Description of stratification phenomena in the fluid reservoirs with first-order chemical reaction

Abstract

In real life applications, summer and winter stratifications prevent the fluid from mixing. This phenomenon has great impact on the phytoplankton (algae) populations, fisheries management and water supply quality, deficiency of dissolved oxygen in the lower region of ponds, lakes and rivers. Thus, our main moto in this article is to highlight the features of mixed convection in Powell–Eyring fluid deformed by an inclined stretchable sheet. Characteristics of heat transfer are exposed via thermal stratification. The solutal stratification and chemical reaction of first order are accounted to elaborate the nature of mass transfer. The coupled non-linear equations with ordinary derivatives are acquired after utilizing suitable transformation. The homotopic approach is adopted to accomplish the convergent series solution. Analysis of various emerging parameters on the fluid’s temperature, velocity and concentration fields is elaborated through graphs. Surface drag, Sherwood and Nusselt numbers are studied through graphical data corresponding to numerous parameters. In conclusion, dominant solutal and thermal stratified parameters are responsible for reduction of concentration and temperature fields, respectively, which enables us to prevent the formation of fluid layers with different density regions.

Introduction

In recent past, non-Newtonian fluids have attracted the researchers and engineers due to its diverse nature. These fluids are comprehensive and have more practical significance. Due to wide spread applications of these fluid, researchers and investigators utilize many constitutive equations in the literature. Some industrial materials such as melts, mud’s, condensed milk, emulsions, soaps, shampoos, molten plastics, food stuffs, paste, and polymeric liquids, nuclear fuel slurries, plasma and mercury, paper coating are the examples of these fluids. Among non-Newtonian fluid models, Powell–Eyring fluid model is one which is capable of predicting the shear thickening and shear thinning phenomena. Different physical aspects of flows of such fluids in the light of practical applications have been examined by researchers. Rahimi et al. ¹ has studied Eyring-Powell fluid flow over a stretching sheet. Hayat et al. ² examined the magnetohydrodynamic Powell–Eyring nanoliquid over a non-linear thicked stretchable sheet. Qayyum et al. ³ examined the behaviour of Powell–Eyring magneto-nanomaterial flow characterized via Newtonian heating and convection. Khan et al. ⁴ studied the mixed convection and magnetic field effects on Powell–Eyring nanoliquid flow over an inclined surface. Hayat et al. ⁵ presented stratification effects on chemically reactive flow of Powell–Eyring fluid by implementing non-Fourier heat flux.

Stratification belongs to variation of density field, which appears due to variation of temperature and concentration differences. In other words, double stratification occurs when variation in densities occurs due to change in temperature and concentration of the fluid flow. The variation in the density of atmosphere can effect the motion of air and water. Stratification is a significant research area and so many scientists used this phenomena in pertaining heat and mass transport processes. In this direction, Ahmad et al. ⁶ disclosed dual stratification in mixed convection radiative flow of sutterby fluid with chemical reaction. Rehman et al. ⁷ have explained stratification phenomena in chemically reactive fluid flow with ohmic heating. Hayat et al. ⁸ exposed features of thermal conductivity on dual stratified fluid flow deformed by stretchable surface. Kandasamy et al. ⁹ depicted magnetohydrodynamic flow of dual stratified nanoliquid past through a plate having pores. Rehman et al. ¹⁰ addressed the properties of heat source (or sink) on convective flow of Powell–Eyring fluid through inclined shrinking cylinder with double stratification.

Features of chemical reaction have various applications in the field of engineering and technology such applications involve in a glass manufacturing industry, food processing, polymer processing and chemical industries. Moreover, in this regard, most studies are accounted to emphasize for exposing the effect of warming (or cooling) over mobile surfaces and often experience in various processes. Sulochana et al. ¹¹ discussed the effects of chemical reaction on mixed convection radiative flow of Casson nanoliquid through porous inclined plate. Hayat et al. ¹² described chemically reactive non-Newtonian fluid flowing numerical procedure. Hayat et al. ¹³ explored the features of modified Fourier law in dual stratified flow with variable fluid property and chemical reaction. Hayat et al. ¹⁴ exposed the chemically reactive Powell–Eyring fluid flow using Cattaneo–Christov model. Hayat et al. ¹⁵ disclosed the effect of mixed convection on chemically reactive fluid flow through stretchable surface in stagnation point region.

Stratification has great importance in our daily life as mixing efficiency of fluid particles is especially important in oceangraphy, management of fisheries, rivers, lakes and ponds. Researchers paid their attention to explore the features of Newtonian and non-Newtonian fluids with prescribed surface temperature, Newtonian heating and convective boundary conditions. However, in view of the above applications of stratification phenomenon in real life, we elaborate convective heat and mass transfer at the boundary with thermal and solutal stratification phenomena. These phenomena cannot be neglected at the surface and inside the boundary layer. Hence, our main theme is to fill this gap with non-Newtonian (Powell–Eyring) fluid. Characteristics of mass and heat are explored via double stratification and chemical reaction. Homotopic technique^16–20 is implemented to solve the obtained non-linear coupled ordinary differential equations. Effects of different physical parameters of interest are discussed in detail through graphs. Features of skin friction, Sherwood and Nusselt numbers are elaborated comprehensively through graphs.

Mathematical modelling

We consider steady and incompressible fluid flow of Powell–Eyring deformed by linearly stretchable sheet. Sheet makes an angle of inclination ${\beta }_1$ with the horizontal. Due to inclined sheet, mixed convection phenomena is considered. Convective heat and mass transfer conditions are implemented to elaborate the features of combined mass and heat phenomena. Variable temperature and concentration are implemented within the heated fluid and surrounding fluid in order to account the thermal and solutal stratifications. Destructive and constructive chemical reactions are incorporated to elaborate the behaviour of mass transfer. The constitutive governing equations for the flow after boundary layer approximations are ²¹

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

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=(\nu +\frac{1}{\rho \beta C_1})\frac{{\partial }^{2}u}{\partial y^2}-\frac{1}{2\rho \beta C_1^3}{\left(\frac{\partial u}{\partial y}\right)}^2\frac{{\partial }^{2}u}{\partial y^2}+$$

$$(({\beta }_T(T-T_{\infty })+{\beta }_c(C-C_{\infty }))g\sin {\beta }_1$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k}{\rho c_p}\frac{{\partial }^{2}T}{\partial y^2}$$

(3)

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D\frac{{\partial }^{2}C}{\partial y^2}-K_1(C-C_{\infty })$$

(4)

with the boundary conditions

$$\begin{array}{c}u=u_w\left(x\right)=\mathrm{cx},v=0,-k\frac{\partial T}{\partial y}=h_f(T_f-T),-D\frac{\partial C}{\partial y}\hfill \\ \multicolumn{1}{c}{=h_c(C_f-C),\mathrm{at}y=0}\\ u\to 0,T\to T_{\infty }(x),C\to C_{\infty }(x)\mathrm{as}y\to \infty \hfill \end{array}$$

(5)

where

$$\begin{array}{c}\multicolumn{1}{c}{T_f(x)=T_0+d_{1}x,C_f(x)=C_0+e_{1}x,T_{\infty }(x)}\\ \multicolumn{1}{c}{=T_0+d_{2}x,C_{\infty }(x)=C_0+e_{2}x}\end{array}$$

In equations (2)–(6), u and v respectively denote velocity components in x- and y-directions, $u_w$ represents stretching velocity, $u_e$ is free stream velocity, $\upsilon$ represents fluid kinematic viscosity, $\rho$ represents density, $T_f$ the heated fluid temperature of fluid, $T_{\infty }$ the variable ambient fluid temperature, g is the gravitational acceleration, $C_f$ the variable concentration of heated fluid, $C_{\infty }$ the variable ambient concentration, ${\beta }_t$ the coefficient of thermal expansion, $C_1$ and $\beta$ are material parameters, ${\beta }_1$ represents angle of inclination, ${\beta }_c$ the coefficient of mass expansion, $K_1$ chemical reaction coefficient, $C_p$ is specific heat of the fluid, $h_f$ and $h_c$ are the heat and mass transfer coefficients, D is diffusion species coefficient, $T_0$ and $C_0$ represent reference temperature and concentration, respectively, k represents thermal conductivity, $d_1,d_2$ , $e_1$ and $e_2$ represent dimensional constants.

We define

$$\begin{array}{c}\xi =y\sqrt{\frac{c}{\upsilon }},u=\mathrm{cxf}\prime (\xi ),v=-\sqrt{c\upsilon }f(\xi )\hfill \\ \multicolumn{1}{c}{\theta \left(\xi \right)=\frac{T-T_{\infty }}{T_f-T_0},\varphi \left(\xi \right)=\frac{C-C_{\infty }}{C_f-C_0}}\end{array}$$

(6)

Equation (1) reduces to identically zero, whereas other governing equations are as follows

$$(1+ϵ)f″\prime +\mathrm{ff}″-f{\prime }^2-ϵ\delta {f″}^{2}f″\prime +\lambda \theta \sin {\beta }_1+{\lambda }_1\varphi \sin {\beta }_1=0$$

(7)

$$\theta ″+\mathrm{Pr}f\theta \prime -\mathrm{Pr}f\prime S_1-\mathrm{Pr}f\prime \theta =0$$

(8)

$$\varphi ″+\mathrm{Scf}\varphi \prime -\mathrm{Scf}\prime \varphi -\mathrm{Sc}{S}_{2}f\prime -\mathrm{Sckr}\varphi =0$$

(9)

The corresponding dimensionless boundary conditions

$$\begin{array}{c}f\prime (0)=1,f(0)=0,\theta \prime (0)=-{\gamma }_1(1-S_1-\theta (0)),\varphi \prime (0)\hfill \\ \multicolumn{1}{c}{=-{\gamma }_2(1-S_2-\varphi (0))}\\ f\prime (\infty )=0,\theta (\infty )=0,\varphi (\infty )=0\hfill \end{array}$$

(10)

where Sc represents Schmidt number, Prandtl number is represented by $\mathrm{Pr}$ , $ϵ$ and $\delta$ represent the dimensionless material parameters, kr represents chemical reaction parameter, ${\gamma }_1$ is thermal Biot number, ${\gamma }_2$ is solutal Biot number, $\lambda$ is thermal buoyancy parameter, ${\lambda }_1$ is solutal buoyancy parameter, $S_2$ and $S_1$ represent solutal and thermal stratified parameter, respectively, and ${\beta }_1$ is an angle of inclination. These quantities can be expressed as follows

$$\begin{array}{c}\mathrm{Pr}=\frac{\upsilon }{\alpha },ϵ=\frac{1}{\mu \beta C_1},\delta =\frac{x^2{c}^3}{2\nu C_1^2},\mathrm{kr}=\frac{K_1}{c},\hfill \\ \multicolumn{1}{c}{\mathrm{Sc}=\frac{\upsilon }{D},\lambda =\frac{g{\beta }_T(T_f-T_0)}{x{c}^2}}\\ {\lambda }_1=\frac{g{\beta }_c(C_f-C_0)}{x{c}^2},S_1=\frac{d_2}{d_1},S_2=\frac{e_2}{e_1},{\gamma }_1=\frac{h_f}{k\sqrt{\frac{c}{\upsilon }}},\hfill \\ \multicolumn{1}{c}{{\gamma }_2=\frac{h_c}{D\sqrt{\frac{c}{\upsilon }}}}\end{array}$$

(11)

Surface drag force, local Nusselt and Sherwood number are as follows

$$C_{\mathrm{fx}}=\frac{{\tau }_w}{\rho u_w^2},N{u}_x=\frac{x{q}_w}{k(T_f-T_{\infty })},S{h}_x=\frac{x{q}_m}{D(C_f-C_{\infty })}$$

(12)

where wall shear stress, heat and fluxes are

$$\begin{array}{c}\multicolumn{1}{c}{{\tau }_w={\tau }_{\mathrm{xy}}=(\mu +\frac{1}{\beta C_1})\left(\frac{\partial u}{\partial y}\right)-\frac{1}{6\beta C_1^3}{\left(\frac{\partial u}{\partial y}\right)}^3,}\\ \multicolumn{1}{c}{q_w=-k\left(\frac{\partial T}{\partial y}\right),q_m=-D\frac{\partial C}{\partial y}}\end{array}$$

(13)

After using equation (13), equation (12) reduced to

$$\begin{array}{c}\multicolumn{1}{c}{C_f{\mathrm{Re}}_x^{1/2}=(1+ϵ)f″(0)-\frac{ϵ}{3}\delta {f″}^3(0),N{u}_x}\\ \multicolumn{1}{c}{=\frac{-\theta \prime (0)}{{\mathrm{Re}}_x^{-1/2}(1-S_1)},S{h}_x=\frac{-\varphi \prime (0)}{{\mathrm{Re}}_x^{-1/2}(1-S_2)}}\end{array}$$

(14)

Reynolds number is represented by $\mathrm{R}{\mathrm{e}}_x=u_{w}x/\upsilon$ .

Homotopic solutions

Homotopy analysis method was developed by Liao,^22,23 which is utilized to obtain analytical solutions of non-linear ordinary and partial differential equations. This technique is useful for solving weak and strong non-linear mathematical problems. It is preferred because (1) it ensures the series convergence, (2) it is independent of small and large parameters and (3) it provides great freedom to choose the base function and linear operator. The initiatory guesses and supporting linear operators are

$$f_0\left(\xi \right)=1-\exp (-\xi )$$

(15)

$${\theta }_0\left(\xi \right)=\frac{(1-S_1){\gamma }_1}{1+{\gamma }_1}\exp (-\xi )$$

(16)

$${\varphi }_0\left(\xi \right)=\frac{(1-S_2){\gamma }_2}{1+{\gamma }_2}\exp (-\xi )$$

(17)

$${\mathcal{L}}_f\left(f\right)=\frac{d^{3}f}{d{\xi }^3}-\frac{\mathrm{df}}{d\xi },{\mathcal{L}}_{\theta }\left(\theta \right)=\frac{d^2\theta }{d{\xi }^2}-\theta ,{\mathcal{L}}_{\varphi }\left(\varphi \right)=\frac{d^2\varphi }{d{\xi }^2}-\varphi$$

(18)

which satisfy the specified properties

$${\mathcal{L}}_f[A_1+A_2\exp (\xi )+A_3\exp (-\xi )]=0$$

(19)

$${\mathcal{L}}_{\theta }[A_4\exp (\xi )+A_5\exp (-\xi )]=0$$

(20)

$${\mathcal{L}}_{\varphi }[A_6\exp (\xi )+A_7\exp (-\xi )]=0$$

(21)

where $A_i(i=1-7)$ are the optional constants.

Zeroth-order problems

$$\begin{array}{c}\multicolumn{1}{c}{(1-q){\mathcal{L}}_f[\tilde{f}(\xi ;q)-f_0\left(\xi \right)]=q{\hslash }_f{\mathcal{N}}_f}\\ \multicolumn{1}{c}{[\tilde{f}(\xi ;q),\tilde{\theta }(\xi ;q),\tilde{\varphi }(\xi ;q)]}\end{array}$$

(22)

$$(1-q){\mathcal{L}}_{\theta }[\tilde{\theta }(\xi ;q)-{\tilde{\theta }}_0\left(\xi \right)]=q{\hslash }_{\theta }{\mathcal{N}}_{\theta }[\tilde{\theta }(\xi ;q),\tilde{f}(\xi ;q)]$$

(23)

$$(1-q){\mathcal{L}}_{\varphi }[\tilde{\varphi }(\xi ;q)-{\tilde{\varphi }}_0\left(\xi \right)]=q{\hslash }_{\varphi }{\mathcal{N}}_{\varphi }[\tilde{\varphi }(\xi ;q),\tilde{f}(\xi ;q)]$$

(24)

$$\tilde{f}\prime (0;q)=1,\tilde{f}(0;q)=0,\tilde{\theta }\prime (0;q)=-{\gamma }_1[1-\tilde{\theta }(0)-S_1]$$

(25)

$$\begin{array}{c}\multicolumn{1}{c}{\tilde{\varphi }\prime (0;q)=-{\gamma }_2[1-\tilde{\varphi }(0)-S_2],}\\ \multicolumn{1}{c}{\tilde{f}\prime (\infty ;q)=0,\tilde{\theta }(\infty ;q)=0,\tilde{\varphi }(\infty ;q)=0}\end{array}$$

(26)

$$\begin{array}{c}{\mathcal{N}}_f[\tilde{f}(\xi ,q),\tilde{\theta }(\xi ;q),\tilde{\varphi }(\xi ;q)]=(1+ϵ)\frac{{\partial }^3\tilde{f}(\xi ;q)}{\partial {\xi }^3}\hfill \\ \multicolumn{1}{c}{+\tilde{f}(\xi ,q)\frac{{\partial }^2\tilde{f}(\xi ;q)}{\partial {\xi }^2}+{\left(\frac{\partial \tilde{f}(\xi ;q)}{\partial \xi }\right)}^2}\\ -ϵ\delta {\left(\frac{{\partial }^2\tilde{f}(\xi ;q)}{\partial {\xi }^2}\right)}^2\frac{{\partial }^3\tilde{f}(\xi ;q)}{\partial {\xi }^3}+\lambda \tilde{\theta }(\xi ,q)\sin \beta 1\hfill \end{array}$$

$$+{\lambda }_1\tilde{\varphi }(\xi ,q)\sin {\beta }_1$$

(27)

$$\begin{array}{c}{\mathcal{N}}_{\theta }[\tilde{f}(\xi ,q),\tilde{\theta }(\xi ;q),]=\frac{{\partial }^2\tilde{\theta }(\xi ;q)}{\partial {\xi }^2}\hfill \\ \multicolumn{1}{c}{+\mathrm{Pr}\tilde{f}(\xi ,q)\frac{\partial \tilde{\theta }(\xi ;q)}{\partial \xi }-\mathrm{Pr}\frac{\partial \tilde{f}(\xi ;q)}{\partial \xi }{S}_1}\\ -\mathrm{Pr}\frac{\partial \tilde{f}(\xi ;q)}{\partial \xi }\tilde{\theta }(\xi ,q)\hfill \end{array}$$

(28)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{N}}_{\varphi }[\tilde{f}(\xi ,q),\tilde{\varphi }(\xi ;q),]=\frac{{\partial }^2\tilde{\varphi }(\xi ;q)}{\partial {\xi }^2}+\mathrm{Sc}{S}_2\tilde{f}\prime (\xi ,q)}\\ \multicolumn{1}{c}{+\mathrm{Sc}\tilde{f}(\xi ,q)\frac{\partial \tilde{\varphi }(\xi ;q)}{\partial \xi }}\end{array}$$

$$-\mathrm{Sc}\frac{\partial \tilde{f}(\xi ;q)}{\partial \xi }\tilde{\varphi }(\xi ,q)-\mathrm{Sckr}\tilde{\varphi }(\xi ,q)$$

(29)

embedding parameter is represented by q which belongs to [0,1] while auxiliary parameters are denoted by ${\hslash }_f,$ ${\hslash }_{\theta },$ and ${\hslash }_{\varphi }$ having non-zero values.

mth-order problems

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

(30)

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

(31)

$${\mathcal{L}}_{\varphi }[{\varphi }_m\left(\xi \right)-{\chi }_m{\varphi }_{m-1}\left(\xi \right)]={\hslash }_{\varphi }{\mathcal{R}}_m^{\varphi }\left(\xi \right)$$

(32)

$$\begin{array}{c}\multicolumn{1}{c}{{f\prime }_m\left(0\right)=0,f(0)=0,{\theta \prime }_m\left(0\right)={\gamma }_1{\theta }_m(0),{\varphi \prime }_m\left(0\right)={\gamma }_2{\varphi }_m(0)}\\ \multicolumn{1}{c}{{f\prime }_m(\infty )=0,{\theta }_m\left(\infty \right)=0,{\varphi }_m\left(\infty \right)=0}\end{array}$$

(33)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{R}}_m^f\left(\xi \right)=(1+ϵ)f″{\prime }_{m-1}+\sum _{k=0}^{m-1}(f{″}_{m-1-k}{f}_k)-f_{m-1}^{\prime 2}}\\ \multicolumn{1}{c}{-ϵ\delta \sum _{k=0}^{m-1}f″{\prime }_{m-1-k}\sum _{l=0}^k({f″}_{k-l}{f″}_l)+\lambda \sin {\beta }_1{\theta }_{m-1}}\end{array}$$

$$+{\lambda }_1\sin {\beta }_1{\varphi }_{m-1}$$

(34)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{R}}_m^{\theta }\left(\xi \right)={\theta ″}_{m-1}+\mathrm{Pr}\sum _{k=0}^{m-1}\left({\theta \prime }_{m-1-k}{f}_k\right)-\mathrm{Pr}{S}_{1}f{\prime }_{m-1}}\\ \multicolumn{1}{c}{-\mathrm{Pr}\sum _{k=0}^{m-1}\left({f\prime }_{m-1-k}{\theta }_k\right)}\end{array}$$

(35)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{R}}_m^{\varphi }\left(\xi \right)={\varphi ″}_{m-1}-\mathrm{Sc}{S}_2{f}_{m-1}+\mathrm{Sc}\sum _{k=0}^{m-1}\varphi {\prime }_{m-1-k}{f}_k}\\ \multicolumn{1}{c}{-\mathrm{Sc}\sum _{k=0}^{m-1}f{\prime }_{m-1-k}{\varphi }_k-\mathrm{Sckr}{\varphi }_{m-1}}\end{array}$$

(36)

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

(37)

For $q=0$ and $q=1$ , one can write

$$f(\xi ;0)=f_0\left(\xi \right),f(\xi ;1)=f\left(\xi \right)$$

(38)

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

(39)

$$\varphi (\xi ;0)={\varphi }_0\left(\xi \right),\varphi (\xi ;1)=\varphi \left(\xi \right)$$

(40)

and with variation of q from 0 to 1 solution starts from initial approximations $f_0\left(\xi \right)$ , ${\theta }_0(\xi )$ and ${\varphi }_0(\xi )$ and approaches to final solutions. Using Taylor’s series and using $q=1$ , we have

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

(41)

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

(42)

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

(43)

The appropriate solutions $f_m,$ ${\theta }_m$ and ${\varphi }_m$ of equations (38)–(40) corresponding to $(f_m^{*},{\theta }_m^{*}\mathrm{and}{\varphi }_m^{*})$ are

$$f_m\left(\xi \right)=f_m^{★}\left(\xi \right)+A_1+A_2{e}^{\xi }+A_3{e}^{-\xi }$$

(44)

$${\theta }_m\left(\xi \right)={\theta }_m^{★}\left(\xi \right)+A_4{e}^{\xi }+A_5{e}^{-\xi }$$

(45)

$${\varphi }_m\left(\xi \right)={\varphi }_m^{★}\left(\xi \right)+A_6{e}^{\xi }+A_7{e}^{-\xi }$$

(46)

Convergence analysis

Homotopy analysis method facilitates us to adopt and confine the region where the homotopic series solution converges. The region in which $\hslash$ curves are parallel to h axis is termed as convergence region. Thus, we have constructed (see Figure 1). It is clear from the figure that allowable ranges of ${\hslash }_f$ , ${\hslash }_{\theta }$ and ${\hslash }_{\varphi }$ are $-1.0\le {\hslash }_f\le -0.1,-1.3\le {\hslash }_{\theta }\le -0.2$ and $-1.4\le {\hslash }_{\varphi }\le -0.1$ .

Figure 1.

$\hslash$ curves for $f″(0),\theta \prime (0),\varphi \prime (0)$

Discussion

This section illustrates the behaviour of velocity, concentration and temperature distributions. Figure 2 reflects the effect of thermal buoyancy parameter ( $\lambda$ ) on horizontal velocity field. Velocity grows for higher thermal buoyancy parameter $\lambda$ . Physically, dominant values of thermal buoyancy parameter result in enhancement in gravitational and buoyancy force which assists the fluid motion. Higher velocity exists at the surface of sheet. Figure 3 illustrates the behaviour of material fluid parameter $ϵ$ on the horizontal velocity component. The fluid velocity raises with an increase of material fluid parameter $ϵ$ . Larger values of material fluid parameter decline the viscous force of the fluid and plate, which provides more fluid deformation. Thus, velocity field enhances. The behaviour of velocity component corresponding to variation in angle of inclination ${\beta }_1$ is illustrated in Figure 4. Dominant values of ${\beta }_1$ result in enhancement of velocity field. Infact, increasing ${\beta }_1$ enhances the buoyancy forces due to dominant gravitational force. Hence, velocity distribution enhances. Figure 5 exhibits the features of Prandtl number $\mathrm{Pr}$ on temperature field. Temperature field decays for larger Prandtl number. Physically, it justifies that enlarge Prandtl number is corresponding to low thermal diffusivity which causes the reduction in temperature. Figure 6 shows the decreasing trend of temperature distribution and associated boundary layer for dominant stratification parameters $S_1$ . Physically, temperature differences decay between fluid at surface and ambient fluid. Hence, low temperature is observed. Figure 7 describes the response of thermal Biot number ${\gamma }_1$ on temperature profile. With increase in thermal Biot number, heat transfer increases and as a result, temperature of the fluid grows. Variation in concentration field due to Schmidt number is depicted in Figure 8. Concentration field reflects decreasing behaviour for larger Schmidt number. Higher value of Schmidt number leads to low mass diffusion and as a result, concentration distribution reduces. Figure 9 demonstrates solutal stratification parameter $S_2$ on concentration distribution. Dominant solutal stratified parameter is responsible for lower concentration. Infact larger $S_2$ reduces the concentration difference between concentration outside the boundary layer and at surface. Hence, concentration field decays. Characteristics of Solutal Biot number ${\gamma }_2$ on concentration profile are demonstrated in Figure 10. Larger solutal Biot number ${\gamma }_2$ leads to higher concentration field. Physically with increment in solutal Biot number, mass transfer coefficient enhances which consequently enhances the concentration distribution. Figure 11 describes the influence of concentration field corresponding to chemical reaction (destructive) parameter $k_r$ . It demonstrates that higher chemical reaction parameter decays the concentration field. Physically higher destructive chemical reaction parameter results in enhancement of destructive reaction rate which consequently decays the reactant of the species. Thus, concentration field decays. Figure 12 represents behaviour of $ϵ$ and solutal buoyancy parameter ${\lambda }_1$ on skin friction ${\mathrm{Re}}_x^{1/2}\mathrm{Cf}$ . It is noted that skin friction coefficient increases with increment in $ϵ$ and ${\lambda }_1$ Dominant $ϵ$ results in decrement of viscosity of the fluid which produces less resistance to the fluid at the surface of the wall. Therefore, skin friction grows. Furthermore, higher values of $\lambda 1$ enhances the rate of mass transfer which is due to an increase in the magnitude of solutal buoyancy force. Thus, skin friction enhances. Nusselt number ${\mathrm{Re}}_x^{1/2}\mathrm{Nu}$ is depicted in Figure 13. It is revealed that Nusselt number increases with $S_1$ and ${\gamma }_1$ . By thermal Biot number, heat transfer rate from the plate to the fluid enhances which results in enhancement of Nusselt number. It is noted that by increasing $S_1$ temperature difference enhances between the plate and ambient fluid. Thus, Nusselt number grows. Figure 14 illustrates the features of Schmidt number Sc and chemical reaction parameter on Sherwood number ${\mathrm{Re}}_x^{1/2}\mathrm{Sh}$ . It is noted that Schmidt number Sc shows increasing trend for higher kr and Sc.

Figure 2.

Analysis of $\lambda$ on $f\prime (\xi )$ .

Figure 3.

Analysis of $ϵ$ on $f\prime (\xi )$ .

Figure 4.

Analysis of ${\beta }_1$ on $f\prime$ .

Figure 5.

Analysis of $\mathrm{Pr}$ on $\theta$ .

Figure 6.

Analysis of $S_1$ on $\theta$ .

Figure 7.

Analysis of ${\gamma }_1$ on $\theta$ .

Figure 8.

Analysis of Sc on $\varphi$ .

Figure 9.

Analysis of $S_2$ on $\varphi$ .

Figure 10.

Analysis of ${\gamma }_2$ on $\varphi$ .

Figure 11.

Analysis of kr on $\varphi$ .

Figure 12.

Analysis of $ϵ$ and ${\lambda }_1$ Cf.

Figure 13.

Response of ${\gamma }_1$ and $S_1$ on Nu.

Figure 14.

Response of Sc and kr on Sh.

Summary

In present analysis, phenomena of thermal and solutal stratifications in Powell–Eyring fluid flow over an inclined plate with convective heat and mass conditions are analysed and elaborated. The important points are summarized as follows:

• Dominant angle of inclination results in enhancement of velocity field due to higher rate of heat transfer.

• Thermal buoyancy parameter results rapid enhancement of velocity field of fluid.

• Thermal and solutal Biot number are responsible for higher temperature and concentration fields, respectively.

• Thermal and solutal stratified parameter significantly reduces the temperature and concentration fields, respectively.

• It is hoped that this study serves as a stimulus in the mathematical modelling of stratification phenomena in practical life applications, such as to control fisheries management and water supply quality, deficiency of dissolved oxygen in the lower region of ponds, lakes and rivers.