Collision of drops with inertia effects in strongly sheared linear flow fields
Résumé
The relative motion of drops in shear flows is responsible for collisions leading to the creation of larger drops. The collision of liquid drops in a gas is considered here. The drops are small enough for the Reynolds number to be low (negligible fluid motion inertia), yet large enough for the Stokes number to be possibly of order unity (non-negligible inertia in the motion of drops). Possible concurrent effects of Van der Waals attractive forces and drop inertia are taken into account. General expressions are first presented for the drag forces on two interacting drops of different sizes embedded in a general linear flow field. These expressions are obtained by superposition of solutions for the translation of drops and for steady drops in elementary linear flow fields (simple shear flows, pure straining motions). Earlier solutions adapted to the case of inertialess drops (by Zinchenko, Davis and coworkers) are completed here by the solution for a simple shear flow along the line of centres of the drops. A solution of this problem in bipolar coordinates is provided; it is consistent with another solution obtained as a superposition of other elementary flow fields. The collision efficiency of drops is calculated neglecting gravity effects, that is for strongly sheared linear flow fields. Results are presented for the cases of a simple linear shear flow and an axisymmetric pure straining motion. As expected, the collision efficiency increases with the Stokes numbers, that is with drop inertia. On the other hand, the collision efficiency in a simple shear flow becomes negligible below some value of the ratio of radii, regardless of drop inertia. The value of this threshold increases with decreasing Van der Waals forces. The concurrence between drop inertia and attractive van der Waals forces results in various anisotropic shapes of the collision cross-section. By comparison, results for the collision efficiency in an axisymmetric pure straining motion are more regular. This flow field induces axisymmetric sections of collision and strong inertial effects resulting in collision efficiencies larger than unity. Effects of van der Waals forces only appear when one of the drops has a very low Stokes number.