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

## TL;DR

This paper presents detailed numerical simulations of incompressible, pressure-dependent viscous flows in a plane slider bearing, highlighting the effects of various parameters and critiquing common viscosity cutoff practices.

## Contribution

It introduces a finite element approach to simulate piezoviscous fluids without lubrication approximation, revealing the impact of pressure-dependent viscosity on flow characteristics.

## Key findings

- Pressure differences across the fluid film due to piezoviscous effects.
- Artificial viscosity cut-offs lead to results dependent on arbitrary parameters.
- Simulation results depend on the variation limit of viscous stress with pressure.

## Abstract

We provide numerical simulations of an incompressible pressure-thickening and shear-thinning lubricant flowing in a plane slider bearing. We study the influence of several parameters, namely the ratio of the characteristic lengths $\varepsilon>0$ (with $\varepsilon\searrow0$ representing the Reynolds lubrication approximation); the coefficient of the exponential pressure--viscosity relation $\alpha^*\geq0$; the parameter $G^*\geq0$ related to the Carreau--Yasuda shear-thinning model and the modified Reynolds number $\mathrm{Re}_\varepsilon\geq0$. The finite element approximations to the steady isothermal flows are computed without resorting to the lubrication approximation. We obtain the numerical solutions as long as the variation of the viscous stress $\mathbf{S}=2\eta(p,\mathrm{tr}\mathbf{D}^2)\mathbf{D}$ with the pressure is limited, say $|\partial\mathbf{S}/\partial p|\leq1$. We show conclusively that the existing practice of avoiding the numerical difficulties by cutting the viscosity off for large pressures leads to results that depend sorely on the artificial cut-off parameter. We observe that the piezoviscous rheology generates pressure differences across the fluid film.