Oseen approximation
= Oseen approximation
{c}
{title2=$\rho\mathbf U\cdot\nabla\mathbf u=-\nabla p+\mu\nabla^2\mathbf u$}
The <Oseen approximation> retains <advection> of a small disturbance by a uniform incident <velocity> $\mathbf U$, but neglects <advection> of the disturbance by itself. Its steady equations are
$$
\rho(\mathbf U\cdot\nabla)\mathbf u=-\nabla p+\mu\nabla^2\mathbf u,\qquad\nabla\cdot\mathbf u=0.
$$
A Stokes disturbance of a <sphere> decays as $Ua/r$, so the ratio of background <advection> to viscous <diffusion> grows as $Ur/\nu$. Thus the approximation resolves the outer region $r\sim\nu/U$, where the <Stokes flow> expansion is nonuniform, while self-convection remains smaller by $a/r$.