Hilbert fluid dynamics equations expressed in Chapman-Enskog pressure tensor and heat current
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Hilbert fluid dynamics equations expressed inChapman-Enskog pressure tensor and heatcurrent
Soderholm, Lars H.
Mekanik, KTH, SE-10044 Stockholm, Sweden [email protected]
The connection between the Chapman-Enskog and Hilbert expansions is investigatedin detail. In particular the fluid dynamics equations of any order in the Hilbertexpansion are given in terms of the pressure tensor and heat current of the Chapman-Enskog expansion
Key words:
Hilbert expansion; Chapman-Enskog expansion.
The most convenient way to calculate heat current and pressure tensor is that of theChapman-Enskog method, see Grad [4]. In particular, the terms up to second andhigher order have been calculated and at least to second order there is an agreedconvention of notation for the terms. But in many occasions the Hilbert methodis preferred because of the difficulties with the Burnett equations, see Sone [6].Already in the Handbuch article [4], Grad made clear that the Chapman-Enskogmethod and Hilbert method are two ways of expressing the same thing. But Gradadds ”to confirm this directly is an intricate task due to the different formalisms”.So, his proof is somewhat abstract. It is the object of the present work to do theexplicit confirmation that Grad does not do.The benefit of this is that the fluid dynamics equations of the Hilbert expansionare given explicitly in terms of the pressure tensor and heat current of the Chapman-Enskog method. This means that results that have been derived by the Chapman-Enskog method directly can be used in the Hilbert expansion. In the interestingpaper [3] by Chekmarev and Chekmareva multiple scale methods are applied in the1ilbert expansion. The results of the present paper should considerably simplify andgeneralize their results to second order. In the paper [5] the results by Chekmarevand Chekmareva were generalized but derived from the Burnett equations. Thiscould be considered unsatisfactory as the Burnett equations are mathematicallyunstable. But here we have been able to show that this result was neverthelesscorrect. We have shown that the fluid dynamics equations of the Hilbert expansionin fact can be obtained from the corresponding equations of the Chapman-Enskogexpansion by expanding the fluid dynamics fields.
We write the Boltzmann equation in dimensionless variables ( ε is the Knudsen num-ber) ε D f = Q ( f, f ) , where the streaming operator is ( F is the force per mass) D = ∂∂t + c · ∂∂ x + F · ∂∂ c . See Cercignani [1]The fluid dynamic fields are given by m Z f d c = ρ,m Z f c d c = ρ v = j , (3.1) m Z f c d c = ρ ( v k B T m ) = e In the Chapman-Enskog expansion is based on the important auxiliary condition,that only the zero order distribution function contributes to the flud dynamics fields.Hence (3.1) is satisfied also with f replaced by f CE . We summarize the fluid dy-namical fields w = ρ v T . The components of w are written w a where a = 1 ...
5. The Chapman-Enskog seriesis written ( ε is the Knudsen number) f CE = f CE ( w, c ) + εf CE ( w, c ) + ... Here, f CEr ( w, c ) stands for a function of w and its r first derivatives and of c , themolecular velocity. In the sequel we don’t write out c explicitly.Let us now assume that the fluid dynamics variables can be expanded in theKnudsen number w = w + εw + ... f CEr ( w, c ) we find f CEr ( w ) = f CEr, ( w ) + εf CEr, ( w , w ) + ... Here f CEr,s is a function of w , w , ...w s and their r first derivatives.We first of all consider the zero order term f CE . When we expand it we obtainto lowest order f CE , ( w ). This is a Maxwellian for the fluid dynamics fields w .In other words, this is the zero order term in the Hilbert expansion f CE , = f H . Let me write the sum of the first r terms in the Chapman-Enskog expansion andintroduce this expansion of the fluid dynamics fields f CE + εf CE + ...ε n f CEr = f CE , + ε ( f CE , + f CE , ) + ...ε r r X s =0 f CEr − s,s + O ( ε r +1 )Thus we obtain ρ = m Z f CE d c = m Z ( f CE + εf CE + ...ε r f CEr ) d c = m Z f CE , d c + εm Z ( f CE , + f CE , ) d c + ... + ε r m Z r X s =0 f CEr − s,s d c + O ( ε s +1 )Identifying the terms we find ρ r = m Z r X s =0 f CEr − s,s d c. (3.2)In the same way we obtain j r = ( ρ v ) r = r X s =0 ρ r − s v s = m Z r X s =0 f CEr − s,s c d c. (3.3)Finally we have e r = [ ρ ( v k B T m )] r = r X s =0 ρ r − s ( s X p =0 v s − p · v p + 3 k B m T s )= m Z r X s =0 f CEr − s,s c d c (3.4)3rom (3.2-3.4) we conclude that if we assume that the fluid dynamics field areexpanded in a series in the Knudsen number, the corresponding terms to order r ofthe fields are given as the projections of r X s =0 f CEr − s,s . (3.5)We shall see that r P s =0 f CEr − s,s is in fact the coefficient f Hr of ε r in the Hilbertexpansion. Let us now consider the terms up to order r in the Chapman-Enskog expression ofthe collision term. We then expand the fluid dynamics fields and obtain r X p + q =0 ε p + q Q ( f CEp , f
CEq )= r X p + q + i + j =0 ε p + q + i + j Q ( f CEp,i , f
CEq,j ) + O ( ε r +1 )In the coefficient of ε r we put p + i = k, q + j = l so that it becomes X k + l = r Q ( f CEk − p,p , f CEl − q,q )= X k + l = r Q ( k X p =0 f CEk − p,p , l X q =0 f CEl − q,q )We obtains sums of the kind (3.5).If we do the corresponding thing in the streaming term we find that the coefficientof ε r − is D r − X p =0 f CEr − − p,p So from the Chapman-Enskog equations of order 0 , , ...r we find that Q ( r X p =0 f CEr − p,p , f CE , ) + Q ( f CE , , r X p =0 f CEr − p,p )= − X k + l = r Q ( r − X p =1 f CEk − p,p , r − X q =1 f CEl − q,q ) (4.6)+ D r − X p =0 f CEr − − p,p r − f Hr − = r − X s =0 f CEr − − s,s . (4.7)We can then write (4.6) as Q ( r X p =0 f CEr − p,p , f H ) + Q ( f H , r X p =0 f CEr − p,p )= − r − X s =1 Q ( f Hr − s , f Hs ) + D f Hr − This means that r X s =0 f CEr − s,s (4.8)satisfies the equation Q ( f Hr , f H ) + Q ( f H , f Hr ) (4.9)= − r − X s =1 Q ( f Hr − s , f Hs ) + D f Hr − for f Hr in the Hilbert expansion. We assume the existence of a solution and itsuniqueness up to a linear combination of 1 , c ,c . But according to (3.2-3.4) thefunction (4.8) also has the correct projections for f Hr on the functions 1 , c ,c . So byinduction this is the function f Hr of the Hilbert expansion. We conclude that thesimple relation between the Chapman-Enskog and Hilbert distribution functions is f Hr = r X s =0 f CEr − s,s . (4.10) Let us to start by writing down the conservation laws. They are ρ ,t + ∇ · j = 0 j ,t + ∇ · J = ρ F (5.11) e ,t + ∇ · j e = F · j Here J = m Z f cc d c = P + ρ vv = P + jv , (5.12)5here P = m Z f c ′ c ′ d c (5.13)and j e = Z mc f c d c (5.14)= e v + q + P · v , q = m Z c ′ f c ′ d c. (5.15)If we instead calculate the moments from f CEr we obtain to zero order J CE = m Z f CE cc d c = P CE + jv , (5.16) P CE = m Z f CE c ′ c ′ d c = p (5.17) j eCE = Z mc f CE c d c (5.18)= e v + q CE + P CE · v , q CE = m Z c ′ f c ′ d c = . (5.19)For r ≥ J CEr = m Z f CEr cc d c = P CEr , (5.20) P CEr = m Z f CEr c ′ c ′ d c (5.21) j eCEr = Z mc f CEr c d c (5.22)= q CEr + P CEr · v , q CEr = m Z c ′ f CEr c ′ d c. (5.23) P CEr and q CEr and hence J CEr and j eCEr are functions of the fluid dynamics fields w and its r first derivatives. This means that we can expand the fluid dynamicsfields in them to obtain P CEr = P CEr, + ε P CEr, + ..., where P CEr,s is a function of w , w , ...w s and its r first derivatives.6he Chapman-Enskog equations to order r are obtained if we make the approx-imations P → r X s =0 ε s P CEs , (5.24) q → r X s =0 ε s q CEs in the expressions for J and j e and substitute into the equations of balance. Theequations are then ρ ,t + ∇ · j = 0 j ,t + ∇ · ( r X s =0 ε s J CEs ) = ρ F (5.25) e ,t + ∇ · ( r X s =0 ε s j eCEs ) = F · j We consider the equation (4.9) of the Hilbert expansion where we now replace r by r + 1. The condition of integrability is that is Z mm c mc / D f Hr d c = 0 . (6.26)We now use the relation (4.10) to express this condition in terms of the Chapman-Enskog distribution function. To that end we need the corresponding integrals f¨orthe Chapman-Enskog terms. To zero order we obtain m Z D f CE d c = ρ ,t + ∇ · j ,m Z D f CEr c d c = j ,t + ∇ · J CE − ρ F , (6.27) m Z c D f CE c d c = e ,t + ∇ · j eCE − F · j . To order r ≥ m Z D f CEr d c = 0 ,m Z D f CEr c d c = ∇ · J CEr , (6.28) m Z c D f CEr c d c = ∇ · j eCEr .
7f we now expand the fluid dynamics fields we obtain m Z D f CE ,s d c = ρ s,t + ∇ · j s ,m Z D f CE ,s c d c = j s,t + ∇ · J CE ,s − ρ s F , (6.29) m Z c D f CE ,s c d c = e s,t + ∇ · j eCE ,s − F · j s . For r ≥ m Z D f CEr,s d c = 0 ,m Z D f CEr,s c d c = ∇ · J CEr,s , (6.30) m Z c D f CEr c d c = ∇ · j eCEr,s . Now, using the expression for f Hr in terms of f CEr − s,s we finally obtain the condi-tions of integrability to order r ρ r,t + ∇ · j r = 0 , j r,t + ∇· r X s =0 J CEr − s,s − ρ r F = , (6.31) e r,t + ∇· r X s =0 j eCE r − s,s − F · j r = 0 . They are the fluid dynamics equations of the Hilbert expansion. Note that the sameequations are obtained if we start with the Chapman-Enskog equations of order r ,expand the fluid dynamics fields and pick out the coefficient of ε r .Let us now as an example consider the linearized Boltzmann equation and findthe Hilbert fluid dynamics equations to second order. We have, see Chapman &Cowling P CE = − µ < ∇ v >, P CE = µ ρT ( ̟ − ̟ ) h∇∇ T i − ̟ µ ρ h∇∇ ρ i , (6.32) q CE = − κ ∇ T, q CE = µ ρ [( θ △ v + ( θ − θ ∇ ( ∇ · v )] . ρ ,t + ρ ∇ · v = 0 ,ρ v ,t = −∇ p + 2 µ ( △ v + 23 ∇ ( ∇ · v )) (6.33) − [ 2 µ ρT ( ̟ − ̟ )] △∇ T + [ ̟ µ ρ ] △∇ ρ , (6.34)3 k B ρ m ( T ,t + T ∇ · v ) = κ △ T − [ 2 µ ρ ( θ − θ )] △ ( ∇ · v ) . Acknowledgements
Discussions with A.V. Bobylev and Y. Sone have been stimulating.
References [1] Cercignani, C. (2000).
Rarefied Gas Dynamics. From Basic Concepts to ActualCalculations . Cambridge: Cambridge University Press.[2] Chapman, S., T.G. Cowling (1970).
The Mathematical Theory of Non-UniformGases . Cambridge: Cambridge University Press.[3] Chekmarev, I.B., Chekmareva, O.M. (2003). Multiple expansion method in theHilbert problem.
Fluid Dynamics : 646.[4] Grad, H., Principles of the kinetic theory of gases (1958). Handbuch der Physik
Band XII. Berlin: Springer-Verlag.[5] Sderholm, L.H. (2007). Stable equations to second order in the fluid dynamicsvariables.