Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops

The coalescence of two viscous liquid drops in an inviscid gas or in a vacuum is studied using the interface formation model. In the very early stages of coalescence during the formation of the 'liquid-bridge' connecting the two drops, this model predicts a moving contact line and a dynamic contact angle. This paper examines the dynamic evolution of this contact angle, and for small Reynolds number and small Capillary number, relevant particularly in micro-fluidics, a non-linear differential equation is derived for the contact angle and solved computationally. It is found that the contact angle evolution can only be evaluated by determining information about the flow away from the contact line. This is a manifestation of so-called hydrodynamic assist, studied experimentally in the context of curtain coating by Blake et al. (1999 Experimental evidence of non-local hydrodynamic influence on the dynamic contact angle. Phys. Fluids, 11, 1995-2007). For small Capillary number and small Reynolds number, the free-surface evolution is determined for the coalescence of two cylinders of equal radius. Finally, some comments are made on experiments in coalescence, as well as on issues arising in a computational solution of the full model described here.