Simple flows of fluids with pressure–dependent viscosities

TLDR

Stokes’ constitutive relation allows viscosity to depend on pressure, a feature that is especially relevant for high‑pressure flows and lubricants. The study investigates unidirectional and two‑dimensional flows of fluids whose viscosity varies with pressure, a topic that has received limited attention. Numerical simulations were performed on two technologically significant two‑dimensional configurations: flow between rotating, coaxial, eccentric cylinders and flow across a slot. The results show that pressure‑dependent viscosity leads to markedly different flow characteristics, requiring special viscosity forms for Couette or Poiseuille flow and producing non‑unique solutions for flow between moving plates, with overall departures from classical Navier–Stokes predictions.

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.