In the Appendix we mainly follow to the short derivation of [105]. More detailed derivation of Michelson Sivashinsky equation can be found in [29].

From the hydrodynamic principles of premixed flame propagation, the diffusion length which usually characterizes the premixed flame thickness

\(l_f\) is thought to be very little if compared to the typical size of the area

*L*. as a result a premixed flame can be approximated to a surface of infinitesimal width separating the fresh unburnt mixture from the burnt combustion products. Stream on either part of the premixed flame is supposed being inviscid and incompressible, a full detailed description which can be given by the Euler equations

$$\begin{aligned} \nabla \cdot \mathrm{v} =0 \end{aligned}$$

(2.91)

$$\begin{aligned} \rho \frac{\partial \mathrm{v}}{\partial t}= \rho \left( \nabla \cdot \mathrm{v}\right) \mathrm{v} =-\nabla p \end{aligned}$$

in which

*v* is the velocity,

*p* is the pressure and

\(\rho \) is the density that is piecewise constant, having values

\(\rho _u\) for the unburnt gas and

\(\rho _b\) for the burnt gas. A premixed flame surface is mathematically displayed by the function

\(\psi (x,t)\) provided by

$$\begin{aligned} \psi (x,t)=y-f(x,t)=0 \end{aligned}$$

(2.92)

in which

\(\psi < 0\) relates to the fresh unburnt mixture and

\(\psi > 0\) is the burnt mixture. The unit normal vector

*n* to the premixed flame surface provided by

\(\mathrm{n}\mathrm{=}\nabla \psi /\left| \nabla \psi \right| \) is directed towards the burnt part. The propagation velocity of the premixed flame surface

\(V_f\), normal to itself, is provided by

\(V_f=-\psi _t/\left| \nabla \psi \right| \). Making use of (

2.92) the expressions for

*n* and

\(V_f\) can be written as

$$\begin{aligned} n=\left( \frac{-f_x}{\sqrt{1+f^2_x}},\frac{1}{\sqrt{1+f^2_x}}\right) \end{aligned}$$

(2.93)

$$\begin{aligned} V_f=\frac{f_t}{\sqrt{1+f^2_x}} {\mathbf \ } \end{aligned}$$

(2.94)

The flow quantities suffer from jump discontinuities throughout the premixed flame surface and even are subject to the Rankine Hugoniot jump relations given below

$$\begin{aligned}{}[[\rho \left( \mathrm{v} \cdot n-V_f\right) ]]=0 \end{aligned}$$

$$\begin{aligned}{}[\left[ n \times \left( \mathrm{v} \times n\right) \right] ]=0 \end{aligned}$$

(2.95)

$$\begin{aligned}{}[[p+\rho \left( \mathrm{v} \cdot n\right) \left( \mathrm{v} \cdot n-V_f\right) ]]=0 \end{aligned}$$

in which

\(v\ =\ (u,v)\) and [[.]] means the jump in the quantity throughout the premixed flame surface. The premixed flame velocity

\(S_f\) is described as the propagation velocity of the surface comparably to the incoming unburnt gas, viz.

\(\left. {S_f=\mathrm{v}}^* \cdot n-V_f\right. \), in which

\({\mathrm{v}}^*=\mathrm{v}|_{\psi =0^{-}}\). Applying (

2.92), the expression for premixed flame velocity

\(S_f\) could be written as

$$\begin{aligned} \left. {S_f=\mathrm{v}}^*\cdot n-V_f\right. = \frac{{{-u}^*f_x+v}^*-f_t}{\sqrt{1+f^2_x}} \end{aligned}$$

(2.96)

$$\begin{aligned} S_f=S_L\left( 1-{\mathtt{J}}_k\right) \end{aligned}$$

(2.97)

in which

\(S_L\) is the laminar premixed flame velocity,

\(\mathtt{J}\) is the Markstein length as well as

\(k=-\nabla \cdot n\) is the mean curvature. Laminar premixed flame velocity

\(S_L\), the propagation velocity of a premixed flame , is a exclusive property of a mixture, showing its reactivity and exothermicity in a provided diffusive medium. The Markstein length

\(\mathtt{J}\) is normally a coefficient of the order of the premixed flame thickness

\(l_f\) and is depending on physico-chemical factors like the thermal expansion coefficient, Lewis number, equivalence ratio and global activation energy of the chemical reaction. The dependence of

\(\mathtt{J}\) on Lewis number is determined by the deficient reactant, i.e. it is dependent on Lewis number of the fuel in lean mixtures as well as Lewis number of the oxidant in rich mixtures.

Equations (

2.91), (

2.95) and (

2.97) admit a simple solution in the form of a planar premixed flame located at

\(y = -S_Lt\). The velocity and pressure across the premixed flame front are piecewise constants given by

$$\begin{aligned} v=\left\{ \begin{array}{cc} 0 &{} y\ <\ -\ S_Lt \\ \left( \sigma -1\right) S_L &{} y\ >\ -\ S_Lt, \end{array} \right. \end{aligned}$$

(2.98)

$$\begin{aligned} p=\left\{ \begin{array}{cc} 0 &{} y\ <\ -\ S_Lt \\ -\left( \sigma -1\right) \sigma _u{S_L}^2 &{} y\ >\ -\ S_Lt, \end{array} \right. \end{aligned}$$

(2.99)

in which

\(\sigma =\rho _u/\rho _b > 1\) will be the thermal expansion coefficient. An asymptotic solution which represents a weakly corrugated premixed flame can be received as a perturbation of the planar premixed flame for the limit of weak thermal expansion, i.e.

\((\rho - 1) \ll 1\). The disturbed premixed flame front can be determined in the form

$$\begin{aligned} y\ =\ -\ S_Lt+\left( \sigma -1\right) \phi \end{aligned}$$

(2.100)

in which

\(\phi = \phi (x, \tau )\) represents the premixed flame front and

\(\tau \ =\ (\sigma - 1)t\) is the scaled time. The speed is scaled as

\(\mathrm{v}=\overline{\mathrm{v}}\,+\,{\left( 1\,-\,\sigma \right) }^{2}\widetilde{\mathrm{v}}\) and pressure as

\(\mathrm{p}\,=\,\overline{\mathrm{p}}\,+\,{\left( 1\,-\,\sigma \right) }^{\mathrm{2}}\widetilde{\mathrm{p}}\) where

\(\overline{\mathrm{v}}\),

\(\mathrm{\ }\overline{\mathrm{p}}\) correspond to the basic state of the planar premixed flame . The premixed flame velocity relations (

2.96) and (

2.97) provide

$$ \frac{-{\left( \sigma -1\right) }^3 \phi _x \tilde{u}+{\left( \sigma -1\right) }^2{{ \tilde{v}}}^*+ S_L-{\left( \sigma -1\right) }^2\phi _\tau }{\sqrt{1+{\left( \sigma -1\right) }^2{\phi _x}^2}} =S_L-\frac{S_L\mathtt{J}\left( \sigma -1\right) \phi _{xx}}{\sqrt{1+{\left( \sigma -1\right) }^2{\phi _x}^2}} $$

where the curvature

\(\nabla \cdot n\approx {-\phi _{xx}}/{\sqrt{1+{\left( \sigma -1\right) }^2{\phi _x}^2}}\). Using the approximation

$$\begin{aligned} \sqrt{1+x^2}\simeq 1+\frac{1}{2}x+ \cdots \end{aligned}$$

$$\begin{aligned} \sqrt{1+{\left( \sigma -1\right) }^2{\phi _x}^2}\simeq 1+\frac{1}{2}{\left( \sigma -1\right) }^2{\phi _x}^2+ \cdots \end{aligned}$$

and keeping terms of

\(O{(\sigma - 1)}^2\) we get

$$S_L+{\left( \sigma -1\right) }^2{\tilde{v}}^*-{\left( \sigma -1\right) }^2\phi _\tau = S_L+{\frac{1}{2}S}_L{\left( \sigma -1\right) }^2{\phi _x}^2-{\mathtt{J}S}_L\left( \sigma -1\right) \phi _{xx}$$

$$\begin{aligned} \phi _\tau + {\frac{1}{2}S}_L{\phi _x}^2-\frac{S_L\mathtt{J}}{\left( \sigma -1\right) }\phi _{xx}-{\tilde{v}}^*=0 \end{aligned}$$

(2.101)

To evaluate

\(v^*\) we will need to solve for the flow field. Prior to substituting the scaled expressions in the Euler equations, it is convenient to move to a coordinate system connected to the planar front. Let

\(\hat{y}\ =\ y-\ S_Lt\) and then

$$\frac{\partial }{\partial t}=\frac{\partial }{\partial t}+ S_L\frac{\partial }{\partial \hat{y}},\ \ \frac{\partial }{\partial \hat{y}}=\frac{\partial }{\partial y}.$$

The Euler equations can be written as

$$\begin{aligned} u_x+v_x=0 \end{aligned}$$

$$\begin{aligned} \rho u_t+\rho uu_x+\rho \left( u+S_L\right) u_y=-p_x, \end{aligned}$$

$$\begin{aligned} \rho v_t+\rho uv_x+\rho \left( v+S_L\right) u_y=-p_y. \end{aligned}$$

Substituting the scaled expressions for pressure and speed in the Euler equations and keeping terms of

\(O{(\sigma - 1)}^2\) we get

$$\begin{aligned} {\tilde{u}}_x+{\tilde{v}}_x=0 \end{aligned}$$

$$\begin{aligned} \rho _uS_L{\tilde{u}}_y=-{\tilde{p}}_x, \end{aligned}$$

$$\begin{aligned} \rho _uS_L{\tilde{v}}_y=-{\tilde{p}}_y. \end{aligned}$$

These equations are valid for

\(\hat{y}\ >\ 0\) and

\(\hat{y}\ <\ 0\) since

\(\rho _b\ ={\rho _u}/{\sigma }={\rho _u}/{\left[ 1\ +\ \left( \sigma - 1\right) \right] }\sim \ \rho _u\left[ 1\ - \left( \sigma - 1\right) \right] \sim \rho _u+\ O(\sigma - 1)\).

The jump conditions throughout the disturbed premixed flame surface are listed by (

2.95). Because the perturbation of the premixed flame surface is small, the jump conditions are transferred throughout

\(y\ =\ 0\) by doing a Taylor expansion around

\(\ y\ =\ 0\). Keeping terms of

\(O{(\sigma - 1)}^2\) we obtain

$$\begin{aligned}{}[[\tilde{v}]]=0, \end{aligned}$$

$$\begin{aligned} \left[ \left[ \tilde{u}\right] \right] =-\phi _x, \end{aligned}$$

(2.102)

$$\begin{aligned}{}[[\tilde{p}]]=0. \end{aligned}$$

Non-dimensionalizing the system with the transverse area of integration as the unit for length,

\(S_L\) as the unit for speed,

\(L/S_L\) as the unit for time and

\(\rho _u{S_L}^2\) as the unit for pressure we get

$$\begin{aligned} {\tilde{u}}_x+{\tilde{v}}_x=0 \end{aligned}$$

(2.103)

$$\begin{aligned} {\tilde{u}}_y=-{\tilde{p}}_x, \end{aligned}$$

(2.104)

$$\begin{aligned} {\tilde{v}}_y=-{\tilde{p}}_y. \end{aligned}$$

(2.105)

as the non-dimensionalized Euler equations. The non-dimensional evolution equation for the premixed flame profile can be described as

$$\begin{aligned} \phi _\tau +\frac{1}{2}{\phi _x}^2-\alpha \phi _{xx}-{\tilde{v}}^*=0 \end{aligned}$$

(2.106)

The parameter

\(\alpha =\frac{\mathtt{J}}{L\left( \sigma -1\right) }\) is the scaled Markstein number and as observed from the expression, is inversely proportional to the transverse area of integration. Equations (

2.104) and (

2.105) jointly provide the following

$$\begin{aligned} {\nabla }^{\mathrm{2}}\hat{p}=0. \end{aligned}$$

(2.107)

The Fourier transform of a function, say

\(h(x, y, \tau )\), is provided by

$$\begin{aligned} {\mathcal F}\left( h\left( x,y,\tau \right) \right) \equiv h_k\left( y,\tau \right) =\int ^{\infty }_{-\infty }{e^{-ikx}h\left( x,y,\tau \right) {{dx}},} \end{aligned}$$

(2.108)

and its inverse is provided by

$$\begin{aligned} h\left( x,y,\tau \right) = \frac{1}{2\pi }\int ^{\infty }_{-\infty }{e^{ikx}h_k\left( y,\tau \right) {{dk}},} \end{aligned}$$

(2.109)

Using Fourier transform of (

2.107) we get

$$\begin{aligned} {\left( {\tilde{p}}_k\right) }_{yy}-k^2{\tilde{p}}_k=0 \end{aligned}$$

(2.110)

which solving on either of the surface results in

$$\begin{aligned} {\tilde{p}}_k=\left\{ \begin{array}{cc} C_1e^{\left| k\right| y} &{} y\ <\ 0 \\ C_2e^{-\left| k\right| y} &{} y\ >\ 0. \end{array} \right. \end{aligned}$$

(2.111)

From (

2.105),

$$\begin{aligned} {\left( {\tilde{v}}_k\right) }_y=\left\{ \begin{array}{cc} -{\left| k\right| C}_1e^{\left| k\right| y} &{} y\ <\ 0 \\ \left| k\right| C_2e^{-\left| k\right| y} &{} y\ >\ 0. \end{array} \right. \end{aligned}$$

(2.112)

Integrating according to y,

$$\begin{aligned} {\tilde{v}}_k=\left\{ \begin{array}{cc} -C_1e^{\left| k\right| y}+C_3 &{} y\ <\ 0 \\ -C_2e^{-\left| k\right| y}+C_4 &{} y\ >\ 0. \end{array} \right. \end{aligned}$$

(2.113)

Because the premixed flame is propagating in a quiescent flow, the speed field in the unburnt gases a long way away from the premixed flame front has a tendency to 0, i.e.

\({\mathop {\lim }_{y\rightarrow -\infty } \tilde{v}\ }=0\) which sets

\(C_3=\ 0\), leading to

$$\begin{aligned} {\tilde{v}}_k=\left\{ \begin{array}{cc} -C_1e^{\left| k\right| y} &{} y\ <\ 0 \\ -C_2e^{-\left| k\right| y}+C_4 &{} y\ >\ 0. \end{array} \right. \end{aligned}$$

(2.114)

From (

2.103),

$$\begin{aligned} {\tilde{u}}_x=-{\tilde{v}}_y\mathrm{,} \end{aligned}$$

$$\begin{aligned} {\tilde{u}}_k={\frac{i}{k}{\tilde{v}}_k}_y. \end{aligned}$$

From (

2.112),

$$\begin{aligned} {\tilde{v}}_k=\left\{ \begin{array}{cc} -{\frac{\left| k\right| }{k}iC}_1e^{\left| k\right| y} &{} y\ <\ 0 \\ \frac{\left| k\right| }{k}C_2e^{-\left| k\right| y} &{} y\ >\ 0. \end{array} \right. \end{aligned}$$

(2.115)

Applying the jump conditions given by (

2.102)–(

2.111), (

2.114) and (

2.115) we get

$$\begin{aligned} C_1=C_2=-\frac{\left| k\right| }{2}\phi _k, \end{aligned}$$

$$\begin{aligned} C_4=0. \end{aligned}$$

Transforming into physical space via inverse Fourier transform, as shown in (

2.109) we obtain

$$\begin{aligned} {\tilde{v}}^*=\frac{1}{4\pi }\int ^{\infty }_{-\infty }{\int ^{\infty }_{-\infty } {{\left| k\right| e}^{ik\left( x-\xi \right) }\phi \left( \xi ,\tau \right) dk{{d\xi =\frac{1}{2}}}}}I\left\{ \phi \right\} \end{aligned}$$

(2.116)

The operator

\(I\left\{ \phi \right\} \) is a linear operator that in Fourier constitutes a multiplication by |

*k*| i.e.

\(I\left\{ {\cos \left( kx\right) \ }\right\} \mathrm{=\ |k|}{\cos \left( kx\right) \ }\). Equation (

2.106) may at this point be written as

$$\begin{aligned} \phi _\tau +\frac{1}{2}{\phi _x}^2-\alpha \phi _{xx}-\frac{1}{2}I\left\{ \phi \right\} =0 \end{aligned}$$

(2.117)

This equation is well-known as the Michelson-Sivashinsky (MS) equation. Its dimensional form will be the following

$$\begin{aligned} \phi _\tau +\frac{1}{2}S_L{\phi _x}^2-\frac{S_L\mathtt{J}}{\left( \sigma -1\right) }\phi _{xx}-\frac{1}{2}I\left\{ \phi \right\} =0 \end{aligned}$$

(2.118)