Tangential and normal partial slip at the liquid-fluid interfaces: application to a small liquid droplet, gas bubble, and aerosol

Abstract

An analytical solution is obtained for the problem of the slow movement of a small drop of a fluid in another immiscible fluid in an infinitely large reservoir with the boundary condition of the normal slip and/or tangential partial slip at the interface. That generalizes the conventional Navier and Maxwellian boundary conditions of partial slip. Normal slip is accompanied by the density gradient in the fluid and is applicable only if one of the phases in contact at the interface is a gas. Although tangential partial slip and the associated generalization of the Hadamard-Rybczynski equation (HRE) have been considered previously, they were done using the friction coefficient formalism. Here, this issue is discussed within the more general formalism of slip lengths. It is proven that each of the two fluids separated by an interface has its own slip length. New equations describing the terminal velocity of gas bubble rise and aerosol falling have been obtained. The result is compared with experiment. It has been shown that the gas density within a rising bubble and around a falling droplet in the air is not uniform. The relative magnitude of the density increment increases with the size of the bubble or aerosol. Presumably, the best applicability of the generalized HRE should be expected for the interface of hydrophobic liquid and hydrophilic one (water and hydrocarbons, water and higher alcohols, in general: aqueous emulsions, water, lipophilic organic liquids and oils, etc.). These are quite important emulsions in practical terms, for example, for the oil industry and medicine. Experimental methods for determining the slip length are considered.