Tham khảo tài liệu 'understanding non-equilibrium thermodynamics - springer 2008 episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 230 8 Theories with Internal Variables equation will be determined later on ơv is the so-called suspension viscous stress related to the stress nv by Lhuillier 1995 nv ơv - J besides the viscous term ơv one observes the presence of a stress J of kinetic origin which is introduced to recover the global momentum equation . Making use of the following identity llp p - 11 V v 1 2c - 1 2 r dt dt 2 J one obtains the following evolution equation for the internal variable é PĨ1 - p-1 Vp-V-ơv f . v 1 2c - 1 2 pc 1 c 2 - -Vv. Set contains some indeterminate unknown quantities like p u ơv q and f whose expressions will be determined from thermodynamics and more particularly from the positive definite property of the entropy production ơs ps V Js 0. Of course at this stage of the discussion the entropy s and the entropy flux Js remain undetermined quantities to be expressed by means of constitutive relations. Let us now examine the consequences issued from the positiveness of ơs. Restrictions Placed by the Second Law of Thermodynamics The total kinetic energy per unit mass can be written as cv2 1 c v2 v2 c 1 c 2. 2 p 2 c f 2 2c c . To keep the usual expression 2v2 for the kinetic energy per unit mass we shall admit that the part of the kinetic energy involving the relative velocity pertains to the internal energy so that u s p c u0 s p c 2c 1 - c 2 where u0 is the local equilibrium energy depending exclusively on the set of equilibrium variables. The corresponding Gibbs equation written in rate form will therefore be given by u Ts p p ----- . Applications 231 Note that the chemical potential is now including a kinetic contribution and is related to the local equilibrium chemical potential Mo du0 dc by M Mo 2 1 - 2c i2. After elimination of u p c and i by means of the evolution equations and we obtain the following entropy balance T pS V Js Tơs V TJs - q - mJ u - v ơv ơv Vu sym -Js VT - J
