Lubrication solution of the axisymmetric Poiseuille flow of a Bingham fluid with pressure-dependent rheological parameters

The lubrication flow of a Bingham plastic in long tubes is modeled using the approach proposed by Fusi and Farina (Appl. Math. Comp. 320, 1-15 (2018)). Both the plastic viscosity and the yield stress are assumed to vary linearly with the total pressure. The resulting nonlinear system of an ordinary differential equation and an algebraic one with unknowns the total pressure and the radius of the unyielded core are solved by using two different techniques. A pseudospectral numerical method utilizing Chebyshev orthogonal polynomials and an analytical perturbation method with the small parameter being the difference of the two dimensionless parameters which are introduced due to the pressure-dependence of the yield stress and the plastic viscosity of the material. The effects of the pressure-dependence of the material parameters on the critical pressure difference required for flow to occur and on the shape of the unyielded core are investigated and discussed.