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

Sivashinsky's (1977) nonlinear integro-differential equation for the shape of corrugated 1-dimensional flames is ultimately reducible to a 2N-body problem, involving the 2N complex poles of the flame slope. Thual, Frisch & Henon (1985) derived singular linear integral equations for the pole density in the limit of large steady wrinkles $(N \gg 1)$, which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise. Here we solve those analytically for isolated crests, next for monocoalesced then bicoalesced periodic flame patterns, whatever the (large-) amplitudes involved. We compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole-decomposition approach. Good agreement is obtained, even for moderately large Ns. The results are extended to give hints as to the dynamics of supplementary poles. Open problems are evoked

Potential-flow models for channelled two-dimensional premixed flames around near-circular obstacles

International audienceThe dynamics of two-dimensional thin premixed flames is addressed in the framework of mathematical models where the flow field on either side of the front is piecewise incompressible and vorticity free. Flames confined in channels with asymptotically straight impenetrable walls are considered. Besides a few free propagations along straight channels, attention is focused on flames propagating against high-speed flows and positioned near a round central obstacle or near two symmetric bumps protruding inward. Combining conformal maps and Green's functions, a regularized generalization of Frankel's integro-differential equation for the instantaneous front shape in each configuration is derived and solved numerically. This produces a variety of real looking phenomena: steady fronts symmetric or not, noise-induced subwrinkles, flashback events, and breathing fronts in pulsating flows. Perspectives and open mathematical and physical problems are finally evoked

Existence conditions and drift velocities of adiabatic flame-balls in weak gravity fields

Combining activation energy asymptotics, suitable scalings and numerical methods, we study how flame-balls move under the action of the free convection that they themselves generate in the presence of a weak, uniform gravity field. Attention is focused on steady configurations (in a suitable reference frame), on an isolated flame-ball of size comparable to what is obtained in the absence of gravity, and on deficient reactants that are characterized by a low Lewis number. For the sake of simplicity, we consider an adiabatic combustion process, in the sense that the radiative exchanges are neglected. This work provides one with: (a) a description of the free-convection field around the flame-ball, along with an asymptotic estimate of the drift velocity; (b) a relationship between the flame-ball radius, strength of gravity and physico-chemical properties of the reactive premixture; (c) extinction conditions, caused by the net effect of heat extraction from the flame-ball to its surroundings by the free-convection field. Hints on generalizations currently under consideration are also given

Low vorticity and small gas expansion in premixed flames

Different approaches to the nonlinear dynamics of premixed flames exist in the literature: equations based on developments in a gas ex- pansion parameter, weak nonlinearity approximation, potential model equation in a coordinate-free form. However the relation between these different equations is often unclear. Starting here with the low vor- ticity approximation proposed recently by one of the authors, we are able to recover from this formulation the dynamical equations usually obtained at the lowest orders in gas expansion for plane on average flames, as well as obtain a new second order coordinate-free equation extending the potential flow model known as the Frankel equation. It is also common to modify gas expansion theories into phenomelogical equations, which agree quantitatively better with numerical simula- tions. We discuss here what are the restrictions imposed by the gas expansion development results on this process

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 2D stationary solutions are trivially obtained by 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. This type of solution is shown to play an important role in the dynamics of the equation with additive noise

Flames with chain-branching/chain-breaking kinetics

A steady plane flame subject to the chain-branching/chain-breaking kinetics A plus X yields 2X, 2X plus M yields 2P plus M is considered for a certain distinguished limit of parameter values corresponding to fast recombination. Here A is the reactant, X the radical, P the product, and M a third body. The activation energy of the production step is very large, while that of the recombination step is small and taken to be zero. The object is to find the 'laminar-flame eigenvalue' DELTA , representing the burning rate, as a function of r, which is essentially the ratio of the two reaction rates. The response function DELTA (r) is described by numerical integration and by asymptotic analysis for r approaches 0, infinity

Flame front propagation V: Stability Analysis of Flame Fronts: Dynamical Systems Approach in the Complex Plane

We consider flame front propagation in channel geometries. The steady state solution in this problem is space dependent, and therefore the linear stability analysis is described by a partial integro-differential equation with a space dependent coefficient. Accordingly it involves complicated eigenfunctions. We show that the analysis can be performed to required detail using a finite order dynamical system in terms of the dynamics of singularities in the complex plane, yielding detailed understanding of the physics of the eigenfunctions and eigenvalues.Comment: 17 pages 7 figure

Flame front propagation I: The Geometry of Developing Flame Fronts: Analysis with Pole Decomposition

The roughening of expanding flame fronts by the accretion of cusp-like singularities is a fascinating example of the interplay between instability, noise and nonlinear dynamics that is reminiscent of self-fractalization in Laplacian growth patterns. The nonlinear integro-differential equation that describes the dynamics of expanding flame fronts is amenable to analytic investigations using pole decomposition. This powerful technique allows the development of a satisfactory understanding of the qualitative and some quantitative aspects of the complex geometry that develops in expanding flame fronts.Comment: 4 pages, 2 figure

Nonlinear equation for curved stationary flames

A nonlinear equation describing curved stationary flames with arbitrary gas expansion $\theta = \rho_{{\rm fuel}}/\rho_{{\rm burnt}}$, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak nonlinearity. It is proved that in the scope of the asymptotic expansion for $\theta \to 1,$ the new equation gives the true solution to the problem of stationary flame propagation with the accuracy of the sixth order in $\theta - 1.$ In particular, it reproduces the stationary version of the well-known Sivashinsky equation at the second order corresponding to the approximation of zero vorticity production. At higher orders, the new equation describes influence of the vorticity drift behind the flame front on the front structure. Its asymptotic expansion is carried out explicitly, and the resulting equation is solved analytically at the third order. For arbitrary values of $\theta,$ the highly nonlinear regime of fast flow burning is investigated, for which case a large flame velocity expansion of the nonlinear equation is proposed.Comment: 29 pages 4 figures LaTe

Determinants of immediate price impacts at the trade level in an emerging order-driven market

The common wisdom argues that, in general, large trades cause large price changes, while small trades cause small price changes. However, for extremely large price changes, the trade size and news play a minor role, while the liquidity (especially price gaps on the limit order book) is a more influencing factor. Hence, there might be other influencing factors of immediate price impacts of trades. In this paper, through mechanical analysis of price variations before and after a trade of arbitrary size, we identify that the trade size, the bid-ask spread, the price gaps and the outstanding volumes at the bid and ask sides of the limit order book have impacts on the changes of prices. We propose two regression models to investigate the influences of these microscopic factors on the price impact of buyer-initiated partially filled trades, seller-initiated partially filled trades, buyer-initiated filled trades, and seller-initiated filled trades. We find that they have quantitatively similar explanation powers and these factors can account for up to 44% of the price impacts. Large trade sizes, wide bid-ask spreads, high liquidity at the same side and low liquidity at the opposite side will cause a large price impact. We also find that the liquidity at the opposite side has a more influencing impact than the liquidity at the same side. Our results shed new lights on the determinants of immediate price impacts.Comment: 21 IOP tex pages including 5 figures and 5 tables. Accepted for publication in New Journal of Physic