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 (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 timeperiodic 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 (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 continuumlimit 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 Ma. A marked difference from the latter is that the normal force either decreases with Kn or, at sufficiently small actuation frequencies, varies nonmonotonically with Kn, reaching a maximum at some intermediate rarefaction conditions.

Nonlinear oscillatory fully-developed rarefied gas flow in plane geometry

“…Simulations have been based on linearized 24,25 and nonlinear 26,[30][31][32]34,35,37 kinetic models, as well as on the Direct Simulation Monte Carlo (DSMC) method. [21][22][23][27][28][29]33,36 It is also noted that steady-state force driven Poiseuille type flows have been investigated [38][39][40][41][42][43] based on kinetic theory and modeling, clarifying certain phenomena and paradox appearing near the continuum regime that cannot be described by the typical hydrodynamic approach. Such phenomena include the nonconstant pressure profile across the channel, the bimodal shape of the temperature profile with a slight shallow at the channel center and the presence of axial heat flow.…”

“…It is noted that although pressure/force driven oscillatory rarefied gas flows have received very little attention, the corresponding shear-driven flows have attracted considerable attention. [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] Rarefied oscillatory flows due to moving boundaries are present in resonating filters, sensors, actuators, and signal processing, 36,37 where the computation of the damping forces is crucial in order to control and optimize the resolution and sensitivity of the signal. Simulations have been based on linearized 24,25 and nonlinear 26,[30][31][32]34,35,37 kinetic models, as well as on the Direct Simulation Monte Carlo (DSMC) method.…”

Effect of heat-flux boundary conditions on the Rayleigh-Bénard instability in a rarefied gas

“…In recent contributions by the authors [20,21], a noniterative procedure for the imposition of a heat-flux condition in a DSMC calculation has been presented. The algorithm was assigned in onedimensional single-and bidirectional setups.…”

“…After completing several computational time-steps, the "numerically real" heat-flux was obtained through inversion of Eq. (21). The boundary conditions were then switched to heat-flux conditions, and the simulation was followed to later times.…”

“…Remarkably, the Kn variation in all cases is nonmonotonic and contains several maxima and minima points along the 0.01 Kn Kn interval. While the eventual decrease in |σ xy | max as Kn → Kn results from the stabilizing effect of gas rarefaction and consequent decrease in convection speed, its increase with Kn at sufficiently lower Kn < Kn reflects the amplifying effect of rarefaction on the boundary shear stress at constant shear rate [21]. Noting the characteristic dimensional shear rate by u * /D * (where u * marks the characteristic convection speed), it is therefore a balance between the simultaneous decrease in u * and D * with Kn (for a fixed mean free path l * ) that determines the nonmonotonic behavior observed in Fig.…”