Linear stability of spiral Poiseuille flow with a radial temperature gradient: Centrifugal buoyancy effects

Abstract

For spiral Poiseuille flow with a radial temperature gradient and radius ratio η=Ri∕Ro=0.5, we have computed complete linear stability boundaries for several values of the rotation rate ratio μ=Ωo∕Ωi, where Ri and Ro are the inner and outer cylinder radii, respectively, and Ωi and Ωo are the corresponding (signed) angular speeds. The effects of gravity are neglected, but the variation of density with temperature induces radial buoyancy effects through the centripetal acceleration term in the radial momentum equation. The analysis extends previous results with no axial flow (Reynolds number Re=0) to the range of Re for which the annular Poiseuille flow is stable and accounts for arbitrary disturbances of infinitesimal amplitude. For μ<η2 and a temperature gradient consistent with the Boussinesq approximation, we show that over the entire range of Re considered the stability boundaries do not differ significantly from those found for the isothermal case by Cotrell and Pearlstein [“The connection between centrifugal instability and Tollmien-Schlichting-like instability for spiral Poiseuille flow,” J. Fluid. Mech. 509, 331 (2004)]. For μ>η2 and Re=0, we show for the first time that the flow is destabilized for any positive value of the radial buoyancy parameter (i.e., β=αoΔT), where ΔT=To−Ti is the temperature difference between the outer and inner cylinder walls, respectively, and αo is the coefficient of thermal expansion. This differs significantly from the isothermal results which show that there is no linear instability for 0⩽Re⩽Remin in the absence of centrifugal buoyancy effects, where Remin is the turning point value in the isothermal results.