I was deriving Low Reynolds Number Flow around a Sphere of radius R placed in uniform stream U. There I got stuck in following PDEs, which are as under -
$\nabla^2 w = \frac{3UR}{2}(\frac{3z^2}{r^5}-\frac{1}{r^3})$
$\nabla^2 v = \frac{3UR}{2}(\frac{3zy}{r^5})$
$\nabla^2 u = \frac{3UR}{2}(\frac{3zx}{r^5})$
where r = $(x^2 + y^2 + z^2)^\frac{1}{2}$
I need their solutions in the form -
$w = U(\frac{3}{4}\frac{Rz^2}{r^3}(\frac{R^2}{r^2}-1)+1-\frac{3}{4}\frac{R}{r}-\frac{1}{4}\frac{R^3}{r^3})$
$v = U(\frac{3}{4}\frac{Rzy}{r^3}(\frac{R^2}{r^2}-1))$
$u = U(\frac{3}{4}\frac{Rzx}{r^3}(\frac{R^2}{r^2}-1))$
I was unable to find any appropriate method to solve them. Any help in solving the PDEs appropriately will be of great use. Thanks!