A spectral collocation scheme for the two dimensional flow of a regularized viscoplastic fluid: Numerical results and comparison with analytical solution

Abstract

In this paper we present a numerical scheme based on spectral collocation method (SCM) for the two dimensional incompressible creeping flow of a non-Newtonian fluid in a symmetric channel of variable width. Afters a suitable scaling of the governing equations and of the boundary conditions, we discretize the problem getting a nonlinear system that is solved via Newton–Raphson method. We focus on two regularized viscoplastic fluids (Bingham, Casson), but the method can be applied to any fluid in which the apparent viscosity depends on the modulus of the strain rate tensor. In the case of small aspect ratio of the channel we explicitly determine the lubrication solution at the leading order and we prove some symmetry properties. We validate the numerical scheme through a comparison with the analytical solution, showing an excellent agreement between the latter and the numerical solution. We consider three types of wall functions (convergent, divergent and non-monotone) and we perform numerical simulations for channels with general aspect ratio. We observe that the symmetries of the lubrication solution are maintained also when the characteristic length and width of the channel are of the same order. We have proved that the monotonicity of the yield surface follows the one of the channel profile when the aspect ratio of the channel is “sufficiently small”. This is no longer true when the latter hypothesis is removed.