Animals are filled with tubes, pushing fluids from one place to another– blood, lymph, mucus. In many cases, the tubes do the pushing by a process called peristalsis, where waves of contraction slide along the tubes,driving the fluid along with them.

Understanding how peristalsis works has become particularly important in recent years with the growing use of in vitro fertilization, because both eggs and sperm are pushed around in the oviducts by peristaltic waves. Sometimes the waves push the egg and sperm together, aiding fertilization, but sometimes they drive the two apart.

Fluid dynamics engineers have analyzed peristalsis for decades, and it would seem that their expertise should be helpful. Unfortunately, most of the analysis doesn't apply very well to biological fluids like the fluid in the oviducts.

The trouble is, mucus (or other polymers) gum up the fluid, making its properties somewhat intermediate between a fluid and a solid. A normal solid,like a rubber band, resists being stretched and produces a force that's linearly proportional to how far it's stretched. A normal fluid, like water,doesn't care how far it's deformed, but responds linearly to the rate of deformation. Fluids that aren't `normal' are called non-Newtonian, and can have some properties of solids. For example, many biological fluids, called viscoelastic fluids, have polymers in solution that stretch like rubber bands,while the fluid around them flows like water. Together they make for complicated, poorly understood non-Newtonian fluids.

To start to understand peristalsis of such fluids, Joseph Teran and Michael Shelley of the Courant Institute at New York University teamed up with Lisa Fauci at Tulane University, and developed a computational simulation of a tube with rhythmically contracting walls. Starting with code they had developed to simulate Newtonian flows, they overlaid a mesh of elastic structures to simulate the rubber band-like polymers. Since a tube is symmetrical around its long axis, they only simulated a two dimensional slice through the middle. In this view, the walls on either side contract in a traveling wave.

To compare their results with previous ones on Newtonian fluids, the researchers had to assess how weird their fluid was. They used a dimensionless number called the Weissenberg number Wi – the ratio between the rate that the polymers spring back and the rate of fluid motion. For a Newtonian fluid such as water, Wi is always zero; for a solid, it's infinity. Teran and his colleagues tested a range from zero to five.

They found some interesting differences with Newtonian fluids. For Newtonian fluids, the higher the amplitude of the peristaltic wave, the more fluid gets pushed along, even if the two sides of the wave meet in the center of the tube. For viscoelastic fluids, this wasn't true at all. With more polymers in solution, high amplitude waves produced little or no net motion,since most of the energy went into squishing or stretching the polymers perpendicular to the tube, not squirting the fluid along the tube. But with low amplitude waves, adding more polymers (to increase Wi) initially decreased the flow, but then, as Wi increased even further, started to increase the flow.

The group's research is still in its early stages, but they've set up a framework for studying non-Newtonian peristalsis. Now they can start to investigate practical questions, such as how are particles, like sperm,transported in such tubes? And for in vitro fertilization, when,relative to the peristaltic wave, would it be best to inject the sperm to obtain the optimal fertilization rate?

Teran, J., Fauci, L. and Shelley, M. (
). Peristaltic pumping and irreversibility of a Stokesian viscoelastic fluid.
Phys. Fluids