i'm wondering where I've messed up with this one, any help would be great.
in 3D cylindrical Polars, a 3D point source at the origin is represented by the Stokes Streamfunction. $$\Psi(r,z)=\frac{\Lambda}{4 \pi}\left(1- \frac{z}{\sqrt{z^2+r^2}}\right)$$ Show that the stokes streamfunction for a source at the origin and a sink at $z=a$ in the limit $a \longrightarrow 0$ is given by $$\Psi(r,z)=\frac{\lambda}{4 \pi}\left(\frac{z-\sqrt{z^2 + r^2}}{z^2+r^2}\right)$$ with $\lambda = \lim_{a \longrightarrow 0} ma$
Express this in spherical polars
after having read Help: Fluid Dynamics And Vertical Dipoles
i tried to adapt the same(ish) method.
From the question we're given the streamfunction, we want to let the sink tend to the origin so i've tried $$\Psi = \lim_{a \longrightarrow 0}\frac{\Lambda}{4 \pi}\left[\left(1- \frac{z}{\sqrt{z^2+r^2}}\right) - \left(1- \frac{(z-a)}{\sqrt{(z-a)^2+r^2}}\right)\right]$$
$$=\lim_{a \longrightarrow 0}\frac{\Lambda}{4 \pi}\left(\frac{(z-a)}{\sqrt{(z-a)^2+r^2}} - \frac{z}{\sqrt{z^2+r^2}}\right)$$ $$=\lim_{a \longrightarrow 0} \frac{\Lambda}{4\pi}\frac{-z \sqrt{a^2-2az+r^2+z^2}~-~a\sqrt{r^2+z^2}~+z \sqrt{r^2+z^2}}{(\sqrt{(z-a)^2+r^2})(\sqrt{z^2+r^2})}$$
which no way i can figure will rearrange to give me what i need, even if i take a factor of a out of it, things cancel and i dont end up with the right result.
any suggestions? Cheers for the help, i appreciate it.