Compound droplet in extensional and paraboloidal flows

Exact analytical solutions are found for the steady state creeping flow in and around a vapor-liquid compound droplet, consisting of two orthogonally intersecting spheres of arbitrary radii (a and b), submerged in axisymmetri c extensional and paraboloidal flows of fluid with viscosity mu((1)). The s olutions are presented in singularity form with the images located at three points: the two centers of the spheres and their common inverse point. The important results of physical interest such as drag force and stresslet co efficient are derived and discussed. These flow properties are characterize d by two parameters, namely the dimensionless viscosity parameter: Lambda=m u((2))/(mu((1))+mu((2))), and the dimensionless parameter: beta=b/a, where mu((2)) is the viscosity of the liquid in the sphere (part of the compound droplet) with radius b. We find that for some extensional flows, there exis ts a critical value of beta=beta(c) for each choice of Lambda in the interv al 0 less than or equal to Lambda less than or equal to 1 such that the dra g force is negative, zero or positive depending on whether beta <beta(c), b eta=beta(c), or beta >beta(c) respectively. For other extensional flows, th e drag force is always positive. The realization of these various extension al flows by simply changing the choice of the origin in our description of the undisturbed flow field is also discussed. In extensional flows where th e drag force is always positive, we notice that this drag force D-e for vap or-liquid compound droplet is maximum when beta approximate to 1 (i.e., two spheres have almost the same radii). Moreover, we find the drag force D-e is a monotonic function of Lambda, i.e., the drag force for vapor-liquid co mpound droplet lies between vapor-vapor and vapor-rigid assembly limits. We also find that the maximum value of the drag in paraboloidal flow depends on the viscosity ratio Lambda and significantly on the liquid volume in the dispersed phase. (C) 2000 American Institute of Physics. [S1070-6631(00)01 310-6].