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Stability of rotating non-smooth complex fluids

Published online by Cambridge University Press:  29 August 2012

Ishan Sharma
Ishan Sharma*
Affiliation:
Mechanics & Applied Mathematics Group, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh 208016, India
*
†Email address for correspondence: [email protected]
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Abstract

We extend the classical energy criterion for stability, the Lagrange–Dirichlet theorem, to rotating non-smooth complex fluids. The stability test so developed is very general and may be applied to most rotating non-smooth systems where the spectral method is inapplicable. In the process, we rigourously define an appropriate coordinate system in which to investigate stability – this happens to be the well-known Tisserand mean axis of the body – as well as systematically distinguish perturbations that introduce angular momentum and/or jumps in the stress state from those that do not. With a view to future application to planetary objects, we specialize the stability test to freely rotating self-gravitating ellipsoids. This is then employed to investigate the stability to homogeneous perturbations of rotating inviscid fluid ellipsoids. We recover results consistent with earlier predictions, and, in the process, also reconcile some contradictory conclusions about the stability of Maclaurin spheroids. Finally, we consider the equilibrium and stability of freely rotating self-gravitating Bingham fluid ellipsoids. We find that the equilibrium shapes of most such ellipsoids are secularly stable to homogeneous perturbations that preserve angular momentum, but not otherwise. We also touch upon the effect of shear thinning on stability.

JFM classification

Granular media: Granular media Non-Newtonian Flows: Plastic materials Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 71 - 99
Copyright
Copyright © Cambridge University Press 2012

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