CAREER: Analysis and Numerics for the Dynamics of Fluids under Magnetic Forces

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

While invisible to the eye, magnetic fields contribute to many phenomena in nature and aspects of daily life, such as ocean tides, solar flares, or electric motors. Additionally, they are used to manipulate materials for industrial applications like precision sensors, liquid-metal cooling of nuclear reactors, or magnetic drug targeting. The goal of this project is to study several mathematical models involving fluids forced by magnetic fields, and develop numerical algorithms for their simulation on a computer. This will benefit practical applications in engineering and in physical and biomedical sciences. Educational components targeting students at the high school, undergraduate and graduate level, are integrated with the research activities. The research program includes projects suitable for graduate and undergraduate research. Curriculum development for undergraduate courses in computational mathematics, and outreach to high school students in the form of an interdisciplinary math and engineering summer camp, will be undertaken. The objective of this project is the mathematical investigation of three important problems motivated from science and engineering. The first is the numerical simulation of liquid crystals subjected to magnetic fields, which are used in LCD screens. The second deals with mathematical aspects of magnetohydrodynamics turbulence in order to gain more insight into the emergence and existence of the magnetic field of the earth. The third problem concerns the question of how experimental data affects mathematical models: Probabilistic tools will be employed to develop algorithms for uncertainty quantification in compressible flows applications. These problems are described mathematically by nonlinear systems of mixed type partial differential equations (PDEs). The mathematical treatment of these systems requires the development of new analytical techniques and innovative algorithms for their simulation. The finite difference and finite volume schemes constructed in this project will be analyzed with mathematical tools such as energy estimates, compensated compactness and relative entropy methods to prove robustness and convergence. 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.