No abstract

Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.

New stationary solutions of the (Michelson) Sivashinsky equation of premixed flames are obtained numerically in this paper. Some of these solutions, of the bicoalescent type recently described by Guidi and Marchetti, are stable with Neumann boundary conditions. With these boundary conditions, the time evolution of the Sivashinsky equation in the presence of a moderate white noise is controlled by jumps between stationary solutions.

An extension of the Frankel equation, coupled to a treatment of reconnections, is used to describe numerically turbulent flames submitted to the hydrodynamic Darrieus-Landau instability. The role played by this instability on the fractal properties of the front is evaluated. ͓S1063-651X͑97͒03506-X͔ PACS number͑s͒: 47.70.Fw, 82.40.Py It has been customary to describe turbulent premixed flames as interfaces propagating normally with a given laminar velocity, and submitted to an imposed turbulent flow field ͓1͔. However, it is well known that there is a retroaction of the flame front, which induces a new velocity field, which in turn will also affect the flame wrinkling.In the case of laminar flames, this retroaction effect leads to the hydrodynamic Darrieus-Landau instability. A nonlinear equation, the Michelson-Sivashinsky equation ͓2͔, has been introduced to describe this instability in the limit where the burnt gases are irrotational and where the flame is close to a plane flame. Despite these simplifying hypotheses, this equation had a lot of success in describing correctly most of the nonlinear features observed in laminar flames ͓3-6͔. Extensions of this equation have also been successful in describing spherically expanding flames ͓7,8͔. This Michelson-Sivashinsky equation has been generalized to obtain a coordinate-free equation describing a closed front, the Frankel equation ͓9͔ ͑an alternative way to incorporate heat release into a lagrangian method has been proposed in ͓10͔͒. A numerical solution of the Frankel equation can be found in ͓11͔, and in ͓12͔ an interesting study of the behavior of laminar flames with a very large radius. We shall be concerned here with the problem of turbulent flame fronts submitted to the Darrieus-Landau instability; a variant of the Frankel equation, coupled to a treatment of reconnections occurring on the front, will be used for this purpose.Let us first recall the usual form of the Frankel equation. Let us note by n the local normal vector at a given point of the flame, which is assumed to have the topology of a closed contour. n will be chosen to point in the direction of flame propagation. The evolution of the front is completely described by specifying the normal velocity of the flame at each point of the contour. This normal velocity is the sum of two terms: a first term u 1 (1Ϫ) represents the flame velocity: it is given by the laminar velocity u 1 corrected by a term proportional to the curvature ͑ being a constant usually called the Markstein length͒. The values u 1 ϭ1 and ϭ0.1 will be used in the simulations presented in this paper. A second term corresponding to the normal velocity caused by hydrodynamics has to be added to the flame velocity, and the Frankel equation ͓9͔ is obtained,where the integration is taken over the contour, and ␣ is a parameter controlling the importance of the hydrodynamic instability. In the case of a circle and ϭ0, the normal velocity V n becomes simply 1, as has been shown in the original article of Frankel ͓9͔. If we consider the g...

No abstract

A nonlinear integral-differential equation describing the cellular front of an overdriven detonation is obtained by an analysis carried out in the neighborhood of the instability threshold. The analysis reveals both an unusual mean streaming motion, resulting from the rotational part of the oscillatory flow, and pressure bursts generated by the crossover of cusps representative of Mach stems propagating on the detonation front. A numerical study of the nonlinear equation exhibits the "diamond" patterns observed in experiments. An overall physical understanding is provided.

No abstract

Abstract.The Zhdanov-Trubnikov equation describing wrinkled premixed flames is studied, using pole-decompositions as starting points. Its one-parameter (1 c 1) nonlinearity generalizes the Michelson-Sivashinsky equation (c=0) to a stronger Darrieus-Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c  1, which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner-Pollaczek polynomials, which numerical results confirm for 10 (over-stabilization) such analytical solutions can yield accurate flame shapes for 0 c 0.6. Open problems are invoked.