A new thin layer model for viscous flow between two nearby non-static surfaces

TL;DR

This paper introduces a new two-dimensional thin layer model for viscous flow between close moving surfaces, which asymptotically aligns with Navier-Stokes solutions and simplifies computations without pre-classifying flow types.

Contribution

The paper develops a unified thin layer model that captures viscous flow between moving surfaces, bridging lubrication and thin fluid layer regimes, reducing computational complexity.

Findings

Model asymptotically matches Navier-Stokes solutions as gap closes.

Limit behavior depends on boundary conditions, leading to lubrication or thin layer models.

Provides a computationally efficient alternative to full Navier-Stokes simulations.

Abstract

We propose a two-dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier-Stokes equations. The leading term of the formal asymptotic expansions of the solutions to the new model and Navier-Stokes equations are solution of the same limit problem, and the type of the limit problem depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is a solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is a solution of a thin fluid layer model. The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two nearby moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication or a thin fluid layer problem, and without the enormous computational effort that would be required to solve the Navier-Stokes equations in such a thin domain.