ON THE SLOW MOTION OF AN INTERFACIAL VISCOUS DROPLET IN A THIN LIQUID LAYER

In order to investigate the influence of the surface viscosity on the type of the how inside and outside a droplet moving in a thin liquid layer, it is essential to compute all hydrodynamical parameters which are important for a better understanding of the hydrodynamical interaction of the thin liquid film and the droplet in it. In the present paper, the problem of a translational slow motion of a droplet with a viscous interface in a liquid layer bounded by viscous liquid-gas interfaces is considered. For low Reynolds and capillary numbers, different values of droplet and film bulk viscosity ratios and surface dilatational and shear viscosities are used in the frame of Newtonian surface rheology. The problem reduces to two dimensions when using the ''two vorticities-one velocity'' formulation of basic how equations. The model equations and boundary conditions, which contain second-order derivatives of the velocity and the vorticity, are solved numerically to provide information on type of how, pressure distribution and drag coefficient. The numerical results reveal the strong influence of the surface viscosity on the motion of the droplet in the viscous liquid layer when the radius of the droplet is of the same order of magnitude as the thickness of the liquid film. The presence of the viscous liquid-gas interface close to the droplet changes the flow pattern inside the droplet considerably when the droplet bulk viscosity is sufficiently higher than the viscosity of the film.