Nonlinear thermal effects in unsteady shear flows of a rarefied gas

Abstract

We study the response of a rarefied gas in a slab to the motion of its boundaries in the tangential direction. Different from previous investigations, we consider boundaries displacements at nonsmall Mach ($\mathrm{Ma}$) numbers, coupling the dynamic and thermodynamic gas states, and deviating the system from its low-velocity isothermal condition. The problem is studied in the entire range of gas rarefaction rates, combining limited case ballistic- and continuum-flow analyses with direct simulation Monte Carlo computations. A nonlinear solution is derived in the ballistic regime for arbitrary velocity profiles and amplitudes. At near-continuum conditions, a slip-flow time-periodic solution is obtained for the case of oscillatory boundary motion, by expanding the flow field in an asymptotic Mach power series. The effect of replacing between isothermal and adiabatic surfaces is examined. The results indicate that, at all Knudsen ($\mathrm{Kn}$) numbers, the thermodynamic fields and normal velocity component are dominated by double-frequency (and descending higher-order even-frequency harmonic) time dependence, different from the fundamental-frequency time dependence dominating the tangential gas velocity. At continuum-limit conditions, this stems from the quadratic viscous dissipation term (negligible at low-Mach conditions), coupling the square of the tangential velocity gradient as a forcing term. System nonlinearity also results in an unsteady normal force acting on the boundaries, overcoming the tangential force with increasing $\mathrm{Ma}$. A marked difference from the latter is that the normal force either decreases with $\mathrm{Kn}$ or, at sufficiently small actuation frequencies, varies nonmonotonically with $\mathrm{Kn}$, reaching a maximum at some intermediate rarefaction conditions.