36 research outputs found
Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension
AbstractWe consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments
Spiral flames
AbstractWe describe computations of periodic and meandering spiral patterns in a reaction-diffusion model of flames
Structure And Dynamics Of Modulated Traveling Waves In Cellular Flames
We describe spatial and temporal patterns in cylindrical premixed flames in
the cellular regime, , where the Lewis number is the ratio of
thermal to mass diffusivity of a deficient component of the combustible
mixture. A transition from stationary, axisymmetric flames to stationary
cellular flames is predicted analytically if is decreased below a critical
value. We present the results of numerical computations to show that as is
further decreased traveling waves (TWs) along the flame front arise via an
infinite-period bifurcation which breaks the reflection symmetry of the
cellular array. Upon further decreasing different kinds of periodically
modulated traveling waves (MTWs) as well as a branch of quasiperiodically
modulated traveling waves (QPMTWs) arise. These transitions are accompanied by
the development of different spatial and temporal symmetries including period
doublings and period halvings. We also observe the apparently chaotic temporal
behavior of a disordered cellular pattern involving creation and annihilation
of cells. We analytically describe the stability of the TW solution near its
onset+ using suitable phase-amplitude equations. Within this framework one of
the MTW's can be identified as a localized wave traveling through an underlying
stationary, spatially periodic structure. We study the Eckhaus instability of
the TW and find that in general they are unstable at onset in infinite systems.
They can, however, become stable for larger amplitudes.Comment: to appear in Physica D 28 pages (LaTeX), 11 figures (2MB postscript
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Studies in nonlinear problems of energy. Final report
The author completed a successful research program on Nonlinear Problems of Energy, with emphasis on combustion and flame propagation. A total of 183 papers associated with the grant has appeared in the literature, and the efforts have twice been recognized by DOE`s Basic Science Division for Top Accomplishment. In the research program the author concentrated on modeling, analysis and computation of combustion phenomena, with particular emphasis on the transition from laminar to turbulent combustion. Thus he investigated the nonlinear dynamics and pattern formation in the successive stages of transition. He described the stability of combustion waves, and transitions to waves exhibiting progressively higher degrees of spatio-temporal complexity. Combustion waves are characterized by large activation energies, so that chemical reactions are significant only in thin layers, termed reaction zones. In the limit of infinite activation energy, the zones shrink to moving surfaces, termed fronts, which must be found during the course of the analysis, so that the problems are moving free boundary problems. The analytical studies were carried out for the limiting case with fronts, while the numerical studies were carried out for the case of finite, though large, activation energy. Accurate resolution of the solution in the reaction zone(s) is essential, otherwise false predictions of dynamical behavior are possible. Since the reaction zones move, and their location is not known a-priori, the author has developed adaptive pseudo-spectral methods, which have proven to be very useful for the accurate, efficient computation of solutions of combustion, and other, problems. The approach is based on a combination of analytical and numerical methods. The numerical computations built on and extended the information obtained analytically. Furthermore, the solutions obtained analytically served as benchmarks for testing the accuracy of the solutions determined computationally. Finally, the computational results suggested new analysis to be considered. A cumulative list of publications citing the grant make up the contents of this report