**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

Information Technology Laboratory

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