Continuum Analysis of Rarefaction Effects on a Thermally Induced Gas Flow

A Maxwell gas confined within a micro cavity with nonisothermal walls is investigated in the slip and early transition regimes using the classical and extended continuum theories. The vertical sides of the cavity are kept at the uniform and environmental temperature T0, while the upper and bottom ones are linearly heated in opposite directions from the cold value T0 to the hot one TH. The gas flow is, therefore, induced only by the temperature gradient created along the longitudinal walls. The problem is treated from a macroscopic point of view by solving numerically the so-called regularized 13-moment equations (R13) recently developed as an extension of Grad 13-moment theory to the third order of the Knudsen number powers in the Chapman-Enskog expansion. The gas macroscopic properties obtained by this method are compared with the classical continuum theory results (NSF) using the first and second order of velocity slip and temperature jump boundary conditions. The gas flow behavior is studied as a function of the Knudsen number (Kn), nonlinear effects, for different heating rates T0/TH. The micro cavity aspect ratio effect is also evaluated on the flow fields in this study.

Comparison of viscosities from the Chapman-Enskog and relaxation time methods

A quantitative comparison between the results of shear viscosities from the Chapman-Enskog and relaxation time methods is performed for selected test cases with specified elastic differential cross sections: (i) the non-relativistic, relativistic and ultra-relativistic hard sphere gas with angle and energy independent differntial cross section, (ii) the Maxwell gas, (iii) chiral pions and (iv) massive pions. Our quantitative results reveal that the extent of agreement (or disagreement) depends very sensitively on the energy dependence of the differential cross sections employed.

On the compressible Taylor–Couette problem

We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing (Ma) of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio (Ma/Kn) (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects.