Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 36E (Fluid Dynamics II)

In a steady axisymmetric flow without swirl, time derivatives and azimuthal derivatives vanish. For a slender high-Reynolds number jet of width $\delta \ll z$, radial viscous derivatives dominate axial ones by $(z/\delta {)}^2$. The transverse momentum balance makes pressure nearly constant across the jet, matching the constant ambient pressure, so the leading axial pressure gradient vanishes. Retaining axial convection and radial diffusion in the Navier-Stokes equation gives the boundary-layer equationThe radial velocity is small but its convective term is of the same order as axial convection because radial gradients are larger.

The incompressibility condition turns the left side of the boundary-layer equation into conservative form:Multiply by $2\pi \rho r$ and integrate from the axis to infinity. Regularity at the axis and decay in the ambient fluid eliminate the radial boundary terms. Thus the momentum flux is constant,Matching the applied force sets this constant to $F$. Scaling convection against radial viscosity gives $U^2/z\sim \nu U/{\delta }^2$, while the flux gives $F\sim \rho U^2{\delta }^2$. Combining them yieldsThe slenderness requirement is $\delta /z\sim \nu \sqrt{\rho /F}\ll 1$, equivalent to a large force-based Reynolds number $\sqrt{F/\rho }/\nu$.

Fix the scaling constants as $U=A/z$, $\delta =Bz$, with $A=F/(\rho \nu )$ and $B=\nu \sqrt{\rho /F}$, so $A{B}^2=\nu$. The Stokes streamfunction is then $\psi =\nu zf(\eta )$. Write $g=f^{\prime }/\eta$. Differentiation givesSubstitution in the boundary-layer equation, cancelling $A^2/z^3$, gives $-(f/\eta )g^{\prime }-g^2=g^{\prime \prime }+g^{\prime }/\eta$. Multiplication by ${\eta }^3$ therefore gives the required similarity equationAxis regularity requires zero radial velocity and a finite smooth axial velocity: choose the streamfunction gauge $f(0)=0$ and require $f(\eta )=c{\eta }^2+O({\eta }^4)$. Equivalently $f^{\prime }(0)=0$, $f^{\prime }/\eta$ has a finite limit, and $(f^{\prime }/\eta {)}^{\prime }\to 0$ at the axis. The constant $c$ is not specified in advance. In the quiescent ambient fluid require $f^{\prime }/\eta \to 0$ and $f^{\prime }-f/\eta \to 0$ as $\eta \to \infty$, so both velocity components vanish. The localized physical jet has finite axial volume flux, corresponding to finite $f(\infty )$.

Finally the prescribed force fixes the amplitude through the finite momentum-flux normalizationThe axis conditions express regularity rather than independently prescribed arbitrary derivatives; the far-field conditions express matching to the resting fluid, and the integral condition supplies the jet strength.