Macroscopic modelling of thermally-induced rarefied gas motion

Recently, an article entitled “Continuum analysis of rarefaction effects on a thermally-induced gas flow” by Hssikou, Baliti, and Alaoui, considered the numerical simulation of the two-dimensional gas flow induced by a temperature gradient imposed along two opposite walls of a rectangular cavity.  The gas motion and the heat transfer process in the domain is modelled in this work with the regularized 13-moment equations (R13; Struchtrup, 2005) developed as an extension of Grad’s 13-moment theory to account for Knudsen number effects not only on the boundaries but also in the fluid bulk. The Knudsen number is determined by the ratio of the molecular mean free path to the characteristic length of the domain.  The phenomena studied in this work bear closed relation with the processes taking place in spaces within micro-electro-mechanical systems (MEMS).

The study modelled the fluid as a Maxwell gas enclosed in a rectangular microcavity with the vertical sidewalls kept at the constant ambient temperature whereas the upper and bottom walls exhibited opposite linear temperature profiles.  The range of Knudsen numbers studied is such that the gas is in the slip and early transition regimes.  The analysis included the non-linear terms in the R13 equations.  The numerical simulation was carried out using the finite difference method.  The paper also considered, for comparison, the Navier-Stokes-Fourier (NSF) equations with velocity slip and temperature jump boundary conditions.

Among the various features shown and discussed by the authors, perhaps the most relevant is the reversal of the flow direction near and along the walls after changing the Knudsen number from 0.05 (slip regime) to 0.3 (early transition regime) predicted by the R13 equations.  In contrast, the NSF equations with modified boundary conditions predicted the same flow direction near the walls for both Knudsen numbers.  This direction is the same predicted by R13 for the lower Knudsen number.  R13 equations are thus able to account for the competing rarefaction effects of thermal transpiration on one hand and thermal stresses on the other. Another significant difference observed is that NSF predicts two vortices vertically oriented near the centre of the cavity whilst R13 predicts only one near the centre.

The results in the article showed the capability of the R13 equations to modelled rarefactions effects in the gas motion in microcavities at both locations, namely, near the boundaries and within the bulk.