Centered shear flow past a partially encapsulated two-dimensional droplet

Analytic solutions for the cylindrical multiphase droplet, consisting of two infinitely long circular cylinders with a contact angle π/2, suspended in a centered shear flow are obtained by the method of reflections. These exact solutions describe the flow fields in the continuous and dispersed phase liquid regions. The flow topologies are sketched for various values of the viscosity ratio and they display interesting features. In the continuous phase, back-flow regions are observed on either side of the droplet for several values of the viscosity ratio. These flow topologies appear to change significantly with respect to viscosity ratio. The streamline topologies in the dispersed phase show the existence of a pair of attached eddies. The bounded streamlines enclosing these eddies intersect on the x-axis yielding a hyperbolic fixed point and a homoclinic orbit—an indication of chaos. Furthermore, the asymptotic analysis leads to a rather surprising conclusion that there is a uniform flow far away from the droplet. The exact results presented here may be useful in validating numerical algorithms and codes on multiphase flow involving complex geometries.