Axisymmetric approach of a solid sphere toward a non-deformable planar slip interface in the normal stagnation flow––development of global rational approximations for resistance coefficients

In this paper a model was developed to describe the hydrodynamic force acting on the particle as it approaches a bubble with a mobile surface in an axisymmetric liquid flow. The particle size was considered to be sufficiently small relative to the bubble size that the bubble surface could be locally approximated to a planar interface. The model incorporated a bispherical coordinate system to derive a stream function for the liquid flow disturbed by the particle. The stream function was then used to calculate the hydrodynamic force acting on a particle of radius, R, as a function of the separation distance, H, from the bubble surface. The force equation was related to the modified Stokes equation to obtain an exact numerical solution for the correction factor, f2. Finally, simplified analytical rational approximations for the whole range of the separation distance are presented, which are in good agreement with the exact numerical result, and can be readily applied to more general mineral flotation applications.