# Low Reynolds number assumption of Oseen type of flow

Momentum balance part of the fluid dynamical equations for Oseen type of flow can be written as (in the reference frame of an object moving with $\mathbf{V}$),

$$\mathbf{V}\cdot \nabla \mathbf{w} = -\frac{\nabla p}{\rho} + \frac{\mu_{\text{l}}}{\rho} \nabla ^2 \mathbf{w}$$

I am particularly interested in an approximate range of Reynolds number for which the Oseen equations make sense. Wikipedia page for example, mentions that it is applicable for very low Reynolds number. From which arguments such condition is derived?

First take a look at the steady Navier-Stokes equations:

$$\nabla\cdot[\rho\vec{u}\otimes \vec{u}] = -{\nabla p} + \mu{\nabla^2}\vec{u} + \vec{f}$$

The way to obtain the Stokes equations used in microhydrodynamics is by assuming that the terms on the right hand side (forcing and/or viscous terms) are much bigger than the convective acceleration on the left hand side. One approximate way of quantifying that assumption is by also assuming the system has some characteristic densities/velocities/velocity gradients/viscosities/length scales and non-dimensionalizing the equation with these. Doing so gives you:

$$\text{Re}\left(\nabla\cdot[\rho^*\vec{u^*}\otimes \vec{u^*}]\right) = -{\nabla p^*} + {\nabla^2}\vec{u^*} + \vec{f^*}$$

where $\text{Re} = \frac{\rho U L}{\mu}$ is the familiar Reynolds number and the quantities defining it are the "characteristic" quantities of the system. Assuming the Reynolds number is tiny means that the left-hand side is negligible according to the approximation, and we don't have to care about it, netting the Stokes equations (back in dimensional form):

$$-{\nabla p} + \mu{\nabla^2}\vec{u} + \vec{f} = 0$$

Problem is, these assumption are pretty bad, and generate a massive amount of paradoxes and inconsistent results! Oseen's idea was to, rather than give up the convective acceleration term, to linearize it by saying the left hand-side terms can be approximated by

$$\nabla\cdot[\rho\vec{u}\otimes \vec{u}]\approx\nabla\cdot[\rho\vec{U}\otimes \vec{u}]$$

where $\vec{U}$ is the characteristic velocity of the system whose scalar form popped up in the Reynolds number; it's usually the velocity of whatever is causing the flow disturbance. This approximation straightforwardly generates the Oseen equations when you plug it into the steady Navier-Stokes equation:

$$\nabla\cdot[\rho\vec{U}\otimes \vec{u}] = -{\nabla p} + \mu{\nabla^2}\vec{u} + \vec{f}$$

In short, the Stokes equations are what you obtain when you neglect convective acceleration, and the Oseen equations are what you get if you try to linearize it.

• Thanks for a detailed answer. I see the importance of Reynolds number in the formulation now. Jun 21, 2018 at 18:18
• how far does Oseen's approximation push the upper boundary on $Re$? Jun 22, 2018 at 9:40
• The Oseen approximation doesn't change at what $Re$ the system becomes undescribable by laminar flow, as that is -almost- always determined experimentally; what it does do is improve the accuracy of the solutions in the low-$Re$ regime. Jun 22, 2018 at 17:29