Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 38A / b / Solution

Use the axisymmetric streamfunction convention

w = \frac{1}{r} \frac{\partial ψ}{\partial r}, u = - \frac{1}{r} \frac{\partial ψ}{\partial z} .

For the similarity solution

ψ = ν z g (η)

,

η = r / z

, regularity on the axis requires

g (0) = g^{'} (0) = 0

, while matching to still ambient fluid requires

g^{'} (η) / η \to 0

η \to \infty

. The given equation has the first integral

η g^{'} - 2 g + \frac{1}{2} g^{2} = C .

The axis conditions give

C = 0

, and separation yields

\frac{d g}{g ( 4 - g )} = \frac{d η}{2 η}, g (η) = \frac{4 a η ^{2}}{1 + a η ^{2}}

for some

a > 0

.

The axial velocity is

w = \frac{ν}{ηz} g^{'} (η) = \frac{8 ν a}{z ( 1 + a η ^{2} ) ^{2}} .

Normalizing by the conserved momentum flux gives

M = \int_{0}^{\infty} r w^{2} d r = 64 ν^{2} a^{2} \int_{0}^{\infty} \frac{η d η}{( 1 + a η ^{2} ) ^{4}} = \frac{32}{3} ν^{2} a,

so

a = 3 M / (32 ν^{2})

. The similarity solution for a laminar round jet is therefore

g (η) = \frac{12 M η ^{2}}{32 ν ^{2} + 3 M η ^{2}} .

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