A triple emulsion droplet in a uniform flow

Abstract

The fluid mechanics problem of a triple emulsion droplet in a uniform creeping flow is theoretically studied. The triple emulsion, submerged in an external fluid 1, is composed of an inner spherical drop (fluid 4), covered by a spherical shell (fluid 3), which is subsequently covered by another spherical shell (fluid 2). In the absence of a body force, eight parameters are governed: the Capillary number (Ca), three viscosity ratios (λ21, λ32, λ43), two radii ratios (K, k), and two-surface tension ratios (M, m). If all surfaces remain spherical and concentric, five parameters (λ21, λ32, λ43, K, k) are sufficient. Expressions for the stream functions and the drag forces were obtained to calculate the terminal velocity of the triple emulsion falling or rising in a stagnant fluid under the influence of gravity. With gravity, four additional parameters are needed: the ratio of the buoyancy force to the external viscous force (B), and three density ratios (γ21, γ31, γ41). To remain concentric, all surfaces must have the same velocity (magnitude and direction). This results in two restricted parameters, but even then, not every combination of parameters leads to a physical solution. Our findings suggest that the terminal velocity of a triple emulsion is equal to or lower than that of a double emulsion, which is subsequently equal to or lower than that of a single drop. Similar to single and double emulsion droplets, no deformation takes place in a concentric spherical triple emulsion subjected to a gravitational field under creeping flow conditions.