The unsteady Boltzmann kinetic equation and non-equilibrium thermodynamics of an electron gas for the Rayleigh flow problem A.M. Abourabia and T.Z. Abdel Wahid Abstract: In the framework of irreversible thermodynamics, the characteristics of the Rayleigh flow problem of a rarified electron gas extracted from neutral atoms is examined and proved to obey the entropic behavior for gas systems. A model kinetic equation of the BGK (Bhatnager–Gross–Krook) type is solved, using the method of moments with a two-sided dis- tribution function. Various macroscopic properties of the electron gas, such as the mean velocity, the shear stress, and the viscosity coefficient, together with the induced electric and magnetic fields, are investigated with respect to both distance and time. The distinction between the perturbed velocity distribution functions and the equilibrium velocity distribution function at different time values is illustrated. We restrict our study to the domain of irreversible thermodynamics proc- esses with small deviation from the equilibrium state to estimate the entropy, entropy production, entropy flux, thermody- namic force, and kinetic coefficient and verify the celebrated Boltzmann H-theorem for non-equilibrium thermodynamic properties of the system. The ratios between the different contributions of the internal energy changes, based upon the total derivatives of the extensive parameters, are predicted via Gibbs’ equation for both diamagnetic and paramagnetic plasmas. The results are applied to a typical model of laboratory argon plasma. PACS Nos: 51.10.+y, 51.30.+i, 45.50.Tn, 47.54.–n, 47.10.ab Re ´sume ´: Dans le cadre de la thermodynamique irre ´versible, nous examinons les caracte ´ristiques de l’e ´coulement de Ray- leigh de gaz rare ´fie ´s d’e ´lectrons extraits d’atomes neutres, connus pour avoir un comportement entropique pour des syste `- mes gazeux. Nous solutionnons une e ´quation d’un mode `le cine ´tique de type BGK (Bhatnager-Gross-Krook) en utilisant la me ´thode des moments avec une fonction de distribution a ` deux queues. Nous e ´tudions, selon la distance et le temps, diffe ´- rentes proprie ´te ´s macroscopiques du gaz d’e ´lectrons, comme la vitesse moyenne, la contrainte de cisaillement, le coeffi- cient de viscosite ´, ainsi que les champs magne ´tique et e ´lectrique induits. Nous illustrons la diffe ´rence a ` divers temps entre la distribution perturbe ´e de vitesse et la distribution de vitesse a ` l’e ´quilibre. Nous restreignons notre e ´tude au domaine des processus de thermodynamique irre ´versible montrant des de ´viations proches de l’e ´tat d’e ´quilibre, afin d’e ´valuer l’entropie, la production d’entropie, le flux d’entropie, la force thermodynamique, le coefficient cine ´tique et nous ve ´rifions le ce ´le `bre the ´ore `me H de Boltzmann pour les proprie ´te ´s de thermodynamique hors d’e ´quilibre du syste `me. Pour les deux cas de plas- mas diamagne ´tique et paramagne ´tique, l’e ´quation de Gibbs pre ´dit les rapports entre les diffe ´rentes contributions des chan- gements d’e ´nergie internes. Nous appliquons les re ´sultats a ` un mode `le typique de plasma d’argon en laboratoire. [Traduit par la Re ´daction] 1. Introduction In Lebon’s article [1], dedicated to Professor I. Gyarmati on his 60th birthday, the author explained that the funda- mental hypothesis underlying classical irreversible thermo- dynamics (CIT) is that of local equilibrium. It postulated that the local and instantaneous relations between the ther- mal and mechanical properties of a physical system are the same as for a uniform system at equilibrium. The local equilibrium hypothesis implies that: 1. the equilibrium is stable, 2. the values defined in classical thermodynamics (thermo- statics) remain significant, 3. the relationships in thermostatics between state variables remain valid outside of equilibrium under the conditions that they are stated locally and at a given instant of time. Thus, entropy outside of equilibrium will depend on dif- ferential form by using the Gibbs equation dUðr;tÞ¼ T ðr;tÞdSðr;tÞ pðr;tÞdr 1 ðr;tÞ ð1Þ Gibbs’ equation plays a key role in the elaboration of the theory. Expression (1) refers to a one-component fluid, per unit mass. A precise limitation on the domain of validity of the local equilibrium hypothesis can be obtained from the kinetic theory of gases. Using the Chapman–Enskog devel- opment, Prigogine established that the hypothesis of local equilibrium is satisfactory on the condition that the distribu- tion function is limited to the first-order term. The main ob- jective of thermodynamics is to determine the changes in Received 3 February 2010. Accepted 22 April 2010. Published on the NRC Research Press Web site at cjp.nrc.ca on 4 June 2010. A.M. Abourabia 1 and T.Z. Abdel Wahid. 2 Faculty of Science, Monoufiya University, Shebine-Elkom 32511, Egypt. 1 Corresponding author (e-mail: am_abourabia@yahoo.com). 2 Corresponding author (e-mail: taha_zakaraia@yahoo.com). 501 Can. J. Phys. 88: 501–511 (2010) doi:10.1139/P10-032 Published by NRC Research Press Can. J. Phys. Downloaded from www.nrcresearchpress.com by 220.249.99.179 on 04/27/11 For personal use only.