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# Potential and stream function of a vortex disk in the presence of a rigid sphere†

## Extract

In Sadowsky and Sternberg(1), elliptic integral representations of axially symmetric flows suggested essentially by Weinstein's work (2) on axially symmetric flows of an ideal incompressible fluid have been considered. The physical and practical significance of their investigation was twofold. First, a more transparent analytic description was obtained than that afforded by the representation through discontinuous integrals of Bessel functions originally used by Weinstein. Secondly, by superposition of such axially symmetric flows and appropriately chosen uniform streams, a variety of technically significant flows around solids, annular bodies and half-bodies of revolution can be constructed. On the other hand, in an attempt to utilize the symmetry properties of the potential field in reference to a spherical boundary, Weiss(3) has derived an (exterior) ‘sphere theorem’ by applying Kelvin's transformation and using the potential function. Butler (4) later obtained, by means of the Stokes stream function, a sphere theorem applicable to axially symmetric flows only. Ludford, Martinek and Yeh(5) found then the ‘interior sphere theorem’ as well as a theorem satisfying the general radiation condition. A general sphere theorem was consequently conceived, valid for all linear boundary conditions, and was recently published by Yeh, Martinek and Ludford (6). The significance of these theorems again lies in their application to physical problems. They often give closed form expressions of the disturbance potential in terms of higher transcendental functions whenever the undisturbed potential is given by means of transcendental functions. Furthermore, when the singularities (discrete or distributed) he near the perturbing spherical boundary the usual treatment by expansion in spherical harmonics leads to solutions in the form of infinite series which are, because of slow convergency, unsuited for numerical computation. For this situation the sphere theorems provide a remedy in the form of neat formulae readily adaptable to numerical work.

## References

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(1)Sadowsky, M. A. and Sternberg, E.Elliptic integral representation of axially symmetric flows. Quart. Appl. Math. 8 (1950), 113.
(2)Weinstein, A.On axially symmetric flows. Quart. Appl. Math. 4 (1948), 429–44.
(3)Weiss, P.On hydrodynamical images, arbitrary irrotational flow disturbed by a sphere. Proc. Camb. Phil. Soc. 40 (1944), 259–61.
(4)Butler, S. F. J.A note on Stokes's stream function for motion with a spherical boundary. Proc. Camb. Phil. Soc. 49 (1953), 169–74.
(5)Ludford, G. S. S., Martinek, J. and Yeh, G. C. K.The sphere theorem in potential theory. Proc. Camb. Phil. Soc. 51 (1955), 389–93.
(6)Yeh, G. C. K., Martinek, J. and Ludford, G. S. S.A general sphere theorem for hydro-dynamics, heat, magnetism and electrostatics. Z. Angew. Math. Mech. 36 (1956), 111–16.
(7)Yeh, G. C. K., Martinek, J. and Ludford, G. S. S.The potential due to certain singularities in the presence of a fixed sphere. J. Soc. Industr. Appl. Math. 3 (1955), 142–52.
(8)Lighthill, M. J.The image system of a vortex element in a rigid sphere. Proc. Camb. Phil. Soc. 52 (1956), 317–21.

# Potential and stream function of a vortex disk in the presence of a rigid sphere†

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