Journal of Fluid Mechanics

Table of Contents - Volume 700 - 2012  

Papers

Release of a viscous power-law fluid over an inviscid ocean

Samuel S. Peglera1 c1, John R. Listera1 and M. Grae Worstera1

a1 Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK

Abstract

We consider the two- and three-dimensional spreading of a finite volume of viscous power-law fluid released over a denser inviscid fluid and subject to gravitational and capillary forces. In the case of gravity-driven spreading, with a power-law fluid having strain rate proportional to stress to the power $n$, there are similarity solutions with the extent of the current being proportional to ${t}^{1/ n} $ in the two-dimensional case and ${t}^{1/ 2n} $ in the three-dimensional case. Perturbations from these asymptotic states are shown to retain their initial shape but to decay relatively as ${t}^{\ensuremath{-} 1} $ in the two-dimensional case and ${t}^{\ensuremath{-} 3/ (n+ 3)} $ in the three-dimensional case. The former is independent of $n$, whereas the latter gives a slower rate of relative decay for fluids that are more shear-thinning. In cases where the layer is subject to a constraining surface tension, we determine the evolution of the layer towards a static state of uniform thickness in which the gravitational and capillary forces balance. The asymptotic form of this convergence is shown to depend strongly on $n$, with rapid finite-time algebraic decay in shear-thickening cases, large-time exponential decay in the Newtonian case and slow large-time algebraic decay in shear-thinning cases.

(Received May 17 2011)

(Reviewed February 10 2012)

(Accepted February 12 2012)

(Online publication April 17 2012)

Key Words:

  • gravity currents;
  • non-Newtonian flows;
  • thin films

Correspondence:

c1 Email address for correspondence: ssp23@cam.ac.uk

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