A new flow model for a thin viscous fluid film between two moving surfaces

TL;DR

This paper introduces a new two-dimensional flow model for viscous fluids between close moving surfaces, which simplifies computations by capturing the asymptotic behavior of Navier-Stokes equations without solving them directly.

Contribution

The novel model unifies lubrication and shallow water flow regimes, reducing computational complexity for thin viscous fluid flows between moving surfaces.

Findings

Model converges to Navier-Stokes solutions as gap narrows

Unified approach handles different boundary conditions

Reduces computational effort compared to Navier-Stokes simulations

Abstract

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