Complex Analysis: Theory and Applications · Problem 10.7.1

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · fluid-flow-unit-circle-complex-potential

Problem

Consider the ideal fluid flow for the complex potential F(z)=A(z+1z)F(z) = A\left(z + \frac{1}{z}\right), where AA is a positive real number. (a) Show that the velocity vector at the point (1,θ)(1, \theta), z=reiθz = re^{i\theta} on the unit circle is given by V(1,θ)=A(1−cos⁡2θ−isin⁡2θ)\mathbf{V}(1, \theta) = A(1 - \cos 2\theta - i\sin 2\theta). (b) Show that the velocity vector V(1,θ)\mathbf{V}(1, \theta) is tangent to the unit circle ∣z∣=1|z| = 1 at all points except −1-1 and +1+1. Hint: Show that V⋅P=0\mathbf{V} \cdot \mathbf{P} = 0, where P=cos⁡θ+isin⁡θ\mathbf{P} = \cos\theta + i\sin\theta. (c) Show that the speed at the point (1,θ)(1, \theta) on the unit circle is given by ∣V∣=2A∣sin⁡θ∣|\mathbf{V}| = 2A|\sin\theta| and that the speed attains the maximum of 2A2A at the points ±i\pm i and is zero at the points ±1\pm 1. Where is the pressure the greatest?

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