Theory Of Asymptotic Expansions Research Articles

ON THE SLOW MOTION OF BODIES IN STRATIFIED AND ROTATING FLUIDS

The flow due to a body moving horizontally in a fluid with moderate vertical density gradient is found in the limit of small Reynolds number and zero diffusivity using the technique of matched asymptotic expansion theory. The solution yields an upstream wake-like behaviour, a thin layer on the rear of the object somewhat analogous to an Ekman layer, and small equatorial regions. There is no wake in this limit downstream of the body. The flow due to a body moved slowly along the axis of a fluid in strong rotation is also considered. The behaviour this time is similar upstream and downstream of the body due to the symmetry of the equations and the difference in the mechanism of propagation of the body force. A general solution is presented for ranges of the parameters analogous to the stratified case, and some general conclusions are drawn concerning the rotation of the body when allowed to move without torsional restraint. When a body is moved slowly along the axis of a rotating fluid, the results of Taylor (1) and Proudman (2) suggest, on the basis of linearized inviscid theory, that the body will carry along with it the fluid inside a cylinder circumscribing the body and having its generators parallel to the axis of rotation. Yih (3), using a model with analogous linearization, obtained similar results for flow with strong density stratification, indicating that a body moved horizontally should carry along the fluid between horizontal planes tangent to and containing the body. Similar effects have been observed in magnetohydrodynamic flows. These theories leave much to be desired in describing the flow, since the solutions have a high degree of arbitrariness. For the cases of rotation and magnetohydrodynamic flows, this arbitrariness has been resolved by consideration of the initial-value problem (Stewartson (4)) or inertia (Ludford (5)). In the present work we investigate how the consideration of viscous effects also eliminates this arbitrariness for the stratified and rotating cases.

Note on the asymptotic expansion of the modified Bessel function of the second kind

ura is usually the least term (in absolute value) of the series. Since the converging factor Rp/up may itself be developed as an asymptotic series, it is convenient to discuss this quantity rather than the remainder term. In a treatment due to Dingle a general theory of asymptotic expansions and their converging factors [2] is applied to the hypergeometric function [3], special cases of which are the modified Bessel functions of the first and second kinds. The formula relevant to the latter case is as follows