= Solution
Use the continuity equation to write the axial momentum equation in conservative form:
$$
\frac{\partial}{\partial z}(rw^2)
+\frac{\partial}{\partial r}(ruw)
=\nu\frac{\partial}{\partial r}\left(r\frac{\partial w}{\partial r}\right).
$$
Integrating from the axis to infinity, regularity at $r=0$ and decay of $u,w,w_r$ as $r\to\infty$ eliminate the boundary terms. Therefore the <conserved momentum flux of a laminar round jet> satisfies
$$
\boxed{\frac{dM}{dz}=0.}
$$
Let $W(z)$ be the centre-line velocity scale and $\delta(z)$ the jet width. Conservation gives $M\sim W^2\delta^2$, hence $W\delta\sim M^{1/2}$. The axial inertial and viscous terms scale as
$$
\frac{W^2}{z}\sim\frac{\nu W}{\delta^2},
$$
so $W\delta^2\sim\nu z$. Combining the two relations gives
$$
\boxed{\delta(z)\sim\frac{\nu z}{M^{1/2}},\qquad W(z)\sim\frac{M}{\nu z}.}
$$
Thus the width grows linearly and the centre-line speed decays as $z^{-1}$. The volume flux scales as
$$
Q(z)=\int_0^\infty rw\,dr\sim W\delta^2\sim\nu z,
$$
so it increases downstream. By <fluid entrainment>, the extra flux must be ambient fluid drawn radially into the jet.
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