Sliding of fine particles on the slip surface of rising gas bubbles: Resistance of liquid shear flows

In this paper a model was developed to describe the shear flow resistance force and torque acting on a fine particle as it slides on the slip surface of a rising gas bubble. The shear flow close to the bubble surface was predicted using a Taylor series and the numerical data obtained from the Navier–Stokes equations as a function of the polar coordinates at the bubble surface, the bubble Reynolds number, and the gas hold-up. 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 Stokes equation for the disturbance shear flows was solved for the velocity components and pressure using series of bispherical coordinates and the boundary conditions at the no-slip particle surface and the slip bubble surface. The solutions for the disturbance flows were then used to calculate the flow resistance force and torque on the particle as a function of the separation distance between the bubble and particle surfaces. The resistance functions were determined by dividing the actual force and torque by the corresponding (Stokes) force and torque in the bulk phase. Finally, numerical and simplified analytical rational approximate solutions for force correction factors for sliding particles as a function of 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 modelling of the bubble–particle interactions.