Petia M. Vlahovska Group

Current teaching

ESAM 311-1 and 2: Methods of Applied Mathematics
Ordinary differential equations; Sturm-Liouville theory, properties of special functions, solution methods including Laplace transforms. Fourier series: eigenvalue problems and expansions in orthogonal functions. Partial differential equations: classification, separation of variables, solution by series and transform methods.

MATH 234 Multiple Integration and Vector Calculus
Cylindrical and spherical coordinates, double and triple integrals, line and surface integrals. Change of variables in multiple integrals; gradient, divergence, and curl.

Upcoming Mathematical modeling modules:

Mathematics of cell shapes
Differential geometry. Curvature-elasticity models of biomembranes. Equilibrium shapes of vesicles and the red blood cell. Shape fluctuations /`flicker spectroscopy.

Fluid-structure interaction in Stokes flows (dynamics of drops, capsules, vesicles.)
Stokes flow. Multipole expansions. Perturbative solutions for nearly spherical particles -spherical harmonics expansion. Vesicle dynamics -bifurcations and chaos. Rheology of suspensions.

Active matter and active fluids
Collective motion of interacting self-propelled particles. The Vicsek Model. Role of hydrodynamics. Synthetic and living active matter.

Electrified interfaces
Equilibrium double layer: Poisson-Boltzmann equation. Electrokinetic phenomena. Poisson-Nernst-Planck equations. Electrohydrodynamic instabilities. Membranes: Donnan equilibrium. Nernst potential. Action potentials.

Particles at interfaces
Shapes of fluid interfaces. Minimal surfaces. Particles at interfaces. Capillary interactions. Contact line motion. Interfacial rheology. Curvature-driven interactions between membrane-bound inclusions (proteins, nanoparticles).

Past teaching

Fluid Mechanics
Junior-level fluid mechanics course. Topic include hydrostatics; mass, momentum, and energy conservation; control volume analysis; Navier-Stokes equations; viscous flow in pipes; lift and drag; compressible flow; and open-channel flows. Laboratory and project.

Complex fluids
Graduate level course introducing disperse systems (colloidal suspensions, emulsions, surfactant solutions, blood) with special attention to the thermodynamics and mechanics of interfaces. The course bridges the physico- chemical and mechanical perspectives in the study of structured fluids.

Heat and mass transfer
Graduate level course providing an unified study of momentum, heat and mass transfer; kinetic theory of transport properties; scaling and order-of-magnitude concepts; analytical and approximate solutions to the equations of change; forced and natural convection; radiation; diffusion in mixtures; simultaneous momentum, heat and mass transfer; Taylor dispersion; transport in electrolyte solutions; special topics (e.g., transport at interfaces, porous media).

Fluid Mechanics
A second part of a two-semester graduate course. It covers topics from incompressible, Newtonian flows (Stokes flow, lubrication theory, free-surface flows, hydrodynamic stability), electrokinetics, geophysical fluid dynamics, and if time permits explores some more specialized topics of current research interest. The emphasis is on basic physics, scaling and nondimensionalization, and approximations that can be used to obtain analytical solutions.

Cellular and molecular biomechanics
Advanced undergraduate/beginning graduate course on the engineering principles of cell design. Topics include elasticity of biopolymers and biomembranes, rheology of cytoskeletal components, molecular motors, cell motility. The course connects cell mechanics to micro- and nano- technology.

Biological Physics
Sophomore-level course introducing physics and engineering approaches to analyze biological problems. Topics include the architecture of biological cells, molecular motion, entropic forces, enzymes and molecular machines, and nerve impulses.