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Stability of rotating non-smooth complex fluids

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.