Oseen equations
In fluid dynamics, the Oseen equations (or Oseen flow) describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
Oseen's work is based on the experiments of G.G. Stokes, who had studied the falling of a sphere through a viscous fluid. He developed a correction term, which included inertial factors, for the flow velocity used in Stokes' calculations, to solve the problem known as Stokes' paradox. His approximation leads to an improvement to Stokes' calculations.
The Oseen equations are, in case of an object moving with a steady flow velocity U through the fluid—which is at rest far from the object—and in a frame of reference attached to the object:
where
The boundary conditions for the Oseen flow around a rigid object are:
with r the distance from the object's center, and p∞ the undisturbed pressure far from the object.
A fundamental property of Oseen's equation is that the general solution can be split into longitudinal and transversal waves.
A solution is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out. The equations become
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