Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 37A / iii / Solution

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.