Pressure driven lubrication flow of a Bingham fluid in a channel: A novel approach

•We investigate the flow of a Bingham fluid in a channel of non uniform width.•We model the unyielded part of the fluid using a global integral approach.•We determine an integro-differential for the pressure and solve it numerically.•We find constraints on the channel profile that predict the breaking of the plug.•We extend the model to the case with pressure-dependent viscosity. In this paper we present a novel approach for modelling the lubrication flow of a Bingham fluid in a channel whose amplitude is non uniform. 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 also extend our model to the case of a pressure-dependent viscosity.