We consider the problem of locating stagnation points in the flow produced by a system of N interacting point vortices in 2-D. There is a general solution, due to a theorem by Siebeck (1864), that the stagnation points are the foci of a certain plane curve of class N -1 that has all lines connecting vortices pairwise as tangents. The theorem uses ideas from algebraic geometry. We have considered in detail the case N =3, for which Siebeck's curve is a conic. The issue of invariance of the topology of streamlines during the motion is being explored.

This research investigates mathematically the propagation of a surface wave over a curved elastic shell and demonstrates that it will couple from the exterior surface to the interior one, propagate along it, and ultimately return to the exterior. Thus, it can sense damage to an interior surface and carry that information to the exterior. Curved structures such as pipelines must be nondestructively inspected for damage at their interior surface while in service. Ultrasonic inspection using this wave provides one way to carry this out. This research also studies the scattering experienced by this wave when it strikes a fatigue crack or encounters a patch of corrosive roughness, both indicators of damage.

Diffraction from a strip when the strip is rigid has many approximate solutions of varying degrees of accuracy. One method is the Wiener-Hopf technique, which yields uniform asymptotic expressions for wide strips. The present research extends this technique to problems where the strip is elastic and therefore capable of interacting with the incident sound wave in a way that makes the interaction quite nonlocal. The problem is formulated as a set of integral equations. Solutions are sought in the form of asymptotic expansions using the thickness of the strip as the small parameter. The remaining set of integral equations is then cast into the form of a matrix Wiener-Hopf problem. This work is part of a U.S.-Pakistan scientific exchange.

Numerical simulation of turbulence has generally relied on highly accurate numerical methods known as spectral methods. However, spectral methods are extremely limited in the geometries in which they can be used and this (among other things) has limited the application of turbulence simulation. A new class of numerical methods is being developed that retains much of the high accuracy of spectral methods, but which is considerably more flexible. The methods are similar to finite-element methods, but use spline expansions, which are more accurate than finite elements, but less flexible. They can be viewed as a compromise between the extreme accuracy of spectral methods and the great flexibility of finite-element methods.

All of the conventional approaches to the plane problem of linearized elastostatics (i.e., plane strain, plane stress, and generalized plane stress) rest on approximations that go beyond those inherent in the underlying linearized theory of elasticity. Here, we use some new results due to D. Gregory concerning the form of the exact three- dimensional solution to the plane problem to investigate these approximations.

This new project seeks to exploit all relevant asymptotic limits for the standard detonation model to construct a wholly analytic theory of detonation stability. New and simplified stability formulations will be given. Evolution equations for the dynamics of the detonation shock are being derived. Exploration of new theory for cellular detonation dynamics continues.