= Solution
Use the idealized fully developed, laminar cylindrical-flow model implied by the fixed cross-section and zero <velocity> at its sides. Take the jet length to be $R$, as for a central source reaching the cloud boundary; if the source is off-centre, the actual path length must replace $R$. Let $s$ denote distance from the jet axis and write the axial <velocity> as $u(s)$. Then $(\mathbf u\cdot\nabla)\mathbf u=0$, the <pressure gradient> is $dp/dz=-p_0/R$, and the axial <Navier-Stokes equation> reduces to
$$
0=\frac{p_0}{R}+\rho\nu\frac1s\frac d{ds}\left(s\frac{du}{ds}\right).
$$
Regularity at $s=0$ excludes a logarithmic term. The imposed <no-slip boundary condition> $u(r_0)=0$ gives the <Hagen-Poiseuille flow> profile
$$
u(s)=\frac{p_0}{4\rho\nu R}(r_0^2-s^2)=U_{\max}\left(1-\frac{s^2}{r_0^2}\right),\qquad U_{\max}=\frac{p_0r_0^2}{4\rho\nu R}.
$$
The <kinetic energy> crossing each area element per unit time is $(\rho u^2/2)u\,dA$, so the <velocity> must be cubed before integration. Thus
$$
P=\frac12\rho\int u^3\,dA=\pi\rho U_{\max}^3\int_0^{r_0}\left(1-\frac{s^2}{r_0^2}\right)^3s\,ds=\frac{\pi\rho U_{\max}^3r_0^2}{8}.
$$
Consequently
$$
\boxed{P=\frac{\pi p_0^3}{512\rho^2\nu^3R^3}\,r_0^8,\qquad A=\frac{\pi p_0^3}{512\rho^2\nu^3R^3}}.
$$
The <mean velocity> is $\bar u=U_{\max}/2$. Replacing the profile by $\bar u$ in $\tfrac12\rho\pi r_0^2\bar u^3$ underestimates the flux by two; the <kinetic energy correction factor> is two for this profile. The radius dependence follows from $u\propto r_0^2$ and cross-sectional area $\propto r_0^2$.
The specified $P$ is outgoing bulk kinetic-energy flux, not the pressure-work rate $p_0Q$, which scales as $r_0^4$. In a fully developed section the profile is independent of $z$ and <pressure> work is dissipated as heat; entrance acceleration and <energy> supply establish the profile. A turbulent or freely entraining astrophysical jet need not have this no-slip parabolic structure. The model assumes a long enough, laminar confined region and neglects entrance losses and <gravity>.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2016/paper-1-jet-profile.png]
{title=Parabolic <Hagen-Poiseuille flow> and its cubic velocity weighting in kinetic-energy flux}
Back to article page
