A rarefied gas flow caused by a discontinuous wall temperature

A flow of a rarefied gas caused by a discontinuous wall temperature is investigated on the basis of kinetic theory in the following situation. The gas is confined in a two-dimensional square container, and the left and right halves of the wall of the container are kept at different uniform temperatures, so that the temperatures of the top and bottom walls are discontinuous at their respective middle points. External forces are assumed to be absent. The steady flow of the gas induced in the container by the effect of the discontinuities is analyzed numerically on the basis of the Bhatnagar-Gross-Krook model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The features of the flow are clarified for a wide range of the Knudsen number. In particular, it is shown that, as the Knudsen number becomes small (i.e., as the system approaches the continuum limit), the maximum flow speed tends to approach a finite value, but the region with appreciable flow shrinks to the points of discontinuity; thus, the overall flow in the container vanishes nonuniformly in the continuum limit. The behavior of the molecular velocity distribution function is also investigated in detail.