Motion of a viscous droplet bisecting a free surface of a semi-infinite micropolar fluid

The Stokesian flow of a spherical-shaped droplet which is halfway immersed in a semi-infinite phase of a micropolar fluid is discussed, the surface of which is assumed to remain flat. This configuration is studied analytically using the stream function formulation in two different settings, when the movement of the droplet perpendicular to the free flat surface of the micropolar fluid and the parallel motion. The interface conditions on the droplet boundary are that the velocity is continuous, the shear stress is continuous, and the microrotation is proportional to the vorticity. Analytical solutions for the stream functions outside and inside the droplet are obtained in each case of the droplet movement. The drag force acting, in each case, on the part of fluid sphere immersed in the micropolar fluid is evaluated. Numerical results for the drag force coefficient versus the relative viscosity, micropolarity parameter and spin parameter are presented both in tabular and graphical forms. The results for the drag coefficient are compared with the available solutions in the literature for the limiting cases.