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

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.

Asymptotic solutions of weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity

“…Such an assumption, however, is valid only at low processing pressures and may introduce significant error when modeling flows involving high pressures or a large pressure range, such as polymer processing, crude oil and fuel oil pumping, fluid film lubrication, microfluidics, and in certain geophysical flows [1][2][3][4].…”

“…In fact, under certain conditions, e.g. in elastohydrodynamics, the dependence of the viscosity on pressure may be several orders of magnitude stronger than that of density [3,17,18]. Málek and Rajagopal [19] reviewed different equations proposed in the literature in order to describe experimental observations on the pressure-dependence of the viscosity.…”

An exact analytical solution for viscoelastic fluids with pressure-dependent viscosity

“…In general however, it depends on the flow conditions and in particular on the shear-rate, the pressure, and, for non-isothermal problems, strongly on the temperature. The effect of the pressure on the viscosity becomes important at a pressure 50 atm approximately [1], while for pressures of the order of 1000 atm the viscosity appears to increase more than an order of magnitude [2,3]. Applications which involve a high pressure difference and/or a large pressure range include polymer processing operations such as extrusion and injection molding [1,[4][5][6], food processing, pharmaceutical tablet manufacturing, crude oil and fuel oil pumping [7], fluid film lubrication [8], journal bearing applications [9], microfluidics [10] and geophysics [11].…”

A Novel Approach to the Description of Constitutive Relations

“…When one deals with a generalization of the Navier-Stokes fluid with a pressure dependent viscosity (usually referred to in the literature as a piezoviscous fluid), one cannot express the stress explicitly in terms of the velocity gradient, but one can express the velocity gradient as a non-linear function of the stress. Such fluids are special cases of the more general implicit fluid model and have been studied in detail in a variety of applications [see Dowson et al (1983); Tran and Suslov (2009);Saccomandi and Vergori (2010); Szeri (2011);Rajagopal et al (2012)]. …”