Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 7 / b / ii

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- Solution ii

Solution

ζ = ξ + i η

,

u - i v = \frac{1}{ζ + 2} = \frac{ξ + 2 - i η}{( ξ + 2 ) ^{2} + η ^{2}} .

Therefore

u = \frac{( ξ + 2 , η )}{( ξ + 2 ) ^{2} + η ^{2}}, ∣ u ∣ = \frac{1}{( ξ + 2 ) ^{2} + η ^{2}} .

The vector field points radially away from $(- 2, 0)$ , with magnitude inversely proportional to distance from that point. It diverges on approach to that boundary point and tends to zero far away. This is the radial potential flow described by the logarithmic complex potential of a two-dimensional potential flow.

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