Nonlinear problem of a spherical suspension

Abstract

We consider in the nonlinear formulation the steady-state motion of an incompressible viscous fluid between two concentric spheres, into the gap between which fluid enters through one hole and leaves through a second. The holes are replaced by a source and sink, after which the boundary conditions are written in terms of the delta function. The delta function is expanded approximately in a finite series in Legendre polynomials. Depending on the number of terms, this series represents holes of various sizes. The solution to the problem is sought by expanding the desired function in a series in powers of the Reynolds number, whose coefficients are expanded in series in associated Legendre functions of the first kind. The velocity field and also the force acting on the inner sphere are found. Numerical computations are presented for holes whose aperture half-angle is 6°.