Tangential and normal partial slip at the liquid-fluid interfaces: An 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 liquid-liquid or gas-liquid 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. At the first time the generalization of the Hadamard-Rybczynski equation (HRE) is discussed within the formalism of two slip lengths. It is proven that each of the two fluids separated by an interface has its own slip length. The slip lengths of the fluids that contact each other through an interface are related by the ratio of the viscosities of these fluids. New equations describing the terminal velocity of gas bubble rise and aerosol falling have been obtained. The theoretical results are in good agreement with the experiments at moderate Knudsen numbers, Kn < 0.25. 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 - hydrocarbons, water - 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.