Flow of a compressible fluid in a rapidly rotating pipe with azimuthally varying wall thermal condition

Abstract

An analysis is made of steady-state flow of a compressible fluid in an infinite rapidly rotating pipe. Flow is induced by imposing a small azimuthally varying thermal forcing at the pipe wall. The Ekman number is small. Analyses are conducted to reveal both the axisymmetric-type and non-axisymmetric-type solutions. The axisymmetric solution is based on the azimuthally averaged wall boundary condition. The non-axisymmetric solution stems from the azimuthally fluctuating part of the wall boundary condition. It is shown that the two-dimensional (uniform in the axial direction) non-axisymmetric solution exists for sigma(gamma - 1)M-2 much greater than O(E-1/3). However, an axially dependent solution is found if sigma(gamma - 1)M-2 less than or similar to O(E-1/3), in which E denotes the Ekman number, M the Mach number, gamma the specific heat ratio and sigma the Prandtl number. The axisymmetric solution prevails over the whole flow region; the two-dimensional non-axisymmetric solution is confined to the near-wall thermal layer of thickness O(E-1/3). As a canonical example, a detailed description is given for the case of a highly conducting wall with differential heating.