Image Treatment of a 2D Vapor-Liquid Compound Droplet in a Linearized Steady Viscous Flow

Abstract

The classical method of images is used to construct closed form exact solutions for the two-dimensional (2D) perturbed flow fields in the presence of a 2D vapor-liquid compound droplet in the limit of low-Reynolds number. The geometry of the multiphase droplet is composed of two overlapping infinitely long cylinders Ca and Cb of radii a and b, respectively, intersecting at a vertex angle π∕2. The composite inclusion has the shape resembling a 2D snowman type of object with a vapor cylinder Ca partly protruded into the cylinder Cb filled with another fluid whose viscosity is different from that of the host fluid. The mathematical problem with this inclusion in the Stokes flow environment is formulated in terms of Stokes stream function with mixed boundary conditions at the boundary of the hybrid droplet. General expressions for the perturbed stream functions in the two phases are obtained in a straightforward fashion using Kelvin’s inversion together with shift and reflection properties of biharmonic functions. Application of our method to other related problems in creeping flow and possible further generalizations are also discussed. The general results are then exploited to derive singularity solutions for the hybrid droplet embedded in (i) a centered shear flow, (ii) a quadratic potential flow, and (iii) an extensional flow past the 2D vapor-liquid compound droplet. The image singularities in each case depend on the two radii of the cylinders, the center-to-center distance, and the viscosity ratio. The exact solutions are utilized to plot the flow streamlines and they show some interesting patterns. While the flow fields exterior to the droplet exhibit symmetrical topological structures, the interior flow fields show existence of free eddies—enclosed in a figure-eight separatrix—and stagnation points (hyperbolic points). The flow characteristics are influenced by the viscosity and radii ratios. Furthermore, the asymptotic analysis leads to a rather surprising conclusion that there is a (subdominant) uniform flow far away from the droplet in all cases. The existence of an origin, the natural center of the drop of the composite geometry, which neutralizes the uniform flow for a particular choice of the physical parameters, is illustrated. This reveals the sensitivity of the geometry in 2D Stokes flow. The present results may be of some interest in models involving a combination of stick and slip boundaries. Moreover, the method discussed here can be useful both as a teaching tool and as a building block for further calculations.