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 $\sigma (\gamma - 1)M^2 \,{\gg}\, O(E^{1 / 3})$. However, an axially dependent solution is found if $\sigma (\gamma - 1)M^2 \,{\lesssim}\,O(E^{1 / 3})$, 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.