A triple emulsion droplet in an extensional flow

Abstract

In this theoretical report we analytically explore the deformation and breakup of an initially spherical triple emulsion drop undergoing an axisymmetric extensional creeping flow. The triple emulsion, suspended in an external fluid (fluid 1), is originally made from an inner spherical drop (fluid 4), engulfed by a spherical shell (fluid 3), which is subsequently engulfed by another spherical shell (fluid 2). The problem is described by eight dimensionless parameters: the external capillary number (Ca), three viscosity ratios (λ21, λ32, λ43), two radii ratios (K, k), and two surface tensions ratios (M, m). When the triple emulsion is subjected to an external uniaxial extensional flow (Ca > 0), the outer interface (21) and the inner drop interface (43) deform into prolate spheroids, while the interface between them (32) deforms into an oblate spheroid. The reverse occurs for an external biaxial flow (Ca < 0), the outer interface (21) and the inner drop interface (43) deform into oblate spheroids and the middle interface (32) becomes a prolate spheroid. Three types of breakup mechanisms are to be expected: when the outer and the middle interfaces are in contact, when the middle and the inner interfaces are in contact, and when the inner drop breaks while the two shell phases remain continuous. Finally, an analytical expression for the effective viscosity of a dilute emulsion containing spherical triple emulsion droplets is suggested. For a triple emulsion which behaves like a solid: a very viscous or a very thin outer shell (λ21→∞ or K→ 1), the effective viscosity reduces to Einstein’s formula, and when the triple emulsion behaves like a single drop: a very thick outer shell such that the inner shell and the inner drop disappear (K→ 0), Taylor’s expression is recovered.