Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 36B / i

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- Solution i

Solution

The steady axial Navier-Stokes equation for the straight-pipe approximation is

0 = G / ρ + ν w^{''} (y)

. No-slip boundary conditions are

w (\pm a) = 0

. Integrating twice gives

u = (0, 0, w (y)), w (y) = \frac{G}{2 ρ ν} (a^{2} - y^{2}) .

Ignoring the radial variation is justified away from the side walls by the given ordering

a ≪ b

.

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