Diffusive–thermal instability or thermo–diffusive instability is an
instrinsic 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 throughou ...
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. ...
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 The Lewis number (Le) 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. The Lewis number puts the thickness of the th ...
s. The instability mechanism that arises here is the same as in
Turing instability
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical comp ...
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. Combusti ...
by
Yakov Zeldovich
Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...
in 1944 to explain the cellular structures appearing in lean
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
flames. Quantitative stability theory for premixed flames were developed by
Gregory Sivashinsky
Gregory I. Sivashinsky (also known as Grisha) is a professor at Tel Aviv University, working in the field of combustion and theoretical physics.
Biography
Sivashinsky was born in Moscow to Israel and Tatiana Sivashinsky. He is married to Terry ...
(1977), Guy Joulin and
Paul Clavin
Paul Clavin is a French scientist at Aix-Marseille University, working in the field of combustion and statistical mechanics. He is the founder of Institute for Research on Nonequilibrium Phenomena (IRPHE).
Biography
Clavin served as the chair ...
(1979) and for diffusion flames by Jong S. Kim (1997).
Dispersion relation for premixed flames
To neglect the influences by hydrodynamic instabilities such as
Darrieus–Landau instability,
Rayleigh–Taylor instability
The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
Drazin (20 ...
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 introudced by
Grigory Barenblatt
Grigory Isaakovich Barenblatt (russian: Григо́рий Исаа́кович Баренблат; 10 July 1927 – 22 June 2018) was a Russian mathematician.
Education
Barenblatt graduated in 1950 from Moscow State University, Department of Mec ...
,
Yakov Zeldovich
Yakov Borisovich Zeldovich ( be, Я́каў Бары́савіч Зяльдо́віч, russian: Я́ков Бори́сович Зельдо́вич; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet physicist of Bel ...
and A. G. Istratov in 1962. With a one-step chemistry model and assuming the perturbations to a steady planar flame in the form
, where
is the transverse coordinate system perpendicular to flame,
is the time,
is the perturbation wavevector and
is the temporal growth rate of the disturbance, the dispersion relation
for one-reactant flames is given implicitly by
[Clavin, P., & Searby, G. (2016). Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press.]
:
where
,
,
is the
Lewis number The Lewis number (Le) 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. The Lewis number puts the thickness of the th ...
of the fuel and
is the
Zeldovich number The Zel'dovich 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 Zel'dovich, who along ...
. This relation provides in general three roots for
in which the one with maximum
would determine the stability character. The stability margins are given by the following equations
:
describing two curves in the
vs.
plane. The first curve is associated with condition
, whereas on the second curve
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
*
Kuramoto–Sivashinsky equation
In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation ...
References
{{DEFAULTSORT:Diffusive-thermal instability
Fluid dynamics
Combustion
Fluid dynamic instabilities