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.

翻译：润滑理论中的雷诺方程与低雷诺数流动的斯托克斯方程是描述惯性可忽略不可压缩流体的两种不同模型。本文研究了雷诺方程对表面几何形状大梯度的敏感性。我们给出了分段线性域中雷诺方程的解析解以及更一般的有限差分解。对于斯托克斯方程，我们采用双调和流函数-速度公式的有限差分解。我们比较了多种阶梯轴承几何构型在润滑极限和斯托克斯极限下的流体速度、压力和阻力。研究发现，雷诺方程的解无法捕捉由大跨膜压力梯度引起的流动分离现象。本文进一步探讨了阶梯轴承中的流动分离与角区回流；我们通过平滑表面几何形状的大梯度来考察润滑近似与斯托克斯近似趋于一致的条件极限。