On axisymmetric creeping flows involving a hybrid droplet

The Stokes flow involving a hybrid droplet submerged in an immiscible liquid is considered. The droplet has a two-sphere geometry with the two spherical surfaces intersecting orthogonally. The types of hybrid droplet considered here include (i) vapor–solid; (ii) vapor–vapor; and (iii) vapor–liquid, with the first two being limiting cases of (iii). It is assumed that the surface tension is sufficiently large so that the interfaces have uniform curvature. In a constructive and simple manner, steady state solutions of Stokes equations involving such configurations are constructed. The theory leading to the general expressions for the flow fields under the above-mentioned limitations exploit the inverse transformation, and the reflection and translation properties of the axisymmetric biharmonic function associated with the Stokes flow. These are cast in the form of a theorem followed by a simple proof. The theorem is then applied to construct closed form singularity solutions for several axisymmetric flow fields disturbed by a hybrid droplet. The different primary flow fields considered here include paraboloidal flow, flow due to a, single stokeslet and a pair of stokelets, and flow due to a potential-dipole. The salient features of the image singularities are discussed in each case. In all cases, the drag force is found to vary significantly with respect to the two radii associated with the two-sphere geometry of the droplet, and the viscosity ratio of the two liquids in the continuous and dispersed phases. In the case of singularity driven flows, the drag is influenced by an additional parameter namely, the location of the singularity. The flow streamlines in some cases show interesting flow patterns. In the case of paraboloidal flows, either a single eddy or a pair of eddies is observed depending on the ratio of the two radii. The sizes and shapes of these eddies vary monotonically with viscosity ratio. For flows due to a single and a pair of stokeslets with the same strength, no eddy is noticed. However, a toroidal eddy appears in the case of a pair of stokelets with opposite strengths. The locations of the stokeslets and the viscosity ratio influence the eddy structure significantly