Most flows and transport processes encountered in nature and industry are nonlinear and give rise to spatio-temporal patterned states, e.g., wave-trains on the surface of flowing water, droplets arrayed along a thin rod, sediment patterns on the sea floor, and Faraday hexagons on a vibrated liquid pool.

These patterns raise several fascinating questions: What is the mechanism that gives rise to these patterns? How do they manage to remain well ordered even in uncontrolled environments? What are the competing forces that set the length and time scales of the pattern? Can the patterned state be suppressed or precipitated by boundary or external forces, or changes to the microscopic constituents of the system? These questions demand attention—even if one were immune to the beauty of pattern formation—because of the significant impact that self-organized structures have on heat/mass transport rates and mechanical forces. 

We have a reasonably clear understanding of how simple patterns (with few active modes) emerge and evolve, at least close to onset. Typically, they arise from instabilities of base solutions that are either unidirectional flows or quiescent states. In such cases, significant progress can be made via analytical methods, such as linear and weakly nonlinear stability analysis, and relatively simple simulations. A part of our work consists of applying this theory to a variety of problems, including interfacial instabilities in engineering systems, mucus flows in lungs, and channeling instabilities in magma. However, we are also deeply interested in the many fundamental issues that remain open, especially with regard to the strongly nonlinear, far-from-onset dynamics, and the (possibly competitive) interaction of distinct instability modes. Our approach is to study these issues in the context of concrete physical systems, while trying to maintain a close connection between mathematical and physical insight.