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Oseen equations

A body with an arbitrary shape is moving in a fluid of density $\rho$ with an velocity $v$$r$ denotes the distance from the body. Further $u$ denotes the velocity of the fluid and $p$ the hydrostatical oressure. Oseen equations for flow due to a moving body at small Reynolds numbers is given as follows:
\begin{eqnarray*}\rho\partial _tu&=&-\rho (v\cdot {\rm grad})u -{\rm grad} p + \mu\Delta u,\\{\rm div} \;u&=&0.\end{eqnarray*}
The second relation models the incompressibility of the fluid. Boundary conditions for a rigid body are given by:
\begin{eqnarray*}u=v \quad\mbox{at$\;$\ the$\;$\ surface$\;$\ of $\;$the$\;$\ b......\quad\mbox{and}\quad p-p_0\to 0\quad\mbox{ as} \quad r\to \infty\end{eqnarray*}
For further information on the model have a look in G.K.Batchelor (2000) An introduction to fluid dynamics Cambridge Mathematical Library. It can be seen as an exterior Dirichlet problem in $\R^3\setminus G$ where $G$ is simple connected domain.
 

This problem is discussed as direct and inverse problem. A single-layer potential approach in a Sobolev setting with a lot of interesting references is treated in:
 

A late reference: R. Kress and Mayer S. (2000) An Inverse Boundary Value Problem for the Oseen Equation MMAS 23, 103-120.


collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg