The collision efficiency of two slightly deformable drops in thermocapillary motion at small Reynolds and Marangoni numbers is determined by a trajectory analysis involving methodology from matched asymptotic expansions. The outer solution for two spherical drops which are nearly touching provides the contact force driving the inner solution. Accurate calculation of the contact force and near-contact motion is aided by new solutions for the mobility functions parallel and normal to the drops' line of centers that are valid at very small separations. Governed by a system of integro-differential equations coupling the flow inside the drops and that within the small gap, the inner solution allows demarcation of the regions of drop coalescence and separation. Apart from the driving force, the thin-film equations are unchanged to leading order from the buoyancy-driven case, since no additional singularity is introduced into the tangential stress by the presence of the finite temperature gradient. The interplay of small deformation, as measured by the capillary number (Ca), and attractive van der Waals forces controls the apparent contact motion. Results for the collision efficiency are mapped out for a range of five dimensionless parameters: Ca, size ratio, drop-to-medium viscosity ratio, drop-to-medium thermal conductivity ratio (k), and a dimensionless Hamaker parameter. Since the only effect on the inner solution of an increase in the thermal conductivity ratio is an increase in the amount of time the drops spend in close approach, it is possible for the collision efficiency of two slightly deformable drops with higher k to be greater than that for two similar drops with lower k. This behavior differs from that of spherical drops, where an increase in thermal conductivity ratio always leads to a decrease in the collision efficiency, as a result of greater hydrodynamic interaction between the spherical drops due to the temperature gradient. In addition, collision efficiencies are provided for a model system of ethyl salicylate (ES) drops in diethylene glycol (DEG). The collision efficiency decreases rapidly with increasing drop size above a critical value, due to the increasing role of deformation in retarding the drainage of the thin film between two drops in close approach. Population dynamics simulations are performed for homogeneous suspensions of the ES/DEG system, showing that slight deformation limits droplet growth due to coalescence in dilute dispersions.
Bibliographical noteFunding Information:
This work was supported by a NASA Graduate Student Researchers Program fellowship and by NASA Grant NAG3-2116. We are also grateful to Alexander Z. Zinchenko for help with the bispherical and multipole solutions for the mobility functions LM and MM, for useful discussions and for providing several source codes. In addition, we thank the Minnesota Supercomputing Institute for the use of their facilities.
- Marangoni motion
- Thermocapillary motion