Oseen approximation ($\rho \mathbf{U}\cdot \nabla \mathbf{u}=-\nabla p+\mu {\nabla }^2\mathbf{u}$)

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 areA 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 $a/r$.

Let $U=|\mathbf{U}|$, $\nu =\mu /\rho$, and $r=|\mathbf{x}|$. A decaying outer disturbance matched to the leading translating sphere in Stokes flow isHere $\Delta \varphi =0$ and $(\nu \Delta -\mathbf{U}\cdot \nabla )\chi =0$ away from zero. The substitution $\chi =h{e}^{\mathbf{U}\cdot \mathbf{x}/(2\nu )}$ reduces the latter equation to $(\Delta -U^2/(4{\nu }^2))h=0$, whose decaying radial solution is proportional to $e^{-Ur/(2\nu )}/r$. In the inner overlap, cancellation of the potential monopole gives $\mathbf{u}\sim -3a[\mathbf{U}+\mathbf{n}(\mathbf{U}\cdot \mathbf{n})]/(4r)$. The potential source's volume flux is $6\pi \nu a$, and its mass flux is $6\pi \mu a$. Downstream, the fluid wake width grows as $\sqrt{\nu x/U}$; its mass deficit balances this source, while its momentum deficit is $6\pi \mu aU$, the Stokes drag law.