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Unformatted text preview: l liquid and spinning at the
same speed as shown in the following diagram: 45 –1 1 2 4 (a) Find a mapping from the above diagram into an annulus, and then map the annulus
into a vertical strip using the logarithm.
(b) Invert both maps to obtain a mapping from the vertical strip onto the given region.
(c) Now obtain the streamlines of the resulting flow.
(d) Using the non-inverted maps, obtain the complex potential and hence the velocity
field, assuming that the outer surfaces of the shafts are rotating at unit speed. 14. Some Interesting Examples of Flows (Based partly on Kreyszig, p. 818)
When a fluid fails to be incompressible at an isolated point (usually when the divergence
at that point is singular) we say it has a source or sink at that point depending on whether
the divergence there—as measured by the total flux out of a small surface—is positive or
negative. The strength of the source is equal to the total flux, if the flux integral exists.2
In physical terms, the flux of a vector field out of a surface measures the volume
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