Effect of Taylor dispersion on the thermo-diffusive instabilities of flames in a Hele–Shaw burner

Abstract

We investigate the effect of Taylor dispersion on the thermo-diffusive instabilities of premixed flames. This is a physically interesting and analytically tractable problem within a relatively unexplored class of problems pertaining to the interaction between Taylor dispersion (or flow-enhanced diffusion) and Turing-like instabilities in reaction–diffusion systems. The analysis is carried out in the Hele–Shaw burner configuration and adopts a constant density and negligible heat-loss assumptions. These simplifying assumptions allow to isolate the effect of Taylor dispersion on flame stability (by switching off the Darrieus–Landau instability and experimentally challenging extinction phenomena) while keeping the problem analytically tractable. Starting from a 3D formulation, depth-averaged equations are first obtained leading to a 2D model which accounts for enhanced diffusion in the flow direction and shows that diffusion is effectively anisotropic. A linear stability analysis of the travelling wave solutions of the 2D problem leads to a simple dispersion relation which generalises a classical one obtained by Sivashinsky to incorporate the effect of the flow Peclet number coupled to that of the mixture's Lewis number. Based on the new dispersion relation, stability-bifurcation diagrams are drawn in terms of the Peclet and Lewis numbers and their physical implications are discussed. In particular, the study clearly demonstrates the ability of Taylor dispersion to significantly affect the flame thermo-diffusive instabilities, whether these are of the cellular or oscillatory types, with the effect on the latter being more pronounced. It is found that Taylor dispersion typically promotes the cellular instability and hampers the oscillatory instability. This is the first stability analysis accounting for Taylor dispersion in the context of combustion and has thus a fundamental value, both in combustion and in other reaction–diffusion areas, independent of the fact that the phenomena predicted may well be difficult to reproduce experimentally.

Keywords:

• Taylor dispersion
• flame instabilities
• Hele–Shaw burner
• travelling-waves in reaction–diffusion systems
• turing instability

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Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Note that formula (2(2) $\frac{{\mathrm{L}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\mathrm{L}\mathrm{e}}=\frac{1+\gamma (1-\alpha {)}^2{\mathrm{P}\mathrm{e}}^2}{1+\gamma (1-\alpha {)}^2{\mathrm{P}\mathrm{e}}^2{\mathrm{L}\mathrm{e}}^2},$(2) ) implies that ${\mathrm{L}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\to \mathrm{L}\mathrm{e}$ as $\mathrm{P}\mathrm{e}\to 0$ and ${\mathrm{L}\mathrm{e}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\to {\mathrm{L}\mathrm{e}}^{-1}$ as $\mathrm{P}\mathrm{e}\to \mathrm{\infty }$

2 Defined as a square root of a complex number, Γ is assumed to lie on the right half-complex plane.

Additional information

Funding

The author is grateful to the EPSRC for financial support under grant [EP/V004840/1].