Effects of chemical reaction and activation energy on Marangoni flow, heat and mass transfer over circular porous surface

Abstract

Effects of heat generation, chemical reaction and activation energy on Marangoni flow, heat and mass transfer over a circular porous surface are investigated. The problem has been formulated in the form of the nonlinear system of PDEs, which are changed to the dimensionless set of ODEs using the similarity replacement. The problem is solved numerically by a method named Legendre–Galerkin. The influences of the different parameters on the velocity, temperature and concentration are discussed and analysed. The local skin friction, local Nusselt number and local Sherwood number are also computed and investigated for different embedded parameters in the problem statements.

Keywords:

• Chemical reaction and activation energy
• circular surface
• heat and mass transfer
• heat generation
• Marangoni flow
• porous medium

1. Introduction

Marangoni convection flow, which is caused by surface tension, may be found in a variety of applications, including vapour bubbles, materials science, crystal formation melting, thin film spreading and welding. During the nucleation and development of vapour bubbles, Marangoni convection occurs by surface tension fluctuations produced by temperature/concentration differences at the bubble skin (see Christopher & Wang, 2001a, 2001b).

Literature research reveals that there is little research on the axisymmetric stream above a surface stretched in the course of its radius. Heat transport across a radially stretched surface has been examined by Ramly, Sivasankaran, and Noor (2017), Soid, Ishak, and Pop (2018) and Butt, Ali, and Mehmood (2017). Makinde, Mabood, Khan, and Tshehla (2016) studied combined effects of heat emission, Brownian flow and magnetic field on boundary layer flow of nanofluid through a surface with an extendable radius, whereas Mustafa, Hayat, and Alsaedi (2012) looked at an axisymmetric stream of nanofluid over a surface with extendable radius. Hayat, Waqas, Shehzad, and Alsaedi (2015) examined the flow of Jeffrey fluid across a surface with an extendable radius as a result of viscous dissipation and Joule heating.

The analysis of heat production in fluids is significant because it is relevant to a variety of physical problems, including chemical processes and dissociating fluids. The heat dispersion and, as a result, the deposition proportion of particles shall be affected by possible heat production processes. This might happen in nuclear reactors, blaze/ignition models, electronics and semiconductors, among other uses. In reality, there are several models of heat transmission in the laminar flow of sticky fluids in the literature. The heat transmission properties in the laminar boundary layer of a viscous fluid above a stretched sheet with viscous diffusion or frictional heating and within heat production were examined by Vajravelu and Hadjinicolaou (1993). Khidir and Sibanda (2014) looked at the volumetric rate of heat production, $q^{*},$ in their investigation such that $$q^{*}=\{\begin{array}{cc}{Q}_0(T-T_{\infty })& \mathrm{for}T\ge T_{\infty }\\ 0& \mathrm{for}T\le T_{\infty }\end{array}$$ where the heat creation/intake constant is $Q_0,$ such that $Q_0> 0$ serves as heat source while $Q_0< 0$ refers to heat sink. This relationship is accurate as a rough estimate of the case of some heat-releasing mechanisms with $T_{\infty }$ as the ambient temperature. They employed $q^{*}=Q_0\left(T-T_{\infty }\right)$ when the input temperature was not less than $T_{\infty }.$

Liu, Liu, and Wu (2022) are considered the thermo-dynamical model for rotating disk electrodes for second-order electrochemical-chemical-electrochemical (ECE) reactions, the effect of concentrations of three species on the current for ECE reaction is theoretically analysed, and the optimal current value is obtained. The magnetic field flow of an electrically conducting micropolar fluid with a radiative heat source and mixed chemically reactive species is considered by Nadeem, Feng, Islam, and He (2022).

He and Abd Elazem (2021) are studied the significance of partial slips and temperature jumps on the heat and mass transfer of a boundary layer nanofluid flowing through a stretched or shrinking surface.

For its extensive variety of practices in civil and chemical/mechanical engineering, convective transit in porous scheme has been a tremendous topic relevance and enthusiasm in modern years. Over the last several years, some investigations on convective heat and mass transport in a permeable media saturated with a nanofluid have been published in the biography. Darcy’s law is inapplicable in many actual scenarios because the porous medium confined by an impervious wall (see Khidir & Sibanda, 2014; Uddin, Kabir, & Alginahi, 2015), has greater stream amounts, and discloses a non-uniform permeability distribution towards barrier area.

Chemical interactions between species with limited Arrhenius energizing energy exist in many chemically reacting systems, with illustrations in geothermic and oil field engineering. Bestman (1990), who described asymptotic solution using perturbation technique to explain influence of the energizing energy on natural transfer of heat through a fluid by molecular motion in a permeable media, published one of the early investigations utilizing the dual chemical changes in flow profile. Bestman (1991) then examined emissive exchange of thermal energy on the flow of a fiery compound in a steep duct using the Arrhenius activation energy.

The purpose of current investigation is to see the way that heat generation controls stream, temperature and mass exchange through a circular expanding surface surrounded by a permeable scheme with chemical reaction/activation energy. This study is a follow-up to and expansion of a previous publication (Rashid & Ibrahim, 2020).

2. Mathematical model

The impact of heat production on two-dimensional, steady and laminar boundary layer Marangoni convictive stream above a circular surface buried in porous material is studied. As depicted from Figure 1, the coordinate system is chosen so that $r$-direction (motion direction) is alongside the radially surface and $z$-direction is right angled to it. The surface is $z=0,$ while the stream is contained in the zone $z>0.$ The flow analysis can be written as follows, assuming a kind of chemical changes with a finite Arrhenius activation energy: (1) $$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,$$(1) (2) $$u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}=\frac{\mu }{\rho }\left(\frac{{\partial }^{2}u}{\partial z^2}\right)-\frac{\mu }{K}u,$$(2) (3) $$u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\alpha \left(\frac{{\partial }^{2}T}{\partial z^2}\right) +Q_o(T-T_{\infty }),$$(3) (4) $$u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z}=D\left(\frac{{\partial }^{2}C}{\partial z^2}\right) -k_r^2{\left(\frac{T}{T_{\infty }}\right)}^n{e}^{(E_a/kT)}(C-C_{\infty }),$$(4) with boundary conditions (5) $$at\mathrm{ }\eta =0:\mathrm{ }\mu \frac{\partial u}{\partial z}=\frac{\partial \sigma }{\partial r},\mathrm{ }w=w_w,\mathrm{ }T=T_{w,\mathrm{ }}C=C_w,$$(5) (6) $$\eta \to \mathrm{\infty }:\mathrm{ }u=0,\mathrm{ }w=0,\mathrm{ }T=T_{\mathrm{\infty }},\mathrm{ }C=C_{\mathrm{\infty }},$$(6) where $u$ and $w$ are the velocities in the $r$ and $z$ directions, respectively, $\mu$ is the dynamic viscosity, $\alpha$ is the thermic diffusion, $\rho$ is the density of fluid, $K$ is the permeability, $T$ is the fluid temperature, $D$ is the solutal diffusivity, ${\left(T/T_{\infty }\right)}^n{e}^{E_a/kT}$ is the amended Arrhenius function, $k$ is the Boltzmann constant, $k_r^2\left(r\right)$ is the chemical change proportional with variable $r^2,$ $-1<n<1$ is exponent fitted rate constant, $\sigma$ is the surface tension, $C_{\infty }$ is the ambient concentration of fluid and $T_w$ and $C_w$ are the surface temperature and concentration, respectively. The surface tension is approximated by the Boussinesq approximation $$\sigma ={\sigma }_0[1-\gamma (T-T_{\mathrm{\infty }})-{\gamma }^{\mathrm{*}}(C-C_{\mathrm{\infty }}\left)\right]$$ where ${\sigma }_0$ is interface surface tension ratio and $$\gamma =-\frac{\partial \sigma }{\partial T},\mathrm{ }{\gamma }^{\mathrm{*}}=-\frac{\partial \sigma }{\partial C}.$$

Designating the variables $$u=-\frac{1}{r}\frac{\partial \psi }{\partial z},\mathrm{ }w=\frac{1}{r}\frac{\partial \psi }{\partial r},\mathrm{ }\psi \left(r,z\right)=\frac{\upsilon r^2}{L}f\left(\eta \right),\mathrm{ }\eta =\frac{z}{L},$$ $$\theta \left(\eta \right)=\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}},\mathrm{ }\varphi =\frac{C-C_{\mathrm{\infty }}}{C_w-C_{\mathrm{\infty }}},\mathrm{ }{T}_w-T_{\mathrm{\infty }}=T_0{r}^2,\mathrm{ }{C}_w-C_{\mathrm{\infty }}=C_0{r}^2,$$ transforms Equations (1)(1) $$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0,$$(1) –(6)(6) $$\eta \to \mathrm{\infty }:\mathrm{ }u=0,\mathrm{ }w=0,\mathrm{ }T=T_{\mathrm{\infty }},\mathrm{ }C=C_{\mathrm{\infty }},$$(6) into (7) $$f^{″\prime }-f^{\prime 2}+f{f}^{″}-\xi f^{\prime }=0,$$(7) (8) $${\theta }^{″}+\mathrm{Pr}\left(f{\theta }^{\prime }-2f^{\prime }\theta +\lambda \theta \right)=0,$$(8) (9) $${\varphi }^{″}+\mathrm{Sc}\mathrm{ }\left(2f{\varphi }^{\prime }-2f^{\prime }\varphi -{ϵ}^2\left(1+n\delta \theta \right)e^{\left(-E/\left(1+\delta \theta \right)\right)}\varphi \right)=0,$$(9) (10) $$\eta =0: f=S, f^{″}\left(0\right)=-2(1+{\delta }^{*}), \theta =1, \varphi =1,$$(10) (11) $$\eta \to \infty : f^{\prime }=0, \theta =0, \varphi =0,$$(11) where $T_0$ and $C_0$ are constants, the primes refer to the derivatives of a function with respect to η, $S=-2w_{w}L/\upsilon$ corresponds to suction $(S>0)$ or blowing $(S<0),$ $\xi =\rho /K$ is the permeability parameter, $\mathrm{Pr}=\nu /\alpha$ is the Prandtl number, $\lambda =Q_{0}L/\alpha$ is the heat-source/sink constant, $\mathrm{Sc}=\nu /D$ is the Schmidt number, $\delta =\left(T_w-T_{\infty }\right)/T_{\infty }$ is the relative temperature variable, $ϵ=k_{r}L/\sqrt{D{C}_0}$ is the dimensionless chemical change parameter, ${\delta }^{*}={\gamma }^{*}{C}_0/\gamma T_0$ is the surface tension parameter, $E=E_a/k{T}_{\infty }$ is the non-dimensional activation energy and $L^3=\mu \upsilon /2 \gamma {\sigma }_0{T}_0.$

Figure 1. Problem diagrammatic representation.

In practical applications, the quantity of physical interest in our case is the local Nusselt Nu and Sherwood number Sh. $$\mathrm{Nu}=-\frac{r{\left(\frac{\partial T}{\partial z}\right)}_{z=0}}{{\kappa }^{\mathrm{*}}(T_w-T_{\mathrm{\infty }})}=-\frac{r}{L}{\theta }^{\prime }\left(0\right),\mathrm{ }\mathrm{Sh}=-\frac{r{\left(\frac{\partial C}{\partial z}\right)}_{z=0}}{C_w-C_{\mathrm{\infty }}}=-\frac{r\sqrt{R_e}}{L}{\varphi }^{\prime }\left(0\right),$$ where $R_e$ is local Reynold number and ${\kappa }^{*}$ is the thermal conductivity.

2.1. The procedure for finding solutions

Using the Legendre–Galerkin method, the issue will be solved asymptotically by combining Equations (7)(7) $$f^{″\prime }-f^{\prime 2}+f{f}^{″}-\xi f^{\prime }=0,$$(7) –(9)(9) $${\varphi }^{″}+\mathrm{Sc}\mathrm{ }\left(2f{\varphi }^{\prime }-2f^{\prime }\varphi -{ϵ}^2\left(1+n\delta \theta \right)e^{\left(-E/\left(1+\delta \theta \right)\right)}\varphi \right)=0,$$(9) with boundary conditions (10)(10) $$\eta =0: f=S, f^{″}\left(0\right)=-2(1+{\delta }^{*}), \theta =1, \varphi =1,$$(10) and (11)(11) $$\eta \to \infty : f^{\prime }=0, \theta =0, \varphi =0,$$(11) . To begin, we will begin by converting the solution domain from $[0, {\eta }_{\infty }]$ to $[-1,1]$ using the linear transformation solution $\eta ={\eta }_{\infty }\left(\kappa +1\right)/2.$ The updated configuration of the system looks like this: (12) $$\frac{8}{{\eta }_{\infty }^3}\frac{d^{3}f}{d{\kappa }^3}+\frac{4}{{\eta }_{\infty }^2}f\left(\kappa \right)\frac{d^{2}f}{d{\kappa }^2}-\frac{4}{{\eta }_{\infty }^2}{\left(\frac{df}{d\kappa }\right)}^2-\frac{2\xi }{{\eta }_{\infty }}\frac{df}{d\kappa }=0,$$(12) (13) $$\frac{4}{{\eta }_{\infty }^2}\frac{d^2\theta }{d{\kappa }^2}+\frac{2\mathrm{P}\mathrm{r}}{{\eta }_{\infty }}f\left(\kappa \right)\frac{d\theta }{d\kappa }-\mathrm{Pr}\left(\frac{2}{{\eta }_{\infty }}\frac{df}{d\kappa }+\lambda \right)\theta \left(\kappa \right)=0,$$(13) (14) $$\frac{4}{{\eta }_{\infty }^2}\frac{d^2\varphi }{d{\kappa }^2}+\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\infty }}f\left(\kappa \right)\frac{d\varphi }{d\kappa }-\mathrm{Sc}\left(\frac{4}{{\eta }_{\infty }}\frac{df}{d\kappa }-{ϵ}^2\left(1+n\delta \theta \left(\kappa \right)\right)e^{-\frac{E}{\left(1+\delta \theta \left(\kappa \right)\right)}}\right)\varphi \left(\kappa \right)=0,$$(14) with conditions (15) $$f\left(-1\right)=S, f^{″}\left(-1\right)=-\frac{{\eta }_{\infty }^2}{2}(1+{\delta }^{*}), \theta (-1)=1, \varphi (-1)=1,$$(15) (16) $$f^{\prime }\left(1\right)=0, \theta \left(1\right)=0, \varphi \left(1\right)=0.$$(16)

Using the Maclurin expansion, the last term in Equation (14)(14) $$\frac{4}{{\eta }_{\infty }^2}\frac{d^2\varphi }{d{\kappa }^2}+\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\infty }}f\left(\kappa \right)\frac{d\varphi }{d\kappa }-\mathrm{Sc}\left(\frac{4}{{\eta }_{\infty }}\frac{df}{d\kappa }-{ϵ}^2\left(1+n\delta \theta \left(\kappa \right)\right)e^{-\frac{E}{\left(1+\delta \theta \left(\kappa \right)\right)}}\right)\varphi \left(\kappa \right)=0,$$(14) may be expressed as follows: (17) $$\begin{array}{c}-\mathrm{Sc}{ϵ}^2\left(1+n\delta \theta \left(\kappa \right)\right)e^{-E/\left(1+\delta \theta \left(\kappa \right)\right)}\varphi \left(\kappa \right)\\ \approx -e^{-E}\text{ Sc }{ϵ}^2\varphi \left(\kappa \right)-\left(15+E\right)e^{-E}\text{ Sc }\delta {ϵ}^2\varphi \left(\kappa \right)\theta \left(\kappa \right)\\ -\frac{1}{2}\left[E\left(28+E\right)e^{-E}\mathrm{ }\mathrm{Sc}{\mathrm{ }\delta }^2{ϵ}^2\right]{\varphi \left(\kappa \right)\theta }^2\left(\kappa \right)+{O\left[\theta \right]}^3\end{array}$$(17)

Believing $f\left(\kappa \right),$ $\theta (\kappa )$ and $\varphi (\kappa )$ are asymptotic solutions of the Legendre function series for system of equations (12)(12) $$\frac{8}{{\eta }_{\infty }^3}\frac{d^{3}f}{d{\kappa }^3}+\frac{4}{{\eta }_{\infty }^2}f\left(\kappa \right)\frac{d^{2}f}{d{\kappa }^2}-\frac{4}{{\eta }_{\infty }^2}{\left(\frac{df}{d\kappa }\right)}^2-\frac{2\xi }{{\eta }_{\infty }}\frac{df}{d\kappa }=0,$$(12) –(14)(14) $$\frac{4}{{\eta }_{\infty }^2}\frac{d^2\varphi }{d{\kappa }^2}+\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\infty }}f\left(\kappa \right)\frac{d\varphi }{d\kappa }-\mathrm{Sc}\left(\frac{4}{{\eta }_{\infty }}\frac{df}{d\kappa }-{ϵ}^2\left(1+n\delta \theta \left(\kappa \right)\right)e^{-\frac{E}{\left(1+\delta \theta \left(\kappa \right)\right)}}\right)\varphi \left(\kappa \right)=0,$$(14) , let’s assume they are: $$f\left(\mathrm{\hslash }\right)\approx \sum _{i=0}^m{c}_i{P}_i\left(\kappa \right),\mathrm{ }\theta \left(\mathrm{\hslash }\right)\approx \sum _{i=0}^m{d}_i{P}_i\left(\kappa \right),\mathrm{ }\varphi \left(\mathrm{\hslash }\right)\approx \sum _{i=0}^m{e}_i{P}_i\left(\kappa \right).$$

Using the Galerkin method and substituting them into Equations (12)(12) $$\frac{8}{{\eta }_{\infty }^3}\frac{d^{3}f}{d{\kappa }^3}+\frac{4}{{\eta }_{\infty }^2}f\left(\kappa \right)\frac{d^{2}f}{d{\kappa }^2}-\frac{4}{{\eta }_{\infty }^2}{\left(\frac{df}{d\kappa }\right)}^2-\frac{2\xi }{{\eta }_{\infty }}\frac{df}{d\kappa }=0,$$(12) –(16)(16) $$f^{\prime }\left(1\right)=0, \theta \left(1\right)=0, \varphi \left(1\right)=0.$$(16) , we get (18) $$\sum _{j=0}^m{c}_j\left[\frac{8}{{\eta }_{\infty }^3}〈\frac{d^3{P}_j\left(\kappa \right)}{d{\kappa }^3},P_{\mathcal{l}}\left(\kappa \right)〉-\frac{2\xi }{{\eta }_{\infty }}〈\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\right]+\frac{4}{{\eta }_{\infty }^2}\sum _{j=0}^m\sum _{i=0}^m{c}_j{c}_i\left[〈P_i\left(\kappa \right)\frac{d^2{P}_j\left(\kappa \right)}{d{\kappa }^2}-\frac{d{P}_i\left(\kappa \right)}{d\kappa }\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\right]=0$$(18) (19) $$\begin{array}{c}\sum _{j=0}^m{d}_j\left[\frac{4}{{\eta }_{\mathrm{\infty }}^2}〈\frac{d^2{P}_j\left(\kappa \right)}{d{\kappa }^2},P_{\mathcal{l}}\left(\kappa \right)〉+\mathrm{Pr}\lambda 〈P_j\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉\right]\\ +\frac{2\mathrm{P}\mathrm{r}}{{\eta }_{\mathrm{\infty }}}\sum _{j=0}^m\sum _{i=0}^m{d}_j{c}_i〈P_i\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa }-P_j\left(\kappa \right)\frac{d{P}_i\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉=0,\end{array}$$(19) (20) $$\begin{array}{c}\sum _{j=0}^m{e}_j\left[\left(\frac{4}{{\eta }_{\mathrm{\infty }}^2}-e^{-E}\text{Sc }{ϵ}^2\right)〈\frac{d^2{P}_j\left(\kappa \right)}{d{\kappa }^2},P_{\mathcal{l}}\left(\kappa \right)〉\right]\\ +\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\mathrm{\infty }}}\sum _{j=0}^m\sum _{i=0}^m{e}_j{c}_i〈P_i\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa }-P_j\left(\kappa \right)\frac{d{P}_i\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\\ -\left(15+E\right)e^{-E}\text{Sc }\delta {ϵ}^2\sum _{j=0}^m\sum _{i=0}^m{e}_j{d}_i〈P_i\left(\kappa \right)P_j\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉\\ -\frac{1}{2}\left(E\left(28+E\right)e^{-E}\mathrm{S}\mathrm{c}{\mathrm{ }\delta }^2{ϵ}^2\right)\sum _{j=0}^m\sum _{i=0}^m\sum _{k=0}^m{e}_j{d}_i{d}_k〈P_i\left(\kappa \right)P_i\left(\kappa \right)P_k\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉=0,\end{array}$$(20) with boundary conditions: (21) $$\begin{array}{c}\sum _{i=0}^m{\left(-1\right)}^j{c}_i=S,\mathrm{ }\sum _{i=0}^m{\left(-1\right)}^j{c}_i\prod _{o=-1}^2\left(i+o\right)=-\frac{{\eta }_{\mathrm{\infty }}^2}{2}\left(1+{\delta }^{\mathrm{*}}\right),\mathrm{ }\sum _{i=0}^m{\frac{j\left(j+1\right)}{2}c}_i=0,\mathrm{ }\sum _{i=0}^m{\left(-1\right)}^j{d}_i=1,\mathrm{ }\sum _{i=0}^m{\left(-1\right)}^j{e}_i=1,\mathrm{ }\sum _{i=0}^{n^{\mathrm{*}}}{d}_j=0,\mathrm{ }\sum _{i=0}^{n^{\mathrm{*}}}{e}_j=0,\end{array}$$(21) where the inner product governed by $$〈\mathcal{M}\left(\kappa \right),\mathcal{F}\left(\kappa \right)〉=\underset{-1}{\overset{1}{\int }}\mathcal{M}\left(\kappa \right)\mathcal{ }\mathcal{F}\left(\kappa \right)d\kappa .$$

To describe the nonlinear system of equations (18)(18) $$\sum _{j=0}^m{c}_j\left[\frac{8}{{\eta }_{\infty }^3}〈\frac{d^3{P}_j\left(\kappa \right)}{d{\kappa }^3},P_{\mathcal{l}}\left(\kappa \right)〉-\frac{2\xi }{{\eta }_{\infty }}〈\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\right]+\frac{4}{{\eta }_{\infty }^2}\sum _{j=0}^m\sum _{i=0}^m{c}_j{c}_i\left[〈P_i\left(\kappa \right)\frac{d^2{P}_j\left(\kappa \right)}{d{\kappa }^2}-\frac{d{P}_i\left(\kappa \right)}{d\kappa }\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\right]=0$$(18) –(20)(20) $$\begin{array}{c}\sum _{j=0}^m{e}_j\left[\left(\frac{4}{{\eta }_{\mathrm{\infty }}^2}-e^{-E}\text{Sc }{ϵ}^2\right)〈\frac{d^2{P}_j\left(\kappa \right)}{d{\kappa }^2},P_{\mathcal{l}}\left(\kappa \right)〉\right]\\ +\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\mathrm{\infty }}}\sum _{j=0}^m\sum _{i=0}^m{e}_j{c}_i〈P_i\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa }-P_j\left(\kappa \right)\frac{d{P}_i\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉\\ -\left(15+E\right)e^{-E}\text{Sc }\delta {ϵ}^2\sum _{j=0}^m\sum _{i=0}^m{e}_j{d}_i〈P_i\left(\kappa \right)P_j\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉\\ -\frac{1}{2}\left(E\left(28+E\right)e^{-E}\mathrm{S}\mathrm{c}{\mathrm{ }\delta }^2{ϵ}^2\right)\sum _{j=0}^m\sum _{i=0}^m\sum _{k=0}^m{e}_j{d}_i{d}_k〈P_i\left(\kappa \right)P_i\left(\kappa \right)P_k\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉=0,\end{array}$$(20) under the constraints stated in equations, we use the notation: (22) $$\mathbf{H} \mathbf{c}+\mathbf{\Psi } \tilde{\mathbf{c}}+\mathbf{\Theta }\overline{\mathbf{c}}+ \mathbf{\Gamma }\check{\mathbf{c}} =\mathbf{b},$$(22) where (23) $$\begin{array}{c}\mathbf{H}=\left(\begin{array}{ccc}{\mathbf{H}}_{\mathbf{f}}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& {\mathrm{ }\mathbf{H}}_{\mathbf{\theta }}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& {\mathrm{ }\mathbf{H}}_{\mathbf{\varphi }}\end{array}\right),\mathbf{\Psi }=\left(\begin{array}{ccc}{\mathbf{\Psi }}_{\mathbf{f}}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& \mathbf{O}\end{array}\right),\mathbf{\Theta }=\left(\begin{array}{ccc}\mathbf{O}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& {\mathbf{\Theta }}_{\mathbf{f}\mathbf{\theta }}& \mathbf{O}\\ {\mathrm{ }\mathbf{\Theta }}_{\mathbf{f}\mathbf{\varphi }}& \mathbf{O}& {\mathbf{\Theta }}_{\mathbf{\theta }\mathbf{\varphi }}\end{array}\right),\mathbf{b}=\left(\begin{array}{c}\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\end{array}\\ {\mathbf{b}}_3\end{array}\right),\\ \mathbf{c}={\left\{\mathrm{span}\left\{c_j\right\},\mathrm{span}\left\{d_j\right\}, \mathrm{span}\left\{e_j\right\}\right\}}^T,\mathrm{ }\tilde{\mathbf{c}}={\left\{\text{span}\left\{c_i{c}_j\right\},\mathrm{ }\text{span}\left\{d_i{d}_j\right\},\mathrm{ }\text{span}\left\{e_i{e}_j\right\}\right\}}^T\\ \overline{\mathbf{c}}={\left\{\mathrm{span}\left\{c_i{e}_j\right\},\mathrm{span}\left\{c_i{d}_j\right\},\mathrm{ }\mathrm{span}\left\{d_i{e}_j\right\}\right\}}^T,\check{\mathbf{c}}=\left\{\mathrm{span}\left\{e_i{d}_j{d}_k\right\}\right\},\\ {\mathbf{b}}_1={\left(000000\cdots S-\frac{{\eta }_{\infty }^2}{2}\left(1+{\delta }^{*}\right)0\right)}^T,\\ {\mathbf{b}}_2={\mathbf{b}}_3={\left(000000\cdots 010\right)}^T,\\ {\mathbf{H}}_{\mathbf{f}}=\left(\begin{array}{c}\frac{{\mathrm{ }\varrho }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ {\left(-1\right)}^j{c}_i\prod _{o=-1}^2\left(i+o\right)\\ \frac{j\left(j+1\right)}{2}\end{array}\right),{\mathbf{H}}_{\mathbf{\theta }}=\left(\begin{array}{c}\frac{{\mathrm{ }\epsilon }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ 1\end{array}\right),\mathrm{ }{\mathbf{H}}_{\mathbf{\varphi }}=\left(\begin{array}{c}\frac{{\mathrm{ }\varpi }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ 1\end{array}\right),\\ \mathbf{\Gamma }=\left\{{\tau }_{i,j,k,\mathcal{l}}\right\}=-\frac{1}{2}\left[E\left(28+E\right)e^{-E}\mathrm{S}\mathrm{c}{ \delta }^2{ϵ}^2\right]〈P_i\left(\kappa \right)P_i\left(\kappa \right)P_k\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉,\end{array}$$(23) (24) $${\varrho }_{j,r}=\frac{8}{{\eta }_{\infty }^3}〈\frac{d^3{P}_j\left(\kappa \right)}{d^3\kappa },P_{\mathcal{l}}\left(\kappa \right)〉-\frac{2\xi }{{\eta }_{\infty }}〈\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(24) (25) $${\epsilon }_{j,r}=\frac{4}{{\eta }_{\infty }^2}〈\frac{d^2{P}_j\left(\kappa \right)}{d^2\kappa },P_{\mathcal{l}}\left(\kappa \right)〉+\mathrm{Pr} \lambda 〈\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(25) (26) $${\varpi }_{j,r}=\left(\frac{4}{{\eta }_{\infty }^2}-e^{-E}\text{Sc }{ϵ}^2\right)〈\frac{d^2{P}_j\left(\kappa \right)}{d^2\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(26) (27) $${\mathbf{\Theta }}_{\mathbf{f}\mathbf{\varphi }}=\left\{{\kappa }_{i,j,r}\right\}=\frac{4\mathrm{S}\mathrm{c}}{{\eta }_{\infty }}〈P_i\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa }-P_j\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(27) (28) $${\mathbf{\Theta }}_{\mathbf{f}\mathbf{\theta }}=\left\{{\mathrm{\Omega }}_{i,j,r}\right\}=\frac{2\mathrm{P}\mathrm{r}}{{\eta }_{\infty }}〈P_i\left(\kappa \right)\frac{d{P}_j\left(\kappa \right)}{d\kappa }-P_j\left(\hslash \right)\frac{d{P}_i\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(28) (29) $${\mathbf{\Theta }}_{\mathbf{\theta }\mathbf{\varphi }}=\left\{{\alpha }_{i, j, r}\right\}=-(15+E)e^{-E}\text{Sc }\delta {ϵ}^2〈P_i\left(\kappa \right)P_j\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉,$$(29) (30) $${\mathbf{\Psi }}_{\mathbf{f}}=\left\{{\omega }_{i,j,r}\right\}=\frac{4}{{\eta }_{\infty }^2}〈P_i\left(\kappa \right)\frac{d^2{P}_j\left(\kappa \right)}{d^2\kappa }-\frac{d{P}_i\left(\kappa \right)}{d\kappa }\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(30)

Fathy (2021) and Fathy, El-Gamel, and El-Azab (2014, 2017) created formulas and theorems for computing equation terms (23)(23) $$\begin{array}{c}\mathbf{H}=\left(\begin{array}{ccc}{\mathbf{H}}_{\mathbf{f}}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& {\mathrm{ }\mathbf{H}}_{\mathbf{\theta }}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& {\mathrm{ }\mathbf{H}}_{\mathbf{\varphi }}\end{array}\right),\mathbf{\Psi }=\left(\begin{array}{ccc}{\mathbf{\Psi }}_{\mathbf{f}}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}& \mathbf{O}\end{array}\right),\mathbf{\Theta }=\left(\begin{array}{ccc}\mathbf{O}& \mathbf{O}& \mathbf{O}\\ \mathbf{O}& {\mathbf{\Theta }}_{\mathbf{f}\mathbf{\theta }}& \mathbf{O}\\ {\mathrm{ }\mathbf{\Theta }}_{\mathbf{f}\mathbf{\varphi }}& \mathbf{O}& {\mathbf{\Theta }}_{\mathbf{\theta }\mathbf{\varphi }}\end{array}\right),\mathbf{b}=\left(\begin{array}{c}\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\end{array}\\ {\mathbf{b}}_3\end{array}\right),\\ \mathbf{c}={\left\{\mathrm{span}\left\{c_j\right\},\mathrm{span}\left\{d_j\right\}, \mathrm{span}\left\{e_j\right\}\right\}}^T,\mathrm{ }\tilde{\mathbf{c}}={\left\{\text{span}\left\{c_i{c}_j\right\},\mathrm{ }\text{span}\left\{d_i{d}_j\right\},\mathrm{ }\text{span}\left\{e_i{e}_j\right\}\right\}}^T\\ \overline{\mathbf{c}}={\left\{\mathrm{span}\left\{c_i{e}_j\right\},\mathrm{span}\left\{c_i{d}_j\right\},\mathrm{ }\mathrm{span}\left\{d_i{e}_j\right\}\right\}}^T,\check{\mathbf{c}}=\left\{\mathrm{span}\left\{e_i{d}_j{d}_k\right\}\right\},\\ {\mathbf{b}}_1={\left(000000\cdots S-\frac{{\eta }_{\infty }^2}{2}\left(1+{\delta }^{*}\right)0\right)}^T,\\ {\mathbf{b}}_2={\mathbf{b}}_3={\left(000000\cdots 010\right)}^T,\\ {\mathbf{H}}_{\mathbf{f}}=\left(\begin{array}{c}\frac{{\mathrm{ }\varrho }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ {\left(-1\right)}^j{c}_i\prod _{o=-1}^2\left(i+o\right)\\ \frac{j\left(j+1\right)}{2}\end{array}\right),{\mathbf{H}}_{\mathbf{\theta }}=\left(\begin{array}{c}\frac{{\mathrm{ }\epsilon }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ 1\end{array}\right),\mathrm{ }{\mathbf{H}}_{\mathbf{\varphi }}=\left(\begin{array}{c}\frac{{\mathrm{ }\varpi }_{j,\mathcal{l}\mathcal{ }}}{{\left(-1\right)}^j}\\ 1\end{array}\right),\\ \mathbf{\Gamma }=\left\{{\tau }_{i,j,k,\mathcal{l}}\right\}=-\frac{1}{2}\left[E\left(28+E\right)e^{-E}\mathrm{S}\mathrm{c}{ \delta }^2{ϵ}^2\right]〈P_i\left(\kappa \right)P_i\left(\kappa \right)P_k\left(\kappa \right),P_{\mathcal{l}}\left(\kappa \right)〉,\end{array}$$(23) –(30)(30) $${\mathbf{\Psi }}_{\mathbf{f}}=\left\{{\omega }_{i,j,r}\right\}=\frac{4}{{\eta }_{\infty }^2}〈P_i\left(\kappa \right)\frac{d^2{P}_j\left(\kappa \right)}{d^2\kappa }-\frac{d{P}_i\left(\kappa \right)}{d\kappa }\frac{d{P}_j\left(\kappa \right)}{d\kappa },P_{\mathcal{l}}\left(\kappa \right)〉,$$(30) . The nonlinear system (22)(22) $$\mathbf{H} \mathbf{c}+\mathbf{\Psi } \tilde{\mathbf{c}}+\mathbf{\Theta }\overline{\mathbf{c}}+ \mathbf{\Gamma }\check{\mathbf{c}} =\mathbf{b},$$(22) has (3n + 3) unidentified variables. Newton’s technique was used to solve this problem. Once the system is solved, the unknowns $c_j,$ $d_j$ and $e_j$ may be found. Finally, the answers to the issue are discovered via the inverse transformation $\kappa =\left(2\eta /{\eta }_{\infty }\right)-1.$

3. Results and discussion

Table 1 lists some asymptotic solutions at some values of the problem parameters while Table 2 recognizes the fluctuations (increasing/decreasing) of amended Nusselt number, $-{\theta }^{\prime }\left(0\right)$ and amended Sherwood number, $-{\varphi }^{\prime }\left(0\right).$

Table 1. Some obtained asymptotic solutions for $m=15.$

	Solutions
$\xi =0.3,$ $E=3,$ $\mathrm{Pr}\mathrm{ }=0.71, n=0.7$ $\mathrm{Sc}=0.7,$ $\lambda =0.3,$ $\delta =0.5,ϵ=0.1$ $S=0.25,$ ${\delta }^{*}=1$	$f\left(\eta \right)=0.25\mathrm{ }+2.13\eta -2{\eta }^2+1.248{\eta }^3-0.58408{\eta }^4+0.2185{\eta }^5-0.06801{\eta }^6+0.01798{\eta }^7-0.004068{\eta }^8+0.00077{\eta }^9-0.000122{\eta }^{10}+0.0000153{\eta }^{11}-0.000001436+9.4\times {10}^{-8}{\eta }^{13}-3.82\times {10}^{-9}{\eta }^{14}+7.2\times {10}^{-11}{\eta }^{15}$
	$\theta \left(\eta \right)=1.-1.451\eta +1.539{\eta }^2-1.351{\eta }^3+0.983{\eta }^4-0.6008{\eta }^5+0.3095{\eta }^6-0.132001{\eta }^7+0.04535-0.012215{\eta }^9+0.00251{\eta }^{10}-0.00038{\eta }^{11}+0.00004208{\eta }^{12}-0.000003107+1.38\times {10}^{-7}{\eta }^{14}-2.8\times {10}^{-9}{\eta }^{15}$
	$\varphi \left(\eta \right)=1\mathrm{ }-2.4428\eta +2.7474{\eta }^2-1.786{\eta }^3+0.653{\eta }^4-0.06236{\eta }^5-0.06525{\eta }^6+0.03678{\eta }^7-0.008435{\eta }^8+0.0002562{\eta }^9+0.000402{\eta }^{10}-0.000126{\eta }^{11}+0.00002006{\eta }^{12}-0.0000018{\eta }^{13}+9.7\times {10}^{-8}{\eta }^{14}-2.19\times {10}^{-9}{\eta }^{15}$
$\xi =0.3,$ $E=3,$ $\mathrm{Pr}\mathrm{ }=0.71, n=0.7$ $\mathrm{Sc}=0.2,$ $\lambda =0.3,$ $\delta =0.5,ϵ=1$ $S=0.25,$ ${\delta }^{*}=1$	$f\left(\eta \right)=0.25\mathrm{ }+2.136\eta -2.{\eta }^2+1.248{\eta }^3-0.584{\eta }^4+0.2185{\eta }^5-0.06801{\eta }^6+0.017989{\eta }^7-0.004068{\eta }^8+0.0007782{\eta }^9-0.000122{\eta }^{10}+0.0000153{\eta }^{11}-0.000001436{\eta }^{12}+9.42\times {10}^{-8}{\eta }^{13}-3.8\times {10}^{-9}{\eta }^{14}+7.2\times {10}^{-11}{\eta }^{15}$
	$\theta \left(\eta \right)=1\mathrm{ }-1.4518\eta +1.5393{\eta }^2-1.351{\eta }^3+0.983{\eta }^4-0.6008{\eta }^5+0.3095{\eta }^6-0.132001{\eta }^7+0.0453{\eta }^8-0.01221{\eta }^9+0.0025{\eta }^{10}-0.0003844{\eta }^{11}+0.000042{\eta }^{12}-0.000003107{\eta }^{13}+1.3\times {10}^{-7}{\eta }^{14}-2.801\times {10}^{-9}{\eta }^{15}$
	$\varphi \left(\eta \right)=1\mathrm{ }-0.795\eta +0.472{\eta }^2-0.2524{\eta }^3-0.0129{\eta }^4+0.32368{\eta }^5-0.5046{\eta }^6+0.44717{\eta }^7-0.26044{\eta }^8+0.1044{\eta }^9-0.029{\eta }^{10}+0.0057{\eta }^{11}-0.00077{\eta }^{12}+0.0000687{\eta }^{13}-0.00000356{\eta }^{14}+8.25\times {10}^{-8}{\eta }^{15}$

Table 2. Fluctuations of amended Nusselt/Sherwood numbers.

$\xi$	$E$	$\mathrm{Pr}$	$n$	$\mathrm{Sc}$	$\lambda$	$\delta$	$ϵ$	$S$	${\delta }^{*}$	$-\theta \prime (0)$	$-\varphi \prime (0)$
0.3	3	0.71	0.5	0.2	0.3	0.5	0.1	0.25	1	$1.4591219$	$0.7868786$
0.5										$1.4918402$	$0.8101274$
0.7										$1.5307113$	$0.8366417$
0.3	1	0.71	0.5	0.4	0.3	0.5	0.6	0.25	1	$1.4918402$	$1.4156183$
	2									$1.4918402$	$1.3974102$
	3									$1.4918402$	$1.3886989$
0.3	3	0.71	0.5	0.2	0.3	0.5	0.1	0.25	1	$1.4518466$	$0.7815505$
		2.00								$2.9831562$	$0.7815259$
		5.00								$5.3138152$	$0.7815111$
0.3	3	0.71	0	0.2	0.3	0.5	1	0.25	1	$1.4518466$	$0.7924235$
			5							$1.4518466$	$0.8130970$
			10							$1.4518466$	$0.8335676$
0.3	3	0.71	0.5	0.5	0.3	0.5	0.1	0.25	1	$1.4518466$	$1.5245813$
				0.7						$1.4518466$	$1.9236615$
				1.0						$1.4518466$	$2.4427844$
0.3	3	0.71	0.5	0.2	$-$0.3	0.5	0.1	0.25	1	$1.6747993$	$0.7815442$
					0.0					$1.5761187$	$0.7815468$
					0.3					$1.4518466$	$0.7815505$
0.3	3	0.71	0.5	0.2	0.3	0.5	1	0.25	1	$1.4518466$	$0.7945001$
						1.0				$1.4518466$	$0.8054823$
						1.5				$1.4518466$	$0.8182046$
0.3	3	0.71	0.5	0.2	0.3	0.5	0.1	0.25	1	$1.4518466$	$0.7815505$
							1.5			$1.4518466$	$0.8108278$
							2.0			$1.4518466$	$0.8336645$
0.3	3	0.71	0.5	0.2	0.3	0.5	0.1	$-$0.5	1	$1.3782259$	$0.7214591$
								0.0		$1.4204404$	$0.7576137$
								0.5		$1.4918402$	$0.8101274$
0.3	3	0.71	0.5	0.5	0.3	0.5	0.5	0.25	1	$1.4918402$	$0.8101274$
									2	$1.7771937$	$0.9367573$
									3	$1.9975252$	$1.0383026$

The parameter $K$ is a physical measurement of how easily a fluid may pass through a porous media. As a result of this, the permeability parameter, $\xi ={\mu }_{\infty }/aK,$ prevents fluid from flowing through the porous media. As a result, by raising the values of $\xi ,$ the fluid velocity boundary layers are reduced, as seen in Figure 2(a). On the other hand, enhancing the values of $\xi$ growths the width of the thermal profile and the width of concentration profile, as seen in Figures 2(b) and 3(a). Prandtl number is a dimensionless quantity that connects viscosity and thermal conductivity in a fluid. As a result, it assesses the link between the motion of a fluid and its heat transfer capacity. Recognized that a larger Prandtl number fluid has a drop in heat conduction. For larger value of $\mathrm{Pr},$ the fluid thermal boundary layer will be thinner, as shown in Figure 3(b).

Figure 2. Velocity/temperature charts influenced by $\xi .$

Figure 3. Concentration charts influenced by $\xi$ and temperature charts influenced by $\mathrm{Pr},$ respectively.

Heat generation/absorption, $\lambda ,$ is a non-dimensional metric that is dependent on the quantity of heat created or absorbed in the fluid. The fluid temperature is increased as figured out from Figure 4(a) which is anticipated whereas the heat-generating engine enhances heat transferred from surface to fluid and reverse takes place in case of heat absorption. Higher $E$ decreases the amended Arrhenius role and encourages an inventive chemical change, as seen in Figure 4(b). As a consequence, concentration levels have improved.

Figure 4. Temperature charts influenced by $\lambda$ and concentration charts influenced by $E,$ respectively.

The action of changing the non-dimensional power matched rate constant, $n,$ on concentration profile is seen in Figure 5(a). As $n$ is increased, the concentration within thermic profile decreases, causing concentration ramp at sheet to grow. From Figure 5(b), as the temperature relative parameter, $\delta ,$ lowers, the width of concentration profile reduces, bringing on a drop in both the soluble concentration and the mass transmit proportion.

Figure 5. Concentration charts influenced by $n$ and $\delta ,$ respectively.

In contrast, Figure 6(a) depicts the effect of Schmidt number, $\mathrm{Sc},$ on concentration. For bigger values of $\mathrm{Sc},$ there is a downward tendency of concentration. Physically, the concentration and accompanying layers are reduced due to a decrease in mass diffusivity for bigger $\mathrm{Sc}.$ An increase in chemical change proportion, $ϵ,$ lowers concentration inside thermal profile area, as seen in Figure 6(b). The point is elevating chemical change proportion drives mass transmission profile to condense.

Figure 6. Concentration charts influenced by $\mathrm{Sc}$ and $ϵ,$ respectively.

The suction/injection constant, $S,$ had a vital role to govern the friction between fluid and surface influencing the heat flow at surface. Actually, suction ($S>0$) drives fluid stream to stick nearer to surface. Ergo, skin friction swells. Accordingly, the forces of friction among fluid layers are improved and drive the fluid momentum to retard as acknowledged from Figure 7(a). But, the contrary action is noticed for injection ($S<0$). On the authority of Figures 7(b) and 8(a), fluid temperature/concentration is strengthened whilst injection as against suction. Real explanation of this occasion is that the lateral mass flux across surface whilst injection strengthens thermal conductivity and mass transfer of fluid.

Figure 7. Velocity/temperature charts influenced by $S.$

Figure 8. Concentration charts influenced by $S$ and velocity charts influenced by ${\delta }^{*},$ respectively.

One of major goals of this research is to look at the surface tension influence on motion according to the Marangoni convection effect. With both heat and mass transmission, the physical values of $\gamma$ and ${\gamma }^{*}$ represent the gradient of the surface tension. As a result, as the surface tension parameter, ${\delta }^{*},$ increases, the surface tension decreases (according to the negative values of the coefficients $\gamma$ and ${\gamma }^{*}$). Also, lowering the surface tension at the free surface lowers the free surface stiffness and enhances the free surface’s capacity to move. Consequently, as illustrated in Figure 8(b), the velocity increases.

In addition, Figure 9(a,b) shows the impact of surface tension coefficients on heat and mass transmission, respectively. The conditions $T<T_{\infty }$ and $C<C_{\infty }$ are considered to be present throughout the boundary layer. In terms of physics, raising both the temperature and concentration gradients cause more instability at the contact by lowering the surface tension. As illustrated in Figure 9(a,b), raising the surface tension parameter lowers the temperature and concentration profiles.

Figure 9. Temperature/concentration charts influenced by ${\delta }^{*}.$

From Table 2, it is found that the heat transfer at the surface is decreasing with $\xi ,$ $\lambda ,$ $P_r$ and ${\delta }^{*},$ and the mass transfer is decreasing with $\xi , E, n$ $S_c,$ $\lambda$ and ${\delta }^{*}.$

4. Conclusion

The purpose of current investigation is to see the way that heat generation controls stream, temperature and mass exchange through a circular expanding surface surrounded by a permeable scheme with chemical reaction/activation energy. This study is a follow-up to and expansion of a previous publication (Uddin et al., 2015). The findings are visually displayed. It has been discovered that:

1. Permeability characteristic reduces fluid velocity while the surface tension increases it.

2. The thermal profiles reduced by growing Prandtl number. Also, it is increased in the case of heat generation and the reverse takes place in the case of heat absorption.

3. Concentration levels are slightly improved with higher activation energy but the reverse takes place with a higher Schmidt number or higher chemical reaction.

4. Fluid velocity, temperature and concentration are reduced by the suction process while they are raised by the injection one.

5. Surface heat flux is improved by higher permeability, Prandtl number, suction, heat generation or surface tension.

6. Surface mass transfer is improved by higher permeability, Schmidt number, chemical reaction, suction, heat generation or surface tension while it is reduced by higher activation energy.

Nomenclature
$C_{\infty }$	=	the ambient concentration of fluid
$C_w$	=	Solute concentration
$D$	=	Solutal diffusivity
$E$	=	Activation energy parameter
$E_a$	=	Activation energy
$f$	=	Dimensionless velocity
$K$	=	the permeability
$k_r^2$	=	Chemical reaction rate constant
$\mathrm{Nu}$	=	Local Nusselt number
$n$	=	Exponent rate Constant
$\mathrm{Pr}$	=	Prandtl number
$Q_o$	=	Uniform volumatric heat generation
$R_e$	=	Reynolds number
$\mathrm{Sh}$	=	Sherwood number
$\mathrm{Sc}$	=	Schmidt number
$T$	=	Fluid temperature
$T_{\infty }$	=	Ambient temperature
$T_w$	=	Surface temperature
$U_o$	=	Constant
$u$	=	r-component of the fluid velocity
$w$	=	z-component of the fluid velocity
$r,z$	=	Coordinates

Greek Symbols
$\alpha$	=	Thermal diffusivity
$\lambda$	=	Heat generation/absorption parameter
$\delta$	=	Temperature relative parameter
$\xi$	=	Permeability parameter
$\eta$	=	Dimensionless coordinator
$\theta$	=	Dimensionless temperature
$\mu$	=	Dynamic viscosity
$\nu$	=	Kinematic viscosity
$\rho$	=	Fluid density
$\sigma$	=	Chemical reaction parameter
$\phi$	=	Dimensionless concentration
$\psi$	=	Stream function

Subscripts
$w$	=	At the surface of the cylinder
$\infty$	=	Far away from the cylinder