Life in the Slow Lane: Low Reynolds Number and Laminar Flow.

"How to swim"

This was chalked on the board by Prof Olmsted in yesterday's Biophysics lecture. Our hopes of a trip to the swimming pool were quickly dashed. In fact the topic was about the motion of cells and micro-organisms in biology. In particular, if you happen to be such a micro-organism how do you get around in order to seek out nutrients to sustain yourself? With just a simple set of physical apparatus - how would you do it?

It turns out on the micro-scale life isn't as simple as the local swimming baths. If I wanted to do some lengths, I might well choose the breaststroke.It works like jet propulsion, with each stroke I push water backwards giving myself a kick in the right direction. After each kick I will glide under my own inertia, gradually slowing down due to the drag force. There is a balance between inertial and viscous forces. What if I took a swim a bath of gloopy treacle? The viscous forces would be so great that with each kick I wouldn't glide at all. The force from my kick would move me forwards by an arms length and then - freeze - nothing.

The Reynolds number is the ratio between the inertial force and viscous force, and also the ratio between the time it takes me to stop after my stroke and the time it takes me to travel one body's length. It is used in aerodynamics and fluid dynamics by the incredibly smart guys and gals who build our planes, formula 1 cars and submarines. The maths shows that the Reynolds number (Re) is given by the density of the stuff you're swimming in times your velocity times your characteristic length divided by the viscosity (gloopiness) of the stuff you're swimming in. So a low Reynolds number means you're in a thick liquid, or can't travel fast enough to successfully get anywhere (not being a strong swimmer I can relate to the latter).

A consequence of viscous forces dominating over inertial ones is the time reversal of your motion from reversible applied forces, or reciprocal swimming strokes. If you are in the low Reynolds number domain, and do a stroke forwards, and repeat your motion in reverse, you will end up at the same position you started. This seems an obvious statement but check out the jaw dropping demonstration in the YouTube clip. It was done in the 70s by a guy called Geoffrey Taylor. Ignoring the fact he had trouble counting to 5, it's a wonderful clip.

Cellular scales produce an incredibly small Re due to the short length scales and low velocities achievable. So small in fact that it is effectively zero. This is the situation I mentioned with the treacle: turns out bacteria can't do the breaststroke. How do they get around it? Thankfully it turns out cells are pretty smart, and thanks to evolution they aren't ones to shirk a challenge. The solution is a non reciprocal method of swimming. (your stroke forward has different dynamics to the stroke back, or recovery stroke).

One solution that has evolved is the Asymmetric Cilia. They are small whip like appendages which can generate net thrust. The cilia sweeps from left to right like an arm, then bends by an angle for the recovery stroke. Many cells use this method to get from A to B in order to eat nutrients which otherwise diffuse out of reach. Other stationary cells use the cilia as a sort of arm to channel food and liquid into their path.

Another is the perhaps more well known rotary flagellum. It's an amazing piece of biological nanotechnology which consists of a helical rod (imagine like a curly pigs tail) which is rotated about its axis by a motor and results in a net force along the axis of the helix proportional to the radius and angular frequency of rotation. This is how the tail of the sperm cell propels it around, although I think most people imagine the tail to work like that of a tadpole flapping from side to side. The fact is this just wouldn't work! (reciprocal motion) It is truly wonderful how evolution has tackled an apparently huge problem with such ingenuity that any modern day engineer would be proud.