A Little History

It is generally held that the modern study of chaos began with Edward Lorenz’s seminal paper, “Deterministic nonperiodic flow”, in J. Atmos. Sci. 20, 130 (1963). In it he developed a set of three coupled, first-order, nonlinear equations as a simple model of convection rolls in the atmosphere.

Professor Lorenz in 2005.  He passed away in 2008 at the age of 90.

A few years later, Willem Malkus and Lou Howard at MIT built a waterwheel whose equations of motion reduced to a special case of the Lorenz equations. At first, even Malkus did not believe that fluid convection could produce the kind of complex motion that Lorenz’s equations predicted, telling him, “Ed, we know – we know very well – that fluid convection doesn’t do that at all” [see Gleick’s book Chaos, p. 31]. Malkus built his waterwheel in part to prove to skeptics that complex behavior was possible in such a system.

Willem Malkus (front right) and Lou Howard (front left), founding members of the Geophysical Fluid Dynamics Program at Woods Hole Oceanographic Institute. Malkus’s MIT waterwheel design, from Strogatz, <b>Nonlinear Dynamics and Chaos</b>, Westview Press, 2001, p. 303

The ISU Waterwheel

In the fall semester of 2002, I gave students in my PHY 270 Experimental Physics class an opportunity to design and build a project of their choosing, and they chose to build a chaotic waterwheel. We soon found out, however, that published accounts of the experimental waterwheel were rare, and real design parameters were hard to extract from demonstration projects described on the web. So we first wrote computer simulation code to solve the waterwheel’s equations of motion (in SI units) so we could place reasonable bounds on the experimental parameter space we wanted to explore. Then we began to build…

To make the wheel as balanced and symmetric as possible, we vacuum-formed the base of the wheel from 1/8 inch thick polycarbonate. The wooden form used to make the base is shown below.

Wooden Form Base

Since the wooden form was machined on a rotary table in a mill, the base (and subsequent machining of pockets for placement of the cylindrical cells) resulted in a very rotationally balanced wheel. Each cone-shaped divot in the polycarbonate base was drilled out to accept a nylon screw, and a thin hole was drilled in the length of each screw to provide a leak from each cell. We also created a vacuum-formed trough to place around the top of the cells to ensure that all the input water stream was directed to the cells. We then added a thin aluminum ring to the outer edge of the base (for magnetic braking, to be explained later), and the finished wheel is shown below.

The ISU waterwheel and its first two students:  Valerie Hackstadt and Vanessa Grabowski.

To stiffen the structure, we later added triangular polycarbonate plates from the base to a small central hub. A steel shaft was fixed to this hub, and at first roller bearings were used to hold the shaft in place in the finished apparatus, which is shown below.

Finished Apparatus