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Buoyancy-driven coalescence of slightly deformable drops. (English) Zbl 0907.76019

The paper deals with the problem to predict collision efficiency of non-Brownian liquid drops creepingly sedimenting \((Re\ll 1)\) under gravity in another surrounding unbounded fluid of different density and slightly different dynamic viscosity. In the model, two spherical drops of different size are considered. Under the given conditions, the two drops are first approaching to another. It is further assumed that in the apparent contact of two slightly deformable drops short-range van der Waals attraction forces and reacting (disjoining) small-gap lubrication forces are simultaneously present, together with viscous forces. This situation is different from the case when the drops are assumed undeformable and van der Walls forces are taken into account or not.
The time-dependent problem is first decomposed into an outer solution (an overall force balance without van der Waals forces gives the joining force component along the line of drop centers and the force component normal to it), and an inner solution for the thin-film lubrication flow in the quasi-near contact. A matched asymptotic expansion is applied for obtaining the initial condition for solving the inner problem. The four dimensionless parameters entering into the governing equations are: the ratio of radii of two drops, the dynamic viscosity ratio of two fluids, the capillary number, and the Hamaker number as the ratio of the molecular attraction force to the lubrication force. The last two parameters are considered here much smaller than unity. Dependig on the values of these parameters, the two drops will coalesce or separate. To solve the problem numerically, a novel, stabile and accurate, matrix-iterative method has been developed and successfully applied. The presented method of solution can be viewed as a complementary one to the three-dimensional boundary integral calculation, efficient in different ranges of the governing parameters. Results for specific coalescence efficiency are presented for a given fluid pair.

MSC:

76D45 Capillarity (surface tension) for incompressible viscous fluids
76D08 Lubrication theory
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