Comparison of lubrication theory and Stokes flow models in step bearings with flow separation

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.
Comment: 20 pages, 17 figures