Numerical and asymptotic solutions of axisymmetric Poiseuille flows of yield stress fluids with pressure-dependent rheological parameters

Abstract

The lubrication flow of a Bingham plastic in long tubes of varying radius 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. Under the lubrication approximation a final set of two highly non-linear ordinary differential equations with unknowns the total pressure and the shape (radius) of the yield surface 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 consistency index of the fluid. In the former, ten spectral coefficients are adequate to fully resolve the pressure and yield-surface profiles down to machine accuracy. In the latter, three terms in the perturbation expansions are found analytically, and then are suitably processed using techniques which accelerate the convergence of series. The agreement between the two techniques is excellent. The implications of the pressure-dependence of the material parameters and the applicability windows of the method are also discussed.