# Propagating Wave in Binary Gas Mixture from Boundary of Variable Temperature and Velocity

• K. Yoshimura
• S. Kuwabara
Conference paper

## Introduction

Propagating waves in a binary gas mixture from a boundary of variable temperatures and velocities are discussed on the basis of two Boltzmann equations. Variable temperatures on the boundary could be gotten by using a thin thermal absorbent with fine holes heated by laser lights of variable powers, and variable velocities on a boundary could be gotten by moving gases through a porous media. In this paper, the Hermite expansion method (Grad [1]) of the distribution function would be extended to a binary gas-mixture, i.e. the Herimite expansion method is used to solve two distribution functions for both gases. We have two Boltzmann equations expanded by Hermite functions. The Galerkin method is applied to solve two Boltzmann equations, i.e. the scalar products of Hermite functions and two expanded Boltzmann equations are executed. Thus, a system of partial differential equations for the expansion coefficients, which are functions of position and time, is obtained. The Hermite expansion is taken up to the third order, so that we have twenty coefficients and a system of twenty partial differential equations. Here, the one dimensional flow in space is considered, so that the number of expansion coefficients and differential equations would reduce from twenty to eight. The intermolecular force consisting of an inverse fifth- and inverse third- order power of intermolecular distance. We have applied this method to the relaxation phenomena of the binary gas mixture with different temperatures and velocities, which are both uniform in space ([2]and [3]). In this paper, we are going to discuss propagating waves in the gas mixture. Temperatures, number densities and velocities at x = 0 could be given. We calculate numerically the time developments of velocity, temperature and number density. The initial conditions of $$v_1^{(i)}(x,0), T^{(i)}(x,0), n^{(i)}(x,0)$$ etc. are taken as uniform in 0 ≤ x ≤ ∞ . The boundary conditions at x = 0 are taken as $$T^{(i)}(0,t)=T_b^{(i)}(t),\> v_1^{(i)}(0,t)=u_b^{(i)}(t), \> n^{(i)}(0,t)=n_b^{(i)}(t)$$. Time developments of temperatures, mass velocities and number densities etc. have been calculated.

## Keywords

Boltzmann Equation Galerkin Method Physical Variable Mass Velocity Relaxation Phenomenon
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## References

1. 1.
Grad, H.: Note on N-Dimensional Hermite Polynomials. CPAM 2, 325–330 (1949)
2. 2.
Yoshimura, K., Kuwabara, S.: Relaxation Phenomena of Binary Gas-Mixture with Sizable Difference in their Temperatures and Velocities on the basis of Boltzmann Equations. In: Proc. 27th Int. Symp. Rarefied Gas Dynamics (2010)Google Scholar
3. 3.
Kuwabara, S., Yoshimura, K.: Derivation of Hydrodynamic Equations for Binary Gas Mixture. In: Proc. 27th Int. Symp. Rarefied Gas Dynamics (2010)Google Scholar