A numerical method to simulate interfacial surfactant mechanics within a volume of fluid method has been developed. Two important features of this new method are that it conserves surfactant mass exactly and the form of the equation of state is not restricted, i.e. the relation between surfactant concentration and surface tension can be linear or nonlinear. To conserve surfactant, the surfactant mass and the interfacial surface area are tracked as the interface evolves, and then the surfactant concentration is reconstructed. The algorithm is coupled to an incompressible Navier-Stokes solver that uses a continuum method to incorporate both the normal and tangential components of the surface tension force into the momentum equation.

Numerical simulations demonstrate the effect of surfactant on the dynamics of several problems by comparison to surfactant-free simulations. First, the buoyant rise of a bubble is examined. Next, the evolution of a drop in an extensional flow is studied. Finally, the motion of a drop through a constriction is investigated. In each of these problems surfactant accumulation allows high interface curvature and the formation of small secondary drops or bubbles.

This is a continuation of the learning seminar on
the scattering theory for wave equations.

Public key cryptography is about 25 years old, and relies on number theory. We will discuss Diffie Hellman key exchange and ElGamal encryption, and some recent improvements on them. We show how number theory and algebraic geometry, and in particular the rationality of certain algebraic tori, can be used to give a deeper understanding of these improvements, and to give new cryptosystems.

The motion of an elastic solid inside of an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a coupled PDE system between parabolic and hyperbolic phases, the latter inducing a loss of regularity. In this talk, I will outline the proof of the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. The functional framework employed is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.