective fluxes (16) and production terms (131) as functions of the thirteen basic fields ϱ, v i, p ij and q i, a system of linearized thirteen scalar field equations can be obtained from (106) (109), namely Dϱ + ϱ v i = 0, (13) ϱdv i + p + p ij = 0, (133) x j 3 nkdt + q i + p v i v i + p ij = 0, (134) x j q i Dp ij p v i = p ij, (135) 5 x j x j τ r Dq i + p p ik + 5 ϱ x k k m p T = q i. (136) 3τ r If we restrict ourselves to a five field theory described by the fields of mass density, momentum density and temperature, the corresponding balance equations are given by (13), (133) and (134). In this case the pressure deviator and the heat flux vector are no longer variables, just constitutive quantities. We may use the remaining eight equations (135) and (136) in order to obtain the constitutive equations for the pressure deviator and for the heat flux vector, by considering only the equilibrium values of the pressure deviator p ij = 0 and of the heat flux vector q i = 0 on the left-hand sides of (135) and (136). Hence, only the underlined terms remain, and we obtain p ij = µ v i x j, where µ = pτ r = 5 16 q i = λ T, where λ = 15k 4m pτ r = 75k 64m mkt 1, (137) π Ω (,) mkt 1. (138) π Ω (,) 9 - RTO-EN-AVT-194

23 6 CHAPMAN-ENSKOG-GRAD COMBINED METHOD Equations (137) and (138) represent the laws of Navier-Stokes and Fourier, respectively, and the coefficients of shear viscosity µ and thermal conductivity λ correspond to the first approximation to these coefficients found by applying the Chapman-Enskog method (see (68)). 6 Chapman-Enskog-Grad Combined Method We describe now another method to obtain the constitutive equations for the pressure deviator and for the heat flux vector which combines the features of Chapman-Enskog and Grad s methods. In this method neither a solution of the integral equation is needed as in the Chapman-Enskog method nor the field equations for the moments of the distribution function are used as in Grad s method. To begin with we observe that in the representation f = f (0) (1+ϕ) the deviation of the Maxwellian distribution function ϕ in the Chapman-Enskog method which is a unknown quantity can be written in terms of known quantities. These known quantities can be chosen as the thirteen scalar fields of mass density, hydrodynamic velocity, temperature, pressure deviator and heat flux vector which appear in Grad s distribution function (15), namely, ϕ = β ϱ [p ij C i C j + 45 ( q ic i βc 5 )]. (139) In the framework of the Chapman-Enskog-Grad combined method we insert the representation (139) into the non-homogeneous integral equation of the Chapman-Enskog method (34), so that it becomes an equation for the determination of the pressure deviator p ij and of the heat flux vector q i. This equation reads f (0) { 1 T ( βc 5 ) T C k } v k + βc k C l = β x k x l ϱ p kl I[C k C l ] [( + 8β 5ϱ q k I βc 5 ) C k ]. (140) For the determination of the pressure deviator, we multiply (140) by C i C j and integrate the resulting equation over all values of c, yielding p v i = β m x j ϱ p kl C i C j I[C k C l ]dc. (141) To solve the above equation for the pressure deviator p kl, we introduce the integral I ijkl = C i C j I[C k C l ]dc = A 1 δ ij δ kl + A (δ ik δ jl + δ il δ jk ), (14) which was written in terms of Kronecker s symbol with coefficients A 1 and A. If we contract the integral I ijkl in two different manners, namely, I rrss = 9A 1 + 6A and I rsrs = 3A 1 + 1A we get that I ij kl = 3I rsrs I rrss δ k i δ j l. (143) 15 RTO-EN-AVT

24 6 CHAPMAN-ENSKOG-GRAD COMBINED METHOD We use now the relationship (143) to write (141) as p v i = β m x j 5ϱ p ij C r C s I[C r C s ]dc = β 5ϱ β(0,0) p ij, (144) where β (0,0) is given by (65). Hence, it follows from (144) Navier-Stokes law p ij = µ v i, where µ = 5 mkt 1, (145) x j 16 π Ω (,) is the coefficient of shear viscosity. We determine the heat flux vector by following the same methodology. We multiply (140) by C C i and the integrate of the resulting equation over all values of c, resulting 5 kp m T = 8β 5ϱ q k thanks to the following relationship I ik = C C i I C C i I [( βc 5 ) C k ] dc = 8β 15ϱ α(1,1) q i, (146) [( βc 5 ) ] C k dc = I rr 3 δ ik. (147) Hence, Fourier s law follows from (146) q i = λ T, where λ = 75 k mkt 64 m π 1, (148) Ω (,) is the coefficient of thermal conductivity. Above, we have used the relationship α (1,1) = β (0,0) which was obtained in the Chapman-Enskog method. We can extended this method to obtain the successive approximations to the transport coefficients, by introducing traceless second order tensors p (s) ij ij = 15 (s)!! (s + 5)!! p (s) ms (s) 5 C i C j f dc, q (s) i = 15(s)!! 4β(s + 3)!! and vectors q(s) i ms (s) 3 defined by C i f dc, (149) where p (0) ij = p ij represents the pressure deviator and q (0) i = q i the heat flux vector, while the the higher-order moments p (s) ij (s > 0) and q(s) i (s > 1) do not have standard designations. For the moments of the distribution function characterized by ϱ, v i, T, p (s) ij the distribution function reads { } f = f (0) 1 + β S (s) 5 C i C j p (s) ij ϱ 8β S (s) 3 C i q (s) i 5ϱ so that (140) becomes { f (0) 1 } T T S(1) 3 C k + βs (0) v k 5 C k C l = β x k x l ϱ s=0 8β 5ϱ s=0 s=0 s=0 p (s) kl I [ q (s) k I [ S (s) 3 S (s) 5 and q(s) i, (150) C k C l ] ] C k. (151) 9-4 RTO-EN-AVT-194

25 7 FOURTEEN MOMENT THEORY FOR GRANULAR GASES The multiplication of (151) by βs (s) C i C j and integration of the resulting equation leads to ϱ 5 5 v mβ 3 δ(0,r) i = x j s=0 β (r,s) p (s) ij. (15) If we solve the above equation for the pressure deviator p ij it follows the Navier-Stokes law v i p ij = [µ] p, where [µ] x p = 5 ( ) β 1 (0,0) j 4 mβ 3 ϱ (153) is the coefficient of shear viscosity in the p th successive approximation. In (153) (β 1 ) (0,0) represents the first row and first column of the inverse of the p p matrix β (r,s). If we multiply (151) by βs (s) C i and integrate the resulting equation we get n k T δ(1,r) = β which can be solved for the heat flux vector q i, yielding Fourier s law: s=0 T q i = [λ] p, where [λ] x p = 75 i 16 α (r,s) q (s) i, (154) n k β ( α 1 ) (1,1). (155) The coefficient of thermal conductivity [λ] p in the p th successive approximation is given in terms of the element (α 1 ) (1,1) of the inverse matrix. The successive approximations of the transport coefficients of shear viscosity (153) and thermal conductivity (155) are the same as those which follow from the Chapman- Enskog method. 7 Fourteen Moment Theory for Granular Gases We may characterize a macroscopic state of the granular gas by the fourteen fields of mass density ϱ, hydrodynamic velocity v i, pressure tensor p ij, heat flux vector q i and contracted fourth moment p iijj which are defined by ϱ = mf dc, ϱv i = mc i f dc, p ij = mc i C j f dc, (156) m q i = C C i f dc, p iijj = mc 4 f dc. (157) The balance equations for the fourteen fields are obtained from the Boltzmann equation (78) by multiplying it successively by m, mc i, mc i C j, mc C i / and mc 4 and integrating RTO-EN-AVT

26 7 FOURTEEN MOMENT THEORY FOR GRANULAR GASES the resulting equations. Hence, it follows ϱ t + ϱv i = 0, (158) ϱv i + (ϱv i v j + p ij ) = 0, t x j (159) p ij t + v j v i (p ijk + p ij v k ) + p ki + p kj = P ij, x k x k x k (160) q i t + v j v i (q ij + q i v j ) + p ijk + q j p ki p kj 1 p rr p ij = Q i, x j x k x j ϱ x j ϱ x j (161) p iijj + v i (p iijjk + p iijj v j ) + 8q ik 8 t x j x k ϱ q p ik i = P. x k (16) In the above equations the new moments of the distribution function are defined by m p ijk = mc i C j C k fdc, q ij = C C i C j fdc, p iijjk = mc 4 C k fdc, (163) where q ii = p iijj /. Furthermore, the production terms read P ij = 1 ( ) m Ci 1 Cj 1 + C ic j Ci 1 Cj 1 C i C j f 1 f d (g k) dk dc 1 dc, (164) Q i = 1 m ( ) C 1 Ci 1 + C C i C 1Ci 1 C C i f 1 f d (g k) dk dc 1 dc, (165) P = 1 m ( C C 4 C1 4 C 4) f 1 f d (g k) dk dc 1 dc. (166) We can decompose the balance equation for the pressure tensor (160) in its trace and traceless parts by introducing the traceless tensors p ij and p ijk defined by p ij = p ij + pδ ij, p ijk = p ijk + 5 (q iδ jk + q j δ ik + q k δ ij ), (167) where p = nkt is the hydrostatic pressure. Hence, we obtain from (160) the equations: T t + v T i + ( qi + p v ) i v i + p ij + T ζ = 0, (168) 3nk x j p ij + ( ) v j v i p ijk + p ij v k + p ki + p kj t x k x k x k 3 p v r kr δ ij x k q i p v i = P ij. (169) 5 x j x j Equation (168) is the balance equation for the temperature with ζ denoting the cooling rate (81), while (169) is the balance equation for the pressure deviator. We represent the non-equilibrium distribution function for the fourteen moments as f = f M ( a + ai C i + a ij C i C j + b i C C i + bc 4), (170) where the fourteen coefficients a, a i, a ij, b i and b do not depend on the peculiar velocity. 9-6 RTO-EN-AVT-194