Insight of thermally stratified Jeffrey fluid flow inside porous medium subject to chemical species and melting heat transfer

Abstract

This article concentrates on two-dimensional magnetohydrodynamic stagnation flow of Jeffrey liquid on a nonlinearly stretching sheet which possesses variable thickness. Simultaneous impact of melting as well as thermal stratification is specifically investigated in this study due to their tremendous involvement in plenty of natural and industrial processes. Internal heat generation and presence of chemical species are considered to ponder at heat transfer properties. Series solution has been obtained by solving the developed nonlinear problems. Physical behavior of various controlling parameters such as velocity, thermal, and concentration fields are investigated. It has been found that temperature field decays due to higher intensity of thermal stratification parameter, but thickness of thermal boundary layer boosts up. Larger Deborah number results in incremented velocity field. For uplifted wall thickness parameter, velocity field depreciates. Concentration field declines for enhanced parameters of homogeneous as well as heterogeneous reaction. Moreover, velocity is decreasing function of porosity parameter.

Introduction

No doubt phenomena of melting heat transfer along with solidification have acquired mentionable importance for innovative technological and industrial processes. Several researchers have explored characteristics of melting heat process due to its splendid application in preparation of semiconductor materials, optimal energy utilization, welding procedure, magma solidification, and so on. Roberts ¹ made pioneer effort to describe phenomena of melting process of ice kept in hot stream. Tien and Chao ² disclosed the aspects of forced convection by taking melting process into consideration. Sheikholeslami and Rokni ³ presented a study on heat transfer via melting process in nanofluid flow with magnetic effects. Impact of melting heat phenomena with chemical species on viscoelastic fluid flow was analyzed by Hayat et al. ⁴ Sheikholeslami and Rokni ⁵ discussed melting process and magnetic effects of nanofluid flow considering the Buongiorno model. Farooq et al. ⁶ analyzed features of stretching phenomena in melting process for dual stratified flow of viscous nanofluid. Melting heat transfer along with nonlinear radiative effects in Sisko fluid flow was scrutinized by Soomro et al. ⁷

There exists extensive involvement of homogeneous as well as heterogeneous reactions in abundant systems which possess chemical reactions. Several of these chemical reactions are totally unable to proceed; however, some of them can occur in slow speed except for the case of presence of catalyst. Scientists have found complex interaction between both these types of reactions. Applications of such reactions are found in several biochemical systems, fog formation as well as dispersion, ceramics, polymer production, food processing, and so on. A quite early attempt in this regard was made by Merkin, ⁸ who discussed the influence of homogeneous and heterogeneous reactions in view of boundary layer flow. Hayat et al. ⁹ investigated the effect of melting heat and homogeneous–heterogeneous reaction on Jeffrey fluid flow over a stretching cylinder. Kameswaran et al. ¹⁰ discussed the aspects of chemical species in nanoliquid flow, which is induced when stretching phenomena is executed on sheet. Powell–Eyring fluid flow while taking chemical species into consideration was described by Hayat et al., ¹¹ while Khan et al. ¹² encountered Casson fluid flow along with the aspects of chemical species. Properties of chemical species in three-dimensional (3D) Burger fluid flow were disclosed by Khan et al. ¹³ Moreover, Shaw et al. ¹⁴ analyzed these types of reactions in the flow of micropolar fluid through porous medium.

Plenty of features for stretching flows of non-Newtonian fluids are being still explored by researchers and scientists. This inspiration is because of several applications of such flows in fiber technology, food products, wire coating, pharmaceuticals, physiology, crystal growth, and so on. It is not possible to analyze properties of non-Newtonian fluids by means of single constitutive relationship. Jeffrey model represents the rate-type fluids which interpret relaxation as well as retardation time behaviors. Jeffrey model explains the linear viscoelastic features of fluids which are extensively applicable in the polymer industries. Ahmad and Ishak ¹⁵ discussed magnetohydrodynamic flow of Jeffrey liquid over a vertical stretching sheet which is entrenched in porous medium. Das et al. ¹⁶ investigated the influence of slip condition in the presence of radiation and melting process for the flow of Jeffrey liquid. Some important aspects of Soret and Dufour effects for Jeffrey flow have been addressed by Khan et al. ¹⁷ while taking Newtonian heating into consideration. Dalir ¹⁸ studied the attributes of entropy generation and mechanism of heat transfer for the case of Jeffrey liquid flow induced because of a stretching sheet. Gaffar et al. ¹⁹ scrutinized radiative aspects for the flow of Jeffrey fluid over vertical plate in semi-infinite domain.

Hayat et al. ²⁰ obtained series solutions by employing homotopy analysis method (HAM) for investigating Burgers fluid flow caused due to stretching sheet. By extending their work, Hayat et al. ²¹ discussed features of Newtonian heating in the flow of Burgers fluid while retaining stagnation point. In this study, the authors operated HAM to attain series solution. Liao ²² employed HAM for the case of nonlinear differential equations. Hayat et al. ²³ used HAM to analyze stratification effects with slip conditions in Casson fluid flow. Study regarding velocity slip and nanoparticles in the flow of nanofluid was carried out by Zhu et al. ²⁴ by making use of homotopy analysis. Utilizing procedure of HAM, Hayat et al. ²⁵ investigated aspects of Newtonian heating and chemical species in the flow of carbon nanotubes. Waqas et al. ²⁶ introduced HAM in the study where they analyzed Powell–Eyring fluid flow using the Cattaneo–Christov theory. Hayat et al. ²⁷ described flow of carbon nanotubes over a melting surface which possesses variable thickness. Moreover, they obtained series solutions with HAM.

Literature is evident for several attempts that have been made to investigate behavior of non-Newtonian fluids over stretching surfaces. However, researchers seemed to be less focused to study such flows over sheets possessing variable thickness. Due to extensive utilization of stretching phenomena in several industrial procedures, we have employed nonlinear stretching process to study flow properties. Here, we have tried to explore combined features of melting process and thermal stratification in Jeffrey fluid flow toward a sheet which possesses thickness of variable nature and further is embedded in porous medium. Internal heat generation and homogeneous–heterogeneous reaction have also been considered to characterize heat transfer analysis. Stagnation point has been retained here. Utilizing appropriate transformations, the subsequent two-dimensional (2D) nonlinear momentum, energy and concentration equations have been transformed into Ordinary differential equations (ODEs) of highly nonlinear nature. Series solutions have been obtained by making use of the homotopic analysis procedure. Velocity, thermal, and concentration profiles are investigated for several pertinent variables involved in the considered problem.

Mathematical formulation

Steady and incompressible 2D Jeffrey fluid flow toward variable thickness surface embedded in porous medium is considered here. In the present situation, surface is taken at $y=A_1(x+b_1{)}^{\frac{1-n}{2}}$ , where A₁ is very small. Analysis is carried out by employing nonlinear stretching phenomena. Combined aspects of melting and thermal stratification are taken into consideration. Characteristics of heat transfer are reconnoitered by utilizing homogeneous–heterogeneous reaction and internal heat generation. Ambient temperature T_∞ is dominant as compared to the surface temperature T_m.

Similar diffusion coefficients for both species have been taken here. For cubic autocatalysis, an advanced model has been followed regarding homogeneous reaction ⁸

$$A+2B\to 3B,rate=k_{c}a{b}^2$$

Representation of first-order catalytic isothermal reaction is

$$A+B\to 3B,\mathrm{rate}=k_{s}a$$

Obviously, A and B interpret chemical species; however, their concentrations are signified by a and b, respectively. Moreover, rate constants are characterized by $k_c$ and $k_s$ . 2D flow along with usual boundary suppositions can be interpreted as follows

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U_e\frac{d{U}_e}{\mathrm{dx}}+\frac{\nu }{1+\lambda }}\\ \multicolumn{1}{c}{[\frac{{\partial }^{2}u}{\partial y^2}+{\lambda }_1(\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}+u\frac{{\partial }^{3}u}{\partial x\partial y^2}-\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3})]}\\ -\frac{\nu \epsilon }{k^{*}}(u-U_e)\hfill \end{array}$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={\alpha }_1\frac{{\partial }^{2}T}{\partial y^2}+\frac{Q_0}{\rho c_p}(T-T_m)$$

(3)

$$u\frac{\partial a}{\partial x}+v\frac{\partial a}{\partial y}=D_A\frac{{\partial }^{2}a}{\partial y^2}-K_{1}a{b}^2$$

(4)

$$u\frac{\partial b}{\partial x}+v\frac{\partial b}{\partial y}=D_B\frac{{\partial }^{2}b}{\partial y^2}+K_{1}a{b}^2$$

(5)

and involved boundary conditions are

$$\begin{array}{c}u=U_w\left(x\right)=U_0(x+b_1{)}^n,T=T_m\left(x\right)=T_0+a_1(x+b_1)\hfill \\ D_A\frac{\partial a}{\partial y}=K_{s}a,D_B\frac{\partial b}{\partial y}=-K_{s}a\mathrm{at}y=A_1(x+b_1{)}^{\frac{1-n}{2}}\hfill \\ k\left(\frac{\partial T}{\partial y}\right)=\rho [{\lambda }_2+c_s(T_m-T_s)]v_{y=A_1{(x+b_1)}^{\frac{1-n}{2}}}\hfill \\ u\to U_e\left(x\right)=U_{\infty }(x+b_1{)}^n,T=T_{\infty }(x)=T_0+a_2(x+b_1),\hfill \\ \multicolumn{1}{c}{a\to a_0,b\to 0\mathrm{when}y\to \infty }\end{array}$$

(6)

We have chosen u for the interpretation of velocity component in x direction and v for y direction; $\nu$ represents kinematic viscosity $(\mu /\rho )$ ; $\lambda$ stands for ratio of relaxation time to retardation time; ${\lambda }_1$ expresses retardation time; $U_w$ , U_e, k*, and $\epsilon$ represent stretching velocity, free stream velocity, permeability, and porosity of medium, respectively; $Q_0$ symbolizes heat generation coefficient; $D_A$ and $D_B$ are coefficients of diffusion species A and B, respectively; $U_0$ and $U_{\infty }$ are the reference velocities; $A_1$ , $b_1$ , $a_0$ , $a_1$ , and $a_2$ are dimensional constants; homogeneous and heterogeneous reaction rate constants are signified by $K_1$ and $K_s$ ; n the power index; $c_s$ represents specific heat; $\rho$ is the fluid density; ${\alpha }_1$ is the thermal diffusivity; T and k depict temperature and thermal conductivity of fluid; ${\lambda }_2$ denote latent heat; and $T_s$ is the temperature of solid surface.

Using suitable transformations in the following form

$$\begin{array}{c}\multicolumn{1}{c}{\psi =\sqrt{\frac{2}{n+1}\nu U_0{(x+b_1)}^{n+1}}F(\xi ),}\\ \multicolumn{1}{c}{\xi =\sqrt{\frac{n+1}{2}\frac{U_0}{\nu }{(x+b_1)}^{n-1}}y}\\ \multicolumn{1}{c}{u=U_0(x+b_1{)}^{n}F\prime (\xi ),v=-\sqrt{\frac{n+1}{2}\nu U_0{(x+b_1)}^{n-1}}}\\ \multicolumn{1}{c}{(F(\xi )+\xi \frac{n-1}{n+1}F\prime (\xi ))}\\ \multicolumn{1}{c}{\mathrm{\Theta }\left(\xi \right)=\frac{T-T_m}{T_{\infty }-T_0},G\left(\xi \right)=\frac{a}{a_0},H\left(\xi \right)=\frac{b}{a_0}}\end{array}$$

(7)

Equation (1) leads to automatic satisfaction of incompressibility condition, while next four equations (2)–(5) take form

$$\begin{array}{c}F″\prime +\frac{2n}{n+1}(1+\lambda )[A^2-{F\prime }^2]+(1+\lambda )\mathrm{FF}″+\beta \hfill \\ \multicolumn{1}{c}{\left[\begin{array}{c}\left(\frac{3n-1}{2}\right){F″}^2\hfill \\ \multicolumn{1}{c}{+(n-1)F\prime F″\prime }\\ \multicolumn{1}{c}{-\frac{n+1}{2}F\prime F″″}\end{array}\right]}\\ -\frac{2(1+\lambda )}{n+1}P(F\prime -A)=0\hfill \end{array}$$

(8)

$$\mathrm{\Theta }″-\frac{2}{n+1}\mathrm{Pr}(\mathrm{\Theta }F\prime +\mathrm{SF}\prime -\delta \mathrm{\Theta })+\mathrm{Pr}\mathrm{\Theta }\prime F=0$$

(9)

$$G″-\frac{2\mathrm{ScK}}{n+1}G{H}^2+\mathrm{ScFG}\prime =0$$

(10)

$$H″+\frac{2\mathrm{ScK}}{{\delta }_1(n+1)}G{H}^2+\frac{\mathrm{Sc}}{{\delta }_1}H\prime F=0$$

(11)

corresponding boundary conditions can be interpreted in the following way

$$\begin{array}{c}M\mathrm{\Theta }\prime (\alpha )+\mathrm{Pr}(F(\alpha )+\alpha \frac{n-1}{n+1})=0\hfill \\ F\prime (\alpha )=1,F\prime (\infty )=A\hfill \end{array}$$

(12)

$$\begin{array}{c}\mathrm{\Theta }(\alpha )=0,\mathrm{\Theta }(\infty )=1-S\hfill \\ G\prime \left(\alpha \right)=\sqrt{\frac{2}{n+1}}{K}_{s}G\left(\alpha \right)\hfill \end{array}$$

(13)

$$G(\infty )=1,H\prime \left(\alpha \right)=-\frac{1}{{\delta }_1}\sqrt{\frac{2}{n+1}}{K}_{s}G\left(\alpha \right),H(\infty )=0$$

(14)

where prime symbolizes differentiation with respect to “ $\xi$ ” and $\alpha =A_1\sqrt{\frac{n+1}{2}\frac{U_0}{\nu }}.$ Let (a) $F(\xi )=f(\xi -\alpha )=f(\eta )$ , (b) $\mathrm{\Theta }(\xi )=\theta (\xi -\alpha )=\theta (\eta )$ , (c) $G(\xi )=g(\xi -\alpha )=g(\eta )$ , and (d) $H(\xi )=h(\xi -\alpha )=h(\eta )$ , we reach at

$$\begin{array}{c}f″\prime +\frac{2n}{n+1}(1+\lambda )[A^2-{f\prime }^2]+(1+\lambda )\mathrm{ff}″\hfill \\ \multicolumn{1}{c}{+\beta \left[\begin{array}{c}\left(\frac{3n-1}{2}\right){f″}^2\hfill \\ \multicolumn{1}{c}{+(n-1)f\prime f″\prime }\\ \multicolumn{1}{c}{-\frac{n+1}{2}\mathrm{ff}″″}\end{array}\right]}\\ -\frac{2(1+\lambda )}{n+1}P(f\prime -A)=0\hfill \end{array}$$

(15)

$$\theta ″-\frac{2}{n+1}\mathrm{Pr}(\theta f\prime +\mathrm{Sf}\prime -\delta \theta )+\mathrm{Pr}\theta \prime f=0$$

(16)

$$g″-\frac{2\mathrm{ScK}}{n+1}g{h}^2+\mathrm{Scfg}\prime =0$$

(17)

$$h″+\frac{2\mathrm{ScK}}{{\delta }_1(n+1)}g{h}^2+\frac{\mathrm{Sc}}{{\delta }_1}h\prime f=0$$

(18)

$$M\theta \prime (0)+\mathrm{Pr}(f(0)+\alpha \frac{n-1}{n+1})=0,f\prime (0)=1,f\prime (\infty )=A$$

(19)

$$\begin{array}{c}\theta (0)=0,\theta (\infty )=1-S\hfill \\ g\prime \left(0\right)=\sqrt{\frac{2}{n+1}}{K}_{s}g\left(0\right),g\left(\infty \right)=1\hfill \end{array}$$

(20)

$$h\prime \left(0\right)=-\frac{1}{{\delta }_1}\sqrt{\frac{2}{n+1}}{K}_{s}g\left(0\right),h(\infty )=0$$

(21)

Pr expresses the Prandtl number, parameters for the strength of homogeneous and heterogeneous reaction are signified by K and $K_s$ . ${\delta }_1$ depicts the ratio of coefficients of mass diffusion, A is the ratio velocity parameter, $\beta$ is the local Deborah number, $\delta$ denotes the parameter for heat generation (internal), Sc represents the Schmidt number, P represents the porosity parameter, $\alpha$ represents the parameter for wall thickness, M is the melting parameter, and S is the thermal stratification parameter. Mathematically

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Pr}=\frac{{\mu }_f{c}_p}{k},{\delta }_1=\frac{D_B}{D_A},A=\frac{U_{\infty }}{U_0},\beta ={\lambda }_1{U}_0{(x+b_1)}^{n-1},}\\ \multicolumn{1}{c}{\mathrm{Sc}=\frac{\nu }{D_A}}\\ \multicolumn{1}{c}{\delta =\frac{Q}{\rho c_p{U}_0},K=\frac{K_1{a}_0^2(x+b_1)}{U_w},K_s=\frac{k_s}{D_A}\sqrt{\frac{v(x+b_1)}{U_w}},}\\ \multicolumn{1}{c}{P=\frac{\nu \epsilon (x+b_1)}{k^{*}{U}_w}}\\ \multicolumn{1}{c}{\alpha =A_1\sqrt{\frac{n+1}{2}\frac{U_0}{v}},M=\frac{c_p(T_{\infty }-T_m)}{{\lambda }_2+c_s(T_m-T_s)},S=\frac{a_1}{a_2}}\end{array}$$

(22)

It can be noticed that melting parameter takes its form by the union of Stefan numbers for liquid $(c_p(T_{\infty }-T_m))/{\lambda }_2$ and solid $(c_s(T_m-T_s))/{\lambda }_2$ phases, respectively. Hiemenz ²⁸ developed a classical equation which in this case can be attained when there exists no melting process, that is, M = 0 in equation (6). We have taken diffusion coefficients $D_A$ and $D_B$ to be equal (i.e. ${\delta }_1=1$ ). Hence

$$g\left(\eta \right)+h\left(\eta \right)=1$$

(23)

By making use of equations (17), (18), and (24), we attain

$$g″-\frac{2\mathrm{ScK}}{n+1}g{(1-g)}^2+\mathrm{Scfg}\prime =0$$

(24)

along

$$g\prime \left(0\right)=\sqrt{\frac{2}{n+1}}{K}_{s}g\left(0\right),g\left(\infty \right)\to 1,\mathrm{when}\eta \to \infty$$

(25)

Mathematically

$$C_f=\frac{{\tau }_w}{\rho U_w^2},\mathrm{Nu}=\frac{(x+b_1)q_w}{k(T_{\infty }-T_m)}$$

(26)

where ${\tau }_w$ and $q_w$ stand for shear stress and heat flux of wall, which are interpreted as

$${\tau }_w=\mu {\left(\frac{\partial u}{\partial y}\right)}_{y=A_1{(x+b_1)}^{\frac{1-n}{2}}},q_w=-k{\left(\frac{\partial T}{\partial y}\right)}_{y=A_1{(x+b_1)}^{\frac{1-n}{2}}}$$

(27)

Respective dimensionless expression is given as follows

$$\begin{array}{l}R{e}_x^{1/2}{C}_f=\frac{1}{1+\lambda }\\ \left[{\left(\frac{n+1}{2}\right)}^{1/2}{F}^{″}\left(0\right)+\beta \left[\begin{array}{c}{\left(\frac{n+1}{2}\right)}^{1/2}\left(\frac{3n-1}{2}\right)F^{\prime }\left(0\right)F^{″}\left(0\right)+\\ \left(\frac{n+1}{2}\right)F^{\prime }\left(0\right)F^{‴}\left(0\right)-\\ {\left(\frac{n+1}{2}\right)}^{3/2}F\left(0\right)F^{‴}\left(0\right)\end{array}\right]\right]\end{array}$$

(28)

$${\mathrm{Re}}_x^{-1/2}N{u}_x=-\sqrt{\frac{n+1}{2}}\theta \prime \left(0\right)$$

Local Reynolds number is given below in its mathematical form

$$R{e}_x=\frac{U_w(x+b_1)}{\nu }$$

Homotopic procedure

Homotopic analysis method is utilized to construct series solution for considered problem. Prominent advantage of this method is the provision of choice to select initial guesses and linear operators and utilizing this choice, we have selected them

$$\begin{array}{c}\multicolumn{1}{c}{f_0\left(\eta \right)=A\eta +(1-A)(1-\exp (-\eta ))}\\ \multicolumn{1}{c}{-(\frac{(1-S)M}{\mathrm{Pr}}+\alpha \frac{n-1}{n+1})}\end{array}$$

(29)

$${\theta }_0(\eta )=(1-S)(1-\exp (-\eta ))$$

(30)

$$g_0(\eta )=1-\frac{1}{2}\exp (-\sqrt{\frac{2}{n+1}}{K}_s\eta )$$

(31)

$${\mathsf{L}}_f\left(f\right)=\frac{d^{3}f}{d{\eta }^3}-\frac{\mathrm{df}}{d\eta },{\mathsf{L}}_{\theta }\left(\theta \right)=\frac{d^2\theta }{d{\eta }^2}-\theta ,{\mathsf{L}}_g\left(g\right)=\frac{d^{2}g}{d{\eta }^2}-g$$

(32)

Next, we attain below stated operators

$${\mathsf{L}}_f[{\hat{Z}}_1+{\hat{Z}}_2\exp (\eta )+{\hat{Z}}_3\exp (-\eta )]=0$$

(33)

$${\mathsf{L}}_{\theta }[{\hat{Z}}_4\exp (\eta )+{\hat{Z}}_5\exp (-\eta )]=0$$

(34)

$${\mathsf{L}}_g[{\hat{Z}}_6\exp (\eta )+{\hat{Z}}_7\exp (-\eta )]=0$$

(35)

where arbitrary constants are denoted by $\hat{Z}$ _i , i = 1–7.

Deformation of zeroth order

$$(1-{\xi }^{*}){\mathsf{L}}_f[\hat{f}(\eta ;{\xi }^{*})-f_0\left(\eta \right)]={\xi }^{*}{\hslash }_f{\mathsf{N}}_f\left[\hat{f}(\eta ;{\xi }^{*})\right]$$

(36)

$$\begin{array}{c}\multicolumn{1}{c}{(1-{\xi }^{*}){\mathsf{L}}_{\theta }[\hat{\theta }(\eta ;{\xi }^{*})-{\theta }_0\left(\eta \right)]}\\ \multicolumn{1}{c}{={\xi }^{*}{\hslash }_{\theta }{\mathsf{N}}_{\theta }[\hat{f}(\eta ;{\xi }^{*}),\hat{\theta }(\eta ;{\xi }^{*})]}\end{array}$$

(37)

$$\begin{array}{c}\multicolumn{1}{c}{(1-{\xi }^{*}){\mathsf{L}}_g[\hat{g}(\eta ;{\xi }^{*})-g_0\left(\eta \right)]={\xi }^{*}{\hslash }_g{\mathsf{N}}_g}\\ \multicolumn{1}{c}{[\hat{f}(\eta ;{\xi }^{*}),\hat{g}(\eta ;{\xi }^{*})]}\end{array}$$

(38)

$$\begin{array}{c}\multicolumn{1}{c}{M\hat{\theta }\prime (0;{\xi }^{*})+⪻(\hat{f}(0;{\xi }^{*})+\alpha \frac{n-1}{n+1})=0,}\\ \multicolumn{1}{c}{\hat{f}\prime (0;{\xi }^{*})=1,\hat{f}\prime (\infty ;{\xi }^{*})=A}\end{array}$$

(39)

$$\hat{\theta }(0;{\xi }^{*})=0,\hat{\theta }(\infty ;{\xi }^{*})=1-S$$

(40)

$$\hat{g}\prime (0;{\xi }^{*})=\sqrt{\frac{2}{n+1}}{K}_s\hat{g}(0;{\xi }^{*}),\hat{g}(\infty ;{\xi }^{*})=1$$

(41)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathsf{N}}_f\left[\hat{f}(\eta ,{\xi }^{*})\right]=\frac{{\partial }^3\hat{f}(\eta ;{\xi }^{*})}{\partial {\eta }^3}+\frac{2n}{n+1}(1+\lambda )}\\ \multicolumn{1}{c}{[A^2-{\left(\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }\right)}^2]}\\ +(1+\lambda )\left(\hat{f}(\eta ;{\xi }^{*})\frac{{\partial }^2\hat{f}(\eta ;{\xi }^{*})}{\partial {\eta }^2}\right)\hfill \\ \multicolumn{1}{c}{+\beta \left[\begin{array}{c}\left(\frac{3n-1}{2}\right){\left(\frac{{\partial }^2\hat{f}(\eta ;{\xi }^{*})}{\partial {\eta }^2}\right)}^2\hfill \\ \multicolumn{1}{c}{+(n-1)\left(\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }\frac{{\partial }^3\hat{f}(\eta ;{\xi }^{*})}{\partial {\eta }^3}\right)}\\ \multicolumn{1}{c}{-\left(\frac{n+1}{2}\right)\left(\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }\frac{{\partial }^4\hat{f}(\eta ;{\xi }^{*})}{\partial {\eta }^4}\right)}\end{array}\right]}\\ -\frac{2(1+\lambda )}{n+1}P(\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }-A)\hfill \end{array}$$

(42)

$$\begin{array}{c}{\mathsf{N}}_{\theta }[\hat{f}(\eta ,{\xi }^{*}),\hat{\theta }(\eta ;{\xi }^{*})]=\frac{{\partial }^2\hat{\theta }(\eta ;{\xi }^{*})}{\partial {\eta }^2}\hfill \\ -\frac{2\mathrm{Pr}}{n+1}[\hat{\theta }(\eta ;{\xi }^{*})\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }+S\frac{\partial \hat{f}(\eta ;{\xi }^{*})}{\partial \eta }-\delta \hat{\theta }(\eta ;{\xi }^{*})]\hfill \\ \multicolumn{1}{c}{+\mathrm{Pr}\frac{\partial \hat{\theta }(\eta ;{\xi }^{*})}{\partial \eta }\hat{f}(\eta ;{\xi }^{*})}\end{array}$$

(43)

$$\begin{array}{c}{\mathsf{N}}_g[\hat{f}(\eta ,{\xi }^{*}),\hat{g}(\eta ;{\xi }^{*})]=\frac{{\partial }^2\hat{g}(\eta ;{\xi }^{*})}{\partial {\eta }^2}-\frac{2\mathrm{ScK}}{n+1}\hat{g}(\eta ;{\xi }^{*})\hfill \\ \multicolumn{1}{c}{-\frac{2\mathrm{ScK}}{n+1}{\left(\hat{g}(\eta ;{\xi }^{*})\right)}^3}\\ +\frac{4\mathrm{ScK}}{n+1}{\left(\hat{g}(\eta ;{\xi }^{*})\right)}^2+\mathrm{Sc}\hat{f}(\eta ;{\xi }^{*})\frac{\partial \hat{g}(\eta ;{\xi }^{*})}{\partial \eta }\hfill \end{array}$$

(44)

where ${\xi }^{*}$ having range [0,1] symbolizes embedding parameter, while ${\hslash }_f$ , ${\hslash }_{\theta }$ , and ${\hslash }_g$ (which are all non-zero) denote auxiliary parameters.

Deformation of the mth order

$${\mathsf{L}}_f[f_m\left(\eta \right)-{\chi }_m{f}_{m-1}\left(\eta \right)]={\hslash }_f{\mathsf{R}}_m^f\left(\eta \right)$$

(45)

$${\mathsf{L}}_{\theta }[{\theta }_m\left(\eta \right)-{\chi }_m{\theta }_{m-1}\left(\eta \right)]={\hslash }_{\theta }{\mathsf{R}}_m^{\theta }\left(\eta \right)$$

(46)

$${\mathsf{L}}_g[g_m\left(\eta \right)-{\chi }_m{g}_{m-1}\left(\eta \right)]={\hslash }_g{\mathsf{R}}_m^g\left(\eta \right)$$

(47)

$$M{{\theta }^{\prime }}_m(0)+Pr\left(f_m(0)+\alpha \frac{n-1}{n+1}\right)=0,{f^{\prime }}_m\left(0\right)=1,{f^{\prime }}_m\left(\infty \right)=A$$

(48)

$${\theta }_m\left(0\right)=0,{\theta }_m\left(\infty \right)=1-S$$

(49)

$$g{\prime }_m\left(0\right)=\sqrt{\frac{2}{n+1}}{K}_s{g}_m\left(0\right),g_m\left(\infty \right)=1$$

(50)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathsf{R}}_m^f\left(\eta \right)=f″{\prime }_{m-1}+\frac{2n}{n+1}(1+\lambda )}\\ \multicolumn{1}{c}{(A^2(1-{\chi }_m)-\sum _{k=0}^{m-1}f{\prime }_{m-1-k}f{\prime }_k)}\\ +(1+\lambda )\sum _{k=0}^{m-1}{f}_{m-1-k}{f″}_k\hfill \\ \multicolumn{1}{c}{+\beta \left[\begin{array}{c}\left(\frac{3n-1}{2}\right)\sum _{k=0}^{m-1}{f″}_{m-1-k}{f″}_k\hfill \\ \multicolumn{1}{c}{+(n-1)\sum _{k=0}^{m-1}f{\prime }_{m-1-k}f″{\prime }_k}\\ \multicolumn{1}{c}{-\left(\frac{n+1}{2}\right)\sum _{k=0}^{m-1}{f}_{m-1-k}f\prime ″{\prime }_k}\end{array}\right]}\\ -\frac{2(1+\lambda )}{n+1}P(f{\prime }_{m-1}-A(1-{\chi }_m))\hfill \end{array}$$

(51)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathsf{R}}_m^{\theta }\left(\eta \right)={\theta ″}_{m-1}-\frac{2\mathrm{Pr}}{n+1}}\\ \multicolumn{1}{c}{[\sum _{k=0}^{m-1}\left({\theta }_{m-1-k}f{\prime }_k\right)+Sf{\prime }_{m-1}-\delta {\theta }_{m-1}]}\\ \multicolumn{1}{c}{+\mathrm{Pr}\sum _{k=0}^{m-1}\theta {\prime }_{m-1-k}{f}_k}\end{array}$$

(52)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathsf{R}}_m^g\left(\xi \right)={g″}_{m-1}-\frac{2\mathrm{ScK}}{n+1}{g}_{m-1}-\frac{2\mathrm{ScK}}{n+1}}\\ \multicolumn{1}{c}{\sum _{k=0}^{m-1}{g}_{m-1-k}\sum _{l=0}^k{g}_{k-1}{g}_l}\\ +\frac{4\mathrm{ScK}}{n+1}\sum _{k=0}^{m-1}{g}_{m-1-k}{g}_k+\mathrm{Sc}\sum _{k=0}^{m-1}{f}_{m-1-k}g{\prime }_k\hfill \end{array}$$

(53)

$${\chi }_m=\{\begin{array}{c}0,m\le 1\hfill \\ \multicolumn{1}{c}{1,m>1}\end{array}$$

(54)

${\xi }^{*}=0$ and 1 imply

$$\hat{f}(\eta ;0)=f_0\left(\eta \right),\hat{f}(\eta ;1)=f\left(\eta \right)$$

(55)

$$\hat{\theta }(\eta ;0)={\theta }_0\left(\eta \right),\hat{\theta }(\eta ;1)=\theta \left(\eta \right)$$

(56)

$$\hat{g}(\eta ;0)=g_0\left(\eta \right),\hat{g}(\eta ;1)=g\left(\eta \right)$$

(57)

For ${\xi }^{*}=0$ to ${\xi }^{*}=1$ , the subsequent values of $\hat{f}(\eta ;{\xi }^{*})$ , $\hat{\theta }(\eta ;{\xi }^{*})$ , and $\hat{g}(\eta ;{\xi }^{*})$ convert previous starting solutions $f_0(\eta )$ , ${\theta }_0(\eta )$ , and $g_0(\eta )$ to the attained final solutions $f(\eta )$ , $\theta (\eta )$ , and $g(\eta )$ , respectively. For Taylor series at ${\xi }^{*}=1$ , we attain

$$f\left(\eta \right)=f_0\left(\eta \right)+\sum _{m=1}^{\infty }{f}_m\left(\eta \right)$$

(58)

$$\theta \left(\eta \right)={\theta }_0\left(\eta \right)+\sum _{m=1}^{\infty }{\theta }_m\left(\eta \right)$$

(59)

$$g\left(\eta \right)=g_0\left(\eta \right)+\sum _{m=1}^{\infty }{g}_m\left(\eta \right)$$

(60)

We have presented the attained general solutions $(f_m,{\theta }_m,\mathrm{and}{g}_m)$ in the form of special solutions $(f_m^{*},{\theta }_m^{*},\mathrm{and}{g}_m^{*})$ as

$$f_m\left(\eta \right)=f_m^{*}\left(\eta \right)+{\hat{Z}}_1+{\hat{Z}}_2{e}^{\eta }+{\hat{Z}}_3{e}^{-\eta }$$

(61)

$${\theta }_m\left(\eta \right)={\theta }_m^{*}\left(\eta \right)+{\hat{Z}}_4{e}^{\eta }+{\hat{Z}}_5{e}^{-\eta }$$

(62)

$$g_m\left(\eta \right)=g_m^{*}\left(\eta \right)+{\hat{Z}}_6{e}^{\eta }+{\hat{Z}}_7{e}^{-\eta }$$

(63)

Using boundary conditions, we write

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{Z}}_3={\hat{Z}}_4={\hat{Z}}_6=0,{\hat{Z}}_1=-f_m^{*}\left(0\right)-{\frac{\partial f_m^{*}\left(\eta \right)}{\partial \eta }|}_{\eta =0}}\\ \multicolumn{1}{c}{-\frac{M}{\mathrm{Pr}}({\theta }_m^{*}\left(0\right)+{\frac{\partial {\theta }_m^{*}\left(\eta \right)}{\partial \eta }|}_{\eta =0})}\\ \multicolumn{1}{c}{{\hat{Z}}_2={\frac{\partial f_m^{*}\left(\eta \right)}{\partial \eta }|}_{\eta =0},{\hat{Z}}_5=-{\theta }_m^{*}\left(0\right)}\\ \multicolumn{1}{c}{{\hat{Z}}_7=\frac{1}{(\sqrt{\frac{2}{n+1}}{K}_s+1)}({\frac{\partial g_m^{*}\left(\eta \right)}{\partial \eta }|}_{\eta =0}-\sqrt{\frac{2}{n+1}}{K}_s{g}_m^{*}\left(0\right))}\end{array}$$

(64)

Convergence analysis

By utilizing HAM, one can feel great ease in convergence region adjustment for series solutions. Problem under consideration constitutes a part of $\hslash$ curves which are displayed in Figures 1 and 2. Concluded allowable ranges are $-1.3\le {\hslash }_f\le -0.1,$ $-1.3\le {\hslash }_g\le -0.3,$ and $-1.3\le {\hslash }_{\theta }\le -0.5$ (Figure 3).

Figure 1.

Geometry of problem.

Figure 2.

The h-curves for f(0) and g(0).

Figure 3.

The h-curve for θ(0).

The $\hslash$ curves for $f(0),$ $g(0)$ , and $\theta (0)$ when $n=0.3$ , $M=\lambda =0.4$ , $\alpha =A=\mathrm{Sc}=S=\delta =0.2$ , $\beta =0.1$ , $K=K_s=0.5$ , $P=1.0$ , and $\mathrm{Pr}=1.2$ .

Discussion

Characteristics of numerous parameters have been analyzed here on thermal $\theta (\eta )$ , velocity $f\prime (\eta )$ , and concentration $g(\eta )$ fields. Figure 4 presents how local Deborah number $\beta$ influences on velocity field $f\prime (\eta )$ . Velocity and thickness of momentum boundary layer grow up for strength of Deborah number $\beta$ . Higher Deborah number $\beta$ results in stronger elastic characteristics which ultimately enhances $f\prime (\eta )$ . Figure 5 depicts rise in temperature field due to intense parameter of heat generation $\delta$ . As expected, the fluid attains more heat during process of heat generation which results in growth of thermal profile. However, thermal boundary layer exhibits opposite trend in this case. Figure 6 illustrates the way porosity parameter P behaves for $f\prime (\eta )$ . Depreciation in velocity as well as thickness of associated boundary layer is noted for enhanced porosity parameter P. Porosity enhancement leads to permeability which ultimately causes decrease in velocity distribution. Figures 7 and 8 portray the decline in concentration profiles for enhanced homogeneous reaction parameter K as well as heterogeneous reaction parameter K_s. Consumption of reactants during homogeneous as well as heterogeneous reaction processes physically justifies the decline in concentration profiles. However, associated boundary layer thickness gets strengthened for both K and $K_s$ . Figure 9 displays enhanced velocity profile for increasing melting parameter M. Strength of melting process relates to higher convection phenomena from fluid at higher temperature toward melting surface due to which velocity increases, while Figure 10 displays that as a result of enhancement in M, temperature gets decreased. Rate of heat transfer shows progressive behavior for higher M from fluid at higher temperature toward the surface which is melting, and it ultimately causes reduction in thermal field. However, associated boundary layer is thicker in this case. Figure 11 demonstrates increment in concentration field for larger Schmidt number. However, opposite attitude of concentration boundary layer thickness is noted here. Figure 12 reveals that velocity field shows growing up trend for greater ratio velocity parameter A. However, thickness of momentum boundary layer declines for both the cases, that is, for $A>1$ as well as $A<1$ , whereas momentum boundary layer vanishes for A = 1, which is due to same velocity of both wall and the fluid. Figure 13 displays temperature field corresponding to thermal stratification parameter S. Temperature field decays due to higher intensity of thermal stratification parameter S, while thicker thermal boundary layer is observed for larger S. Figure 14 demonstrates recessive attitude of $f\prime (\eta )$ for the strength of wall thickness parameter $\alpha$ . As $\alpha$ increases, deformation rate depreciates and it results in decrement of velocity. Figure 15 depicts impact of skin friction coefficient on wall thickness parameter $\alpha$ and Pr. Skin friction deteriorates with the increment in $\alpha$ as well as Pr. Figure 16 shows characteristics of Nusselt number Nu. Higher Pr gives rise to Nu; however, strength of thermal stratification parameter S shows opposite trend in this regard.

Figure 4.

Pictorial view of $\beta$ against f′.

Figure 5.

Pictorial view of $\delta$ against $\theta$ .

Figure 6.

Pictorial view of P against f′.

Figure 7.

Pictorial view of K against g.

Figure 8.

Pictorial view of $K_s$ against g.

Figure 9.

Pictorial view of M against f′.

Figure 10.

Pictorial view of M against $\theta$ .

Figure 11.

Pictorial view of Sc against g.

Figure 12.

Pictorial view of A against f′.

Figure 13.

Pictorial view of S against $\theta$ .

Figure 14.

Pictorial view of $\alpha$ against f′.

Figure 15.

Pictorial view of $\mathrm{Pr}\mathrm{and}\alpha$ against C_f.

Figure 16.

Pictorial view of $\mathrm{Pr}\mathrm{and}S$ against Nu.

Concluding remarks

Aspects of chemical species and melting process have been addressed in flow of Jeffrey liquid toward a sheet of variable thickness. Effect of heat generation (internal) has also been considered. Concluding remarks are summarized as follows:

• Velocity field grows up for progressive Deborah number $\beta$ . Stronger elastic characteristics take place due to enhanced $\beta$ which causes incremented velocity field.

• For uplifted wall thickness parameter $\alpha$ velocity field depreciates. Deformation rate declines for increasing $\alpha$ which justifies decrement in the velocity from physical point of view.

• Velocity profile declines for higher porosity parameter P. Incremented porosity leads to weaker permeability which ultimately reduces velocity.

• Temperature profile decays, whereas velocity field shows opposite behavior for rise in melting parameter M. Rate of heat transfer shows progressive behavior for higher M from fluid at higher temperature toward the surface which is melting, and it ultimately causes reduction in thermal field. Higher melting parameter gives rise to convection from heated fluid to melting surface which tends to enhance velocity of fluid.

• Higher heat generation parameter $\delta$ leads to growth in temperature distribution. More heat is transferred to the fluid during strength of heat generation process which ultimately boosts the temperature of fluid.

This study has several motives and prospects for future research work. Stagnation point flows of non-Newtonian fluids along with melting heat transfer can be elaborated for different aspects like nonlinear stratified flows over a cone or cylinder with variable thickness. Application of the Cattaneo–Christov theory along with heat and mass transfer and chemical aspects in stagnation point flows could also be a possible extension for future work.