Pressure Driven Lubrication Flow of a Bingham Fluid in a Channel: A Novel Approach

In this paper we present a novel approach for modelling the lubrication flow of a Bingham plastic in a channel with non uniform walls. The novelty consists in deriving the rigid plug equation using an integral approach based on Newton’s second law, where the unyielded part is treated as an evolving non material volume. Such an approach leads to an integro- differential equation for the pressure that can be solved with an iterative procedure. We prove that a true unyielded plug exists even when the maximum width variation is not “small” and we find constraints on the amplitude of the channel that prevent the plug from “breaking”. We show numerical results and comparisons with results obtained with different approaches. We also show how to extend our model in the case of a pressure-dependent viscosity.