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

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