Special Lefschetz Center for Dynamical Systems Seminar
Abstract: A new family of exact solutions is analyzed, which model 2D circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, whose fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates.
Brown University
Joint Materials/Solid Mechanics Seminar Series
Abstract: Intergranular pressure solution (IPS) could be compared to Coble creep with the addition of the action of fluids within the grain boundary. The overall compaction of porous rocks due to IPS results from the dissolution of minerals within contact regions and the diffusive transport through the grain boundary of the dissolved species towards the fluid-filled pore space. The evolution of the solid-fluid interfaces within the grain boundary is governed by the phase transformation between the non-hydrostatically stressed elastic solid and the fluid with a driving force which is the local jump in chemical potential. IPS is one the main deformation mechanisms during the early diagenesis of sedimentary rocks.
The objective of this talk is to review some of the mechanical issues concerning the modeling of pressure solution, as a phase transformation within the grain boundary, at the length scale of an aggregate and, finally, on the scale of sedimentary basins. At the scale of contacts within the grain boundary, a finite-element scheme has been developed [1] to capture the solid-fluid kinetically driven phase transformation. It is shown how the Helmholtz free energy entering the driving force for phase transformation is responsible for setting the time necessary to undercut asperities and to wet the grain boundary. At the scale of aggregates, phenomenological models of IPS have been proposed, assuming periodic grain structure, providing the relative role of the characteristic time for kinetics and diffusion in the overall compaction law. Possible extensions of such contact models for arbitrary shaped grains will be discussed. The periodic microstructure is then used to describe the time evolution of the representative volume element and to study the early compaction of a thick sedimentary layer, embedded in a fixed hydrostatic pressure and geothermal temperature field [2].
[1] Ghoussoub J. and Y.M. Leroy, Solid-fluid phase transformation within grain boundaries during compaction by pressure solution, Journal of the Mechanics and Physics of Solids, 49, 2385-2430, 2001.
[2] Lehner F.K. and Y.M. Leroy, Sandstone compaction by intergranular pressure solution. Chapter 3 in Mechanics of Fluid Saturated Rocks, Edts Y. Gueguen and M. Bouteca, International Geophysics Series, Academic press, 2004.
*Clark B. Millikan Visiting Professor, Aeronautics, Caltech, 2003-04
Center for Fluid Mechanics Seminar
Abstract: Turbulent reactive flows are prevalent in combustion, the chemical process industry and elsewhere. Predictive computational tools are sought to aid in the design of reactive-flow devices, to improve performance, and to shorten the design cycle. But the physics and chemistry of turbulent reactive flows is challenging: turbulence causes substantial property fluctuations over a broad range of scales; and the chemical reaction rates are typically complex, non-linear functions of the thermochemical properties. Several recent advances have facilitated the modeling and computation of turbulent reactive flows. In PDF methods, the turbulent fluctuations are fully represented through the joint probability density function (PDF) of the thermochemical and flow variables, for which a modeled transport is solved by a particle method. Various dimension reduction schemes have been developed to simplify the chemistry, and storage-retrieval algorithms have been developed which greatly reduce the computational cost of implementing chemical reactions. Some calculations of turbulent flames are presented to illustrate the capabilities of these computational methodologies.
Brown University
Joint Materials/Solid Mechanics Seminar Series
The University of Michigan, Ann Arbor, MI 48109-2140 | |
Martensitic Transformations in Shape Memory Alloys | |
Abstract: Some of the most interesting and technologically important solid-solid transformations are the first order diffusionless transformations that occur in certain ordered multi-atomic crystals. These include the reconstructive martensitic transformations (where no group-subgroup symmetry relationship exists between the phases) found in steel and ionic compounds such as CsCl, as well as the thermally-induced, reversible, proper (group-subgroup relationships exist) martensitic transformations that occur in shape memory alloys such as NiTi. Shape memory alloys are especially interesting, for engineering applications, due to their strong thermomechanical (multi-physics) coupling. The mechanism responsible for these temperature-induced transformations is a change in stability of the crystal's lattice structure as the temperature is varied.
To model these changes in lattice stability, a continuum- level thermoelastic energy density for a bi-atomic multilattice is derived from a set of temperature- dependent atomic potentials. The Cauchy-Born kinematic assumption is employed to ensure, by the introduction of internal atomic shifts, that each atom is in equilibrium with its neighbors. Stress-free equilibrium paths as a function of temperature are numerically investigated, and an asymptotic analysis is used to identify the paths emerging from \multiple bifurcation" points that are encountered. The stability of each path against all possible bounded perturbations is determined by calculating the phonon spectra of the crystal (Bloch-wave method). The advantage of this approach is that the stability criterion includes perturbations of all wavelengths instead of only the long wavelength information that is available from the stability investigation of homogenized continuum models. The above methods will be reviewed, and results corresponding to both reconstructive and proper martensitic transformations will be presented. Of particular interest is the prediction of a transformation that has been experimentally observed in CuAlNi, AuCd, and other shape memory alloys.
Brown Analysis Seminar
Special Lecture Series
Brown University
Graduate School, Ph.D. Dissertation Talk
Special Lecture Series
PDE Seminar
Abstract: The inverse scattering approach linearizes the Nonlinear Schroedinger (NLS) equation by associating it with a linear 2 by 2 first order ODE eigenvalue problem, the Zakharov-Shabat (ZS) system. The scattering information of ZS constitutes the input to a Riemann-Hilbert problem (RHP) in the plane of the spectral variable whose solution produces the solution to NLS. Thus, the RHP plays a role that is analogous to the role of the Fourier integral in linear PDE. (This is joint work with A. Tovbis and X. Zhou.)
We will introduce the above methodology and will outline
the rigorous asymptotic methods we have developed
that lead to the following results:
1) Proof of existence and basic properties of the first
breaking curve in space-time above which a phase transition
(break) occurs.
2) Post-break structure of the solution.
3) Proof that for pure radiation no further breaks occur.
4) Rigorous error estimate, and
5) Rigorous asymptotics for the large time behavior of the
system in the pure radiation case.
<--- 2004 Index