Nonlinear thermal effects in unsteady shear flows of a rarefied gas

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.

Figure 1

Time variations (in period $t_p=2\pi /\omega$ units) of the deviations of $T_{w_1}$, ${\rho }_{w_1}$, and ${\rho }_{w_2}$ from their equilibrium initial values for an adiabatic wall setup with $\mathrm{Ma}=0.8$ and $\omega =5$. The solid and dashed curves present the ballistic limit ($\mathrm{Kn}\to \infty$) and DSMC $\mathrm{Kn}=72$ results, respectively. The dash-dotted line shows the tangential wall velocity at $y=-1/2$, multiplied by 0.01 for scaling.

Figure 2

Spatial variations of the (a) tangential velocity, (b) density deviation, (c) temperature deviation, and (d) normal heat flux, for the adiabatic wall (blue) and isothermal wall (red) setups with $\mathrm{Ma}=0.8$ and $\omega =5$. The curves (solid blue and dashed red lines) and symbols (blue crosses and red circles) present the ballistic analytical and $\mathrm{Kn}=72$ DSMC predictions at period time, respectively.

Figure 3

Spatial variations of the (a) tangential velocity, (b) density deviation, (c) temperature deviation, and (d) normal heat flux, for the adiabatic wall (blue) and isothermal wall (red) setups with $\mathrm{Ma}=0.3,\omega =0.1$ and $\mathrm{Kn}=0.09$. The curves (solid blue and dashed red lines) and symbols (blue crosses and red circles) present slip flow analytical results and DSMC predictions at period time, respectively.

Figure 4

Spatial variations of the normal velocity for the adiabatic wall (blue) and isothermal wall (red) setups with (a) $\mathrm{Ma}=0.8,\omega =5$, and $\mathrm{Kn}=72$; (b) $\mathrm{Ma}=0.3,\omega =0.1$, and $\mathrm{Kn}=0.09$. The curves (solid blue and dashed red lines) and symbols (blue crosses and red circles) present analytical [collisionless in (a); slip flow in (b)] results and DSMC predictions at period time, respectively.

Figure 5

Variations with $\mathrm{Kn}$ of the amplitudes of the (a, c) tangential and (b, d) normal forces per unit area on the adiabatic $y=-1/2$ (blue) and isothermal $y=1/2$ (red) boundaries for (a, b) $\mathrm{Ma}=0.3$ and $\omega =0.1$; (c, d) $\mathrm{Ma}=0.8$ and $\omega =1$. The symbols show DSMC results for the $y=-1/2$ (blue crosses) and $y=1/2$ (red circles) walls. The solid lines present continuum- and ballistic-limit predictions. Black curves are presented in cases where the adiabatic and isothermal walls results coincide. The black dashed line in (c) shows the linearized isothermal approximation for the tangential stress in the ballistic limit.

Figure 6

Variations with $\mathrm{Ma}$ of the amplitudes of the (a) tangential and (b) normal forces per unit area on the adiabatic $y=-1/2$ (blue) and isothermal $y=1/2$ (red) boundaries at the indicated $(\mathrm{Kn},\omega )$ combinations. The symbols show DSMC results for the $y=-1/2$ (blue crosses) and $y=1/2$ (red circles) walls. The solid lines present continuum- and ballistic-limit predictions. Black curves are presented where the adiabatic and isothermal walls results coincide. The black dashed line in (a) shows the linearized isothermal approximation for the tangential force in the ballistic limit at $\omega =1$.

Figure 7

Spectral densities of the tangential velocity (black solid lines) and temperature deviation (dashed red curves) signals at the gap middle point, $y=0$, for an adiabatic wall setup with $\mathrm{Kn}=0.09,\omega =1$ and (a) $\mathrm{Ma}=0.8$; (b) $\mathrm{Ma}=5.$ The spectra are calculated based on DSMC predictions and are scaled by their maximal values.