Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

Abstract

We examine stability of fully developed isothermal unidirectional plane PoiseuilleCouette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in [1,2]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.

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)]. …”

Unsteady flows of fluids with pressure dependent viscosity in unbounded domains

“…[21] investigate the possibility to find explicit solutions for flows between two infinite parallel plates with no-slip boundary conditions and for the viscosities of two types: (i) ν(p) = exp(αp) and (ii) ν(p, |D| 2 ) = α p |D| r−2 for r ∈ (1, 2 . (See also relevant study [47] where however some imprecise statements are made, and [50] focused on some stability issues.) Vasudevaiah and Rajagopal [53] considered the fully developed flow in a pipe dealing with a fluid that has a viscosity that depends on the pressure and shear rate and were able to obtain explicit exact solutions for the problem.…”