Journal of Fluid Mechanics


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

K. R. Rajagopala1, G. Saccomandia2 and L. Vergoria3 c1

a1 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

a2 Dipartimento di Ingegneria Industriale, Università di Perugia, via G. Duranti, 06125, Italy

a3 Dipartimento di Matematica, Università del Salento, Strada Prov. Lecce-Arnesano, 73100 Lecce, Italy


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.

(Received May 31 2011)

(Reviewed January 30 2012)

(Accepted May 23 2012)

(Online publication July 06 2012)

Key Words:

  • lubrication theory;
  • shallow water flows;
  • wave breaking


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