Diffusive–thermal Instability
   HOME

TheInfoList



OR:

Diffusive–thermal instability or thermo–diffusive instability is an intrinsic flame instability that occurs both 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 and in
diffusion flame In combustion, a diffusion flame is a flame in which the oxidizer and fuel are separated before burning. Contrary to its name, a diffusion flame involves both diffusion and convection processes. The name diffusion flame was first suggested by ...
s and arises because of the difference in the diffusion coefficient values for the fuel and heat transport, characterized by non-unity values of
Lewis number In fluid dynamics and thermodynamics, the Lewis number (denoted ) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer. ...
s. The instability mechanism that arises here is the same as in Turing instability explaining chemical morphogenesis, although the mechanism was first discovered in the context of
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 ...
by
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 ...
in 1944 to explain the cellular structures appearing in lean
hydrogen Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
flames. Quantitative stability theory for premixed flames were developed by Gregory Sivashinsky (1977),
Guy Joulin Guy Joulin is a French scientist at Aix-Marseille University who works in the field of combustion. Biography Guy Joulin obtained his PhD degree from University of Poitiers in 1979 under the supervision of Paul Clavin. Joulin is the re ...
and Paul Clavin (1979) and for diffusion flames by Jong S. Kim and Forman A. Williams (1996,1997).


Dispersion relation for premixed flames

To neglect the influences by hydrodynamic instabilities such as
Darrieus–Landau instability 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 and Lev Landau. It is a key Combustion instability#Classification of comb ...
,
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 ...
etc., the analysis usually neglects effects due to the thermal expansion of the gas mixture by assuming a constant density model. Such an approximation is referred to as diffusive-thermal approximation or thermo-diffusive approximation which was first introduced by Grigory Barenblatt,
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 ...
and A. G. Istratov in 1962. With a one-step chemistry model and assuming the perturbations to a steady planar flame in the form e^, where \mathbf_\bot is the transverse coordinate system perpendicular to flame, t is the time, \mathbf is the perturbation wavevector and \omega is the temporal growth rate of the disturbance, the dispersion relation \sigma(k) for one-reactant flames is given implicitly byClavin, P., & Searby, G. (2016). Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press. :2\Gamma^2(\Gamma-1) + l (\Gamma-1 - 2 \sigma) = 0 where \Gamma=\sqrt, l\equiv (Le-1)/\beta, Le is the
Lewis number In fluid dynamics and thermodynamics, the Lewis number (denoted ) is a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer. ...
of the fuel and \beta is the
Zeldovich number The Zeldovich number is a dimensionless number which provides a quantitative measure for the activation energy of a chemical reaction which appears in the Arrhenius exponent, named after the Russian scientist Yakov Borisovich Zeldovich, who along ...
. This relation provides in general three roots for \sigma in which the one with maximum \Re\ would determine the stability character. The stability margins are given by the following equations :8k^2 + l + 2 =0, \quad 256 k^4 +(-6l^2+32l+256)k^2 -l^2+8l + 32=0 describing two curves in the l vs. k plane. The first curve is associated with condition \Im\=0, whereas on the second curve \Im\\neq 0. The first curve separates the region of stable mode from the region corresponding to cellular instability, whereas the second condition indicates the presence of traveling and/or pulsating instability.


See also

*
Turing pattern The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomousl ...
*
Darrieus–Landau instability 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 and Lev Landau. It is a key Combustion instability#Classification of comb ...
* Kuramoto–Sivashinsky equation *
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 ...
*
Double diffusive convection Double diffusive convection is a fluid dynamics phenomenon that describes a form of convection driven by two different density gradients, which have different rates of diffusion. Convection in fluids is driven by density variations within them u ...


References

{{DEFAULTSORT:Diffusive-thermal instability Fluid dynamics Combustion Fluid dynamic instabilities