# Physical Applied Mathematics

This area has two complementary goals:

- to develop new mathematical models and methods of broad utility to science and engineering; and
- to make fundamental advances in the mathematical and physical sciences themselves.

Our department has made major advances in each of the following areas. We've
developed a theoretical framework to describe the *induced-charge mechanism*
for nonlinear electro-osmotic flow. Our work in *biomimetics* focuses on
elucidating mechanisms exploited by insects and birds for fluid transport on a
micro-scale. These and other activities in *digital microfluidics and
nanotechnology* have applications in biologically inspired materials such
as a unidirectional super-hydrophobic surface, and devices such as the `lab-on-a-chip'
and micropumps. The theory of *transport phenomena*} provides a variety of
useful mathematical techniques, such as continuum equations for collective motion,
efficient numerical methods for many-body hydrodynamic interactions, measures of
chaotic mixing, and asymptotic analysis of charged double layers. *Nanophotonics*
is the study of electromagnetic wave phenomena in media structured on the same
lengthscale as the wavelength, and is an active area of study in our group, for example
to allow unprecedented control over light from ultra-low-power lasers to hollow-core
optical fibers. New mathematical tools may be useful here, to give rigorous theorems
for optical confinement and to understand the limit where quantum and atomic-scale
phenomena become significant. *Granular materials* provide challenging problems
of collective dynamics far from equilibrium. The intermediate nature (between solid and fluid)
of dense granular matter defies traditional statistical mechanics and existing continuum
models from fluid dynamics and solid elasto-plasticity. Despite two centuries of research
in engineering, no known general continuum model describes flow fields in multiple
situations (say, in silo drainage and in shear cells), let alone diffusion or mixing of discrete
particles. A fundamental challenge is to derive continuum equations from microscopic mechanisms,
analogous to collisional kinetic theory of simple fluids. On a far larger scale, we have also
been remarkably successful in unraveling some of the curious *dynamics of galaxies*.