Journal of Fluid Mechanics


Roll waves in mud

N. J. BALMFORTH a1 and J. J. LIU a2
a1 Departments of Mathematics and Earth & Ocean Science, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
a2 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA

Article author query
balmforth nj   [Google Scholar] 
liu jj   [Google Scholar] 


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.

(Received December 3 2003)
(Revised April 13 2004)