Wednesdays at 4 PM in Eckhart 202.
This is a joint work with Weinan E from the Courant Institute. We consider experiments by Tanaka on spinodal decomposition of a semi dilute solution of a polymer in a small--molecule solvent. The observed pattern formation is qualitatively different from spinodal decomposition of a mixture of two small--molecule liquids. Tanaka suggests to use a model by Doi and Onuki for his experiments. The main feature of this continuum level model is that it takes into account the effect of transient entaglements of the polymer molecules: The polymeric component is modelled as a viscoelastic material. We claim that the model by Doi and Onuki has to be modified in order to reproduce the observed pattern formation at least qualitatively. The modification is in the stress--strain rate relation for the polymeric component. We hope to convince the audience of the necessity of the modification by presenting numerical simulations and a heuristic analysis which is based on the gradient flow structure of the model.
- Anatoly Neishtadt, Space Research Institute, Moscow
"On calculation of stability loss delay time for dynamical bifurcation"In classical bifurcation theory the behavior of systems, depending on parameter, is considered for values of the parameter close to some critical, bifurcational one. In theory of dynamical bifurcations the parameter is changing slowly in time and passes through the value, which would be critical in classical static theory. Some phenomena, arising here, are drastically different from predictions, derived by static approach. Let at bifurctional value of the parameter the equilibrium loses its asymptotic linear stability, but remains nondegenerate. In analytic systems the stability loss is inevitably delayed: the phase points remain near the unstable equilibrium for a long time after bifurcation; during this time the parameter changes by a quantity of order 1. Such delay is not in general found in nonanalytic (even infinitely smooth) systems. The talk is devoted to estimates of delay time. The delay time is controlled by behavior of solutions in the plane of complex time.
- October 21
- Peter Constantin, University of Chicago
Bounds for Dissipation in Infinite Prandtl Number Convection.We will show how one can obtain upper bounds for the average dissipation in active scalar equations, and in particular the infinite Prandtl number Boussinesq equation. We will show how lower order perturbations, such as rotation, can have a significant and nontrivial effect.
- October 28
- Rich McLaughlin, University of North Carolina-Chapel Hill
Anelastic Mixing: Transport by Weakly Compressible FlowMany studies to date have focused upon the turbulent diffusion of a passive scalar in the presence of an incompressible fluid flow. This is quite natural as a means for understanding mixing in turbulent environments for which the fluid in question satisfies a zero divergence constraint. However, for many physical environments, the fluid density is not constant, and may admit a non-trivial adiabatic steady-state density profile leading to non-zero flow divergence constraints. Such is the case when considering the atmosphere over moderately large vertical scales. We'll first review a homogenization theory for calculating enhanced diffusion coefficients in the case of periodic incompressible flows which demonstrates complicated flow parameter dependence. Then, we'll discuss the extension of this theory to a simplified flow model aimed at describing fluid flow in the presence of large-scale density variation. We compare the predictions of this theory with the incompressible theory and demonstrate some interesting differences in the effective bulk transport of the scalar quantity specifically due to the combined effects of the variable density profile and small-scale fluid motion.
- November 4
- Lenya Ryzhik, University of Chicago
Level set structure in some turbulent combustion problemsTurbulent flows have pronounced effect on combustion processes. We study two models of turbulent combustion: G equation and reaction-diffusion KPP type equations. It turns out that the sturcture of the level sets for the two models is significantly different. We use our resutls for the area of the level sets to obtain a non-rigorous upper bound for the effective front speed. We also obtain rigorous lower bounds on the burning rate in the homogenization regime. This is an ongoing joint work with Peter Constantin and Alexander Kiselev.
- November 11
- Vered Rom-Kedar, Weizmann Institute of Science
Big islands appearing in Near-ergodic smooth flowsWe consider the motion of a particle in a very steep two dimensional potential. In the limit of infinite steepness of the potential the particle travels with a constant speed in a region, undergoing elastic collisions at the region's boundary - namely it performs a billiard motion. When the boundary is concave, causing neighboring trajectories to diverge upon reflection, the billiard is called scaterring. Due to the divergence instability the scattering billiards are ergodic and mixing systems. Thus, these have been suggested as a first step models for substantiating the basic assumption of statistical mechanics - the ergodic hypothesis of Boltzmann. We prove the existence and derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in physically natural smooth Hamiltonian approximations of such scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, {\it universality} classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter $\eps \goto 0$, islands of {\it power-law} size in $\eps$ appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect. Joint work with Dr. D. Turaev.
- 4:15pm, November 19 (rescheduled to Thursday)
- David Sattinger, University of Minnesota
"Soliton Collisions in the Ion Acoustic Plasma Equations"Numerical experiments involving the interaction of two solitary waves of the ion acoustic plasma equations are described. An exact 2-soliton solution of the relevant KdV equation was fitted to the initial data, and excellent agreement was maintained throughout the entire interaction, demonstrating that the KdV approximation is quantitatively accurate for the interaction of solitary waves in plasmas at low amplitudes.
- November 25
- To be announced
- December 2
- Yuan-nan Young, University of Chicago
Layer Dynamics in Doubly-Diffusive ConvectionLayers are seen in the sedimenting experiments, where an initially linear profile of sedeiment concentration evolves to a staircase. Similar phenomena are also found in douly-diffusive slot convection, where in addition to an initial, linear concentration profile, a lateral temperature difference is imposed across the slot. In this talk, I will present a linear and weakly nonlinear analysis on such a system, and provide an explanation for the formation of layers . In addition, I will show computational results from two-dimensional simulations using a pseudospectral method. Through the computations, we are able to quantify layers and describe their nonlinear dynamics. Finally, We will provide a discussion on the sources of similarities/differences of the layer formation between these two systems: laterally driven slot convection and stirred stratified fluid. This is a joint work with R. Rosner.
- December 9
- Graeme Milton, University of Utah
Exact relations and fast numerical schemes for compositesThis is joint work with David Eyre, Yury Grabovsky and Dan Sage. Typically, the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for ( or linking) effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data, and for evaluating the merit of various approximation schemes. Classic examples include, Hill's formulae for the effective bulk modulus of a two-phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two-phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two-dimensional polycrystalline material. Here we present the first systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for tensor subspaces satisfying a certain algebraic condition. One of many new exact relations is for the effective shear modulus of a class of three-dimensional polycrystalline materials. The series expansions which prove useful for establishing exact relations also turn out to have especially rapid convergence properties, and provide the foundation for a fast numerical scheme for computing the response of composites.
- December 16
- Mary Pugh, University of Pennsylvania
Long-wave instabilities in thin film equations --- blow-up, saturation, and steady-statesHocherman and Rosenau conjectured that longwave unstable Cahn-Hilliard type interface models develop finite time singularities when the nonlinearity in the destabilizing term grows faster at larger amplitudes than nonlinear effects in the stabilizing term. Andrea Bertozzi and I show that for a subclass of their general model, corresponding to a family of equations often used to model thin films in a lubrication context, that in fact the destabilizing term can be stronger (by up to a power of two in the nonlinear diffusion coefficient) yet the solution still remains globally bounded. We show that this alternate scaling can be easily explained by a conservation of volume constraint and we prove this bound rigorously using energy methods, an entropy dissipation, and interpolation inequalities. Finally we show that when the interface equations permit globally bounded solutions, a weak solution existence theory can be established following similar arguments developed by the authors in recent publications. We also study finite-time blow-up. Richard Laugesen and I study nontrivial steady states of these equations; when do they exist, are they unique, are the linearly stable? We find that one cannot arbitrarily specify the length and area of a steady state, and when there is a steady state with the specified length and area, it may be nonuniqe. We address similar questions about specifying the area and contact angles, etc.
Winter 1999
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