I was wondering where I could get a detailed account (either in print or online) on applying a Neumann/mixed Boundary condition along the $r=0$ axis in an axially symmetric geometry. Though this is a very common task I have always struggled to find a reasonable account of it. Piecing together sparce and poorly referenced notes I believe that in 1D polar coordinates where:$$\nabla^2 u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r}$$one gets, by Taylor expansion at the origin,:$$\nabla^2 u \rvert_{r=0} \simeq 2\frac{\partial^2 u}{\partial r^2}\rvert_{r=0}$$when $u$ is symmetric about the $r=0$ axis.I would like to more information on higher order derivatives, best practice, non-symetric distributions ... etc. Also I would be very keen to find some info on best practice for applying this boundary condition in FEM analysis.
Neumann BCs in cylindrical geometry (FEM)
finite element;boundary conditions