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Roll waves in mud

Balmforth, N. J. and Liu, J. J. (2004) Roll waves in mud. Journal of Fluid Mechanics, 519 . pp. 33-54. ISSN 0022-1120.

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The stability of a viscoplastic fluid film falling down an inclined plane is explored, with the aim of determining the critical Reynolds number for the onset of roll waves. The Herschel–Bulkley constitutive law is adopted and the fluid is assumed two-dimensional and incompressible. The linear stability problem is described for an equilibrium in the form of a uniform sheet flow, when perturbed by introducing an infinitesimal stress perturbation. This flow is stable for very high Reynolds numbers because the rigid plug riding atop the fluid layer cannot be deformed and the free surface remains flat. If the flow is perturbed by allowing arbitrarily small strain rates, on the other hand, the plug is immediately replaced by a weakly yielded ‘pseudo-plug’ that can deform and reshape the free surface. This situation is modelled by lubrication theory at zero Reynolds number, and it is shown how the fluid exhibits free-surface instabilities at order-one Reynolds numbers. Simpler models based on vertical averages of the fluid equations are evaluated, and one particular model is identified that correctly predicts the onset of instability. That model is used to describe nonlinear roll waves.

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Additional Information:"Reprinted with the permission of Cambridge University Press." Received December 3 2003, Revised April 13 2004, Published Online 29 October 2004 This work began at the Geophysical Fluid Dynamics Summer Study Program (Woods Hole Oceanographic Institution), which is supported by the National Science Foundation and the Office of Naval Research. We thank the participants, and especially Ian Frigaard for discussions. The work was completed whilst N. J.B. was visiting Massachusetts Institute of Technology; he thanks the Mathematics Department for hospitality.
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Deposited On:28 Feb 2006
Last Modified:02 Oct 2019 22:49

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