Exact Non-Local Hydrodynamics Predict Rarefaction Effects

Abstract

We combine the theory of slow spectral closure for linearized Boltzmann equations with Maxwell's kinetic boundary conditions to derive non-local hydrodynamics with arbitrary accommodation. Focusing on shear-mode dynamics, we obtain explicit steady state solutions in terms of Fourier integrals and closed-form expressions for the mean flow and the stress. We demonstrate that the exact non-local fluid model correctly predicts several rarefaction effects with accommodation, including the Couette flow and thermal creep in a plane channel.

Figures (3)

• Figure 1: Real part (red) and imaginary part (blue) of the Fourier multiplier $\hat{s}$ for the BGK model \ref{['eq:sBGK']}. For $k\tau>\sqrt{\frac{\pi}{2}}$ (critical wave number, dashed vertical line), the real part vanishes while the imaginary part becomes $-\frac{1}{2 k\tau}\mathrm{i}$ (dashed blue line).
• Figure 2: Normalized stress for the planar Couette flow ($\tau=1$, $\alpha=1$). Solid red line with markers: present rarefied hydrodynamics; Black markers: tabulated numerical calculation for the BGK model barichello2001unified.
• Figure 3: Normalized flow rate for the thermal creep flow ($\tau=1$, $\alpha=0.5$). Solid red line with markers: present rarefied hydrodynamics; Black markers: tabulated numerical calculation for the BGK model barichello2001unified.