I have the Laplace's equation in 2D axisymmetric coordinate:
$$\nabla^2 f=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{\partial^2f}{\partial z^2}=0,$$
where $f(r,z)$ is a scalar function of radial coordinate $r$ and axial coordinate $z$.
I need to impose a constant gradient boundary condition at infinity. After consideration, I came up with the following two kinds of conditions:
Neumann BC.1
\begin{equation} \lim_{l \to +\infty}\frac{\partial f}{\partial r}= \lim_{l \to +\infty}\frac{\partial f}{\partial z}=c, \quad \mbox{with} \quad l=(r^2+z^2)^{1/2}; \end{equation}
Neumann BC.2 \begin{equation} \lim_{l \to +\infty}\frac{\partial f}{\partial \mathbf{n}} =\lim_{l \to +\infty}\frac{rf_r+zf_z}{l}=c, \quad \mbox{with} \quad l=(r^2+z^2)^{1/2}, \end{equation} where $\mathbf{n}=\frac{1}{l}(r,z)$ is the unit normal of the circle $l=(r^2+z^2)^{1/2}$.
I got confused which one is correct or they are equivalent each other? Thanks!