Comparison of Extended Lubrication Theories for Stokes Flow

TL;DR

This work evaluates extended lubrication theories for low-Reynolds-number flows in thin-gap geometries by formulating a velocity-adjusted extension (VA-ELT) that enforces incompressibility and flux constraints. The authors compare VA-ELT, TG-ELT, perturbed lubrication theory (PLT), Reynolds equation, and full Stokes solutions across logistic-step and triangular-slider geometries, revealing that extended and perturbed models improve accuracy for small-to-moderate surface gradients but can over-correct at large gradients and fail to capture Stokes-like corner eddies. Key findings show VA-ELT generally enhances velocity accuracy and PLT can offer superior pressure accuracy for negative texturing, though model performance depends strongly on the height variation magnitude and the length-scale ratio $\varepsilon$. The results guide practitioners in selecting reduced-order lubrication models for microfluidic and tribological applications and clarify the breakdown regimes where full Stokes or more sophisticated treatments are required.

Abstract

Lubrication theory leverages the assumption of a long and thin fluid domain to formulate a linearized approximation to the Navier-Stokes equations. Models in extended lubrication theory consider relaxations of the thin film assumption, leading to the inclusion of higher order terms. However, such models are sensitive to large surface gradients which lead the assumptions of the model to break down. In this paper, we present a formulation of extended lubrication theory, and compare our models with several existing models, along with the numerical solution to the Stokes equations. The error in pressure and velocity is characterized for a variety of fluid domain geometries. Our results indicate that the new solution is suitable for a wide range of geometries. Nonetheless, the magnitude of surface variation and the length scale ratio are both important factors influencing the accuracy of the extended lubrication theory models.

Figures (8)

• Figure 1: Schematic of the logistic step.
• Figure 1: The Stokes solutions to the logistic step with $\lambda = 32$ (left) and the BFS (right). The logistic step at large $\lambda$ exhibits similar patterns of corner flow recirculation and $\frac{\partial p}{\partial y}$ as in the BFS.
• Figure 2: Pressure and velocity solutions for the logistic step with $\lambda = 8$. The ELT and $\varepsilon^k$-PLT solutions overestimate $\frac{\partial p}{\partial y}$ in the vicinity of the surface variation, leading to flow recirculation which is not observed in the Stokes solution at this moderate $\lambda$.
• Figure 3: Relative percent error in velocity $(u,v)$ and pressure $p(x,y)$ compared with the Stokes solution for the logistic step at varying slopes, $\lambda/4 = \max |\frac{dh}{dx}|$. For $2 < \lambda \le 6$, the VA-ELT solution has the smallest error in velocity and the $\varepsilon^2$-PLT solution has the smallest error in pressure.
• Figure 4: Schematic of the triangular slider