Professor Philip Yecko was awarded a National Science Foundation (NSF) research grant by the Computational Mathematics program of the Division of Mathematical Sciences.It is titled "Transport of inertial particles in time-dependent and inertial flows"

Abstract
Objects in moving fluids rarely go with the flow. Instead they may sink, swim or steer in order to reach a destination, or they may respond to other influences, including their own sizes and shapes. Taking advantage of the ways that real objects interact with flows enables a wide range of important technologies. On small scales, micro robots may be steered inside the human body to perform surgery. On the largest scales, ocean drifters may efficiently monitor currents, marine life or global weather patterns. Each example presents challenges originating from the complex fluid flow patterns and from the difficulty in planning the most efficient navigation strategy. This project concentrates on the challenges of positioning autonomous vehicles in the ocean, where unpredictable and variable currents, seasonal variability, weather events, and other random influences must also be accounted for. Computer models of fluid flows and mathematical models of control will be combined to find optimal strategies to position autonomous ocean vehicles. Laboratory experiments will use precisely tuned fluid flows and remotely controlled particles to capture the important effects of the vehicles' mass, size, and shape. The project payoff is significant in that a better monitored ocean is advantageous to fishing and shipping, the military, and environmental monitoring. The project will involve and support undergraduate and graduate students in leading-edge research. Significantly, the student population at Montclair State University, and in particular, the Department of Mathematical Sciences, includes a substantial proportion who are members of groups underrepresented in STEM disciplines (including women and minorities) and the research program will leverage existing programs directed to these students. The outcome of the research will be disseminated through seminars, presentations at meetings, and publications in peer-reviewed journals.

In this project computer models of fluid flows and mathematical models of transport and control will be combined to find optimal control strategies for autonomous ocean vehicles, which will be modeled both as inertial and non-inertial objects. Laboratory experiments on precisely tuned flows and magnetically controlled particles will be used both to validate and guide the investigations. The goal is to use experimental and computed flow fields to identify critical transport features and integrate these features into control algorithms that optimally position particles. This project will improve transport control capabilities by developing models for transport and control of inertial objects in canonical flows subject to time-dependent and stochastic perturbations. Flow data will be generated by the numerical simulation of gyre flows, jets, and boundary currents. Inertial particles will be modeled directly using a state-of-the-art interfacial multi-phase numerical code. Laboratory experiments have been designed so that similar flows can be generated by reconfiguring the geometric forcing devices. Through high resolution particle imaging velocimetry (PIV) and particle tracking, experimental flow fields and their transport properties will be correlated with those of the associated model flows. Additionally, control strategies will be implemented using ferromagnetic tracer particles and magnetic pulses. To gain insight into the flow transport properties, flow and tracer data will be analyzed using a variety of geometric and probabilistic methods including finite-time Lyapunov exponents, inertial particle models, almost invariant sets, and finite-time coherent sets. These techniques will directly result in the ability to identify loitering regions and their boundaries and to determine maximal transport rates. This information will be leveraged to develop simple predictive models of transport and trajectory control that can be efficiently adapted to emergent applications.