Scaling Laws and Optimal Design in Some Problems of Continuum Mechanics

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The problems under investigation in this project are at the forefront of the study of energy-driven pattern formation and more generally the Calculus of Variations. The project is interdisciplinary by design - the problems cut across the fields of materials science and fluid dynamics - and is motivated by both practical and futuristic engineering applications. In materials science, for instance, there is the potential for new biologically-inspired and growth-based design mechanisms for building highly flexible but controlled solid membranes. Regarding fluid dynamics, besides the potential for the design of new and extremely efficient heat exchangers, the investigation is motivated by the broader scientific challenge of deriving the scaling laws of naturally occurring transport in turbulent fluids. As experience shows, these questions are very difficult to solve on a computer by applying conventional numerical techniques. There are oftentimes many nearly optimal answers, and each can be rather complex. Rigorous mathematical analysis presents the possibility of knowing that certain patterns are truly optimal overall: such information cannot be gleaned using other non-rigorous approaches. Thus, the project is an important contribution to the larger scientific community interested in understanding why or how such complex patterns achieve global optimality. This project consists of two main parts: (a) The first class of problems concerns the mechanics of thin elastic sheets, such as naturally occurring ones like leaves and flowers, but also man-made versions which can be orders of magnitude thinner than a standard sheet of paper. The thinner an elastic sheet becomes the more easily it can be deformed, in particular through bending forces, but extracting from such intuitive physical principles the mathematical mechanisms behind the vast array of wrinkling, folding, and crumpling patterns found in experiments is a clear and difficult challenge. The investigator seeks to derive through rigorous mathematical analysis the effective, coarse-grained models governing optimal elastic patterns in the vanishing thickness limit. A key task is to obtain the scaling law of the minimum energy in any natural parameters, and then to ask which patterns can possibly attain such energy scaling laws. (b) The second class of problems regards the design of incompressible fluid flows for achieving optimal heat transfer. While hot objects submerged in fluid media do cool on their own, it is natural to ask whether the efficiency of heat transfer can be significantly enhanced by intelligent stirring of the surrounding fluid. Oftentimes, any stirring protocol achieves some enhancement of heat transfer, but determining which strategies attain maximal heat transfer overall is an intriguing and open question. In the advection-dominated limit, which can be reached by increasing the amount of power available with which to stir, such optimal strategies exhibit patterns remarkably similar to those from problems of "energy-driven pattern formation" in mathematical materials science. The investigator is developing a precise and general link between these apparently related classes of problems, and is exploring its ramifications for the study of the scaling laws of naturally occurring turbulent transport.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.