Simple flows of fluids with pressure–dependent viscosities

Abstract

In his seminal paper on fluid motion, Stokes developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particularly well suited to modelling flows of many fluids at high pressures and is relevant to several flow situations involving lubricants. Fluid models in which the viscosity depends on the pressure have not received the attention that is due to them, and we consider unidirectional and two-dimensional flows of such fluids here. We note that solutions can have markedly different characteristics than the corresponding solutions for the classical Navier-Stokes fluid. It is shown that unidirectional flows corresponding to Couette or Poiseuille flow are possible only for special forms of the viscosity. Furthermore, we show that interesting non-unique solutions are possible for flow between moving plates, which has no counterpart in the classical Navier-Stokes theory. We also study, numerically, two two-dimensional flows that are technologically significant: that between rotating, coaxial, eccentric cylinders and a flow across a slot. The solutions are found to provide interesting departures from those for the classical Navier-Stokes fluid.

On Fully Developed Flows of Fluids with a Pressure Dependent Viscosity in a Pipe

“…Such multiplicity of solutions is not possible in the classical NavierStokes model. The analysis of Hron, Málek and Rajagopal [11] suggests that similar departure from the Navier-Stokes solution is possible in other flow situations, and we investigate such a possibility in this paper.…”

“…When γ = 2 and µ(p) is constant, the model reduces to the classical Navier-Stokes model (1.4). Recently Hron, Málek and Rajagopal [11] have studied the flow of a fluid modeled by (7) between two parallel plates as well as between two rotating cylinders. They have shown that the solution of the governing equation for the problem considered for such a fluid is significantly different from the solution for the Navier-Stokes equation.…”

Pressure-dependent surface viscosity and its surprising consequences in interfacial lubrication flows

“…While this geometry has been studied numerically for piezoviscous fluids [22], this example highlights the ease with which Newtonian results may be adapted to surfactant-laden interfaces with arbitrary η s ( ) relations. The 2D analog of the journal bearing (Fig.…”

“…3D studies of such "piezoviscous" liquids have used exponential and power-law η(p) relations, particularly in the contexts of polymer melt processing [20,21] and lubricating oils under exceedingly high pressures [22,23]. Such flows may depart from the classical Newtonian solutions [22,24], e.g., the extrusion flux of polymer melts may slow dramatically under high pressures [21].…”

Lubricant flow in thin-film elastohydrodynamic contact under extreme conditions

“…It seems that perspective approaches to study the phenomena are molecular dynamic simulation [11,12] and solution of NavierStokes full system depending whether the solution is beyond or within continuum approach. In the latter case, the same speed profile was obtained by solution of Navier-Stokes equations for fluid with pressure dependent viscosity and simple Couette flow [27,28].…”