On the particle inertia-free collision with a partially contaminated spherical bubble

The collision between a contaminated spherical bubble and fine particles in suspension is considered for rp/rb ≪ 1 (rp being the radius of the particles in suspension and rb the radius of the bubble). The collision probability or efficiency is defined as the number of particles colliding the bubble surface to the number of particles initially present in the volume swept out by the bubble. In this note we show that the collision probability can be expressed as Pc(rp/rb,Re) = g(rp/rb)f(Re) for both mobile and immobile interfaces. For partially contaminated bubbles a linear or quadratic dependency in rp/rb is found depending on the level of contamination and the value of rp/rb. These behaviors are given by the flux of particles near the surface which is controlled by the tangential velocity for mobile interfaces and by the velocity gradient for immobile interfaces. The threshold value (rp/rb)th between the rp/rb and (rp/rb)2 evolution is shown to vary as sinn(Re)(θclean/n(Re))sin(3θclean/4), θclean being the angle describing the front clean part of the bubble and n(Re) varying from n = 2 to n = 1 from small to large Reynolds number.