Stream function, flow separation and force equation for stagnation flow passing a small solid sphere touching a rising gas bubble

This paper considers an axisymmetric stagnation flow past a small solid sphere touching an air bubble, which is significantly larger than the particle but smaller than the capillary length so that the deformation can be neglected. The disturbed flow due to the presence of the particle at the bubble surface was modelled by considering an axisymmetric stagnation point flow about a sphere and a plane. The stream function for the disturbed flow was derived based on the tangent-sphere coordinates. It is shown that the flow separation having an infinite set of nested ring vortices exists in a finite region enclosing the contact point. The presence of such vortices might have considerable influence on the bubble-particle attachment as well as on the transport of surfactants during the bubble-particle interaction. The force equation was obtained from the derived stream function and was equated to the modified Stokes drag force equation, and then an expression for the correction factor was obtained. The model for the correction factor includes the effect of particle size, and predicts finite force values at zero separation distance between the particle and bubble surfaces. The models presented in this paper should provide a better estimate for calculating the normal (stagnation flow) fluid force acting on particles in contact with bubbles in the multiphase flows found in mineral flotation and other metallurgical operations