Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 37A (Fluid Dynamics II)

Axisymmetric incompressibility is ${\partial }_r(r{u}_r)+r{\partial }_z{u}_z=0$. Multiply the radial momentum equation by $r$ and use this to put advection in conservative form:Integrating through the layer givesThe boundary terms vanish because radial velocity and its shear decay into the surrounding fluid; an entrainment velocity need not itself vanish as long as its product with $u_r$ does. Hence

Let $U(r)$ be the radial velocity scale. Advection-viscosity balance gives $U^2/r\sim \nu U/{\delta }^2$, while the conserved momentum flux gives $F\sim U^{2}r\delta$. Eliminating $U$ yields $\delta /r\sim ({\nu }^2/F{)}^{1/3}$, a constant, so $\delta =Cr$ and $U\propto r^{-1}$.

The streamfunction automatically satisfies incompressibility. With the proposed form it giveswhere $u_r,u_z$ are the two velocity components. Differentiation in the momentum equation cancels the terms proportional to $\eta f^{\prime }{f}^{\prime \prime }$ and leavesChoose $A{C}^2=\nu$. The momentum integral is $F=A^{2}C\int f^{\prime 2}d\eta$, so choosing $A^{2}C=F$ makes its normalization one. ThereforeReflection symmetry gives $u_z=0$ and ${\partial }_z{u}_r=0$ at zero. Taking the streamfunction's odd representative makes $f(0)=0$, and symmetry gives $f^{\prime \prime }(0)=0$; far-field decay gives $f^{\prime }\to 0$. The factor $f$ multiplying $f^{\prime \prime }$ is present in the original PDF but missing from the TeX aid; the direct substitution confirms the PDF equation.

The thin-layer condition isIntegrate the differential equation once: $(f^{\prime \prime }+f{f}^{\prime }{)}^{\prime }=0$. Since $f(0)=f^{\prime \prime }(0)=0$, its integration constant is zero. A second integration gives $f^{\prime }+f^2/2=b$. The outward-jet solution with $b>0$ and $f(0)=0$ isUsing the given fourth-power integral in the normalization gives $1=4k^4[4/(3k)]=16k^3/3$, so $k=(3/16{)}^{1/3}$. Thus the similarity profile of an axisymmetric radial viscous free jet isIt is symmetric, outward, decays in $z$, and has exactly the conserved flux $F$. It describes the region beyond the small injection slit where the stated boundary-layer approximation applies; its formal $r=0$ singularity is outside that approximation.