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{{merge from|Oseen's approximation|date=October 2012}}
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 able to derive improved expressions for the viscous flow around a sphere in 1911, improving on [[Stokes law]] towards somewhat higher Reynolds numbers.<ref name=Batch240/> Also, Lamb derived—for the first time—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/>
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:<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]],
*&nabla; is the [[gradient]] operator, and
*&nabla;<sup>2</sup> 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''<sub>∞</sub> the undisturbed pressure far from the object.
 
== Notes ==
{{reflist}}
 
== References ==
*{{ Citation
| first1=Carl Wilhelm
| last1=Oseen
| author1-link=Carl Wilhelm Oseen
| title=Über die Stokes'sche formel, und über eine verwandte 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
}}
 
[[Category:Fluid dynamics]]
[[Category:Equations of fluid dynamics]]

Latest revision as of 19:16, 17 November 2014


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