Mathematical Methods in the Physical Sciences · Problem 10.15

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · vortex-flow-circles

Problem

In Problem 14, the streamlines were the images of

v = \text{const.}

Consider the flow (over the whole plane, that is, with no boundaries) with streamlines

u = \text{const.}

This flow may be described as two vortices rotating in opposite directions. Sketch a number of streamlines indicating the direction of the velocity with arrows. Since a boundary is a streamline, a flow is not disturbed by inserting a boundary along a streamline. Insert two circular boundaries corresponding to

u = a

and

u = -a

. Show that the velocity through the narrow neck (say at

z = 0

) is greater than the velocity elsewhere (say at

z = i

). You can simplify your calculation of the velocity by showing that the result in Problem 8 holds here also.

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