  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.
