Exact solution of the unsteady Krook kinetic model and nonequilibrium thermodynamic study for a rarefied gas affected by a nonlinear thermal radiation field

摘要

A development of the previous paper (J. Non-Equilib. Thermodyn. 36, 75 (2011)) is introduced. The nonstationary Krook kinetic equation model for a rarefied gas affected by nonlinear thermal radiation field is solved, instead of the stationary equation. In a frame comoving with the fluid, analytically the Bhatnager-Gross-Krook model kinetic equation is applied. The travelling wave solution method is used to get the exact solution of the nonlinear partial differential equations. These equations were produced from applying the moment method to the unsteady Boltzmann equation. Now we should solve nonlinear partial differential equations in place of nonlinear ordinary differential equations, which represent an arduous task. The unsteady solution gives the problem a great generality and more applications. The new problem is investigated to follow the behavior of the macroscopic properties of the gas, such as the temperature and concentration. They are substituted into the corresponding two-stream maxiwallian distribution functions permitting us to investigate the nonequilibrium thermodynamic properties of the system (gas particles + the heated plate). The entropy, entropy flux, entropy production, thermodynamic forces, and kinetic coefficients are obtained. We investigate the verification of the Boltzmann H-theorem, Le Chatelier principle, the second law of thermodynamic and the celebrated Onsager's reciprocity relation for the system. The ratios between the different contributions of the internal energy changes based upon the total derivatives of the extensive parameters are estimated via the Gibbs formula. The results are applied to helium gas for various radiation field intensities due to different plate temperatures. Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.