Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 7 / i

Table of contents
- Solution i
- a i
  - Solution a
- b i
  - Solution b

Solution

The complex potential of a two-dimensional potential flow combines the velocity potential

u

with the stream function

v

. For its holomorphic representation, the Cauchy-Riemann equations are

u_{x} = v_{y}

and

u_{y} = - v_{x}

.

a

Solution

Since

V = \nabla u

, its divergence is the Laplacian of the velocity potential. Using the Cauchy-Riemann equations and equality of mixed partial derivatives gives

\nabla \cdot V = u_{xx} + u_{yy} = (v_{y})_{x} - (v_{x})_{y} = 0.

Thus this is an incompressible flow, and

u

is a harmonic function.

b

Solution

The derivative of a holomorphic function can be evaluated using an increment along the real axis:

f^{'} (z) = u_{x} + i v_{x} = u_{x} - i u_{y} .

Identifying the components of the gradient

V = \nabla u

gives

f^{'} (z) = V_{x} - i V_{y} .

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