PDE and Applied Mathematics Seminar[ Edit ]
Moderator: Amy Novick-Cohen
Adoption of new products that mainly spread through word-of-mouth is a classical problem in Marketing. In this talk, I will use agent-based models to study spatial (network) effects, temporal effects, and the role of heterogeneity, in the adoption of solar PV systems.
In this talk I will discuss a model for auto-ignition of fully developed free round turbulent jets consisting of oxidizing and chemically reacting components.I will present the derivation of the model and present results of its mathematical analysis.
The derivation of the model is based on well established experimental fact that the fully developed free round turbulent jets, in a first approximation, have the shape
of a conical frustum. Moreover, the velocity as well as concentrations fields within such jets, prior to auto-ignition, assume self-similar profiles and can be viewed as prescribed. Using these facts as well as appropriately modified
Semenov-Frank-Kamenetskii theory of thermal explosion I will derive an equation that describes initial stage of evolution of the temperature field within the jet.
The resulting model falls into a general class of Gelfand type problems.
The detailed analysis of the model results in a sharp condition for auto-ignition of free round turbulent jets in terms of principal physical and geometric parameters involved in this problem. This is a joint work with M.C. Hicks and U.G. Hegde of NASA Glenn Research Center.
In his famous 1900 ICM address Hilbert proposed his famous list of problems for the 20th century. Among these was his 6th problem which was less clearly formulated than the others but dealt with a rigorous derivation of the macroscopic equations of continuum mechanics from the available microscopic theory of his time, i.e. statistical mechanics and specifically Boltzmann's kinetic theory of gases. The problem has drawn attention from analysts over the years and even Hilbert himself made a contribution. In this talk I will note how an exact summation of the Chapman-Enskog expansion for the Boltzmann equation due to Ilya Karlin ( ETH) and Alexander Gorban (Leicester) can be used to represent solutions of the Boltzmann equation and then show that these solutions CANNOT converge the classical balance laws of mass, momentum, and energy associated the Euler equation of compressible gas dynamics. Hence alas Hilbert's program (at least with respect to gas dynamics) has a negative outcome.