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Muller A History of Thermodynamics The Doctrine of Energy and Entropy phần 9

Tham khảo tài liệu 'muller a history of thermodynamics the doctrine of energy and entropy phần 9', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Extended Thermodynamics 259 V is a characteristic speed and la and da are the left and right eigenvalues of the matrix in the one-dimensional field equations ịa ỉ 2_n . fit dx. The solution of the Bernoullicquationreads . A 0 ois i i-A0 f -i so that A t remains finite unless the initial amplitude A 0 is large. In general - for arb oạty solutions osteadof moreln acieration waves -the condition for smoolhuoblionn dnotdodsivolyknuwn.Hiere exists a sufficient condition for smoothness65 which however is not necessary. Characteristic Speeds in Monatomic Gases We recall the generic eqooolons nflransferintuekineticthoosynf gaoes cf. Chap. 4. and appl OhiftT npoluaomifi n wfocitycoiOiXiTOi ho settug . In tots iuwTtftu nu ofuaunequaegov ofbgOance for moments u . uc c . .c . fác of the distribution function f which read o . n ui i0 w io .i.AtO. dt 4 1 vti n e Since each index may assume the values 1 2 3 there are J 7 7 7 n 6 N i N 2 N 3 equations. These equations fit into the formal framework of extended thermodynamics see above but they are simpler. Indeed on the left hand side there is only cwoiioio co i toTo Bf ir -foe oshono - whichOs noO explicitly related to the fields UjA .h I 1 .N . 65 s. Kawashima Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws orl r iplo. atiibo Oroorndingu 11 nhf Roful Soocly ol lidinbuif 11 A 106 1987 . 260 8 Thermodynamics of Irreversible Processes Therefore the results of the previous sections may be carried over to the present case in particular the exploitation of the entropy inequality. That inequality reads according to the kinetic theory of gases cf. Chap. 4 -k ln -dcj 77 ica ln -dcj 0. dt 1.1 eTjchmAJ ob t The exploitation makes use of the Lagrange multipliers Arp o 1 2 . .N andthemomentcharactnr pf the de mtpr tag ll lu eeifi rlieh that the distribution function has the form - 4 IX .-Y SO that the scalar and geclng ooleiiliils may in leheiee as N h M P E1 1 111--1 and h a -kYịc exp I J-Vf n A Y i.pci