Solution

Use the continuity equation to write the axial momentum equation in conservative form:

\frac{\partial}{\partial z} (r w^{2}) + \frac{\partial}{\partial r} (r u w) = ν \frac{\partial}{\partial r} (r \frac{\partial w}{\partial r}) .

Integrating from the axis to infinity, regularity at

r = 0

and decay of

u, w, w_{r}

r \to \infty

\frac{d M}{d z} = 0.

Let

W (z)

be the centre-line velocity scale and

δ (z)

the jet width. Conservation gives

M \sim W^{2} δ^{2}

, hence

W δ \sim M^{1/2}

. The axial inertial and viscous terms scale as

\frac{W ^{2}}{z} \sim \frac{ν W}{δ ^{2}},

W δ^{2} \sim ν z

. Combining the two relations gives

δ (z) \sim \frac{ν z}{M ^{1/2}}, W (z) \sim \frac{M}{ν z} .

Thus the width grows linearly and the centre-line speed decays as

z^{- 1}

. The volume flux scales as

Q (z) = \int_{0}^{\infty} r w d r \sim W δ^{2} \sim ν z,

so it increases downstream. By fluid entrainment, the extra flux must be ambient fluid drawn radially into the jet.

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