Laminar boundary layer flow of a combustible gas over a semi-infinite porous surface

Abstract

Laminar boundary layer flow of a combustible gas under mixed convection over a semi-infinite porous surface is analyzed theoretically and numerically. The governing equations are transformed into a set of partial differential equations which are solved numerically by using finite difference method. To assess the accuracy of numerical results, we also solve these equations by using perturbation method for low frequency range and asymptotic method for high frequency range. Numerical computations have been carried out for several values of parameters involved in the equations, namely the Prandtl number, the Schmidt number, the wall temperature and the mass flux for injected fuel. Results show that the momentum, thermal and concentration boundary layers increase owing to the increase in the mass flux, but increasing Prandtl number reduces the thermal boundary layer and thickens the momentum and concentration boundary layers. Moreover, an expression for the ignition distance involving the pertinent parameters is derived using large activation energy asymptotics. It is found that the ignition distance considerably diminishes upon increasing the above parameters.

1 Introduction

Ignition of a reacting boundary layer flow over a porous surface has received noticeable attention due to its wide range of application in various fields. Practical problems of this phenomenon arise in flame stabilization in combustor [1], internal combustion engines, oil and gas recovery, hydrogen production, household heating system and burners, air heating process, vehicle heating system, combustion of liquid fuels, etc.

Sheu and Lin [2] solved the ignition phenomenon of steady non-premixed boundary layer equations with constant mass fluxes of fuel injected from hot wedge-type porous walls by the method of large activation energy asymptotics. They compared numerical results with the asymptotic result for the purpose to verify the qualitative validity of asymptotic results. The dependence of ignition mechanism on system parameters was also discussed. Law and Law [1] provided an analysis of thermal ignition of a steady laminar boundary layer flow of a combustible mixture over a hot flat plate by using asymptotic analysis with high activation energies. They derived solution for the inner diffusive reactive region near the wall and applied a Damkohler number to get boundary conditions for the outer diffusive convective region.

Sheu and Lin [3] investigated the ignition criterion of a combustible gas flowing over a wedge at a high surface temperature. An explicit expression for ignition mechanism including the effect of the Prandtl number and wedge angle was derived. Numerical results were illustrated to expose the relationship between ignition and relevant system parameters. The problem of ignition in the boundary layer on a porous flat plate was described by Suzuki [4]. A series expansion method was applied to solve unsteady non-similar boundary layer equations. The ignition and development of combustion had been examined for various physical parameters.

Tsuji [5] studied the ignition condition of an ignitable mixture over a porous flat plate. Fuel was injected uniformly into the boundary layer from the porous surface. An iterative technique was used to obtain the approximate solution of the boundary layer equations. Dooley [6] used iteration method to solve the non-similar boundary layer equation of a combustible gas mixture to determine the ignition distance. Marble and Adamson [7] examined an analysis of chemical reaction in a laminar mixing region. They solved boundary layer equation by using the series expansion method. The influences of various physical parameters on the ignition distance were also investigated.

Garcia-Ybarra and Treviño [8] determined the thermal diffusion effects of combustible mixtures of hydrogen–air within the development of a boundary layer on a hot flat plate. They considered the large value of plate temperature and large activation energy to investigate the ignition temperature as well as ignition distance. The governing equations were reduced to a single integro-differential equation for the concentration of atomic hydrogen and the final ignition distance was illustrated. Theoretical investigation of reacting catalytic surface in non-premixed stagnation-point flow over a porous plate was investigated by Sheu et al. [9]. The relationship of reacting surface, as well as the temperature of fuel mixture at ignition also analyzed graphically under the influences of the injected mass flux, the strain rate of flow, the Prandtl number, Pr, and the Schmidt number, Sc. The critical ignition temperature and the minimum ignition temperature profile were also systematically discussed.

Sheu and Sun [10] numerically examined ignition delay of stagnation-point oxidizing flows over a wall with the injection of fuel. The fourth-order accurate Runge–Kutta–Fehlberg scheme with a local integration error equal to 10⁻⁵ was used in this work. In addition, the effects of flow strain rate, Lewis numbers and Prandtl number on ignition delay were investigated.

Shangpeng et al. [11] studied the thermal ignition of combustible boundary layer flows over the hot surface of a wedge or cone with an unheated starting length theoretically and numerically. A locally similar solution was derived with large activation energy. Relations between ignition distance and various physical parameters have also been depicted. The effect of catalytic surface reactions on the gas-phase ignition of premixed boundary layer flows was determined by Sheu and Lin [12]. Here, ignition criterion had been examined through the Prandtl number, the Schmidt number, the wedge angle and the surface temperature. The regions of ignition and of absence of ignition were also discussed graphically for different values of wedge angle, surface temperature and Prandtl number.

From the above survey of the literature, it is observed that several analytical methods have been developed to obtain approximate solutions, such as, series expansion [4, 7], iteration [6], activation energy asymptotic [11, 13], etc. It is worthy of mentioning that all of these studies have not compared their analytical results with other numerical approximation methods. To validate the numerical solutions, a comparison with analytical results is important.

The purpose of this study is to solve the governing equations utilizing the finite difference method and to assess the accuracy of numerical results with the solutions of perturbation and asymptotic method. It shows a good agreement among the solutions. The numerical results are illustrated in terms of streamlines, isotherms and isolines of fuel and oxidizer by using different values of Prandtl number, Schmidt number, wall temperature and mass flux for injected fuel. Relations between the ignition distance of a combustible gas and various physical parameters are also elucidated.

2 Mathematical formulation

A steady, two-dimensional, laminar boundary layer flow of a combustible gas over a semi-infinite horizontal heated porous surface under mixed convection is considered. Figure 1 shows the geometry of this problem schematically, which is placed in a combustible gas of ambient temperature $T_{\infty }$. We assume that the plate is heated to a constant temperature T_w and provide a source of energy of ignition, where T_w > $T_{\infty }$. Heat is supplied to the fluid by diffusion, convection and reaction term from the surface. The fuel is injected from the lower side of the porous surface at a constant mass flux of fuel m_w with a uniform high temperature and mixed with the oxidizer. The specific heats at constant pressure of the species are considered to be constant. Thermal radiation, Soret and Dufour effects are neglected. It is assumed that the reaction between the fuel F and oxidizer O leading to the formation of some product can be represented by a one step overall irreversible reaction.

$${\text{Fuel}}\,\left[ {\text{F}} \right] + \nu_{O} \;{\text{Oxidizer}}\,\left[ {\text{O}} \right] \to {\text{Product}}.$$

(1)

Fig. 1

Schematic diagram of the physical system

Furthermore, the combustible gas is a mixture of ideal gases so that the values of ρμ, ρλ, ρD_F and ρD_O are taken to be constant with the approximations ρμ = ρ_∞μ_∞ and ρλ = ρ_∞λ_∞. All of the above assumptions are made based on the Refs. [1,2,3, 6, 14].

Based on the above thoughts, the appropriate governing equations for reacting boundary layer flow under mixed convection are as follows [2, 3, 15]:

$$\frac{\partial }{\partial x}\left( {\rho u} \right) + \frac{\partial }{\partial y}\left( {\rho v} \right) = 0$$

(2)

$$\rho \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = \frac{\partial }{\partial y}\left( {\mu \frac{\partial u}{\partial y}} \right) + \rho g\beta \left( {T - T_{\infty } } \right)$$

(3)

$$\rho c_{p} \left( {u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}} \right) = \frac{\partial }{\partial y}\left( {\lambda \frac{\partial T}{\partial y}} \right) + qW_{F} w$$

(4)

$$\rho \left( {u\frac{{\partial Y_{F} }}{\partial x} + v\frac{{\partial Y_{F} }}{\partial y}} \right) = \frac{\partial }{\partial y}\left( {\rho D_{F} \frac{{\partial Y_{F} }}{\partial y}} \right) - W_{F} w$$

(5)

$$\rho \left( {u\frac{{\partial Y_{O} }}{\partial x} + v\frac{{\partial Y_{O} }}{\partial y}} \right) = \frac{\partial }{\partial y}\left( {\rho D_{O} \frac{{\partial Y_{O} }}{\partial y}} \right) - \nu_{O} W_{O} w$$

(6)

$$P = \rho RT$$

(7)

Reaction rate term used in Eqs. (4)–(6) is expressed by

$$w = BT^{\gamma } \left( {\frac{{\rho Y_{F} }}{{W_{F} }}} \right)^{{n_{F} }} \left( {\frac{{\rho Y_{O} }}{{W_{O} }}} \right)^{{n_{O} }} \exp \left( { - \frac{E}{RT}} \right)$$

(8)

subject to the following boundary conditions

$$u = 0,\quad \rho v = m_{w} ,\quad T = T_{w} ,\quad \rho D_{F} \frac{{\partial Y_{F} }}{\partial y} = m_{w} Y_{F} ,\quad \rho D_{O} \frac{{\partial Y_{O} }}{\partial y} = m_{w} Y_{O} \quad {\text{at}}\quad y = 0$$

(9)

$$u = U_{0} ,\quad T \to T_{\infty } ,\quad Y_{F} \to Y_{F,\infty } ,\quad Y_{O} \to Y_{O,\infty } \quad {\text{as}}\quad y \to \infty .$$

(10)

Here, u and v are the velocity components in the x and y directions, respectively, ρ indicates the density, P the pressure, T the temperature, Y_F and Y_O are mass fraction of fuel and oxidizer, respectively, μ the viscosity, λ the thermal conductivity, W_F and W_O are the molecular weights of fuel and oxidizer. D_i (i = F, O) mass diffusivity of the combustible gas mixture within the boundary layer. q the heat of combustion per unit mass of fuel, ν_O the stoichiometric coefficient of oxidizer, R the gas constant, g is the acceleration due to gravity, β is the volumetric expansion coefficient, B the pre-exponential factor, n_F and n_O is the reaction order for fuel and oxidizer, E the activation energy, c_p the specific heat at constant pressure m_w the constant mass flux of injected fuel. The subscripts $\infty$ and w denote the conditions in the free stream and at the wall, respectively. U₀ is the free stream velocity, Y_F,∞ and Y_O,∞ are the ambient fuel and oxidizer mass fractions, respectively.

We consider the following group of transformations to nondimensionalize Eqs. (2)–(6) and (9)–(10) [2]:

$$\begin{aligned} & \xi = m_{w} \left( {\frac{g}{{\rho_{\infty } \mu_{\infty } U_{0}^{3} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} x,\quad \eta = \left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } x}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} \int_{0}^{y} {\rho \,{\text{d}}y^{\prime } } ,\quad \theta = \frac{{T - T_{\infty } }}{{T_{w} - T_{\infty } }},\, \\ & \psi = \left( {\rho_{\infty } \mu_{\infty } U_{0} x} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} f\left( {\xi ,\eta } \right),\quad y_{F} = Y_{F} ,\quad y_{O} = {{Y_{O} } \mathord{\left/ {\vphantom {{Y_{O} } \chi }} \right. \kern-0pt} \chi },\quad \tilde{T} = \frac{{c_{p} T}}{q} \\ \end{aligned}$$

(11)

where χ = ν_OW_O/W_F.

The stream function ψ is defined in the following way

$$\rho u = \frac{\partial \psi }{\partial y},\quad \rho v = - \frac{\partial \psi }{\partial x}$$

(12)

that identically satisfy the continuity Eq. (2).

Applying Eqs. (11) and (12) on Eqs. (2)–(6), the momentum, energy and concentration equations become,

$$f^{\prime \prime \prime } + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}ff^{\prime \prime } - \frac{\xi }{{2\left( {1 + \xi } \right)}}f^{\prime 2} + \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta = \xi \left( {f^{\prime } \frac{{\partial f^{\prime } }}{\partial \xi } - f^{\prime \prime } \frac{\partial f}{\partial \xi }} \right)$$

(13)

$$\frac{1}{\Pr }\theta^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}f\theta^{\prime } = \xi \left( {f^{\prime } \frac{\partial \theta }{\partial \xi } - \theta^{\prime } \frac{\partial f}{\partial \xi }} \right) - \frac{{\varLambda \xi \left( {1 + \xi } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \omega }}{{m_{w} \delta }}$$

(14)

$$\frac{1}{{Sc_{F} }}y_{F}^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}fy_{F}^{\prime } = \xi \left( {f^{\prime } \frac{{\partial y_{F} }}{\partial \xi } - y_{F}^{\prime } \frac{\partial f}{\partial \xi }} \right) + \frac{{\varLambda \xi \left( {1 + \xi } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \omega }}{{m_{w} A}}$$

(15)

$$\frac{1}{{Sc_{O} }}y_{O}^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}fy_{O}^{\prime } = \xi \left( {f^{\prime } \frac{{\partial y_{O} }}{\partial \xi } - y_{O}^{\prime } \frac{\partial f}{\partial \xi }} \right) + \frac{{\varLambda \xi \left( {1 + \xi } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} \omega }}{{m_{w} A}}$$

(16)

the dimensionless reaction term is given by

$$\omega = \left( {1 + \delta \theta } \right)^{1 + \gamma - n} y_{F}^{{n_{F} }} y_{O}^{{n_{O} }} \exp \left[ { - \frac{{\theta_{A} }}{1 + \delta \theta }} \right]$$

(17)

where n = n_F + n_O and the dimensionless parameters are defined by

$$\begin{aligned} & \Pr = \frac{{\mu c_{p} }}{\lambda },\quad {\text{Sc}}_{F} = \frac{\mu }{{\rho D_{F} }},\quad {\text{Sc}}_{O} = \frac{\mu }{{\rho D_{O} }},\quad \delta = \frac{{T_{w} - T_{\infty } }}{{T_{\infty } }}, \\ & \theta_{A} = \frac{E}{{RT_{\infty } }},\quad A = \frac{q}{{c_{p} \tilde{T}_{\infty } }},\quad \varLambda = \left( {\frac{{\rho_{\infty } \mu_{\infty } U_{0} }}{g}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} AB\left( {\frac{q}{{c_{p} }}} \right)^{\gamma - n} \left( {\frac{P}{{RW_{F} }}} \right)^{n - 1} \nu_{0}^{{n_{O} }} T_{\infty }^{1 + \gamma - n} . \\ \end{aligned}$$

(18)

Here, Pr and Sc_F and Sc_O are the Prandtl number and Schmidt numbers for the fuel and oxidizer, δ is the dimensionless surface temperature, θ_A is the dimensionless activation energy, A is the dimensionless heat release parameter and Λ is the dimensionless reaction parameter.

Finally, the boundary conditions become

$$\begin{aligned} & f = - \frac{{4\left( {1 + \xi } \right)}}{2 + 3\xi }\left[ {\frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} + \xi \frac{\partial f}{\partial \xi }} \right],\quad f^{\prime } = 0,\quad \theta = 1, \\ & y_{F}^{\prime } = {\text{Sc}}_{F} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F} ,\quad y_{O}^{\prime } = {\text{Sc}}_{O} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O} \quad {\text{at}}\quad \eta = \, 0 \\ \end{aligned}$$

(19)

$$f^{\prime } \to \left( {1 + \xi } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,\quad \theta \to 0,\quad y_{F} \to y_{F,\infty } ,\quad y_{O} \to y_{O,\infty } \;{\text{as}}\;\eta \to \infty$$

(20)

Here, the prime (′) denote the differentiation with respect to η.

3 Asymptotic analysis

In the present analysis, we are interested to study the ignition in the gas phase for large values of activation energy. The development of the analysis will follow the guidelines established by Law and Law [1].

3.1 Chemically frozen limit

The solutions in the chemically frozen limit for a standard asymptotic method are first required. In the case $\theta_{A} \to \infty$, the flow field is completely frozen and the reaction terms are negligible; then, the problem is governed by

$$f^{{\prime {\prime \prime }}} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}ff^{{\prime \prime }} - \frac{\xi }{{2\left( {1 + \xi } \right)}}f^{\prime 2} + \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta = \xi \left( {f^{\prime } \frac{{\partial f^{\prime } }}{\partial \xi } - f^{{\prime \prime }} \frac{\partial f}{\partial \xi }} \right)$$

(21)

$$\frac{1}{\Pr }\theta^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}f\theta^{\prime } = \xi \left( {f^{\prime } \frac{\partial \theta }{\partial \xi } - \theta^{\prime } \frac{\partial f}{\partial \xi }} \right)$$

(22)

$$\frac{1}{{{\text{Sc}}_{F} }}y_{F}^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}fy_{F}^{\prime } = \xi \left( {f^{\prime } \frac{{\partial y_{F} }}{\partial \xi } - y_{F}^{\prime } \frac{\partial f}{\partial \xi }} \right)$$

(23)

$$\frac{1}{{{\text{Sc}}_{O} }}y_{O}^{{\prime \prime }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}fy_{O}^{\prime } = \xi \left( {f^{\prime } \frac{{\partial y_{O} }}{\partial \xi } - y_{O}^{\prime } \frac{\partial f}{\partial \xi }} \right)$$

(24)

subject to the conditions

$$\begin{aligned} & f = - \frac{{4\left( {1 + \xi } \right)}}{2 + 3\xi }\left[ {\frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} + \xi \frac{\partial f}{\partial \xi }} \right],f^{\prime } = 0,\quad \theta = 1, \\ & y_{F}^{\prime } = {\text{Sc}}_{F} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F} ,\quad y_{O}^{\prime } = {\text{Sc}}_{O} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O} \quad {\text{at}}\quad \eta = \, 0 \\ \end{aligned}$$

(25)

$$f^{\prime } \to \left( {1 + \xi } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,\quad \theta \to 0,\quad y_{F} \to y_{F,\infty } ,\quad y_{O} \to y_{O,\infty } \quad {\text{as}}\quad \eta \to \, \infty$$

(26)

3.2 Weakly reactive state

Because of large activation energy, it is expected that ignition would occur near the high-temperature wall ($\eta$ → 0). Thus, we introduce the following coordinate transformation

$$\zeta = \int_{0}^{\eta } {\exp \left[ { - \Pr \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}\int_{0}^{\eta } {f\,{\text{d}}\eta } } \right]} {\text{d}}\eta /C,$$

(27)

where

$$C = \int_{0}^{\infty } {\exp \left[ { - \Pr \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}\int_{0}^{\eta } {f\,{\text{d}}\eta } } \right]} {\text{d}}\eta$$

(28)

Using this transformation into the energy Eq. (14), we obtain

$$\frac{{\partial^{2} \theta }}{{\partial \zeta^{2} }} = \Pr Q\xi \left( {Q\frac{{\partial f^{\prime } }}{\partial \zeta }\frac{\partial \theta }{\partial \xi } - \frac{\partial f}{\partial \xi }\frac{\partial \theta }{\partial \zeta }} \right) - \frac{\Pr \varLambda }{{m_{w} \delta }}Q^{2} \xi \left( {1 + \xi } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \omega$$

(29)

where

$$Q\left( \eta \right) = C\exp \left[ { - \Pr \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}\int_{0}^{\eta } {f\,{\text{d}}\eta } } \right]$$

(30)

The boundary conditions take the form

$$\theta = 1\quad {\text{at}}\quad \zeta = 0\quad {\text{and}}\quad \theta = 0\quad {\text{at}}\quad \zeta = 1.$$

(31)

According to Eq. (27), as $\eta$ → 0 the transformed coordinate ζ is

$$\zeta = {{\left[ {\eta - \Pr \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}f^{{\prime \prime }} \left( {\xi ,0} \right)\eta^{2} /2 + O\left( {\eta^{3} } \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {\eta - \Pr \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}}f^{{\prime \prime }} \left( {\xi ,0} \right)\eta^{2} /2 + O\left( {\eta^{3} } \right)} \right]} C}} \right. \kern-0pt} C} .$$

(32)

When $\eta$ → 0 and ζ → 0, we consider the stretched inner coordinate as

$$\sigma = - C\theta_{fz,w}^{\prime } \zeta /\varepsilon .$$

(33)

Here, $\varepsilon$ is a perturbation parameter and $\varepsilon = {{T_{w}^{2} } \mathord{\left/ {\vphantom {{T_{w}^{2} } {\theta_{A} }}} \right. \kern-0pt} {\theta_{A} }}T_{\infty } \left( {T_{w} - T_{\infty } } \right)$ and $\theta_{fz,w}^{\prime } = {{\partial \theta_{fz} \left( {\xi ,0} \right)} \mathord{\left/ {\vphantom {{\partial \theta_{fz} \left( {\xi ,0} \right)} {\partial \eta }}} \right. \kern-0pt} {\partial \eta }}$. The magnitude of $\theta_{fz,w}^{\prime }$ is negative for the problem of interest.

The temperature and species in the inner region are expanded as follows:

$$\theta_{in} = \theta_{f} + \varepsilon \phi + O\left( {\varepsilon^{2} } \right) .$$

(34)

In terms of the coordinate σ, the inner expansion (34) becomes

$$\begin{aligned} \theta_{in} & = \theta_{w} + \frac{{\partial \theta \left( {\xi ,0} \right)}}{\partial \eta } + O\left( {\eta^{2} } \right) + \varepsilon \phi + O\left( {\varepsilon^{2} } \right) \\ & = \theta_{w} + \varepsilon \left( {\phi - \sigma } \right) + O\left( {\varepsilon^{2} } \right) \\ \end{aligned}$$

(35)

After substituting Eqs. (35) into (29), as $\eta$ → 0, we have

$$\begin{aligned} & \frac{{\partial^{2} \phi }}{{\partial \sigma^{2} }} - \Pr \xi \left[ {\varepsilon \frac{1}{{\phi_{f,w}^{\prime } }}\frac{{\partial f\left( {\xi ,0} \right)}}{\partial \xi }\left( {\frac{\partial \phi }{\partial \sigma } - 1} \right) - \varepsilon^{3} \frac{{f^{{\prime \prime }} \left( {\xi ,0} \right)}}{{\phi_{f,w}^{\prime 3} }}\sigma \frac{\partial \phi }{\partial \sigma }} \right] = \\ & \quad - \frac{\varepsilon \Pr }{{\phi_{f,w}^{\prime 2} }}\xi \left( {1 + \xi } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \frac{\varLambda }{{m_{w} \delta }}\left( {\frac{q}{{c_{p} }}} \right)^{n - \gamma } \left( {\frac{{\phi_{w} }}{{\phi_{\infty } }}} \right)^{1 + \gamma - n} \frac{{\varPhi_{F}^{nF} \varPhi_{o}^{no} W_{F}^{no} }}{{W_{o}^{no} }}\exp \left( { - \frac{E}{{R\phi_{w} }}} \right)\exp \left( {\phi - \sigma } \right) \\ \end{aligned}$$

(36)

Noticeably, the diffusion term $\frac{{\partial^{2} \phi }}{{\partial \sigma^{2} }}$ of Eq. (36) is dominating than the other term in left-hand side. In Eq. (36), the diffusion term is balanced with the reaction term at a weakly reactive state. So Eq. (36) is reduced to

$$\frac{{{\text{d}}^{2} \tau }}{{{\text{d}}\sigma^{2} }} = - \frac{1}{2}\Delta \exp \left( {\phi - \sigma } \right)$$

(37)

where

$$\Delta = \frac{2\varepsilon \Pr }{{\phi_{f,w}^{\prime } }}\xi \left( {1 + \xi } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \frac{\varLambda }{{m_{w} \delta }}\left( {\frac{q}{{c_{p} }}} \right)^{n - \gamma } \left( {\frac{{\phi_{w} }}{{\phi_{\infty } }}} \right)^{1 + \gamma - n} \frac{{\varPhi_{F}^{nF} \varPhi_{o}^{no} W_{F}^{no} }}{{W_{o}^{no} }}\exp \left( { - \frac{E}{{R\phi_{w} }}} \right)$$

(38)

The boundary conditions are

$$\phi = 0\quad {\text{at}}\quad \sigma = 0,\quad \frac{{\partial^{2} \phi }}{{\partial \sigma^{2} }} \to 0\quad {\text{as}}\quad \sigma \to \infty$$

(39)

The last boundary condition is obtained with the help of the inner and outer solutions. The matching technique is identical to that performed by Law and Law [1] and Rangle et al. [16].

3.3 Ignition criterion

According to Ref. [13], it is deduced that for Δ < 1 there exist two solutions, but for Δ > 1 there is no solution. This characteristic suggests that the ignition occurs when Δ = 1. It has to be mentioned that DamkÖhler number is defined as the ratio of the chemical effect to transport effect. From this point of view, the ignition criterion is obtainable by substituting $\Delta_{I} = 1$ [13]. In a nutshell, ignition is expected to occur when the reaction rate is equal to the convective mass transport rate. Considering

$$M = \frac{2\varepsilon \Pr }{{\phi_{f,w}^{\prime 2} }}\frac{\varLambda }{{m_{w} \delta }}\left( {\frac{q}{{c_{p} }}} \right)^{n - \gamma } \left( {\frac{{\phi_{w} }}{{\phi_{\infty } }}} \right)^{1 + \gamma - n} \frac{{\varPhi_{F}^{nF} \varPhi_{o}^{no} W_{F}^{no} }}{{W_{o}^{no} }}\exp \left( { - \frac{E}{{R\phi_{w} }}} \right)$$

(40)

from Eq. (38), we obtain

$$\Delta_{I} = M\xi (1 + \xi )^{ - 1/2}$$

(41)

After introducing dimensional coordinate variable x_I in Eq. (41), it becomes

$$x_{I} = \frac{{1 + \left( {1 + 4M^{2} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{2M^{2} }}\frac{1}{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } U_{0}^{3} }}{g}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$

(42)

Here, x_I is the first ignition location.

Ignition occurs simultaneously over the whole surface if the wall temperature T_w exceeds a critical value. Otherwise, the lack of ignition over the whole surface is observed. Thus, this critical temperature is a minimum wall temperature to achieve ignition. The magnitude of critical temperature depends on system parameters.

4 Solution methodologies

4.1 Numerical solutions for all values of ξ

In order to analyze the influences of non-similar flow and consumption of reactants, the numerical study is based on Eqs. (13)–(16).

Equations (13)–(16) are solved by using finite difference method. In this process, we first transform the partial differential Eqs. (13)–(16) into a system of second-order differential equations by taking the following assumption:

$$f^{\prime } = u,\quad f = v$$

(43)

Substituting Eq. (43) into Eqs. (13)–(16), we obtain

$$u^{{\prime \prime }} + p_{1} vu^{\prime } - \frac{1}{2}p_{2} u^{2} + p_{2} \frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta = \xi \left( {u\frac{\partial u}{\partial \xi } - u^{\prime } \frac{\partial v}{\partial \xi }} \right)$$

(44)

$$\frac{1}{\Pr }\theta^{{\prime \prime }} + p_{1} v\theta^{\prime } = \xi \left( {u\frac{\partial \theta }{\partial \xi } - \theta^{\prime } \frac{\partial v}{\partial \xi }} \right) - p_{3} \frac{\varLambda \omega }{{m_{w} \delta }}$$

(45)

$$\frac{1}{{{\text{Sc}}_{F} }}y_{F}^{{\prime \prime }} + p_{1} vy_{F}^{\prime } = \xi \left( {u\frac{{\partial y_{F} }}{\partial \xi } - y_{F}^{\prime } \frac{\partial v}{\partial \xi }} \right) + p_{3} \frac{\varLambda \omega }{{m_{w} A}}$$

(46)

$$\frac{1}{{{\text{Sc}}_{O} }}y_{O}^{{\prime \prime }} + p_{1} vy_{O}^{\prime } = \xi \left( {u\frac{{\partial y_{O} }}{\partial \xi } - y_{O}^{\prime } \frac{\partial v}{\partial \xi }} \right) + p_{3} \frac{\varLambda \omega }{{m_{w} A}}$$

(47)

Then, the boundary conditions are

$$\begin{aligned} & v\left( {\xi ,0} \right) = - \frac{1}{{p_{1} }}\left[ {\frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} + \xi \frac{\partial f}{\partial \xi }} \right],\quad u\left( {\xi ,0} \right) = 0,\quad \theta \left( {\xi ,0} \right) = 1, \\ & y^{\prime}_{F} \left( {\xi ,0} \right) = {\text{Sc}}_{F} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F} \left( {\xi ,0} \right), \\ & y_{O}^{\prime } \left( {\xi ,0} \right) = {\text{Sc}}_{O} \frac{{\left( {m_{w} \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }}\left( {\frac{{U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O} \left( {\xi ,0} \right) \\ \end{aligned}$$

(48)

$$u\left( {\xi ,\infty } \right) \to \left( {1 + \xi } \right)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0pt} 2}}} ,\quad \theta \left( {\xi ,\infty } \right) \to 0,\quad y_{F} \left( {\xi ,\infty } \right) \to y_{F,\infty } ,\quad y_{O} \left( {\xi ,\infty } \right) \to y_{O,\infty }$$

(49)

Here, we consider

$$p_{1} = \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)}},\quad p_{2} = \frac{\xi }{{\left( {1 + \xi } \right)}},\quad p_{3} = \frac{\xi }{{\left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}$$

(50)

Now Eqs. (44)–(47) subject to the boundary conditions (48)–(49) are discretized using central difference approximation along η-direction while backward difference is used for ξ-direction. Thus, we have a system of tri-diagonal algebraic equations in the following form:

$$B_{k} W_{i,j - 1} + A_{k} W_{i,j} + C_{k} W_{i,j + 1} = D_{k}$$

(51)

Here, the subscript k (= 1, 2, 3, 4) indicates the momentum equation, the energy equation and the species equations as well as the functions f, θ, y_F and y_O, respectively. Also i (= 1, 2, 3,…, M) and j (= 1, 2, 3,…, N) are the corresponding grid points in ξ and η-directions, respectively. The coefficients A_k, B_k, C_k, D_k are given as below

For the momentum equation, we have

$$A_{1} = 2 + \left( {\frac{1}{2}\left( {\Delta \eta } \right)^{2} + \xi \Delta \eta } \right)u_{2,j}$$

(52)

$$B_{1} = - 1 + \frac{\Delta \eta }{2}\left( {p_{{}} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(53)

$$C_{1} = - 1 - \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(54)

$$D_{1} = \Delta \eta \left( {\xi u_{2,j} u_{1,j} + p_{2} \Delta \eta \frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta_{2,j} } \right)$$

(55)

From energy equation, we obtain

$$A_{2} = \frac{2}{\Pr } + \xi \Delta \eta u_{2,j} - p_{3} \frac{\varLambda \omega }{{m_{w} \delta \theta_{2,j} }}$$

(56)

$$B_{2} = - \frac{1}{\Pr } + \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(57)

$$C_{2} = - \frac{1}{\Pr } - \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(58)

$$D_{2} = \Delta \eta \xi u_{2,j} \theta_{1,j}$$

(59)

For the concentration equation of fuel, we find

$$A_{3} = \frac{2}{{{\text{Sc}}_{F} }} + \xi \Delta \eta u_{2,j} + p_{3} \frac{\varLambda \omega }{{m_{w} Ay_{F,2,j} }}$$

(60)

$$B_{3} = - \frac{1}{{{\text{Sc}}_{F} }} + \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(61)

$$C_{3} = - \frac{1}{{{\text{Sc}}_{F} }} - \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(62)

$$D_{3} = \Delta \eta \xi u_{2,j} y_{F,1,j}$$

(63)

Finally, from the concentration equation of oxidizer, we have

$$A_{4} = \frac{2}{{{\text{Sc}}_{O} }} + \xi \Delta \eta u_{2.j} + p_{3} \frac{\varLambda \omega }{{m_{w} Ay_{O,2.j} }}$$

(64)

$$B_{4} = - \frac{1}{{{\text{Sc}}_{O} }} + \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2,j} - v_{1,j} }}{\Delta \xi }} \right)$$

(65)

$$C_{4} = - \frac{1}{{{\text{Sc}}_{O} }} - \frac{\Delta \eta }{2}\left( {p_{1} v_{2,j} + \xi \frac{{v_{2.j} - v_{1.j} }}{\Delta \xi }} \right)$$

(66)

$$D_{4} = \Delta \eta \xi u_{2,j} y_{O,2,j}$$

(67)

For fixed values of j, the tri-diagonal Eq. (51) is solved for i (= 1, 2, 3,…, M) by using the well-known Thomas Algorithm [17]. Then, the result is represented in j directions. Until the steady state comes, the solutions are determined at every steps of ξ direction from ξ = 0.0. The convergence property for the numerical solutions is chosen in such a way that the difference between the values of the function $f\left( {\xi ,\eta } \right)$ in two consecutive iterations is less than 10⁻⁵. In the ξ-direction, the simulation is started from ξ = 0.0 and then out marched to the point ξ = 80.

4.2 Perturbation method for small values of ξ

Relatively near the leading edge, i.e., for small ξ, series solution of Eqs. (13)–(16) may be obtained by using perturbation method treating x as a perturbation parameter. Hence, we expand the functions f (ξ, η), θ (ξ, η), y_F (ξ, η) and y_F (ξ, η) in powers of ξ^1/2, that is,

$$\begin{aligned} & f\left( {\xi ,\eta } \right) = \sum\limits_{m = 0}^{\infty } {\xi^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}}} f_{m} \left( \eta \right)} ,\quad \theta \left( {\xi ,\eta } \right) = \sum\limits_{m = 0}^{\infty } {\xi^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}}} \theta_{m} \left( \eta \right)} , \\ & y_{F} \left( {\xi ,\eta } \right) = \sum\limits_{m = 0}^{\infty } {\xi^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}}} y_{F,m} \left( \eta \right)} ,\quad y_{O} \left( {\xi ,\eta } \right) = \sum\limits_{m = 0}^{\infty } {\xi^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2}}} y_{O,m} \left( \eta \right)} \\ \end{aligned}$$

(68)

Substituting the above expansion into Eqs. (13)–(16) and (19)–(20) equating the various powers of ξ up to O (ξ), we obtain the following sets of equations

Equations of O(ξ⁰):

$$\begin{aligned} & f_{0}^{\prime \prime \prime } + \frac{1}{2}f_{0} f_{0}^{\prime \prime } = 0 \\ & \frac{1}{\Pr }\theta_{0}^{\prime \prime } + \frac{1}{2}f_{0} \theta_{0}^{\prime } = 0 \\ & \frac{1}{{{\text{Sc}}_{F} }}y_{F,0}^{\prime \prime } + \frac{1}{2}f_{0} y_{F,0}^{\prime } = 0 \\ & \frac{1}{{{\text{Sc}}_{O} }}y_{O,0}^{\prime \prime } + \frac{1}{2}f_{0} y_{O,0}^{\prime } = 0 \\ \end{aligned}$$

(69)

$$\begin{aligned} & f_{0} = 0,\quad f_{0}^{\prime } = 0,\quad \theta_{0} = 1,\quad y_{F,0}^{\prime } = 0,\quad y_{O,0}^{\prime } = 0\quad {\text{at}}\quad \eta = 0 \\ & f_{0}^{\prime } \to 1,\quad \theta_{0} \to 0,\quad y_{F,0} \to y_{F,\infty } ,\quad y_{O,0} \to y_{O,\infty } \quad {\text{as}}\quad \eta \to \infty \\ \end{aligned}$$

(70)

Equations of O(ξ^1/2):

$$\begin{aligned} & f_{1}^{{\prime {\prime \prime }}} + \frac{1}{2}\left( {f_{0} f_{1}^{{\prime \prime }} + 2f_{1} f_{0}^{{\prime \prime }} } \right) - \frac{1}{2}f_{0}^{\prime } f_{1}^{\prime } = 0 \\ & \frac{1}{\Pr }\theta_{1}^{\prime \prime } + \frac{1}{2}\left( {f_{0} \theta_{1}^{\prime } + 2f_{1} \theta_{0}^{\prime } } \right) - \frac{1}{2}f_{0}^{\prime } \theta_{1} = 0 \\ & \frac{1}{{{\text{Sc}}_{\text{F}} }}y_{F,1}^{\prime \prime } + \frac{1}{2}\left( {f_{0} y_{F,1}^{\prime } + 2f_{1} y_{F,0}^{\prime } } \right) - \frac{1}{2}f_{0}^{\prime } y_{F,1} = 0 \\ & \frac{1}{{{\text{Sc}}_{\text{O}} }}y_{O,1}^{\prime \prime } + \frac{1}{2}\left( {f_{0} y_{O,1}^{\prime } + 2f_{1} y_{O,0}^{\prime } } \right) - \frac{1}{2}f_{0}^{\prime } y_{O,1} = 0 \\ \end{aligned}$$

(71)

$$\begin{aligned} & f_{1} = - 2\left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} ,\quad f_{1}^{\prime } = 0,\quad \theta_{1} = 0, \\ & y_{F,1}^{\prime } = {\text{Sc}}_{F} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F,0} ,\quad y_{O,1}^{\prime } = {\text{Sc}}_{O} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O,0} \quad {\text{at}}\quad \eta = 0 \\ & f_{1}^{\prime } \to 0,\quad \theta_{0} \to 0,\quad y_{F,1} \to 0,\quad y_{O,1} \to 0\quad {\text{as}}\quad \eta \to \infty \\ \end{aligned}$$

(72)

Equations of O(ξ):

$$\begin{aligned} & f_{2}^{\prime \prime \prime } + \frac{1}{2}\left( {f_{0} f_{2}^{\prime \prime } + 2f_{1} f_{1}^{\prime \prime } + 3f_{2} f_{0}^{\prime \prime } } \right) - \frac{1}{2}\left( {f_{0}^{\prime 2} + f_{1}^{\prime 2} } \right) + \frac{1}{4}f_{0} f_{0}^{\prime \prime } - f_{0}^{\prime } f_{2}^{\prime } + \frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta_{0} = 0 \\ & \frac{1}{\Pr }\theta_{2}^{\prime \prime } + \frac{1}{2}\left( {f_{0} \theta_{2}^{\prime } + 2f_{1} \theta_{1}^{\prime } + 3f_{2} \theta_{0}^{\prime } } \right) - \frac{1}{2}f_{1}^{\prime } \theta_{1} + \frac{1}{4}f_{0} \theta_{0}^{\prime } - f_{0}^{\prime } \theta_{2} + \frac{\varLambda \omega }{{m_{w} \delta }} = 0 \\ & \frac{1}{{{\text{Sc}}_{F} }}y_{F,2}^{\prime \prime } + \frac{1}{2}\left( {f_{0} y_{F,2}^{\prime } + 2f_{1} y_{F,1}^{\prime } + 3f_{2} y_{F,0}^{\prime } } \right) - \frac{1}{2}f_{1}^{\prime } y_{F,1} + \frac{1}{4}f_{0} y_{F,0}^{\prime } - f_{0}^{\prime } y_{F,2} - \frac{\varLambda \omega }{{m_{w} \delta }} = 0 \\ & \frac{1}{{{\text{Sc}}_{O} }}y_{O,2}^{\prime \prime } + \frac{1}{2}\left( {f_{0} y_{O,2}^{\prime } + 2f_{1} y_{O,1}^{\prime } + 3f_{2} y_{O,0}^{\prime } } \right) - \frac{1}{2}f_{1}^{\prime } y_{O,1} + \frac{1}{4}f_{0} y_{O,0}^{\prime } - f_{0}^{\prime } y_{O,2} - \frac{\varLambda \omega }{{m_{w} \delta }} = 0 \\ \end{aligned}$$

(73)

$$\begin{aligned} & f_{2} = 0,\quad f_{2}^{\prime } = 0,\quad \theta_{2} = 0,\quad y_{F,2}^{\prime } = {\text{Sc}}_{F} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F,1} , \\ & y_{O,2}^{\prime } = {\text{Sc}}_{O} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O,1} \quad {\text{at}}\quad \eta = 0 \\ & f_{2}^{\prime } \to - \frac{1}{2},\quad \theta_{0} \to 0,\quad y_{F,2} \to 0,\quad y_{O,2} \to 0\quad {\text{as}}\quad \eta \to \infty \\ \end{aligned}$$

(74)

where

$$\omega = \left( {1 + \delta \theta } \right)^{1 + \gamma - n} y_{F,0}^{{n_{F} }} y_{O,0}^{{n_{O} }} \exp \left[ { - \frac{{\theta_{A} }}{1 + \delta \theta }} \right]$$

Solutions of above set of equations up to O(ξ) are obtained numerically using the nonlinear shooting method.

4.3 Asymptotic method for large values of ξ

In this section, attention has been given to the behavior of the solutions to Eqs. (13)–(16) when ξ is large. Here, a series solution in the high frequency range, utilizing the limiting solution as the zeroth order approximation, is sought; for this reason, the following transformations are introduced

$$\begin{aligned} & Y = \left( {1 + \xi } \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} \eta ,\quad f\left( {\xi ,\eta } \right) = \left( {1 + \xi } \right)^{{ - {3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-0pt} 4}}} h\left( {\xi ,Y} \right),\quad \theta \left( {\xi ,\eta } \right) = \theta \left( {\xi ,Y} \right) \\ & y_{F} \left( {\xi ,\eta } \right) = y_{F} \left( {\xi ,Y} \right),\quad y_{O} \left( {\xi ,\eta } \right) = y_{O} \left( {\xi ,Y} \right) \\ \end{aligned}$$

(75)

Equations (13)–(16) become

$$\begin{aligned} & \frac{{\partial^{3} h}}{{\partial Y^{3} }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)^{2} }}h\frac{{\partial^{2} h}}{{\partial Y^{2} }} - \frac{\xi }{{2\left( {1 + \xi } \right)^{2} }}\left( {\frac{\partial h}{\partial Y}} \right)^{2} + \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta = \frac{\xi }{{\left( {1 + \xi } \right)^{2} }} \\ &\times \left[ {\left( { - \frac{1}{2}\left( {\frac{\partial h}{\partial Y}} \right)^{2} + \frac{\partial h}{\partial Y}\frac{{\partial^{2} h}}{{\partial Y^{2} }}\frac{Y}{{4\left( {1 + \xi } \right)}}} \right.} \right. + \left. {\frac{\partial h}{\partial Y}\frac{{\partial^{2} h}}{\partial \xi \partial Y}} \right) - \left. {\left( {\left. { - \frac{3}{4}\frac{\partial \theta }{\partial Y}h + \frac{\partial \theta }{\partial Y}\frac{\partial h}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}} + \frac{\partial \theta }{\partial Y}\frac{\partial h}{\partial \xi }} \right)} \right.} \right] \\ \end{aligned}$$

(76)

$$\begin{aligned} & \frac{1}{\Pr }\frac{{\partial^{2} \theta }}{{\partial Y^{2} }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)^{2} }}h\frac{\partial \theta }{\partial Y} = \frac{\xi }{{\left( {1 + \xi } \right)^{2} }}\left[ {\left( {\frac{\partial h}{\partial Y}\frac{\partial \theta }{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}}} \right.} \right. + \left. {\frac{\partial h}{\partial Y}\frac{\partial \theta }{\partial \xi }} \right) \\ & - \left. {\left( {\left. { - \frac{3}{4}h\frac{\partial \theta }{\partial Y} + \frac{\partial \theta }{\partial Y}\frac{\partial h}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}} + \frac{\partial \theta }{\partial Y}\frac{{\partial^{2} h}}{\partial \xi \partial Y}} \right)} \right.} \right] - \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\varLambda \omega }{{m_{w} \delta }} \\ \end{aligned}$$

(77)

$$\begin{aligned} & \frac{1}{{{\text{Sc}}_{F} }}\frac{{\partial^{2} y_{F} }}{{\partial Y^{2} }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)^{2} }}h\frac{{\partial y_{F} }}{\partial Y} = \frac{\xi }{{\left( {1 + \xi } \right)^{2} }}\left[ {\left( {\frac{\partial h}{\partial Y}\frac{{\partial y_{F} }}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}}} \right.} \right. + \left. {\frac{\partial h}{\partial Y}\frac{{\partial y_{F} }}{\partial \xi }} \right) \\ & - \left. {\left( {\left. { - \frac{3}{4}h\frac{{\partial y_{F} }}{\partial Y} + \frac{{\partial y_{F} }}{\partial Y}\frac{\partial h}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}} + \frac{{\partial y_{F} }}{\partial Y}\frac{{\partial^{2} h}}{\partial \xi \partial Y}} \right)} \right.} \right] + \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\varLambda \omega }{{m_{w} A}} \\ \end{aligned}$$

(78)

$$\begin{aligned} & \frac{1}{{{\text{Sc}}_{O} }}\frac{{\partial^{2} y_{O} }}{{\partial Y^{2} }} + \frac{2 + 3\xi }{{4\left( {1 + \xi } \right)^{2} }}h\frac{{\partial y_{O} }}{\partial Y} = \frac{\xi }{{\left( {1 + \xi } \right)^{2} }}\left[ {\left( {\frac{\partial h}{\partial Y}\frac{{\partial y_{O} }}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}}} \right.} \right. + \left. {\frac{\partial h}{\partial Y}\frac{{\partial y_{O} }}{\partial \xi }} \right) \\ & - \left. {\left( {\left. { - \frac{3}{4}h\frac{{\partial y_{O} }}{\partial Y} + \frac{{\partial y_{O} }}{\partial Y}\frac{\partial h}{\partial Y}\frac{Y}{{4\left( {1 + \xi } \right)}} + \frac{{\partial y_{O} }}{\partial Y}\frac{{\partial^{2} h}}{\partial \xi \partial Y}} \right)} \right.} \right] + \frac{\xi }{{\left( {1 + \xi } \right)}}\frac{\varLambda \omega }{{m_{w} A}} \\ \end{aligned}$$

(79)

For large value of ξ, Eqs. (76)–(79) take the following form

$$\frac{{\partial^{3} h}}{{\partial Y^{3} }} + \frac{\delta }{{m_{w} }}\left( {\frac{{\rho_{\infty } \mu_{\infty } g}}{{U_{0} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \theta = 0$$

(80)

$$\frac{1}{\Pr }\frac{{\partial^{2} \theta }}{{\partial Y^{2} }} + \frac{\varLambda \omega }{{m_{w} \delta }} = 0$$

(81)

$$\frac{1}{{{\text{Sc}}_{F} }}\frac{{\partial^{2} y_{F} }}{{\partial Y^{2} }} - \frac{\varLambda \omega }{{m_{w} A}} = 0$$

(82)

$$\frac{1}{{{\text{Sc}}_{O} }}\frac{{\partial^{2} y_{O} }}{{\partial Y^{2} }} - \frac{\varLambda \omega }{{m_{w} A}} = 0$$

(83)

with the boundary conditions

$$\begin{aligned} & h = - \frac{4}{3}\left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} \xi ,\quad \frac{\partial h}{\partial Y} = 0,\quad \theta = 1, \\ & \frac{{\partial y_{F} }}{\partial Y} = {\text{Sc}}_{F} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{F} ,\quad \frac{{\partial y_{O} }}{\partial Y} = {\text{Sc}}_{O} \left( {\frac{{m_{w}^{2} U_{0} }}{{\rho_{\infty } \mu_{\infty } g}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} y_{O} \quad {\text{at}}\quad Y = 0 \\ \end{aligned}$$

(84)

$$\frac{\partial h}{\partial Y} \to 1,\quad \theta \to 0,\quad y_{F} \to y_{F,\infty } ,\quad y_{O} \to y_{O,\infty } \quad {\text{as}}\quad Y \to \quad \infty$$

(85)

Since ξ is large, the functions h (ξ, Y), θ (ξ, Y), y_F (ξ, Y) and y_F (ξ, Y) are expanded in a power series in negative powers of ξ.

Solutions of above set of equations are obtained numerically using the nonlinear shooting method.

5 Results and discussion

The governing conservation equations of a two-dimensional, steady, laminar boundary layer flow over a semi-infinite horizontal heated porous surface under mixed convection have been elucidated by using the finite difference method. To verify the accuracy of numerical results, we also solve the equations with the help of the perturbation method for small values of ξ, and asymptotic series expansion method for large values of ξ.

The numerical computations are carried out for several values of parameters involved in the equations, namely the Prandtl number (Pr), the Schmidt number (Sc) and the mass flux for injected fuel (m_w). The computed results are explained by plotting and corresponding physical reasoning are also given.

It should be mentioned that the data values taken in the present study are n_F = 0.15, n_O = 1.65, γ = 0.75, W_F = 58 g/mol, W_O = 32 g/mol, ν_O = 1.65, Y_F,∞ = Y_O,∞ = 0.2, p = 1 atm, q = 4.2 × 10⁷ J/kg, c_p = 1046 J/(kg K) [2], Pr = 0.7, Sc_F = Sc_O = 0.65, θ_A = 34.3, T_∞ = 293 K, T_w/T_∞ = 3.9 [14], g = 9.8 m/s², B = 3.4 × 10⁸, μ_∞ = 10⁻⁵ kg/(ms), m_w = 0.005 [2] and Λ = 1.84 × 10⁵.

5.1 Variation of velocity, temperature, fuel and oxidizer concentration profiles for different physical parameters

In Fig. 2a–d the velocity, temperature, fuel and oxidizer concentration profiles are shown to make the comparison among the numerical results, the perturbation solutions for the low frequency range of ξ and the asymptotic solutions for the high frequency range of ξ. It is clear from the figures that the solutions are in good agreement.

Fig. 2

Comparison of a velocity, b temperature, c fuel and d oxidizer concentration against η for different values of ξ

The influence of the Prandtl number, Pr, on the velocity, temperature, fuel concentration and oxidizer concentration is demonstrated in Fig. 3a–d, while Sc = 0.65, m_w = 0.005 and ξ = 80. It is seen that the increase in the Prandtl number, Pr, the momentum and thermal boundary layer thicknesses decrease. Prandtl number is defined as the ratio of the momentum diffusivity to the thermal diffusivity. So the value of Pr is increased with the increase in the momentum diffusivity or the decrease in the thermal diffusivity. For this reason, the increase in the momentum diffusivity leads to the increase in the momentum boundary layer thickness and the decrease in the thermal diffusivity which gives a thin thermal boundary layer thickness.

Fig. 3

Effect of the Prandtl number, Pr, on a velocity, b temperature, c fuel and d oxidizer concentration against η when ξ = 80.0

Figure 4a–d illustrates the response of different values of Schmidt number, Sc, on the velocity, temperature, fuel concentration and oxidizer concentration, while Pr = 0.7 and ξ = 80. It is found from the figures that for the higher values of Sc the momentum boundary layer and thermal boundary layer remain almost unaffected. The Schmidt number (Sc_F = μ/ρD_F, Sc_O = μ/ρD_O) is a dimensionless number defined as the ratio of kinematic diffusivity and mass diffusivity. Thus, the value of Sc becomes high for an increase in kinematic viscosity or a decrease in mass diffusivity. Consequently, the concentration profile corresponding to the small value of the Schmidt number is found to be higher.

Fig. 4

Effect of the Schmidt number, Sc, on a velocity, b temperature, c fuel and d oxidizer concentration against η when ξ = 80.0

The dimensionless velocity, temperature and concentration profiles are depicted in Fig. 5a–d, respectively, for different values of fuel injection parameter m_w while Pr = 0.7, Sc = 0.65 and ξ = 80. For the increase in external fuel injection, m_w, the peak value of the velocity decreases and the concentration increases because due to the injection both boundary layer thicknesses decrease. On the other hand, the reverse scenario is observed for the thermal boundary layer. That is, with increasing value of m_w, the value thermal boundary layer thicknesses increase.

Fig. 5

Effect of mass flux for injected fuel, m_w on a velocity, b temperature, c fuel and d oxidizer concentration against η when ξ = 80.0

5.2 Influence of different physical parameters on streamline, isotherms and isolines of concentration

Figure 6a–d shows how variations in the Prandtl number affect the streamlines, isotherms and isolines of concentration when Sc = 0.65, ξ = 40 and m_w = 0.005. The value of Pr is inversely proportional to the thermal diffusion coefficient. An increase in the value of Pr enhances the decay of the diffusion coefficient. For this reason, the temperature boundary layer thickness decreases. On the other hand, an increase in the value of the Prandtl number thickens the momentum as well as concentration boundary layers.

Fig. 6

Effect of Prandtl number, Pr, on a streamlines, b isotherms, c isolines for fuel and d isolines for oxidizer concentration

The influence of the Schmidt number, Sc, on the streamlines, isotherms, isolines of fuel and oxidizer concentration is demonstrated in Fig. 7a–d, while Pr = 0.7, m_w = 0.005 and ξ = 40.

Fig. 7

Effect of Schmidt number, Sc, on a streamlines, b isotherms, c isolines for fuel and d isolines for oxidizer concentration

The value of Sc is inversely proportional to the mass diffusion coefficient. So, an increment of Schmidt number produces a reduction of the diffusion coefficient and which acts in thinning the concentration boundary layer thickness. It is also found from Fig. 7a and b that for higher values of Sc, the momentum boundary layer and thermal boundary layer remain almost unaffected.

The effect of the injection of fuel, m_w on the streamlines, isotherms, isolines of fuel and oxidizer concentration is exhibited in Fig. 8a–d, while Pr = 0.7, Sc = 0.65 and ξ = 40. Results indicate that the momentum and concentration boundary layers become higher for larger values of m_w. Moreover, the thermal boundary layer also increases with an increase of injected fuel.

Fig. 8

Effect of mass flux for fuel, m_w, on a streamlines, b isotherms, c isolines for fuel and d isolines for oxidizer concentration

5.3 Influence of different physical parameters on ignition distance profile

The dimensionless ignition distance is defined as $X_{I} /X_{I,\xi = 0}$. Here, $X_{I,\xi = 0}$ is the ignition distance for Pr = 0.7, Sc = 0.65 and T_∞ = 293 K.

The Prandtl number is defined as the dimensionless ratio between momentum diffusivity and thermal diffusivity. From the physical point of view, for higher values of Pr, the heat generated from the chemical reaction near the surface is transported away from it at a smaller rate. The result of $X_{I} /X_{I,\xi = 0}$ versus Pr is presented in Fig. 9. According to this figure, the ignition distance decreases with an increase of Pr. So, the ignition is achieved more easily for greater Pr.

Fig. 9

Effect of Prandtl number, Pr, on ignition distance

According to Fig. 10, the greater values of wall temperature result in the growth of the ignition region. A slightly increased wall temperature results in a substantially decreased ignition distance for large activation energy.

Fig. 10

Influence of wall temperature on ignition distance

An increment of mass flus, m_w, produces an increment of the reaction rate and which acts in decreasing the ignition distance. Figure 11 illustrates the impact of mass flux for fuel on ignition distance.

Fig. 11

Effect of mass flux for fuel, m_w, on ignition distance

6 Conclusions

In this study, a steady laminar boundary layer flow of a combustible gas over a semi-infinite horizontal heated porous surface under mixed convection has been examined. The dimensionless governing equations are solved numerically using finite difference method. In order to valid the numerical results we also solved these equations by using perturbation method for low frequency range and the asymptotic method for high frequency range. It is found that the thermal ignition distance decreases because of the increase in different physical parameters, such as, Prandtl number, wall temperature and mass flux for injecting fuel. Moreover, from results and discussion section, we conclude that

• 1. The temperature boundary layer thickness decreases with an increase in the value of the Prandtl number, while the momentum and concentration boundary layer increase.

• 2. An increment of Schmidt number the concentration boundary layer thickness decreases, but the momentum boundary layer and thermal boundary layer remain almost unaffected.

• 3. The momentum and concentration boundary layers become higher for larger values of injected fuel. Moreover, the thermal boundary layer increases for increasing value of injected fuel.