TITLE: Should water at the scale of a cell feel more like tar?
QUESTION [64 upvotes]: The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary manipulations show that if you zoom in by a factor of $\lambda$, then you expect viscosity to scale as $\lambda^{\frac{3}{2}}$. So, for example, if you zoom in to the length scale of a cell, you expect viscosity to be around a million times larger than humans experience it.
This is not observed, however, which makes sense since we expect the components of a cell to move around extremely quickly. (EDIT: this is observed - see answer - my initial google searches were untrustworthy, damn google). Nonetheless, the calculation above suggests that they feel like they are moving through one of the most viscous fluids imaginable.
What then is the mechanism that prevents this? I have seen some explanations through the ideas of 'microviscosity' and 'macroviscosity' in the physics community, but I couldn't find much of a theoretic backing for them.
I'm wondering if there is a more mathematical explanation, perhaps directly from the Navier-Stokes equation itself (seems unlikely), or something from a kinetic theory point of view? For example some kind of statistical model of water molecules that reproduces the result?

REPLY [16 votes]: You may be interested in Shapere, A., and F. Wilczek. 1987. Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58: 2051–2054 where they use gauge theory to describe micro-swimming.  Because the Stokes equation - the infinite viscosity limit of Navier-Stokes - is linear, it allows us  to define a connection for the principal G bundle:
(located shapes) --> (unlocated shapes). Here G is the group of rigid motions of space,  a located shape is (say) a volume-preserving embedding of the ball into usual 3-space, and  the space of unlocated shapes is the quotient space of the space of located shapes by the action of G.  Think of the ball as the cell (parmecium, E Coli, cyanobacterium, ..) which wants to move.   A swimming stroke is then  a loop in the space of unlocated shapes.  The resulting holonomy for the Stokes connection is computed by solving the Stokes equation with zero boundary data at infinity.  Shapere in his thesis estimates the curvature at the embedding which is a round ball, and thereby investigates ``infinitesimal swimming motions''.  Some of this story can also be found in my book, A Tour of SubRiemannian Geometry.