Macro-size drop encapsulation

Viscoplastic fluids do not flow unless they are sufficiently stressed. While in some flows this leads to unwanted features, this property can also be exploited in order to produce novel flow features. One example of such flows are visco-plastically lubricated (VPL) flows, in which a viscoplastic fluid is used to stabilize the interface in a multi-layer flow, far beyond what might be expected for a typical viscous-viscous interface. Here we extend this idea by considering the encapsulation of droplets within a viscoplastic fluid, for the purpose of transportation, e.g. in pipelines. The main advantage of this method, compared to others that involve capillary forces is that significantly larger droplets may be stably encapsulated, governed by the length scale of the flow and yield stress of the encapsulating fluid. We explore this setup both analytically and computationally. We show that sufficiently small droplets are held in the unyielded plug of the Poiseuille flow. As the length or radius of the droplets increase the carrier fluid eventually yields, potentially breaking the encapsulation. We study this process of breaking and give estimates for the limiting size of droplets that can be encapsulated.