An integral boundary layer equation for film flow over inclined wavy bottoms

TL;DR

This paper derives a new integral boundary layer equation for film flow over inclined wavy bottoms using a Galerkin method, capturing eddies and instabilities, and validates its accuracy through numerical results.

Contribution

It introduces a second order weighted residual integral boundary layer equation for wavy incline flows, including eddy modeling and instability analysis, which was not previously available.

Findings

Model accurately describes flow and eddies in wavy bottoms.

Identifies a short wave instability in laminar flow over wavy surfaces.

Numerical validation confirms model's wide-range applicability.

Abstract

We study the flow of an incompressible liquid film down a wavy incline. Applying a Galerkin method with only one ansatz function to the Navier-Stokes equations we derive a second order weighted residual integral boundary layer equation, which in particular may be used to describe eddies in the troughs of the wavy bottom. We present numerical results which show that our model is qualitatively and quantitatively accurate in wide ranges of parameters, and we use the model to study some new phenomena, for instance the occurrence of a short wave instability for laminar flows which does not exist over flat bottom.