Correction factor of the Stokes force undergone by a sphere in the axis of a cylinder in uniform and Poiseuille flows

To contribute to the existing knowledge of the hydrodynamic force exerted on a spherical particle placed in the axis of a cylinder, at small Reynolds numbers, the influence of the uniform and Poiseuille flows on the wall correction factor are numerically and asymptotically investigated. The Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear coordinates. These equations are solved using a finite differences method. The generation of the grid was carried out by the singularities method. The accuracy of the numerical code is tested through comparison with theoretical and experimental results. In both cases we numerically calculated the separate contributions of the pressure and viscosity forces. In concentrated regime these numerical calculations are in very good agreement with those obtained by asymptotic expansions. This analysis allowed us to show the prevalence of the pressure term over the viscosity one in the lubrication regime contrary to what happened for the dilute regime. All our numerical and asymptotical results compared with those of Bungay et al. (Int. J. Multiphase Flow 1, 25-56 (1973)) seem to give a response to this problem argued for a long time.