232 research outputs found
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 , 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 , 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 the new equation gives the true solution to the
problem of stationary flame propagation with the accuracy of the sixth order in
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 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
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