Motion of an infinite elliptic cylinder in fluids with constant vorticity

Abstract

An extension of Kirchoff’s theory of the motion of solid bodies in irrotationally moving liquids to the case of motion in liquids in which a vorticity is present does not exist. Only a few isolated cases of such motion are known. Bearing on the consideration of this paper, there is an important work by Taylor which expresses the additional pressure effect on a system of cylinders moving in a perfect liquid without rotation when the whole system is rotated uniformly about an axis. Taylor’s theory reduces the problem of such motion to one of irrotational motion. In the present paper the motion of a perfect liquid having constant vorticity, and in which a cylinder of any cross-section is moving in any manner, has been considered. The pressure integral can be obtained in a simple form, referred to axes fixed in the body, which is very suitable for calculation. It is shown, whenever the pure potential motion of the liquid for the rotation of the cylinder and the solution of a definite potential problem or the corresponding Green’s function can be found, the formula can be applied to calculate the motion of the cylinder in liquids with constant vorticity. Two important cases of constant vorticity are uniform shear motion along a direction and uniform rotation about an axis. In the present paper the former case is considered in detail for an elliptic cylinder. The case of uniform rotation being covered by Taylor’s result it is only verified that the present method gives the same result as Taylor’s formulae. There are some simple free motions of an elliptic cylinder in a liquid with uniform shear motion which have been discussed in the paper. 2—Equations of Motion Referred to Axes Fixed in the Body and the Pressure Integral It is first necessary to write down the equations of motion referred to a system of axes fixed in the body having both translation and rotation. These equations are obtained below following a method of Taylor.