On the lubrication paradox and the use of regularisation methods for lubrication flows

In this paper we complete the analysis of the flow of a Bingham fluid along a wavy-walled channel. We confirm numerically the results of [I. Frigaard, D. Ryan, Flow of a visco-plastic fluid in a channel of slowly varying width, J. Non-Newt. Fluid Mech. 123 (2004) 67–83], that there is a true rigid plug for a sufficiently small amplitude long wavelength perturbation of the uniform plane channel. For larger amplitude width perturbations the plug breaks and we provide a comparison between analytical estimates for the breaking amplitude and numerical computations using the augmented Lagrangian method. Plug breaking arises due to the creation of significant extensional stresses within the unyielded part of the fluid. When the plug breaks the extensional stresses govern the structure of the flow within the pseudo-plug. We provide a complete asymptotic solution for the flow in this situation, and compare against numerical computations.In the second part of the paper, we study the efficacity of viscosity regularisation methods for computing this type of flow. We show that accurate resolution of yield surface positions requires proper control of the numerical residuals and having a favourable ratio of regularisation error to critical strain rate. We show that failure to manage these errors means that regularisation methods can produce completely erroneous predictions of the yield surfaces, e.g. predicting broken plug regions when they are in fact intact. For general lubrication-type flows one does not have an estimate of these errors.