Comparison of Lubrication Theory and Stokes Flow in Corner Geometries with Flow Separation

Abstract

The Reynolds equation from lubrication theory and the Stokes equations for zero Reynolds number flows are distinct models for an incompressible fluid with negligible inertia. Here we investigate the sensitivity of the Reynolds equation to large surface gradients, and explore flow recirculation in corner geometries in comparison to the Stokes equation. We compare the solutions for the Reynolds and Stokes equations in the backward facing step (BFS), the regularized BFS, and the lid-driven triangular cavity. For the BFS variations listed above, we compute the error in terms of the average pressure drop through the channel and show how the error increases with increasing expansion ratio and with increasing magnitude of surface gradients. We further investigate the phenomenology of corner flow recirculation that arises in the Stokes solutions. In particular, we observe that occluding the corner separated region in the Stokes solution to the BFS does not disrupt the bulk flow characteristics.

Figures (21)

• Figure 1: Schematic of the backward facing step with expansion ratio $\mathcal{H}=H_{\text{in}}/H_{\text{out}}$.
• Figure 1: Convergence of the numerical Stokes solution $\psi$ for the BFS and the triangular cavity. The BFS solution converges with order $\mathcal{O}(\Delta x^2)$ and the triangular cavity converges with order between $\mathcal{O}(\Delta x^2)$ and $\mathcal{O}(\Delta x)$
• Figure 2: Pressure contours for the BFS. Curved contours in the Stokes solutions indicate significant pressure gradients at the step. For a particular $\mathcal{H}$, $\Delta p$ is smaller for the Reynolds solution versus the Stokes solution.
• Figure 3: Pressure contours for the BFS surrounding the step tip, Stokes solutions. The maximum pressure in the domain occurs at the step tip.
• Figure 4: Relative percent error between the Stokes and Reynolds solution to the BFS. Error in the average pressure drop $\Delta p$, in the $l_2$ norm of pressure, and in the $l_2$ norm of velocity, increases with increasing expansion ratio $\mathcal{H}$.