Effect of small heat release and viscosity on thermal-diffusive instability

Keigo Wada

doi:10.1038/s41598-021-99163-6

Abstract

The linear stability of a thermal reaction front has been investigated based on the thermal-diffusive model proposed by Zel’dovich and Frank-Kamenetskii, which is called ZFK model. In the framework of ZFK model, heat-conduction and mass-diffusion equations are treated without the effect of hydrodynamic flow. Then, two types of instability appear: cellular and oscillatory instabilities. The cellular instability has only positive real part of growth rate, while the oscillatory instability is accompanied with non-zero imaginary part. In this study, the effect of heat release and viscosity on both instabilities is investigated asymptotically and numerically. This is achieved by coupling mass-conservation and Navier–Stokes equations with the ZFK model for small heat release. Then, the stable range of Lewis number, where the real part of growth rate is negative, is widened by non-zero values of heat release for any wavenumber. The increase of Prandtl number also brings the stabilization effect on the oscillatory instability. However, as for the cellular instability, the viscosity leads to the destabilization effect for small wavenumbers, opposed to its stabilization effect for moderate values of wavenumber. Under the limit of small wavenumber, the property of viscosity becomes clear in view of cut-off wavenumber, which makes the real part of growth rate zero.

Highlights

The linear stability of a thermal reaction front has been investigated based on the thermal-diffusive model proposed by Zel’dovich and Frank-Kamenetskii, which is called ZFK model
In the framework of ZFK model, the effect of hydrodynamic flow is successfully decoupled from the heat and mass-diffusion equations
The linear stability analysis of the thermal-diffusive model without hydrodynamic flow has been performed for one r eactant[17], for two r eactants[18], for the case of heat loss[19] and in the context of ignition temperature[20]

Summary

Methods

Where the non-dimensional variables, ρ , V = (U, V , W) , T, PM and Y are the density of mixture, velocity field, temperature, pressure and mass fraction of a deficient species which is consumed through the exothermic chemical reaction. The dependent variables are made dimensionless by use of those of fresh mixture at a position far from a flame front, where the flow field is assumed to be steady planar, denoted by ρ−∞ , SL , T−∞ , P−∞ and Y−∞. In the far field from a flame front on the unburned side, where the flow is assumed to be steady planar, all quantities become unity due to the definition (6). From (29), by use of (20) and (28), we readily find that the steady planar mass flux is unity on both sides of reaction front. The other governing Eqs. (5), (13)–(15) are written down, outside the reaction layer ( ξ < 0 , ξ > 0 ), as dWdP 4 d2Wdξ

Pr dξ 2

Conclusions