Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2016 / Paper 1 / 3 / c

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

(u \cdot \nabla) u = 0

, the pressure gradient is

d p / d z = - p_{0} / R

, and the axial Navier-Stokes equation reduces to

0 = \frac{p _{0}}{R} + ρ ν \frac{1}{s} \frac{d}{d s} (s \frac{d u}{d s}) .

(1)

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 ρ ν R} (r_{0}^{2} - s^{2}) = U_{m a x} (1 - \frac{s ^{2}}{r _{0}^{2}}), U_{m a x} = \frac{p _{0} r _{0}^{2}}{4 ρ ν R} .

(2)

The kinetic energy crossing each area element per unit time is

(ρ u^{2} /2) u d A

, so the velocity must be cubed before integration. Thus

P = \frac{1}{2} ρ \int u^{3} d A = π ρ U_{m a x}^{3} \int_{0}^{r_{0}} (1 - \frac{s ^{2}}{r _{0}^{2}})^{3} s d s = \frac{π ρ U _{m a x}^{3} r _{0}^{2}}{8} .

(3)

Consequently

P = \frac{π p _{0}^{3}}{512 ρ ^{2} ν ^{3} R ^{3}} r_{0}^{8}, A = \frac{π p _{0}^{3}}{512 ρ ^{2} ν ^{3} R ^{3}} .

(4)

The mean velocity is

\overset{u}{ˉ} = U_{m a x} /2

. Replacing the profile by

\overset{u}{ˉ}

\frac{1}{2} ρ π r_{0}^{2} \overset{u}{ˉ}^{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_{0} Q

, 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.