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

Abstract

The boundary conditions prescribing the constant traction or the so-called do-nothing conditions are frequently taken on artificial boundaries in the numerical simulations of steady flow of incompressible fluids, despite the fact that they do not facilitate a well-posed problem, not allowing to establish the standard energy estimates. In a pursuit to understand better the possible consequences of using these conditions, we present a particular set of examples of flow problems, where we find none or two analytical or numerical solutions. Namely, we consider problems where the flow connects two such artificial boundaries. In the simple case of the isotropic radial flows where both steady and unsteady analytical solutions are derived easily, we demonstrate that while for some (large) boundary data all unsteady solutions blow up in finite time, for other data (including small or trivial) the unsteady flows either converge asymptotically to one of two steady solutions, or blow up in finite time, depending on the initial state. We then document the very same behavior of the numerical solutions for planar flow in a diverging channel. Finally, we provide an illustrative example of two steady numerical solutions to the flow in a three-dimensional bifurcating tube, where the inflow velocity is prescribed at the inlet, while the two outlets are treated by the do-nothing boundary condition.

A review of implicit algebraic constitutive relations for describing the response of nonlinear fluids

“…[71] studied the flow of such a fluid due to the rolling of a rigid cylinder on an elastic cylinder, a boundary value problem that has relevance to the flows in elastohydrodynamics. [72][73][74][75][76][77][78] and several others have studied specific boundary value problems which have relevance to several problems in elastohydrodynamic lubrication.…”

An adaptive finite element method for the inequality-constrained Reynolds equation

“…This has led to the development of alternative Reynolds-type cavitation models, see, e.g., Elrod-Adams [10], Vijayaraghavan-Keith [11], Bayada et al [12,13,14,15], Almqvist et al [16], Garcia et al [17], Mistry et al [18], for different models and their applications. It has also been recently shown, within the context of fluids with pressure dependent viscosity, by Lanzendörfer et al [19] that a solution based on using a cut-off value for the pressure to determine the region of cavitation is sensitive to the value of the cut-off. Nevertheless, it is obvious that any cavitation model derived from continuum mechanics involves an initially unknown region of cavitation and therefore it is mandatory to consider numerical methods that automatically capture the location of the cavitation region and adapt to the resulting free boundaries.…”