The slow translation of a sphere in a rotating viscous fluid

Abstract

The motion of a sphere along the axis of rotation of an incompressible viscous fluid that is rotating as a solid mass is investigated by means of numerical methods for small values of the Reynolds and Taylor numbers. The Navier–Stokes equations governing the steady axisymmetric flow can be written as three coupled, nonlinear, elliptic partial differential equations for the stream function, vorticity and rotational velocity component. Two numerical methods are employed to solve these equations. The first is the method of series truncation in which the dependent variables are expressed as series of orthogonal Gegenbauer functions and the equations of motion are then reduced to three coupled sets of ordinary differential equations, which are integrated numerically subject to their boundary conditions. In the second method, specialized finite–difference techniques of solution are applied to the two-dimensional partial differential equations. These techniques employ finite-difference equations with coefficients that depend upon the exponential function; a particularly suitable form of approximation for use in calculating numerical solutions is obtained by expanding the exponential coefficients in powers of their exponents.Calculated results obtained by the two methods are in good agreement with each other. The calculations have been carried out according to theoretical assumptions that simulate the experiments of Maxworthy (1965) in which the sphere experiences no resultant torque exerted by the surrounding fluid and is free to rotate with constant angular velocity. Numerical estimates of this angular velocity and of the drag exerted by the fluid on the sphere are found to agree well with the experimental results for Reynolds and Taylor numbers in the range from zero to unity. The results for small values of the Reynolds number are also consistent with theoretical work of Childress (1963, 1964) which is valid as the Reynolds number tends to zero.