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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.

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Hint: $$\psi~=~\frac{1}{4\pi\mu}\det[{\bf F},{\bf x}](\ln|{\bf x}|-1). $$

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  • $\begingroup$ That's quite the hint! Is there a source for that, or can you give a derivation? $\endgroup$
    – Jay Lemmon
    Commented Apr 26, 2021 at 16:18
  • $\begingroup$ Well, let me first ask you this: Can you confirm that it is a solution? $\endgroup$
    – Qmechanic
    Commented Apr 26, 2021 at 16:38
  • $\begingroup$ I'm working on it now but my vector calculus is a little rusty. Oh!, I should say this isn't for school or anything, I'm trying to learn this on my own. $\endgroup$
    – Jay Lemmon
    Commented Apr 26, 2021 at 16:42
  • $\begingroup$ Okay, I've verified that the corresponding velocity field is the same as the stokeslet. I had to break it in to components to get there, but I suspect there's a straightforward derivation? $\endgroup$
    – Jay Lemmon
    Commented Apr 26, 2021 at 17:02
  • $\begingroup$ Well, integration is more difficult than differentiation. $\endgroup$
    – Qmechanic
    Commented Apr 26, 2021 at 17:25

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