UNIFYING LOCOMOTION

Conferences are often a source of inspiration, but few can pinpoint the exact moment as well as Adrian Bejan. In September 2004, Bejan, an engineer from Duke University, was invited to a conference discussing allometric scaling in Ascona, Switzerland. He remembers that the meeting was stimulating and peppered with lively debate, but when he heard Jim Marden's talk about the recently identified force-mass relationship found in natural and man-made motors, Bejan realised that he could explain Marden's startling results with a simple physical law; the constructal law. Meeting over coffee after Marden's talk, the two quickly realised that they had struck up a remarkable collaboration. By the end of the three-day meeting, the pair had derived a theory that could account for several aspects of both flight and walking locomotion in respect to an animal's mass, but could they extend the theory to explain swimming too (p. 238)?

For engineers, the constructal law is one of the founding principles of design. In its simplest form it states `for a flow system to persist, it must morph over time in such a way as to consume the least amount of energy while achieving the most' and a flow system can be anything from a river basin to herds of migrating animals. `The constructal law's success in engineering is evident' says Bejan, `engineers apply it intuitively'; but its application to biological design seems to have been less obvious. Applying the theory to various biological systems has allowed him to make a wide range of accurate physiological predictions, but could it explain the way the three major forms of locomotion scale relative to an animal's mass?

Bejan explains that he had already tackled flight from a constructal theory perspective before he met Marden. Considering the sum of energetic costs incurred by birds as they flap to overcome gravity, while propelling themselves forward against air resistance, Bejan optimised the system by minimising the sum of these losses and found that the speed and force output that he derived corresponded well with the values measured for birds and insects over a wide range of sizes. The constructal theory explained many aspects of flight locomotion.

Inspired by their new collaboration, Bejan and Marden began focusing their attention on running. Breaking running motion down into a set of horizontal and vertical movements, the team derived a set of equations based on vertical and horizontal energy losses and again minimised the sum of these losses as a function of size. The results were surprisingly good considering the simplicity of the theory. The team's calculated running speeds and stride frequencies showed a remarkable agreement with values measured from running animals.

But integrating swimming into the constructal theory of locomotion proved much harder. Bejan recalls that it took several months of brain wracking before the theory presented itself. The breakthrough came when the team began thinking of the way a fish moves through water. They realised that instead of lifting itself up through the water, the fish must push a fish-sized parcel of water over itself due to the incompressibility of water; `the proverbial light bulb had gone on' recalls Bejan. As soon as the team began considering the fish from this new perspective, the constructal theory of swimming fell into place.

Bejan adds that `zoology has known for decades that the relationship between speed and body mass is the same for flyers runners and swimmers, but this can no longer be treated as a coincidence'. The constructal theory seems to have unified three very disparate forms of locomotion.