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Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane

Published online by Cambridge University Press:  06 July 2012

K. R. Rajagopal ,
G. Saccomandi  and
L. Vergori
K. R. Rajagopal
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA
G. Saccomandi
Affiliation:
Dipartimento di Ingegneria Industriale, Università di Perugia, via G. Duranti, 06125, Italy
L. Vergori*
Affiliation:
Dipartimento di Matematica, Università del Salento, Strada Prov. Lecce-Arnesano, 73100 Lecce, Italy
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

JFM classification

Low-Reynolds-number flows: Lubrication theory Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 173 - 189
Copyright
Copyright © Cambridge University Press 2012

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