Interaction between two rigid spheres moving in a micropolar fluid with slip surfaces

Abstract

The quasi-steady translating motion of two rigid spheres immersed in an incompressible micropolar fluid with slip surfaces is studied. The two rigid spheres are of distinct radii translating with different uniform velocities in a direction parallel to the line connecting their centers. A Basset type of slip conditions for microrotation at the interface between fluid and spheres are employed with the common slip conditions for velocity. For creeping motion, a regular general solution is formulated by superposing fundamental results in two systems of spherical coordinates located at spheres centers. Using a collocation scheme, the boundary conditions are enforced to be fulfilled at the boundary of the spheres. Calculations for the non-dimensional drag acting on each particle are found with fast convergence for various values of the parameters involved in this study. The problem of steady rotational motion of two rigid spheres about a common line that joins their center is also considered in this work. The slip boundary conditions on each sphere are also used. The same collocation scheme, as the translational problem, is employed to find the total torque experienced by the fluid on each particle. The precision of our collocation scheme has been proven with known results available in the literature.