tailieunhanh - Thermodynamics Kinetics of Dynamic Systems Part 9

Tham khảo tài liệu 'thermodynamics kinetics of dynamic systems part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Thermohydrodynamics Where Do We Stand 229 Vx E - dL -P0 Ê0 dt Vx B P0 Jt. 6 Notice that the currents Jq Tv J m or Jc require to be expressed in terms of the state variables pe Pi u and e through still unknown constitutive equations through eqs. 2 and 3 . In order to simplify the problem several approximations are introduced. Firstly the temperatures of electrons and ions is taken to be the same Ti Te T a fact which is readily justified Moratto García-Colín 2011 . Next we define the hydrodynamic time tH and the hydrodynamic lenght 1h as follows kft being Boltzmann s constant and -1 Va J 2k a 1 e V mx Max VA r t I A r t I 7 8 1 v l H where A is P u or T. We now assume that both 1h and Th are the characteristic length and time scales of E and B. Then several approximations follow Balescu 1988 García-Colín Dagdug 2009 s ÍV u IV x B 0 in anonrelativisticplasma. at c2 This is a tricky approximation. By eq. 6 it implies that V x B Po Jt or that 9 10 11 12 V Jt 0. But since total charge is conserved the balance equation Q V JT 0 .9Q . . . must be obeyed which implies that d 0. Consistency may be achieved if local neutrality is satisfied namely Balescu 1988 ne r t - n r i ne r t nt r t 13 in which case Q tt 0 in all hydrodynamic equations. . . 9E This leads to the so called pre-Maxwell s equations I -dt I 0 and Q E is supressed. Moreover it is also assumed that since Q ÍV 0 then Qu Jc 14 which implies that pre-Maxwell s equations read as V E 0 15 Vx E - dB Vx B P0 Jt 16 230 Thermodynamics - Kinetics of Dynamic Systems subject to the condition that V Jc 0. This fact is very often ignored or by-passed in many treatises on the subject. If we assume that Jc is determined through Ohm s law which as we will see below is already incorrect then Jc q E -1 u X b Q u X B 17 R R which implies that eq. 16 - V X B Q u X B 18 or V u X B 0 19 which is consistent with eq. 11 . But also V u X B B V X u - u V X B 20 or using eq. 18 B V X u Q u u X B 0. 21 This implies that in general V X u 0