Abstract
The Reynolds equation from lubrication theory and the Stokes equations for
low Reynolds number flows are distinct models for an incompressible fluid with
negligible inertia. Here we investigate the sensitivity of the Reynolds
equation to large gradients in the surface geometry. We present an analytic
solution to the Reynolds equation in a piecewise-linear domain alongside a more
general finite difference solution. For the Stokes equations, we use a finite
difference solution for the biharmonic stream-velocity formulation. We compare
the fluid velocity, pressure, and resistance for various step bearing
geometries in the lubrication and Stokes limits. We find that the solutions to
the Reynolds equation do not capture flow separation resulting from large
cross-film pressure gradients. Flow separation and corner flow recirculation in
step bearings are explored further; we consider the effect of smoothing large
gradients in the surface geometry in order to recover limits under which the
lubrication and Stokes approximations converge.