Mathematical and Numerical Study of a Dusty Knudsen Gas Mixture

Abstract

We consider a mixture composed of a gas and dust particles in a very rarefied setting. Whereas the dust particles are individually described, the surrounding gas is treated as a Knudsen gas, in such a way that interactions occur only between gas particles and dust by means of diffuse reflection phenomena. After introducing the model, we prove the existence and the uniqueness of the solution and provide a numerical strategy for the study of the equations. At the numerical level, we focus our attention on the phenomenon of energy transfer between the gas and the moving dust particles.

Appendix: Numerical Treatment of the Diffuse Reflexion Mechanism

We describe here the treatment of the boundary condition (2.7) on the moving boundary $\varGamma ^{t}$, adapted from the treatment of a diffuse reflexion condition on a fixed boundary at temperature $T_{p}$ (see [14]). For the sake of simplicity, we normalize the mass $m$ and the Boltzmann constant $k_{B}$. If a numerical particle hits the boundary $\varGamma ^{t}$ at $x$, then its velocity will be generated by sampling from the half Maxwellian $h(v-c(t,x))$, where

where $\ell$ is the dimension ($\ell \in \{2,3\}$). Because of its normalization, $h$ is a probability density. If we note $v= (v_{1}, v _{\perp } )$, with $v_{1} = v\cdot n_{x}$, then $h$ can be written as the product $h(v)=h_{1}(v_{1}) \, h_{2} (v_{\perp })$, with

The repartition function $\psi _{1}(s) = 1- \exp (-{s^{2}}/{2 T _{p}} )$ of $h_{1}$ is invertible, and

$$ \psi _{1}^{-1}(u )= \sqrt{-2 T_{p} \, \ln (1-u)}. $$

Assume now $l=2$. As we cannot explicitly express the repartition function $\psi _{2}$ of $h_{2}$, we use the classical trick to simulate a Gaussian random variable, by working with polar variables—$(w_{1},w_{2}) \rightarrow (R,\theta )$, with $w_{1}=R\cos (\theta )$ and $w_{2}= R \sin (\theta )$—in the joint density of $(w_{1},w_{2})$:

$$ \psi _{2}(w_{1}) \psi (w_{2})\, \mathrm{d}w_{1}\, \mathrm{d}w_{2} = \frac{1}{T_{p}} R \exp \bigl(-R ^{2}\bigr)\,\mathrm{d}R \frac{\mathrm{d}\theta }{2\pi }. $$

This allow to invert the repartition function of the variable $R$. We then randomly compute $V_{r} = (V_{r1}, V_{r\perp })$ following the law $h$ by sampling three independent random variables $U_{1}$, $U_{2}$ and $U_{3}$, following a uniform law on $[0,1]$, and set

$$ V_{r1} = \sqrt{-2 T_{p} \, \ln (1-U_{1})}, \quad \mbox{and} \quad V_{r\perp } = \sqrt{-2 T_{p} \ln (1-U_{2})} \cos (2\pi U_{3}). $$

Finally the velocity of the numerical particle after reflexion on boundary $\varGamma ^{t}$ is set to

$$ v_{ar} = c(t,x) + ( V_{r1}, V_{r\perp } ). $$