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In [[fluid dynamics]], the ”’Oseen equations”’ (or ”’Oseen flow”’) describe the flow of a [[viscosity|viscous]] and [[incompressible]] [[fluid]] at small [[Reynolds number]]s, as formulated by [[Carl Wilhelm Oseen]] in 1910. Oseen flow is an improved description of these flows, as compared to [[Stokes flow]], by the partial inclusion of [[Navier–Stokes equations#Convective acceleration|convective acceleration]].<ref name=Batch240>Batchelor (2000), §4.10, pp. 240–246.</ref>

Using the Oseen equation, [[Horace Lamb]] was capable to derive improved expressions for the viscous flow around a sphere, improving on [[Stokes law]] towards somewhat higher Reynolds numbers.<ref name=Batch240/> As well as for the first time deriving a solution for the viscous flow around a circular cylinder.<ref name=Batch240/>

The Oseen equations are, in case of an object moving with a steady [[velocity]] ”’U”’ through the fluid — which is at rest far from the object — and in a [[frame of reference]] attached to the object:<ref name=Batch240/>

:<math>
\begin{align}
-\rho \mathbf{U}\cdot\nabla\mathbf{u} &= -\nabla p\, +\, \mu \nabla^2 \mathbf{u},
\\
\nabla\cdot\mathbf{u} &= 0,
\end{align}
</math>

where
*”’u”’ is the disturbance velocity induced by the moving object, ”i.e.” the total flow velocity in the frame of reference moving with the object is –”’U”’+”’u”’,
*”’p”’ is the [[pressure]],
*”ρ” is the [[density]] of the fluid,
*”μ” is the [[dynamic viscosity]],
*∇ is the [[gradient]] operator, and
*∇² is the [[Laplace operator]].

The boundary conditions for the Oseen flow around a rigid object are:

:<math>
\begin{align}
\mathbf{u} &= \mathbf{U} & & \text{at the object surface},
\\
\mathbf{u} &\to 0 & & \text{and} \quad p \to p_{\infty} \quad \text{for} \quad r \to \infty,
\end{align}
</math>

with ”r” the distance from the object’s center, and ”p”_∞ the undisturbed pressure far from the object.

== References ==
*{{ Citation
| first1=Carl Wilhelm
| last1=Oseen
| author1-link=Carl Wilhelm Oseen
| title=Über die Stokes’sche formel, und über eine verwadte Aufgabe in der Hydrodynamik
| year=1910
| journal=Arkiv för matematik, astronomi och fysik
| volume=vi
| issue=29
}}
*{{ Citation
| last1=Batchelor
| first1=George
| author1-link=George Batchelor
| title=An introduction to fluid dynamics
| publisher=[[Cambridge University Press]]
| edition=second paperback
| series=Cambridge Mathematical Library
| isbn=978-0-521-66396-0
| id={{MathSciNet | id = 1744638}}
| year=2000
}}

== Notes ==
{{reflist}}

[[Category:Fluid dynamics]]

{{fluiddynamics-stub}}