Reciprocity relations in flows of a rarefied gas between plane parallel walls with nonuniform surface properties

Abstract

Flows of a rarefied gas between plane parallel walls with nonuniform surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary whose accommodation coefficient varies periodically in the longitudinal direction. Four fundamental flows are studied, namely, Poiseuille flow, thermal transpiration, Couette flow, and the heat transfer problem. These flow problems are numerically studied based on the linearized Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation over a wide range of the mean free path and the parameters characterizing the distribution of the accommodation coefficient. The flow fields, the mass and heat flow rates through a cross section or the wall surfaces, and the tangential force acting on the wall surfaces are studied. Due to the nonuniform surface properties, a longitudinal motion of the gas induces a local heat transfer through the wall surface in Poiseuille, thermal transpiration, and Couette flows; a temperature difference between the walls induces a motion of a gas and a local tangential stress on the walls in the heat transfer problem. However, the heat flow rate through the wall surface in the first three flow problems and the tangential force acting on the wall surface in the heat transfer problem vanish if integrated over one period. No net mass flow is induced in the heat transfer problem. Six reciprocity relations among the flow rates in the aforementioned four flows are numerically confirmed. Among the six relations, both hand sides of three relations vanish. The background of this phenomenon is discussed based on the flow field of the gas.