Further remarks on simple flows of fluids with pressure-dependent viscosities

Abstract

Suslov and Tran [3] recently revisited the study carried out by Hron et al. [1] and they on the basis of their analysis claim that some of conclusions concerning one specific example, amongst the many considered by Hron et al. We have reexamined both papers, and we find that whether or not velocity profiles with inflection points exist depends on the class of functions to which the pressure belongs. If the pressure field is allowed to be discontinuous, which is in keeping with the class of functions to which pressure belongs to in the study of Hron et al.[1], such inflectional profiles are possible. However, if one requires the pressure field to be continuous then as Suslov and Tran [3] claim, such inflectional profiles are not possible. We provide a detailed explanation for this phenomenon that goes beyond the discussion presented in the paper by Suslov and Tran [3], and concerns subtle mathematical issues. Among other results we show that the solution with the inflectional profile is-interestingly-not a weak solution of the governing equations.Concerning the non-uniqueness of the solution, we show that if we explicitly-instead of assuming that constants are fixed by an unknown procedure-specify a procedure for fixing all the integration constants in the solution, for example by fixing the pressure at two points or fixing the pressure gradient and the pressure at one point, we get a unique solution to the problem, provided all relevant quantities are continuous. On the other hand, if we relax the assumption on continuity, we can get multiple solutions.

Numerical simulations of an incompressible piezoviscous fluid flowing in a plane slider bearing

“…This was well documented for the Couette and Poiseuille plane flows, see e.g. [6,23,24]. In fact, for the exponential pressure-viscosity relation η = e α p no such unidirectional flow can be found (except, interestingly enough, the case with a cross-flow pressure gradient due to the gravitational force, see [32]).…”

“…This desired behaviour has been guaranteed rigorously in [22] after making additional requirements which, however, do not cover realistic viscosity (15) at large pressures. The result in [22] stems from intensive research devoted to the notion and existence of a weak solution for incompressible fluids with pressureand shear rate-dependent viscosity, see [11,12,16,28] (see also [13,24,25,37] and the references therein). One of the assumptions embodied in the theoretical framework requires in particular that for certain constant C, see the concerned results for details 5 .…”

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].…”

On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient

“…Suslov and Tran [40] are indeed correct if one were to restrict oneself to determining classical solutions to the problem wherein the pressure field is assumed to be continuous. However, as pointed out later by Hron et al [39], if one allows for discontinuities in the pressure field, then the a solution such as that obtained by Hron et al [39] is admissible (see Hron et al [41] for a detailed discussion of the nature of various solutions to the problem). However, Suslov and Tran [40] are totally justified in their claim as Hron et al [39] did not have any discussion whatsoever about the class of solutions that were being sought for the problem, classical solution, weak solution, etc.…”