A fast numerical scheme for the Poiseuille flow in a concentric annulus

Abstract

A fast numerical scheme is proposed to determine the velocity field of an incompressible fluid in a concentric annulus under a constant pressure gradient. The idea behind the scheme is to find the radius R in the annulus where the shear stress becomes zero. In the region from the inner wall at R=κ to R, the shear rate is positive, while it is negative from this radius to the outer wall at r=1. Integrating the velocity field from the inner wall, where it is zero, one determines its value at R. This acts as the initial value for the integration of the shear rate over the second region, where the velocity must decrease to zero at the outer wall. Choosing a value for R, iterations continue to find its optimal value till the velocity on the outer wall vanishes to within an acceptable error term, which is 10−10 here. The numerical method chosen here delivers this result within 5 to 10 iterations for generalised Newtonian and PTT fluids. For viscoplastic fluids, instead of finding the optimal value for R which lies within the plug, one has to find its counterpart r1. Noting that the shear stress equals the yield stress at r1 and that the shear rate is positive from κ to r2 one finds the velocity at this radius. Since the width of the plug r2−r1 within the fluid is known and the velocity is constant across the plug, one integrates the velocity field from r2 to the outer wall at r=1 till the velocity approaches zero to within the chosen error term. Once again, the number of iterations to find the velocity field in Bingham and Herschel-Bulkley fluids is small and lies between 5 and 8. Finally, application of the numerical scheme to determine the velocity field in helical flows is also suggested.