Effect of insoluble surfactants on drainage and rupture of a film between drops interacting under a constant force.

The deformation, drainage, and rupture of an axisymmetrical film between colliding drops in the presence of insoluble surfactants under the influence of van der Waals forces is studied numerically at small capillary and Reynolds numbers and small surfactant concentrations. Constant-force collisions of Newtonian drops in another Newtonian fluid are considered. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier–Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection–diffusion equation, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). The mathematical problem is solved by a finite-difference method on a nonuniform mesh at the interfaces and a boundary-integral method in the drops. The whole range of the dispersed to continuous-phase viscosity ratios is investigated for a range of values of the dimensionless surfactant concentration, Peclét number, and dimensionless Hamaker constant (covering both nose and rim rupture). In the limit of the large Peclét number and the small dimensionless Hamaker constant (characteristic of drops in the millimeter size range) a fair approximation to the results is provided by a simple expression for the critical surfactant concentration, drainage being virtually uninfluenced by the surfactant for concentrations below the critical surfactant concentration and corresponding to that for immobile interfaces for concentrations above it

Drainage and rupture of partially mobile films during coalescence in liquid-liquid systems under a constant interaction force

A numerical description is developed of the process of film drainage and rupture during drop coalescence in liquid-liquid systems. The model is based on various simplifying approximations, corresponding to the regime in which interfacial mobility is controlled by the drop viscosity (partially mobile films). The constant force approach of a drop relative to another drop or to a free interface, taking into account the van der Waals attractive forces, is considered in this paper. The results are presented and compared with a simple analytical model proposed earlier for effective critical film-rupture thickness, which is found to provide a good first approximation

Velocity of long bubbles in oscillating vertical pipes

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