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Non-Newtonian fluid displacements in horizontal narrow eccentric annuli: effects of slow motion of the inner cylinder

Published online by Cambridge University Press:  02 June 2010

M. CARRASCO-TEJA  and
I. A. FRIGAARD
M. CARRASCO-TEJA
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, CanadaV6T 1Z2
I. A. FRIGAARD*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, CanadaV6T 1Z2 Department of Mechanical Engineering, University of British Columbia, 2054-6250 Applied Science Lane, BC, CanadaV6T 1Z4
*
†Email address for correspondence: [email protected]
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Abstract

We study non-Newtonian fluid displacements in horizontal narrow eccentric annuli in the situation where the inner cylinder is moving. This represents a practically important extension of the model analysed by Carrasco-Teja et al. (J. Fluid Mech., vol. 605, 2008, pp. 293–327). When motion of the inner cylinder is included, the Hele-Shaw model closure becomes significantly more complex and extremely costly to compute, except for Newtonian fluids. In the first part of the paper we address the model derivation and closure relations. The second part of the paper considers the limit of large buoyancy number, in which the interface elongates along the annulus. We derive a lubrication-style model for this situation, showing that the leading-order interface is symmetric. Rotation of the inner cylinder only affects the length of the leading-order interface, and this occurs only for non-Newtonian fluids via shear-thinning effects. At first order, casing rotation manifests in an asymmetrical ‘shift’ of the interface in the direction of the rotation. We also derive conditions on the eccentricity, fluid rheology and inner cylinder velocity, under which we are able to find steady travelling wave displacement solutions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 653 , 25 June 2010 , pp. 137 - 173
Copyright
Copyright © Cambridge University Press 2010

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References

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