Fundamentals

Low Reynolds Number Flow

Low Reynolds number flow can be a very interesting topic.  Low Reynolds number flow (Re <<1) is also called Stokes flow.  At very low Reynolds number, the Navier-Stokes equations can be greatly simplified.  Fluid mechanics at human length scales, such as swimming, is generally not very low Reynolds number.  Developments in microfluidics, nanotechnology, and biomimicry has increased the frequency with which engineers encounter low Reynolds flows problems.  Because humans often encounter fluids at moderate of high Reynolds numbers, our intuition can deceive us.  Two of the most basic results of low Reynolds flow is that it is fully reversible and independent of time.

As a refresher, the Reynolds number is the ratio of inertial to viscous forces and is given by

    \[ \mbox{Re}=\frac{\rho v d}{\mu} \]

where \rho is the fluid density, v is the velocity, d is a characteristic length such as a diameter, and mu is the viscosity.  The Reynolds number can also be thought of as the ratio of the momentum diffusion rate to the viscous diffusion rate.  At Reynolds numbers less than approximately 2000, the flow is laminar.  For Reynolds numbers greater than approximately 4000, the flow is turbulent.The Navier-Stokes equations which govern fluid mechanics can be simplified greatly for very small Reynolds number (\mbox{Re} \to 0).

    \[ -\nabla p + \mu \nabla^2 \bf{u}+\bf{f}=0 \]

where \bf{u} is the velocity vector, p is the pressure, and \bf{f} is the body force vector.   It should be noticed that this is no longer explicitly a function of time because the d/dt terms have vanished.

The video lecture by G.I Taylor on Low Reynolds flow offers many good explanations.  This video is part of a video lecture series featuring Ascher Shapiro and other renowned fluid mechanics experts.  The video demonstrating reversible flow is quite simple and clear.

A classic paper by Edward M. Purcell (a Nobel prize winning physicist at Harvard) entitled “Life at Low Reynolds Number” is available online for free.  In this paper, he helps explain some of the non-intuitive fluid behaviors.  In particular, he develops the well-known “scallop theorem” for swimming at low Reynolds number.  In short, a body composed of two links connected by a pivot such as a scallop shell cannot swim at low Reynolds number.  The body must have at least three rigid links or have a flexible tail like body to create asymmetric motions.  Figure 1.3 of the thesis by J. Hussong at TU Delft presents the different methods of asymmetry.  These asymmetries were originally described in the paper by S.N. Khaderi et al (available here also from TU Deflt).  Brennen at Caltech has an online textbook which has a section on low Reynolds number locomotion by a variety of mechanisms.