The size effect and the non-local Boltzmann transport equation in orifice and disk geometry

Abstract

A variational approximation is obtained for the resistance (or total entropy production) of a circular orifice in a plane diaphragm at which the scattering is specular. It is found that the Ohm's law prediction must be multiplied by a correction factor of the order of unity and added to the `small orifice' Knudsen contribution. The results for the intermediate régime can be applied to the transport of electricity and heat by electrons and to phonons which are dominated by elastic scattering (e.g. by crystal imperfections). Results are also obtained for the case when the diaphragm separates materials of different mean free path. It is shown that the above problems are equivalent to the determination of the spreading resistance of a small disk-shaped electrode. The mathematical method is based on the formulation of an integral equation over the region of the disk with a kernel containing the Green function for the infinite medium Boltzmann equation.