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AbstractAbstract

[en] The phenomenon of oscillatory necking in the stretching of polyethylene films is described. We propose an extension of a model of Barenblatt for isothermic necking in polymers, and we show that oscillatory necking may arise, for example, in polymers for which the effect of diffusion of stresses exceeds that of diffusion of oriented material. In such polymers, at sufficiently low temperatures, uniform necking is no longer stable and self-oscillations are observed

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Quarterly of Applied Mathematics; ISSN 0033-569X; ; v. 37(1); p. 23-34

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Boundary value

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J. Math. Phys. (N.Y.); v. 16(4); p. 846-854

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[en] We consider the problem of a Brownian particle confined in a potential well of forces, which escapes the potential barrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin's stochastic differential equation. We consider potential wells with many transition states and compute the expected exit time of the particle from the well as well as the probability distribution of the exit points. Our method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. Our results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice

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SIAM Journal of Applied Mathematics; ISSN 0036-1399; ; v. 35(3); p. 604-623

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[en] Steady, planar propagation of a condensed phase reaction front is unstable to disturbances corresponding to pulsating and spinning waves for sufficiently large values of a parameter related to the activation energy. This paper considers the nonlinear evolution equations for the amplitudes of the pulsating and spinning waves in a neighborhood of a double eigenvalue of the problem linearized about the steady, planar solution. In particular, near a degenerate Hopf bifurcation point, closed branches of solutions which represent new quasi-periodic modes of combustion are described

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Bayliss, A.; Matkowsky, B.J.

Funding organisation: National Science Foundation, Washington, DC (United States)

Dynamical issues in combustion theory

Funding organisation: National Science Foundation, Washington, DC (United States)

Dynamical issues in combustion theory

AbstractAbstract

[en] In this paper problems in gaseous combustion and in gasless condensed phase combustion are studied both analytically and numerically. In gaseous combustion we consider the problem of a flame stabilized on a line source of fuel. The authors find both stationary and pulsating axisymmetric solutions as well as stationary and pulsating cellular solutions. The pulsating cellular solutions take the form of either traveling waves or standing waves. Transitions between these patterns occur as parameters related to the curvature of the flame front and the Lewis number are varied. In gasless condensed phase combustion both planar and nonplanar problems are studied. For planar condensed phase combustion we consider two models: accounts for melting and does not. Both models are shown to exhibit a transition from uniformly to pulsating propagating combustion when a parameter related to the activation energy is increased. Upon further increasing this parameter both models undergo a transition to chaos: by intermittency and by a period doubling sequence. In nonplanar condensed phase combustion the nonlinear development of a branch of standing wave solutions is studied and is shown to lead to relaxation oscillations and subsequently to a transition to quasi-periodicity

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Fife, P.C. (Utah Univ., Salt Lake City, UT (United States). Dept. of Mathematics); Linan, A. (Ciudad Universitaria, Madrid (Spain)); Williams, F. (California Univ., San Diego, La Jolla, CA (United States)); 257 p; ISBN 0-387-97583-7; ; 1991; p. 1-36; Springer-Verlag New York Inc; New York, NY (United States); Springer-Verlag New York Inc., 175 Fifth Ave., New York, NY 10010 (United States)

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Book

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AbstractAbstract

[en] The authors employ singular perturbation methods to examine the generalized Langevin equation which describes the dynamics of a Brownian particle in an arbitrary potential force field, acted on by a fluctuating force describing collisions between the Brownian particle and lighter particles comprising a thermal bath. In contrast to models in which the collisions occur instantaneously, and the dynamics are modeled by a Langevin stochastic equation, they consider the situation in which the collisions do not occur instantaneously, so that the process is no longer a Markov process and the generalized Langevin equation must be employed. They compute expressions for the mean exit time of the Brownian particle from the potential well in which it is confined

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[en] The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time

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[en] The authors consider nonadiabatic premixed flame propagation in a long cylindrical channel. A steadily propagating planar flame exists for heat losses below a critical value. It is stable provided that the Lewis number and the volumetric heat loss coefficient are sufficiently small. At critical values of these parameters, bifurcated states, corresponding to time-periodic pulsating cellular flames, emanate from the steadily propagating solution. The authors analyze the problem in a neighborhood of a multiple primary bifurcation point. By varying the radius of the channel, they split the multiple bifurcation point and show that various types of stable periodic and quasi-periodic pulsating flames can arise as secondary, tertiary, and quaternary bifurcations. Their analysis describes several types of spinning and pulsating flame propagation which have been experimentally observed in nonadiabatic flames, and also describes additional quasi-periodic modes of burning which have yet to be documented experimentally

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[en] We introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for Markov jump processes. Our methods are applied directly to the integral equation for the mean first passage time and do not involve the use of diffusion approximations. An absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a Wiener-Hopf problem in the neighborhood of the boundary. A model of unimolecular dissociation is considered to illiustrate our method

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[en] We employ singular perturbation methods to examine the generalized Langevin equation (GLE) with state and memory dependent friction. The GLE describes the dynamics of a Brownian particle in an arbitrary potential field, acted on by a fluctuating force characterizing the collisions between the particle and the thermal bath. We consider the situation in which the collisions are not restricted to occur instantaneously and additionally these interactions are nonlinear. This leads to a non-Markovian description of the dynamics of the particle. We compute explicit analytical expressions for the mean exit time tau of the Brownian particle from the potential well in which it is confined. Thus we derive expressions for the activation rate kappa from the well, which is inversely proportional to tau. In the examples considered, the effect of the state dependent friction is to increase tau and thus to lower kappa

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