IBM Research Division, T.J. Watson Research Center
Yorktown Heights, NY 10598
My experience with the Math Problems in Industry have been overwhelmingly favorable. I have presented problems that spanned a range of level of difficulty. From messy and unmodelled to “pre-digested.” Among the messy ones: I recall presenting the infamous “loss of lubrication problem” that has its origin in head disk interface issues of hard disk drives. At that time the disks were brown disks having a lubricated particulate magnetic coating. Some engineering systems that are “lubricated for life” gradually lose their lubricant and I thought that this was quite analogous to door hinges. In addition to the interesting math that was discussed during that particular workshop, I also appear to have sensitized some RPI faculty to the state of the door hinges in their houses, on campus and wherever interesting lubricant spray patterns can be observed! A second messy problem, also derived from the hard drive industry, dealt with the fundamental changes in the flow pattern that can be observed when one gradually adds fine particles to a shearing lubricant flow. The flow changes from linear shear flow to a flow involving “rolls” and “balls.”
I found that it is a good idea to bring several messy problems to avoid getting stuck too long.
All of the math problems I presented were in the general area of fluid dynamics and arose during my research in various printing technologies and hard disk drive technology. It is surprising to some that these areas deal at all with fluid dynamics, but they turn out to be prolific sources of mathematical problems involving classical equations such the Reynolds lubrication equation. We have become so good at solving this equation that we are now tempted to try synthesis problems. Such problems are also called design problems in the sense that we would like to find the best configuration of a physical system that performs some task in the best possible way. In practice, solutions to design problems are tackled with extensive numerical simulation such as application of genetic algorithms, etc. The workshop used to be too short to do much numerical work, but this has changed rapidly. Often, though, what is really desirable is better modeling. I still hear the late Julian Cole urging us on to “find a better equation.” Adhering to this philosophy, the math workshop has contributed substantially to a deeper understanding of a new class of air bearing designs called Tango, where the better equation is simply ph = constant., describing the relationship between pressure and local bearing gap in an air bearing at infinite bearing number. It turned out that the consequences of this equation had been largely ignored in air bearing design. This insight allows us to “fly” the read/write heads of a disk drive at nearly constant fly height above the disk surface, regardless which data track is being accessed. The workshop also successfully explored the sensitivity of fly height of disk drives to altitude.
Recently, a problem from the area of planar write coil design showed that Laplace’s equation can still be interesting if it is part of a design problem; in this case minimization of the Ohmic resistance, keeping in mind manufacturing constraints.
In addition to the mathematical work going on, the workshop has kept me in touch with mathematicians and academia. I’ve learned a lot from the very gifted participants in the workshop and I hope that I have been able to communicate some of the needs of research in the industrial world.