Peristaltic axisymmetric flow of a Bingham fluid

We model the peristaltic flow of a Bingham fluid in a tube in lubrication approximation. Following the procedure developed in Fusi et al. (2015a) we derive the rigid plug equation using an integral formulation for the balance of linear momentum, modelling the unyielded domain as an evolving non-material volume. The mathematical problem is formulated for the yielded and unyielded part and appropriate boundary conditions are established at the pipe walls and at the yield surface. The zero order approximation leads to a system formed by an integral equation and an algebraic equation for the yield surface and for the plug velocity (which is uniform in space), respectively. Because of the integral approach adopted in the unyielded part of the flow, the leading order approximation does not give rise to the lubrication paradox. The problem is solved numerically and an analytical solution is found when the oscillating wall is given as a small perturbation of the uniform wall.