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Centrifugal Buoyancy Effects on the Linear

Stability of Spiral Poiseuille Flow

David L. Cotrell and Geoffrey B. McFadden

Mathematical & Computational Sciences, ITL

For fluid flow in an annulus driven by the combination of an axial pressure gradient, rotation of the inner and outer cylinders, and a radial temperature gradient (spiral Poiseuille flow with a radial temperature gradient), we are investigating the transition from the simplest flow possible, steady flow with two nonzero velocity components and a radial temperature gradient that vary only with radius, to the next simplest flow possible, steady flow with three nonzero velocity components and a temperature profile that vary in the radial and axial directions. This work is motivated by electrochemical processes in rotating cylinder electrodes, heat transfer in rotating machinery, flow-amplified electrophoretic separations, and vortex flow reactors for shear-sensitive biological systems. This work extends the case of isothermal spiral Poiseuille flow for which D. Cotrell and A. Pearlstein (University of Illinois at Urbana-Champaign) recently computed complete linear stability boundaries for several values of the radius ratio and rotation rate ratio, and shows how the centrifugally-driven instability (beginning with steady or azimuthally-traveling-wave bifurcation of circular Couette flow (flow driven solely by wall motion) connects to a non-axisymmetric Tollmien-Schlichting-like instability of non-rotating annular Poiseuille flow (flow driven solely by an axial pressure gradient). Results for the non-isothermal case show that the stability boundary shifts either up or down depending on the sign of the temperature difference. For the isothermal case, it is also known that in many instances there is no instability for small enough axial flow rates. For the non-isothermal case, however, we have shown that for any nonzero temperature difference between the inner and outer radii, a new non-isothermal mode of instability causes the base state to be destabilized under these conditions.

David L. Cotrell

Mathematical & Computational Sciences Division

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