Effect of weak gravitation on the plane thermal transpiration of a slightly rarefied gas

Abstract

Plane thermal transpiration of a slightly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the fluid-dynamic-type equation and the slip-type boundary condition derived from the Boltzmann equation. We consider the situation where the Knudsen number (the mean free path divided by the channel width) is small and the dimensionless gravity (the channel width divided by the ascent height of the molecules against gravity) is of the order of the square of the Knudsen number. The fluid-dynamic equation is studied by the asymptotic analysis for the slow variation in the longitudinal direction, and the nontrivial leading order solution is obtained analytically. Due to the combination of gravity and the temperature gradient imposed along the channel, a pressure gradient is produced in both the vertical and the horizontal directions. The horizontal pressure gradient induces the flow even in the absence of the gradient of the sectionally averaged pressure. Although the pressure gradient produced is of the higher order of the Knudsen number, the flow induced by the gradient is of the order of the Knudsen number and thus has a relatively finite effect on the thermal transpiration. The velocity profile is considerably different from that of the conventional thermal transpiration due to the effect of weak gravitation. A direct numerical analysis of a flow through a long channel is conducted based on the model Boltzmann equation, and the mechanism of this phenomenon is demonstrated.