What is a flow field generated by a swimming sphere in the fluid? (in low reynolds regime)? What is the flow field generated by a swimming cylinder in this regime? It seems that the question is not clear. So I try to make it better: when an active sphere is moving in a liquid. What is the solution of Stokes equation for flow in the liquid? If an active cylinder is moving in the fluid, what is the solution?
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$\begingroup$ It's a little odd that no one could answer my question! Is it unclear? $\endgroup$– sara njApr 5, 2017 at 22:08
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1$\begingroup$ Yes, the question is unclear. What is it you are asking for? What do you mean by "swimming"? What is the motion of your cylinder or sphere? When you say "low-Re regime", are you asking for a Stokes solution (Limit for $Re\rightarrow0$)? Are you asking for an algebraic expression giving the (2D/3D?) velocity field? For the sphere, is this what you are looking for? $\endgroup$– PirxApr 5, 2017 at 22:15
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$\begingroup$ By the way, the case of the cylinder happens to be tricky, see the Wikipedia article here. It turns out there is no non-trivial solution for the Stokes equations around an infinitely long cylinder, a fact known as Stokes' Paradox. $\endgroup$– PirxApr 5, 2017 at 22:20
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$\begingroup$ I edited the question. The first link doesn't work. (The error is file not found.) what about a short cylinder? @Pirx $\endgroup$– sara njApr 6, 2017 at 4:54
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$\begingroup$ Corrected link $\endgroup$– PirxApr 6, 2017 at 11:38
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1 Answer
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See Pirx link in comments. Recommended book: "Low Reynolds number hydrodynamics". Answer for a sphere from this derivation:
$$ \vec u(\vec r) = \vec v \frac{3R}{4r} (1 + \frac{R^2}{3r^2}) + \vec r\, (\vec v \cdot \vec r) \, \frac{3R}{4r^3} \, (1 - \frac{R^2}{r^2}) $$
where $\vec v$ is spheres velocity.