Past exam of the theoretical mechanics course of the University of Oxford / 2008 / A / 3 / i / Solution

Set $u_{\theta }=0$. Continuity gives ${\partial }_r(r{u}_r)=0$, hence $\boxed{u_r=g(\theta )/r}$. Substituting into radial and angular momentum givesFor example ${\nabla }^2{u}_r-u_r/r^2=g^{\prime \prime }/r^3$, and the radial acceleration is $-g^2/r^3$. Equality of the mixed pressure derivatives now requiresThus the equation for radial viscous flow in a wedge isIt can also be integrated once as $\nu (g^{\prime \prime }+4g)+g^2=\text{constant}$.