Unruled Notebook

We have heard of the Mobius band. A one sided strip, it is a topological peculiarity. Over the decades it has popped up in several places in knowledge-space: the self-induction free Mobius resistor, the Mobius gear, the shape of the trajectory the Solar wind plasma assume in their route to chaos when interacting with the Earth’s magnetic field, the musical arrangement of J. S. Bach’s Crab canon, the space of dyads, to name a few.

What happens when a solid, rigid, Mobius ring is dropped into a fluid, say, water? Will the drag forces acting on it be any different from that for a regular (section of a) solid cylinder? The answer is yes, according to a recent experimental study by Leweke et al. (2009).

In their experimental study they dropped two Mobius bands into water from a height of 114 cm and pictured the entire fall using a digital camera that could traverse along the height with identical velocity. Both bands have identical perimeter to width ratio of 14. One band is made of polyester of diameter 18 mm and width 4 mm weighing 0.034 grams while the other is made of polycarbonate of diameter 45 mm and width 10 mm weighing 1.96 grams. Choosing two bands of different weights is to check for free fall weight effect on the wake-vortex induced vibrations on the band.

A quirk in using a Mobius band is, in principle it has no volume as it is a one sided surface (of negligible thickness). So, while calculating the mass of the fluid the band has displaced, one needs to assume a physical volume containing the band that is the simplest geometrical shape that completely envelops it. This shape used, is a torous. The Reynolds number, another non-dimensional parameter that varies directly with the velocity of the flow, and hence the drag, works out to be 130 and 560 for the smaller and bigger Mobius bands respectively.

Now for the results.

Once released in free fall into water, the Mobius bands align themselves quickly in a direction at about 30 ^° to the vertical, with their blunt edge facing down.

Figure 1:  Stroboscopic visualisations of trajectories of the freely falling small (a) and large (b) Mobius bands. Picture Credit: Ref. [1]

The bands don’t fall down in a straight line, but take a helical trajectory, as seen in the accompanied figure. A mild rotation of the band around the vertical plane is also observed. This direction of rotation depend on the twist direction of the band. The smaller band with a negative twist is observed to spin clockwise while the larger one with positive twist, spins anticlockwise. IN addition to slow spinning and helical traverse, as they fall, the bands also tilt and shake (oscillate) with lateral displacements of several body widths. The angle of oscillation is 42 ^° for the larger band and half of it for the smaller one. This is attributed to be caused by the vortical structures shed from the bluff leading ring element.

Wake generated from the trailing edge is captured using a fluorescent dye. The structure of the wake shown in the accompanied figure is already exhibits a complicated flow even at a low Re = 130. However, the helical trajectory of the fall is observable from the wake.

Figure 2: Dye visualisations of development of wake left behind by the small Mobius band. Picture Credit: Ref. [1]

The vertical distance travelled by the band is shown in white line in the figure. The authors note that the larger wake structure that appears on a later time (third picture in the figure) seems to resemble the wake induced oscillations observed behind a falling sphere, as reported in [2].

Irrespective of such esoteric fluid dynamics analysis, my favorite application of the Mobius band remains to be the mundane one in the picture below.

Figure 3: Mobius band arrangement of motor-crusher fan belt in a rice mill somewhere in Tamil Nadu state of South India

References

1. Leweke, T., Thompson, M., & Hourigan, K. (2009). Motion of a Möbius band in free fall Journal of Fluids and Structures, 25 (4), 687-696 DOI: 10.1016/j.jfluidstructs.2009.04.007
2. Horowitz, M., Williamson, C.H.K., 2008. Critical mass and a new periodic four ring vortex wake mode for freely rising and falling spheres. Physics of Fluids 20, 101701.