Nonlinear dynamics of upward propagating flames.

We study the influence of gravity on the dynamics of upward propagating premixed flames. We show that the role of gravity on the dispersion relation is small, but that the nonlinear effects are large. Using a Michelson Sivashinsky equation modified with a gravity term, it can be observed that the nonlinear dynamics of the crests is greatly influenced by gravity, as well as the final amplitude of the flame.

Nonlinear dynamics of flame fronts with large-scale stabilizing effects.

We investigate the influence of gravity and heat loss on the long-time nonlinear dynamics of premixed flames. We show that even when their influence remains weak in the linear regime they can significantly modify the long-time behavior. We suggest that the presence of such a large-scale stabilizing effect could be responsible for the creation of new cells on the front and the appearance of the strong persistent patterns observed in several recent experimental and numerical studies.

Shapes and speeds of steady forced premixed flames.

Steady premixed flames subjected to space-periodic steady forcing are studied via inhomogeneous Michelson-Sivashinsky (MS) and then Burgers equations. For both, the flame slope is posited to comprise contributions from complex poles to locate, and from a base-slope profile chosen in three classes (pairs of cotangents, single-sine functions or sums thereof). Base-slope-dependent equations for the pole locations, along with formal expressions for the wrinkling-induced flame-speed increment and the forcing function, are obtained on excluding movable singularities from the latter.

Supernovae: an example of complexity in the physics of compressible fluids.

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.

Wrinkled flames and geometrical stretch.

Localized wrinkles of thin premixed flames subject to hydrodynamic instability and geometrical stretch of uniform intensity (S) are studied. A stretch-affected nonlinear and nonlocal equation, derived from an inhomogeneous Michelson-Sivashinsky equation, is used as a starting point, and pole decompositions are used as a tool. Analytical and numerical descriptions of isolated (centered or multicrested) wrinkles with steady shapes (in a frame) and various amplitudes are provided; their number increases rapidly with 1/S>0.

Sivashinsky equation for corrugated flames in the large-wrinkle limit.

Sivashinsky's [Acta Astron. 4, 1177 (1977)] nonlinear integrodifferential equation for the shape of corrugated one-dimensional flames is ultimately reducible to a 2N -body problem, involving the 2N complex poles of the flame slope. Thual, Frisch, and Hénon [J.

Sivashinsky equation in a rectangular domain.

The (Michelson) Sivashinsky equation of premixed flames is studied in a rectangular domain in two dimensions. A huge number of two-dimensional (2D) stationary solutions are trivially obtained by the addition of two 1D solutions. With Neumann boundary conditions, it is shown numerically that adding two stable 1D solutions leads to a 2D stable solution.

Stationary solutions and Neumann boundary conditions in the Sivashinsky equation.

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.