# Solution of Navier-Stokes eq for a single particle?

1. Aug 20, 2012

### DarkLindt

Solution of Navier-Stokes eq for a single particle???

Hi!

I'm reading this paper on fluid dynamics:
http://jcp.aip.org/resource/1/jcpsa6/v50/i11/p4831_s1 [Broken]
Its equation (13) is the velocity distribution around a single bead of radius a subjecting to force fi in solution. (the subscript i is irrelevant here). The bead is located at the origin and $\mathbf\rho^{'}$ is the coordinate for an arbitrary point in space.

Equation (13):
$\mathbf u_{i}(\mathbf\rho^{'}) = (8\pi\eta a)^{-1}\left [ \left ( \frac{a}{\rho ^{'}}+ \frac{1}{3}\frac{a^3}{\rho ^{'3}} \right ) \mathbf f_i + \left ( \frac{a}{\rho ^{'3}}- \frac{a^3}{\rho ^{'5}} \right ) \mathbf f_i \cdot \mathbf\rho^{'}\mathbf\rho^{'} \right ]$

There is not even citation for this equation, it looks like some textbook solution of the Navier-Stokes equation for this simple system. I just want to know how this can be derived? Could anyone provide me some resource to look at???

Thanks sooo much!

Last edited by a moderator: May 6, 2017
2. Aug 20, 2012

### DarkLindt

Re: Solution of Navier-Stokes eq for a single particle???

Ahhh! I found a derivation from another paper: http://jcp.aip.org/resource/1/jcpsa6/v53/i1/p436_s1 [Broken]
The derivation are equation (1)-(7). Equation (7) is equivalent to equation (13) from the post above.
As for the derivation of equation (2) in this paper, one needs to refer to earlier texts of fluid dynamics.

Last edited by a moderator: May 6, 2017