I'm trying to find a source that gives the stream function for a 2D stokeslet. A stokeslet is a solution to unbounded Stokes flow with a point singularity at the origin. I know the velocity field induced by a 2D stokeslet (pdf) at point $\bf{x}$ is
$$ u({\bf x}) = \frac{1}{4 \pi \mu} \left(- {\bf F} \ln|{\bf x}| + ({\bf F} \cdot {\bf x}) \frac{\bf x}{|{\bf x}|^2} \right). \tag{13}$$
where $\bf{F}$ is a point force acting on the fluid at the origin and $\mu$ is the fluid viscosity.
I also know that for a 2D stream function $\psi$ you can get the velocity field by doing:
$$ u_x=-\frac{\partial \psi}{\partial y}, \qquad u_y=\frac{\partial \psi}{\partial x}. $$
However it's not obvious to me how to find a $\psi$ that will produce the velocity field corresponding to the stokeslet and I can't find a source that gives a stream function for the stokeslet.
