1 Introduction

Forced convection is the flow of heat caused by some external applied forces such as a fan,a blower,or a pump. Free convection is a spontaneous flow arising from nonhomogeneous fields of volumetric (mass) forces (gravitational,centrifugal,Coriolis, electromagnetic,etc.). Free-convective flows may be laminar or turbulent. In a gravitational field,the density differences turn into an additional force,i.e.,the buoyancy force besides the viscous force. When the flow is horizontal,the buoyancy force is ignored. However,for vertical or inclined surfaces,the buoyancy force exerts strong impact on the flow field. The heat transfer process driven by the coexisting effects of free and forced convection is named as mixed convection flow. Mixed convection boundary layer flow along a vertical surface has received great attention from both theoretical and practical points of view due to its widespread applications in technical and industrial fields such as electronic devices,nuclear reactors,heat exchanger,solar collectors,solving cooling problems in turbine blades and also in many manufacturing processes. Shehzad et al.^[1] developed the analytic solution for the three-dimensional flow of an Oldroyd-B fluid over a radiative surface with thermophoresis effects. Mukhopadhyay et al.^[2] investigated the convective flow with heat transfer over a porous radiative surface in a Darcy-Forcheimer porous medium. Elbashbeshy et al.^[3] analyzed the effect of magnetic field on unsteady mixed convection flow by an exponentially stretching porous surface. They considered the heat transfer phenomenon in the presence of heat generation/absorption and thermal radiation effects. Bhattacharyya et al.^[4] examined the similarity solution of mixed convective boundary layer flow towards a vertical surface with slip conditions. The analytical solution of mixed convection boundary layer flow of micropolar fluid over a heated shrinking surface was investigated by Rashidi et al.^[5]. Three-dimensional mixed convective flow of viscoelastic fluid towards a stretched porous surface with thermal radiation and convective conditions was considered by Hayat et al.^[6]. Hayat et al.^[7] studied the mixed convective two-dimensional flow due to a vertical porous plate. The flow analysis was carried out in the presence of variable thermal conductivity and convective boundary condition.

A new class of nanotechnology based fluids is nanofluids which fascinate the researchers and scientists because of enhancing physical properties,particularly with respect to heat transfer. Actually,nanofluid is a mixture of nanoparticles and a base fluid (such as water,ethylene glycol,and oils). The nanofluid term was firstly introduced by Choi^[8]. Such fluids are used to enhance the rate of heat transfer of microchips in computers, microelectronics,transportation,fuel cells,food processing, biomedicine,solid state lightening,and manufacturing. Most of the liquids such as water,glycol,oil,and ethylene have low thermal conductivity. To enhance the thermal conductivity of such fluids, the suspended nano-sized metallic particles (titanium,copper,gold, iron,or their oxides) are used in the fluids. Nanoparticles have different shapes such as spherical,rod-like,and tubular. The use of nanofluids spreads over a wide range of fields. Nanofluids serve as a coolant in heat transfer equipments such as electronic cooling systems,heat exchangers,and radiators. The efficiency of polymerase chain reaction can be improved with the use of graphene based nanofluid. Nanofluids have tunable optical properties,and due to these properties,they are used in solar collectors. Nanofluids are also used in biomedical,transportation,microfluids, solid-state lighting,and manufacturing. Turkyilmazoglu and Pop^[9] investigated the combined effects of heat and mass transfer of nanofluids towards a vertical infinite flat surface with thermal radiation. The problem of magnetohydrodynamic (MHD) nanofluid flow with heat transfer characteristics due to a stretching cylinder was studied by Ashorynejad et al.^[10]. Mustafa et al.^[11] examined the unsteady boundary layer flow of nanofluid over an impulsively stretched plate. Rashidi et al.^[12] analyzed the MHD entropy generated flow of nanofluid induced by a rotating porous disk. The study of MHD free convective flow of nanofluid inside an enclosure via effects of Brownian motion and thermophoresis was carried out by Sheikholeslami et al.^[13]. Sheikholeslami et al.^[14] discussed the MHD flow behavior of nanofluid using the Koo-Kleinstreuer-Li (KKL) model.

Stratification is a phenomenon which arises due to temperature variations,concentration differences,or the presence of different fluids,and consequently affects the density of the medium. Practically,when the heat and mass transfer phenomena act simultaneously,it is interesting to study the effect of double stratification (i.e.,stratification of medium with respect to thermal and concentration fields) instead of thermal stratification on the convective transport with nanofluids. Stratified fluids exist in many natural and industrial processes such as thermal stratification of reservoirs and oceans,salinity stratification in estuaries,rivers,groundwater reservoirs,and oceans,heterogeneous mixtures in industrial,food,and manufacturing processing,and density stratification of the atmosphere. Analysis of mixed convection in a doubly stratified medium is an interesting and important problem due to its wide engineering applications through heat rejection into the environment,thermal energy storage systems, and heat transfer from thermal sources such as the condensers of power plants. The effect of double stratification of the medium on the heat and mass removal processes in a fluid is important. Ibrahim and Makinde^[15] analyzed the boundary layer flow of nanofluid with heat transfer phenomenon towards a vertical surface with double stratification. Srinivasacharya and Surender^[16] reported the mixed convective flow of nanofluid saturated in a porous medium over a vertical plate in the presence of double stratification. Rashad et al.^[17] studied natural convection stratified flow of nanofluid through a vertical cylinder with a non-Darcy porous medium. The analysis of convective heat transfer of stratified nanofluid saturated with a non-Darcy porous medium along a vertical surface was carried out by Murthy et al.^[18].

It is noted from the literature review that proper attention has not been given yet to the double stratified viscoelastic nanofluids. Therefore,our aim here is to investigate the MHD flow of double stratified tangent hyperbolic nanofluid induced by a stretched inclined surface. The homotopy analysis method^[19-27] is used to obtain the convergent series solutions of the governing problems. Graphical results are plotted and discussed for various parameters on the velocity,temperature,and concentration profiles. The Nusselt number,the Sherwood number,and the skin friction coefficient are computed and analyzed numerically. Comparison with the existing literature in some limiting cases is found in excellent agreement.

2 Mathematical formulation

We examine the boundary layer flow of an electrically conducting tangent hyperbolic nanoflu-id past an inclined exponentially stretching sheet with mixed convection. The stratification phenomenon is presented in the flow analysis (see Fig. 1). The effects of Brownian motion and thermophoresis are also taken into account. The velocity of the sheet is denoted by $U_{\rm w}(x)=U_{0}{\rm e}^{\frac{x}{L}}$. As a result of the boundary layer approximations,we have the following system of equations:

(1)

(2)

(3)

(4)

with the boundary conditions

(5)

(6)

Here,$u$ and $v$ are the velocity components in the $x$- and $y$-directions,respectively,$\rho $ is the fluid density,$\sigma$ is the electrical conductivity of the fluid,$B_{0}$ is the strength of the magnetic field,$C$ is the fluid concentration,$T$ is the temperature of fluid,$\upsilon $ is the kinematic viscosity, $c_{p}$ is the specific heat,$D_{\mathrm T}$ is the thermophortic diffusion coefficient,$D_{\mathrm B}$ is the Brownian diffusion coefficient,$\tau =\frac{(\rho c_{p}) _{\mathrm p}}{(\rho c_{p}) _{\mathrm f}}$ is the ratio of effective heat capacity of the fluid, $ \kappa $ is the thermal conductivity,$U_{0}$ is the reference velocity,$L$ is the reference length,$T_{\mathrm w}(x)$ is the temperature of surface,$ T_{\infty }(x)$ is the ambient temperature (the temperature far away from the surface),$T_{0}$ is the reference temperature,$C_{\mathrm w}(x)$ is the concentration over the surface,$C_{\infty }(x)$ is the ambient concentration,$C_{0}$ is the reference concentration,$\psi $ is the inclination angle, $\Gamma $ is the time dependent material constant,$n$ is the power law index,i.e.,the flow behaviour index,and $b,$ $c,$ $l,$ and $m$ are dimensionless constants.

We consider

(7)

Substituting Eq.(7) into Eqs.(1)--(6) yields

(8)

(9)

(10)

(11)

(12)

(13)

The dimensionless parameters are defined as follows:

(14)

where $\lambda $ is the mixed convection parameter,$N_{1}$ is the buoyancy parameter,$N_{\mathrm t}$ is the thermophoresis parameter, $N_{\mathrm b}$ is the Brownian motion parameter,$Sc$ is the Lewis parameter,$We$ is the Weissenberg number,$M$ is the magnetic parameter,$Sl$ is the concentration stratification parameter,$St$ is the thermal stratification parameter,and $Pr$ is the Prandtl number. The skin friction coefficient,the local Nusselt number,and the local Sherwood number are defined by

(15)

with $\tau _{xy}=-\mu (\frac{\partial u}{\partial y})_{y=0}$,$ q_{\mathrm w}=K\frac{\partial T}{\partial y}|_{y=0}$,and $j_{\mathrm w}=D\frac{\partial C} {\partial y}|_{y=0}.$

Using Eq.(7) in Eq.(15),we get the following dimensionless forms:

(16)

in which ${Re}_{x}=\frac{U_{0}L{\mathrm e}^{\frac{x}{L}}}{\upsilon }$ is the local Reynolds number.

3 Homotopic solutions

The dimensionless momentum,temperature,and concentration equations have the following initial guesses $(f_{0},\theta _{0},\phi _{0}) $ and auxiliary linear operators $( {\mathcal{L}}_{f},{\mathcal{L}}_{\theta },{\mathcal{L}_{\phi }})$:

(18)

which satisfy the following properties:

(19)

(20)

(21)

where $C_{i}\,(i=1,2,\cdots,8)$ depict the arbitrary constants.

3.1 Zeroth-order deformation equations

We write

(22)

(23)

(24)

(25)

(26)

(27)

The corresponding nonlinear operators ${\mathcal{N}}_{f}(\widehat{f} (\eta ,q),\widehat{\theta}(\eta ,q),\widehat{\phi}(\eta ,q))$, ${\mathcal{N}} _{\theta }( \widehat{f}(\eta ,q),\widehat{\theta}(\eta ,q),\widehat{\phi} (\eta ,q)) $,and ${\mathcal{N}}_{\phi }( \widehat{f}(\eta ,q),\widehat{\theta}(\eta ,q),\widehat{\phi}(\eta ,q))$ are given as follows:

(28)

(29)

(30)

where $q\in \lbrack 0,1]$ represents the embedding parameter,and $\hbar _{f}\neq 0,$ $\hbar _{\theta }\neq 0$,and $\hbar _{\phi }\neq 0$ specify the auxiliary parameters.

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

(31)

(32)

(33)

and for variation of $q$ from $0$ to $1$,$\widehat{f}(\eta ,q)$, $\widehat{\theta} (\eta ,q) $,and $\widehat{\phi}(\eta ,q)$ varies from the initial solutions $f_{0}(\eta )$,$\theta _{0}(\eta) $,and $\phi _{0}(\eta )$ to the final solutions $f(\eta ),$ $\theta (\eta) $,and $\phi (\eta )$,respectively. Using the Taylor series,we acquire

(34)

(35)

(36)

The choice for the value of auxiliary parameter is made so that the convergence of the series (34)--(36) is attained at $q=1$,i.e.,

(37)

(38)

(39)

3.2 ${\mathbf{\tilde m}}$th-order problems

The ${\mathbf{\tilde m}}$th-order deformation problems can be expressed as follows:

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

The general solutions $(f_{\widetilde{m}},\theta _{\widetilde{m}},\phi _{ \widetilde{m}})$ of Eqs.(40)--(45) in terms of special solutions $ ( f_{\widetilde{m}}^{\ast },\theta _{\widetilde{m}}^{\ast },\phi _{\widetilde{m} }^{\ast })$ are defined by

(50)

(51)

(52)

in which the constants $C_{i}\,(i=1,2,\cdots,7)$ through the boundary conditions (43) and (45) have the following values:

(53)

4 Convergence of derived solutions

The auxiliary parameters $\hslash _{f}$,$\hslash _{\theta }$,and $\hslash _{\phi }$ in Eqs.(34)--(36) have a key role in the convergence of the derived series solutions. To select the appropriate values of $\hslash _{f}$,$ \hslash _{\theta }$,and $\hslash _{\phi }$,we plot the $\hslash$-curves at the 18th-order of approximations in Fig. 2. The valid ranges of $\hslash _{i}\,(i=f,\theta ,\phi )$ can be selected from the flat portion of the $\hslash$-curves. The suitable ranges of the auxiliary parameters $ \hslash _{f},$ $\hslash _{\theta }$,and $\hslash _{\phi }$ are found to be $ -1.56\leq \hslash _{f}<-0.3$,$-1.6\leq \hslash _{\theta }<-0.3$,and $ -1.7\leq \hslash _{\phi }ɝ-0.2$, respectively. Table 1 is drawn to see the convergence of the desired series solutions. It is noticed that the 15th-,17th-,and 10th-order of approximations give the convergent solutions for the momentum,temperature,and concentration equations.

5 Discussion

This section is focused on the physical insight of different parameters on the velocity,temperature,and nanoparticle concentration profiles. Figure 3 gives the insight for the influence of the angle of inclination $\psi $ on the velocity profile $f'( \eta) $. It is noted that with the increase in $\psi $,the velocity profile increases. Figure 4 indicates the effect of the power law index $n$ on the velocity profile $ f'(\eta )$. Here,the velocity profile and the associated boundary layer thickness show decreasing behavior for larger values of the power law index $n$. The influence of the mixed convection parameter $\lambda $ on the velocity profile $ f'(\eta )$ is presented in Fig. 5. It shows that the velocity profile enhances through the increase in the mixed convection parameter,as the mixed convection parameter is the ratio of the buoyancy to inertial forces. Hence, for larger mixed convection parameters,the buoyancy force dominates the inertial force which increases the velocity of fluid. Moreover, the momentum boundary layer thickness increases. Figure 6 displays the effect of the buoyancy parameter $N_{1}$ on the velocity profile $f'(\eta )$. It is found that the velocity profile and the momentum boundary layer thickness are increasing functions of $ N_{1}$. The velocity profile shows the emerging behavior near the wall but it increases away from the wall. Figure 7 indicates the effect of the Weissenberg number $We$ on the velocity profile $f'(\eta )$. It is observed that the velocity profile decreases by the increasing $We$. In fact,it is a ratio between the shear rate time and the relaxation time. Hence, for larger Weissenberg numbers $We$,the fluid becomes thicker,and consequently,the velocity and the boundary layer thickness decrease. The effect of the magnetic parameter $M$ on the velocity profile is shown in Fig. 8. It is seen that the velocity profile is a decreasing function of the magnetic parameter $M$. It holds because with the increase in $M$,the Lorentz force increases which produces the retarding effect on the fluid velocity. The temperature for the angle of inclination $\psi $ is displayed in Fig. 9. It is observed that the temperature profile decreases by increasing $\psi $. Figure 10 is plotted to show the influence of the power law index $n$ on the temperature profile. The temperature profile enhances for larger values of the power law index $n$. Figure 11 points out the behavior of mixed convection parameter $\lambda $ on the temperature profile $\theta (\eta )$. It is noted that the temperature and thermal boundary layer decrease with an increase in the mixed convection parameter $% \lambda $,and as a result,the heat transfer rate increases. Figure 12 illustrates the effect of the buoyancy parameter $N_{1}$ on the temperature profile. It is noted that temperature profile enhances for larger buoyancy parameters $N_{1}$. Further,the thermal boundary layer thickness also increases. Figure 13 portrays the behavior of the thermophoresis parameter $N_{\mathrm t}$ on the temperature profile $\theta (\eta ).$ The increase in the thermophoresis parameter $N_{\mathrm t}$ leads to the enhancement of the temperature profile and the thermal boundary layer thickness. The difference between the wall and reference temperatures increases for larger $N_{\mathrm t}$,and the nanoparticles move from hot region to cold region. Hence,the temperature profile increases. The behavior of the Brownian motion parameter $N_{\mathrm b}$ on the temperature profile is drawn in Fig. 14. The temperature profile and the thermal boundary layer are increasing functions of the Brownian motion parameter $N_{\mathrm b}$. This is because that the random motion of the particles enhances by increasing the Brownian motion parameter $N_{\mathrm b}$ and,as a result,the temperature profile increases. The behavior of the magnetic parameter $M$ on the temperature profile is plotted in Fig. 15. It is examined that the temperature and thermal boundary layer thickness increase when $M=0.2$,0.6,1, and 1.4. Clearly,larger magnetic parameter yields larger Lorentz force which causes strong resistance in the fluid motion. Hence, more heat is produced which enhances the temperature profile. Figure 16 displays the influence of the thermal stratification parameter $ St$ on the temperature profile $\theta (\eta )$. It is found that as $St$ increases,the thermal boundary layer thickness and the temperature profile decrease. An increase in $St$ indicates that the temperature difference between the surface and the ambient fluid is reduced. Consequently,the temperature profile decreases. Figure 17 demonstrates the behavior of the Prandtl number $Pr$ on the temperature profile. It is analyzed that the temperature and the thermal boundary layer thickness decrease with an increase in the Prandtl number. In fact,the Prandtl number is the ratio of the momentum diffusivity to the thermal diffusivity. The thermal diffusivity decreases for larger Prandtl numbers. Hence,it causes the reduction in the temperature profile. Variation of the thermophoresis parameter $N_{\mathrm t}$ on the concentration profile is illustrated in Fig. 18. Both the concentration and the solutal boundary layer thickness increase for larger values of the thermophoresis parameter $N_{\mathrm t}$. Thermophoresis is a phenomenon which causes small particles to be driven away from a hot surface towards a cold one. For larger values of the thermophoresis parameter, the hot fluid particles move away from the surface of the sheet,and as a result,the concentration profile increases. Figure 19 plots the concentration profile versus $\eta $ for various values of the Brownian motion parameter $N_{\mathrm b}$. Here,the reduction is found for both the concentration profile and the associated boundary layer thickness. The effect of the Lewis number $Sc$ on the concentration profile is displayed in Fig. 20. With the increase in the Lewis number $Sc$,the concentration profile and the concentration boundary layer thickness are reduced. Since $Sc$ is inversely proportional to the Brownian diffusion coefficient,the increase of $Sc$ causes the reduction of concentration profile. The influence of the solutal stratification on the concentration profile is plotted in Fig. 21. Both the concentration and the associated boundary layer thickness are decreasing functions of the solutal stratification parameter. Table 2 shows the comparison of $-\theta'(\eta )$ with the previous existing data in the limiting case. It is examined that all the results are in good agreement. Table 3 presents the behavior of the skin friction coefficient for various values of the physical parameters. It is observed that the skin friction coefficient increases for larger values of $Sc$,$We$,$M$,$Sl$,$St$,$Pr$, and $n$,while it decreases when $\psi $,$\lambda $,$N_{1}$, $N_{\mathrm t}$,and $N_{\mathrm b}$ increase. Table 4 displays the effect of the physical parameter on the Nusselt and Sherwood numbers. It is found that the Nusselt number increases for larger values of $Sl$,$St$,$Pr $,$X$,$\psi $,$\lambda $,and $N_{1}$. However,$% N_{\mathrm t}$,$We$,$N_{\mathrm b}$,$Sc$,$M$,and $n$ reduce the Nusselt number. Higher values of $N_{\mathrm b}$,$Sc$,$Sl$,$X$, $\psi $,$\lambda $,and $N_{1}$ result in the enhancement of the Sherwood number,while it decreases via larger $% N_{\mathrm t}$,$We$,$M$,$St$, $Pr $,and $n$.

6 Concluding remarks

The mixed convection stratified flow of hyperbolic tangent nanofluid is developed. Main observations of the performed analysis are as follows:

(i)~The velocity profile decreases with $We$,while it increases via $N_{\mathrm b}$.

(ii)~The effects of the Brownian motion parameter on the temperature and concentration profiles are reverse.

(iii)~The mixed convection parameter enhances the velocity field, while it reduces the temperature distribution.

(iv)~The behavior of $N_{\mathrm b}$ on the local Nusselt and Sherwood numbers are opposite.

(v)~The solutal stratification parameter results in the reduction of concentration profile.

(vi)~The increasing $Sc$ shows reduction in the heat transfer coefficient.