Non-Linear Thermal Effects in Unsteady Shear Flows of a Rarefied Gas*

We study the response of a rarefied gas in a slab to the time-harmonic motion of its boundaries in the tangential direction. In difference from previous investigations, we consider boundaries velocities (Uw) of non-small Mach numbers (Ma = Uw/Uth, where Uth marks the mean thermal speed), which deviate the system from its low-velocity isothermal condition. The problem is studied in the entire range of gas rarefaction rates, combining limit case analyses with direct simulation Monte Carlo computations. A non-linear solution is derived in the ballistic regime for arbitrary velocity amplitudes. At near-continuum conditions, a slip-flow time-periodic solution is obtained by expanding the flow variables in a Mach power series. The results indicate that, at all Knudsen numbers (Kn = λ/L, where λ and L mark the molecular mean free path and slab width, respectively), the thermodynamic fields and normal velocity component are characterized by double-frequency time dependence, in difference from the fundamental-frequency time dependence characterizing the tangential gas velocity. System nonlinearity also results in a normal force acting on the boundaries, overcoming the tangential force with increasing Ma.