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The Darrieus–Landau instability, or density fingering, refers to an instability of chemical fronts propagating into a denser medium, named after
Georges Jean Marie Darrieus Georges Jean Marie Darrieus (24 September 1888 – 15 July 1979) was a French aerospace engineering, aeronautical engineer in the 20th century. He invented the Darrieus wind turbine, Darrieus rotor, a wind turbine capable of operating from any di ...
and
Lev Landau Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. He was considered as one of the last scientists who were universally well-versed and ma ...
. It is a key instrinsic flame instability that occurs in
premixed flame A premixed flame is a flame formed under certain conditions during the combustion of a premixed charge (also called pre-mixture) of fuel and oxidiser. Since the fuel and oxidiser—the key chemical reactants of combustion—are available througho ...
s, caused by density variations due to thermal expansion of the gas produced by the
combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion ...
process. In simple terms, stability inquires whether a steadily propagating plane sheet with a discontinuous jump in density is stable or not. The analysis behind the Darrieus–Landau instability considers a planar, premixed
flame A flame () is the visible, gaseous part of a fire. It is caused by a highly exothermic chemical reaction made in a thin zone. When flames are hot enough to have ionized gaseous components of sufficient density, they are then considered plasm ...
front subjected to very small perturbations. It is useful to think of this arrangement as one in which the unperturbed flame is stationary, with the reactants (fuel and oxidizer) directed towards the flame and perpendicular to it with a velocity u1, and the burnt gases leaving the flame also in a perpendicular way but with velocity u2. The analysis assumes that the flow is
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
, and that perturbations are governed by the linearized
Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
, and are thus inviscid. With these considerations, the main result of this analysis is that, if the density of burnt
gases Gas is a state of matter that has neither a fixed volume nor a fixed shape and is a compressible fluid. A ''pure gas'' is made up of individual atoms (e.g. a noble gas like neon) or molecules of either a single type of atom ( elements such ...
is less than that of the reactants (true in practice due to thermal expansion of the gas produced by combustion), the flame front is unstable to perturbations of any
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
. Another result is that the rate of growth of perturbations is inversely proportional to their wavelength; thus small flames (but larger than the characteristic flame thickness) tend to wrinkle, and grow faster than larger ones. In practice, however, diffusive and buoyancy effects that are not taken into account by the analysis of Darrieus and Landau may have a stabilizing effect.


History

Yakov Zeldovich Yakov Borisovich Zeldovich (, ; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet people, Soviet Physics, physicist of Belarusians, Belarusian origin, who is known for his prolific contributions in physical Physical c ...
notes that
Lev Landau Lev Davidovich Landau (; 22 January 1908 – 1 April 1968) was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. He was considered as one of the last scientists who were universally well-versed and ma ...
generously suggested this problem for him to investigate, and Zeldovich however, made calculation errors which led Landau himself to complete the work.


Dispersion relation

If disturbances to the steady planar flame sheet are of the form e^, where \mathbf_\bot is the transverse coordinate system that lies on the undisturbed stationary flame sheet, t is the time, \mathbf is the wavevector of the disturbance and \sigma is the temporal growth rate of the disturbance, then the dispersion relation is given by :\frac = \frac\left(\sqrt-1\right) where S_L is the laminar burning velocity (or, the flow velocity far upstream of the flame in a frame that is fixed to the flame), k=, \mathbf, and r=\rho_u/\rho_b is the ratio of burnt to unburnt gas density. In combustion, r>1 always and therefore the growth rate \sigma>0 for all wavenumbers. This implies that a plane sheet of flame with a burning velocity S_L is unstable for all wavenumbers. In fact,
Amable Liñán Amable Liñán Martínez (born 1934 in Noceda de Cabrera, Castrillo de Cabrera, León, Spain) is a Spanish aeronautical engineer working in the field of combustion. Biography He holds a PhD in Aeronautical Engineering from the Technical Uni ...
and Forman A. Williams quote in their book that: "... ''in view of laboratory observations of stable, planar, laminar flames, publication of their theoretical predictions required courage on the part of Darrieus and Landau".''


With buoyancy

If buoyancy forces are taken into account (in others words,
Rayleigh–Taylor instability The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an Interface (chemistry), interface between two fluids of different densities which occurs when the lighter fluid is pushing the hea ...
is considered) for planar flames perpendicular to the gravity vector, then some stability can be anticipated for flames propagating vertically downwards (or flames held stationary by an upward flow), since in these cases, the denser unburnt gas lies beneath the lighter burnt gas mixture. Of course, for flames propagating upwards or those held stationary by downward flow, both the Darrieus–Landau mechanism and the Rayleigh–Taylor mechanism contributes to the destabilizing effect. The dispersion relation when buoyance forces are included becomes :\frac = \frac\left sqrt-1\right where g>0 corresponds to gravitational acceleration for flames propagating downwards and g<0 corresponds to gravitational acceleration for flames propagating upwards. The above dispersion implies that gravity introduces stability for downward propagating flames when k^>l_=S_^2r/g, where l_b is a characteristic buoyancy length scale. For small values of r-1, the growth rate becomes :\frac =\frac(r-1) +\cdots \quad k^\ll l_b, :\frac =\frac(r-1) \left(1-\frac\right) +\cdots \quad k^\sim l_b, :\frac =-\frac(r-1) +\cdots \quad k^\gg l_b.


Limitations

Darrieus and Landau's analysis treats the flame as a plane sheet to investigate its stability with the neglect of diffusion effects, whereas in reality, the flame has a definite thickness, say the laminar flame thickness k^\sim \delta_L=D_T/S_L, where D_T is the
thermal diffusivity In thermodynamics, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It is a measure of the rate of heat transfer inside a material and has SI, SI units of m2/s. It is an intensive ...
, wherein diffusion effects cannot be neglected. Accounting for the flame structure, as first envisioned by George H. Markstein, is found to result in stabilized flames for small wavelengths k^\sim \delta_L, except when the fuel diffusion coefficient and thermal diffusivity differ from each other significantly, leading to the so-called (
Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical compute ...
) diffusive-thermal instability. Darrieus–Landau instability manifests in the range \delta_L\ll k^\ll l_b for downward propagating flames, and \delta_L\ll k^ for upward propagating flames.


Under Darcy's law

The classical dispersion relation was based on the assumption that the hydrodynamics is governed by
Euler equations In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
. In strongly confined systems such as a Hele-Shaw cell or in porous media, hydrodynamics however, is governed by Darcy's law. The dispersion relation based on Darcy's law was derived by J. Daou and P. Rajamanickam,Daou, J., & Rajamanickam, P. (2025). Hydrodynamic instabilities of propagating interfaces under Darcy's law. Physical Review Fluids. and reads: :\frac = \frac + \frac\frac - \frac \frac where r = \rho_u/\rho_b>1 is the density ratio, m=(\mu_u/\kappa_u)/(\mu_b/\kappa_b)<1 is the ratio of friction factor which involves viscosity \mu and permeability \kappa (in Hele-Shaw cells, \kappa_u=\kappa_b=h^2/12, where h is the cell width, so that m=\mu_u/\mu_b is simply the viscosity ratio). V is the speed of a uniform imposed flow. When V>0, the imposed flow opposes flame propagation and when V<0, it aids flame propagation. As before, g>0 corresponds to downward flame propagation and g<0 to upward flame propagation. The three terms in the above formula, respectively, corresponds to Darrieus–Landau instability (density fingering),
Saffman–Taylor instability The Saffman–Taylor instability, also known as viscous fingering, is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium or in a Hele-Shaw cell, described mathematically by Philip Saffman and ...
(viscous fingering) and
Rayleigh–Taylor instability The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an Interface (chemistry), interface between two fluids of different densities which occurs when the lighter fluid is pushing the hea ...
(gravity fingering), all in the context of Darcy's law. The Saffman–Taylor instability is specific to confined flames and does not exist in unconfined flames.


See also

*
Michelson–Sivashinsky equation In combustion, Michelson–Sivashinsky equation describes the evolution of a premixed flame front, subjected to the Darrieus–Landau instability, in the small heat release approximation. The equation was derived by Gregory Sivashinsky in 1977, who ...
*
Clavin–Garcia equation Clavin–Garcia equation or Clavin–Garcia dispersion relation provides the relation between the growth rate and the wave number of the perturbation superposed on a planar premixed flame, named after Paul Clavin and Pedro Luis Garcia Ybarra, who d ...


References

{{DEFAULTSORT:Darrieus-Landau instability Fluid dynamics Combustion Fluid dynamic instabilities Lev Landau