Slender-body theory for particles of arbitrary cross-section in Stokes flow

Abstract

A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUε, where U is a measure of the undisturbed velocity and ε = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of ε2 are presented, together with iterative relations which allow their determination to the order of any power of ε. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of ε, and, for a cylindrical body, to the order of ε3.