To most, the action of manually mixing two immiscible fluids or materials is an afterthought. For the vast majority, it is simply a mundane task that occasionally needs doing, and often very little thought goes into whether there is a ‘right’ or ‘wrong’ way to accomplish it. But to Professor Gautam Iyer of the Carnegie Mellon University Department of Mathematical Sciences and a growing number of mathematicians and material scientists, there is a steadily increasing consensus that there are, in fact, correct and incorrect ways to go about mixing materials when efficiency is the goal. In this week’s installment of the AMS Department’s colloquium series, Prof. Iyer provided a brief look into some of his research on optimizing material mixing and fluid stirring techniques via mathematics.
The first step in accomplishing this, according to Iyer, was to specify the material being mixed and to consider it as diffusion-free, or something that will not disperse and spread on its own without being physically mixed. This material is then noted mathematically by designating it with a passive scalar value that will be used in later calculations. In a similar way, the fluid being mixed into must be designated as incompressible, and is then assigned a flow value on a predetermined torus structure. Next, the relationship between these two specified values is studied under constraint of an enstrophy condition. This enstrophy constraint essentially a limit placed upon the vorticity (vector field variation describing the ‘spin’ of a fluid at a certain point) of the velocity field in the given system.
Iyer then explained how once these conditions were in place, it was then possible to derive a theoretical result that stated the entire mixing system is bounded at its lowest parts by some exponential function of time. This exponential function essentially models the ‘distinctness’ of the materials being stirred, which would theoretically decrease the more it was mixed, resulting in an exponential decay function for the system. The theoretical model also suggested that the rate of exponential decay varied from situation to situation, depending upon numerous factors determined by the initial material dataset. When analyzed through the use of numerical simulations, all aspects of this theory were confirmed, given that the ‘mixing’ occurred for a significant amount of time (longer than a minimum time that must be determined for each individual situation).
Lastly, Iyer talked about how this was a relatively young branch of fluid and material mechanics with very little previous literature on the subject. Due to this, Iyer explained that this study was based heavily upon theory, and that while it is certainly a step in the right direction, there are still several lingering questions and issues that should be addressed. Namely, further research into how the presence of walls (rigid surfaces such as containers on the edges of the materials being mixed) affects the efficiency of mixing among many other things would be beneficial, as the derived theory would suggest that walls may actually lower the rate of exponential decay. As part of his and his team’s future work, Iyer plans on delving deeper into this fairly new field and try to answer some these remaining questions on the subject.