Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies

Abstract

The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlehs, by which the rotational motion of a given body can be represented.Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient CM = M/(8πμω0ab2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity ω0 about its axis in a fluid of viscosity μ) of various body shapes so far investigated is found to lie between $\frac{2}{3}$ and 1, usually very near unity for not extremely slender bodies.For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.