Magnetohydrodynamic flow of squeezed Maxwell nano-fluid with double stratification and convective conditions

Abstract

This article is made to discuss the effect of magnetohydrodynamic on squeezing flow of the nanoparticles between two confined boundaries. Constitutive expressions are utilized for convectively heated Maxwell nano-fluid using Buongiorno’s model in the mathematical development of considered flow problem. Here system of transport equations incorporates the combined impacts of thermophoresis diffusion, Brownian diffusion, and double stratification. Non-linear coupled ordinary differential equations are transformed by employing suitable similar transformations. The formulated non-linear system is evaluated successfully via convergent approach, that is, Homotopy analysis method. The graphical analysis is carried out for different active physical flow parameters. Nusselt and Sherwood numbers are also treated graphically. The results portray that the velocity profile shows cross flow behavior for increasing values of material parameter. Moreover, for dominant values of Biot number, convective effects dominant on plate surface and help the temperature and nanoparticles concentration to increase rapidly near the surface.

Introduction

Many fluids correspond to industrial and engineering applications cannot be treated through Navier–Stokes equations. Such fluids include polymer solutions, paints, certain lubricants and oils, suspension and colloidal solutions, coating of clay, and many others. Among such fluids that can be described in various physical structures, there is no single model that discloses all the salient features of non-Newtonian fluids. In general, non-Newtonian fluids are categorized into differential, integral, and rate type fluids. Thus, Maxwell fluid includes in a simple class of fluids termed as rate type fluids. Properties of relaxation time are described conveniently through this model. Such effects may not be predicted via differential type non-Newtonian fluids. Flow of electrically conducting non-Newtonian fluid has achieved great importance over the years. Practically, we are dealing with a flow of conducting fluid which reveals dissimilar behavior under the effect of magnetic forces. For such cases, magnetohydrodynamic (MHD) feature is also needed in considered flow. It is expected that MHD analysis of nano-fluids has significant role in switches and optical gratings, separation of ink’s float, magnet is utilized to move up the particles in bloodstream to tumor in cancer therapy. Characteristics of magnetic field effects on stagnant flow of convectively heated Maxwell nano-fluid induced by stretchable sheet are explored by Ibrahim. ¹ Sui et al. ² exhibited the properties of heat and mass transport processes via non-Fourier and non-Fick fluxes in Maxwell nano-fluid flow through stretchable sheet with velocity slip. Saleem et al. ³ inspected the properties of modified law of Fourier on flow and heat transfer of MHD Maxwell liquid through stretchable surface. Reddy et al. ⁴ exhibited slip analysis on convectively heated Maxwell nanomaterial flow deformed by exponentially stretched sheet with magnetic effects. Liu and Liu ⁵ discussed the features of Maxwell fluid flow deformed by stretched sheet having variable thickness under boundary layer assumption. Imran et al. ⁶ described the variation of Newtonian heating and velocity slip on MHD Maxwell fluid flow through vertical sheet which is accelerated exponentially. Jafarimoghaddam ⁷ explored the magnetic effects on the porous flow of Maxwell nano-fluid caused by bi-directional stretchable plan surface. Exact perturbed solutions were computed for basic equations.

In recent past, the studies via Brownian motion and thermophoresis diffusion correspond to convectively heated nano-fluids are relatively not much attended. Due to this reason, researchers show great interest in nano-fluids with these effects. Thus, nanoparticles and the base fluid are mixed together to form a new variety of energy transport fluid termed as nano-fluid. The nano-fluids have been very helpful in number of engineering and industrial processes because of its remarkable enhancement in thermal conductivity. Unconventional characteristics of nano-fluids make such fluids potentially pertinent in different processes involving heat transport analysis such as hybrid-powered engines, fuel cells, and micro-electronics. Hussain et al. ⁸ exposed the combined impacts of thermal radiation and mixed convection in MHD stratified flow of Maxwell nano-fluid. Sheikholeslami et al. ⁹ described the characteristics of heat transfer in MHD flow of nano-fluid using Buongiorno model. Madhu et al. ¹⁰ explained the radiative flow of a Maxwell MHD nano-liquid through stretchable surface. Sreedevi et al. ¹¹ explored the radiative analysis of chemically reactive nano-fluid flow over non-linear and linear stretchable surface. Khan et al. ¹² exposed the variation of mass flux condition on flow of radiative Maxwell nano-fluid over stretchable cylinder with heat source or sink. Irfan et al. ¹³ disclosed the features of convective heating in Maxwell magneto nano-liquid flow over a shrinking cylinder with heat sink/source. Jaimala et al. ¹⁴ exposed the influence of zero mass flux condition (at both surfaces) on Darcy Maxwell nanomaterial flow utilizing law of macroscopic filtration. Shen et al. ¹⁵ exhibited the behavior of different particle shapes on fractional Maxwell magneto nano-fluid flow with non-Fourier heat flux.

Recently, convective heat and mass transfer analysis is acquired much importance among researchers and engineers because of its impact on heat and mass transport properties on the surfaces and as a consequence the quality of final manufacturing industrial products. This analysis under convective boundary condition in magnetic field effects is important in many processes include gas turbine, material drying, nuclear plants, transpiration cooling process, thermal energy storage. Therefore, it seems pertinent to have convective boundary condition in considered flow problem rather taking isothermal or isoflux wall conditions. Ibrahim ¹⁶ investigated the convective effect on stagnant flow of hydro-magnetic Maxwell nano-fluid over stretchable surface with assumption of induced magnetic effects. Stretching effects on convectively heated and concentrated Carreau nano-fluid flow through cylinder is communicated by Khan et al. ¹⁷ Responses of convective surface conditions on radiative Powell–Eyring nano-fluid flow under boundary layer settled on stretching surface are portrayed by Hayat et al. ¹⁸ Magnetic effects on porous flow of convectively heated nano-fluid deformed by exponentially stretched sheet with entropy generation are depicted by Shit et al. ¹⁹ Heat source (or sink) effects on Oldroyd-B nano-fluid flow through stretchable plate subjected to convective surface conditions is illustrated by Gireesha et al. ²⁰ Jyothi et al. ²¹ described the features of convective boundary conditions on radiative CNT-water (single and multi-wall) nano-fluid flow saturated in rotating stretchable disks with magnetic field. Hayat et al. ²² have presented the entropy generation analysis for convectively heated CNT-water nano-liquid flow over a bi-directional stretched plane surface saturated in non-Darcian medium. Srinivasacharya and Bindu ²³ studied the slip effect on micro-polar fluid flow, convective heating and mass transfer analysis over cylinder with entropy generation.

The aim of concern exploration is to examine the MHD squeezed Maxwell nano-fluid flow through parallel two plates. The lower plate is fixed while upper plate squeezes toward lower plate with particularize unsteady velocity distribution. Mathematical illustration of consider flow problem involves domination of Brownian motion, thermophoresis diffusion, and convective boundary condition with heat and mass transfer parameters in stratified heat and mass transport phenomenon. Non-linear differential system is constructed to acquire convergent series solutions via homotopic technique Homotopy analysis method (HAM).^24–29 Graphical results are investigated for embedded physical parameters as well as for Nusselt and Sherwood numbers.

Mathematical modeling

Let us assume that electrically conducting Maxwell fluid with nanoparticles is being squeezed between two parallel plates. The flow is assumed laminar, incompressible, unsteady and two dimensional. The stretching velocity of fixed lower plate (at $y=0$ ) is $\mathrm{ax}/(1-\gamma t)$ and it is stretching in positive x-direction in its own plane. The plate at $y=0$ is assumed to be porous in which the fluid flows with injection (or suction) velocity $-V_0/1-\gamma t$ . The upper plate (at $y=h(t)$ ) is moving toward the lower plate with vertical velocity $-(\gamma /2)\sqrt{\nu /a(1-\gamma t)}$ . The gap width between two plates in the minimal separation region is $h(t)=\sqrt{\nu (1-\mathrm{rt})/a}$ . Magnetic field effects are addressed based on the assumption of low magnetic Reynolds number so that the applied magnetic field $B_0/\sqrt{(1-\gamma t)}$ is utilized in y-direction. Here we choose Cartesian coordinates system $(x,y)$ in this two-dimensional model in such a way that $x-\mathrm{axis}$ and $y-\mathrm{axis}$ are selected along and normal to the stretching sheet. The schematic diagram of the flow geometry is given in Figure 1. Heat and mass transfer features involve Brownian motion and thermophoresis. Thermal and solutal stratifications are encountered. Convective conditions are incorporated at the surface of wall for temperature and concentration. All the thermo-physical features are constant. The governing flow equations with heat and mass transfer take the form ¹²

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

(1)

$$\begin{array}{c}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\lambda (\frac{{\partial }^{2}u}{\partial t^2}+2u\frac{{\partial }^{2}u}{\partial t\partial x}+2v\frac{{\partial }^{2}u}{\partial t\partial y}+2\mathrm{uv}\frac{{\partial }^{2}u}{\partial x\partial y}+u^2\frac{{\partial }^{2}u}{\partial x^2}+v^2\frac{{\partial }^{2}u}{\partial y^2})\hfill \\ =-\frac{1}{\rho }\frac{\partial p}{\partial x}-\frac{\lambda }{\rho }(\frac{{\partial }^{2}p}{\partial t\partial x}+u\frac{{\partial }^{2}p}{\partial x^2}+v\frac{{\partial }^{2}p}{\partial x\partial y}-\frac{\partial u}{\partial x}\frac{\partial p}{\partial x})+\nu (\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2})-\frac{\sigma B_0^2}{1-\gamma t}u\hfill \\ -\frac{\sigma B_0^2\lambda }{1-\gamma t}(\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial y})\hfill \end{array}$$

(2)

$$\begin{array}{c}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+\lambda (\frac{{\partial }^{2}v}{\partial t^2}+2u\frac{{\partial }^{2}v}{\partial t\partial x}+2v\frac{{\partial }^{2}v}{\partial t\partial y}+2\mathrm{uv}\frac{{\partial }^{2}v}{\partial x\partial y}+u^2\frac{{\partial }^{2}v}{\partial x^2}+v^2\frac{{\partial }^{2}v}{\partial y^2})\hfill \\ =-\frac{1}{\rho }\frac{\partial p}{\partial y}-\frac{\lambda }{\rho }(\frac{{\partial }^{2}p}{\partial t\partial y}+u\frac{{\partial }^{2}p}{\partial x\partial y}+v\frac{{\partial }^{2}p}{\partial y^2}-\frac{\partial v}{\partial y}\frac{\partial p}{\partial y})+\nu (\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2})\hfill \end{array}$$

(3)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha (\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})+\tau D_B(\frac{\partial C}{\partial x}\frac{\partial T}{\partial x}+\frac{\partial C}{\partial y}\frac{\partial T}{\partial y})+\frac{\tau D_T}{T_h}({\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2)$$

(4)

$$\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_B(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2})+\frac{D_T}{T_h}(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})$$

(5)

Here u and v represent velocity components in x and y directions, respectively. $\lambda$ is relaxation time, $\rho$ is fluid density, p is pressure, $\nu$ is kinematics viscosity, $B_0$ is constant applied magnetic field, $\sigma$ is electric conductivity, $\gamma$ is positive dimensional constant, T is fluid temperature, $\alpha$ is thermal diffusivity, $\tau$ is ratio of the effective heat capacity of the nanoparticles to heat capacity of fluid, C is fluid concentration, $D_B$ and $D_T$ represent Brownian diffusion and thermophoresis coefficient, respectively.

Figure 1.

Flow geometry.

The boundary conditions are as follows

$$\begin{array}{c}u=U_w=\frac{\mathrm{ax}}{1-\gamma t},v=\frac{-V_0}{1-\gamma t},\mathrm{at}y=0\hfill \\ u=0,v=v_h=\frac{\mathrm{dh}}{\mathrm{dt}}=-\frac{\gamma }{2}\sqrt{\frac{\nu }{a(1-\gamma t)}},\hfill \\ \multicolumn{1}{c}{T=T_h\left(x\right),C=C_h\left(x\right)\mathrm{at}y=h\left(t\right)}\\ {K\frac{\partial T}{\partial y}|}_{y=0}=-h_f[T_f-T],{D_B\frac{\partial C}{\partial y}|}_{y=0}\hfill \\ \multicolumn{1}{c}{=-h_f^{*}[C_f-C]\mathrm{at}y=0}\end{array}$$

(6)

where

$$\begin{array}{c}{T}_f\left(x\right)=T_0+\frac{\mathrm{dx}}{1-\gamma t},T_h\left(x\right)=T_0+\frac{d_{1}x}{1-\gamma t}\hfill \\ C_f\left(x\right)=C_0+\frac{\mathrm{ex}}{1-\gamma t},C_h\left(x\right)=C_0+\frac{e_{1}x}{1-\gamma t}\hfill \end{array}$$

(7)

Here $T_h$ and $C_h$ denote temperature and concentration of upper plate, respectively. $T_f$ and $C_f$ are convective fluid temperature and concentration, respectively. $T_0$ termed as reference temperature which shows the temperature of the surfaces, $C_0$ is reference concentration, $h_f$ and $h_f^{*}$ are convective heat and mass transfer coefficient, respectively. K is the thermal conductivity, $V_0<0$ indicates injection, and $V_0>0$ shows suction velocity; $a,d,d_1,e$ , and $e_1$ are positive dimensional constants.

We introduce the similarity solutions

$$\begin{array}{c}\eta =\frac{y}{h\left(t\right)},\Psi =\sqrt{\frac{a\nu }{1-\gamma t}}\mathrm{xf}\left(\eta \right),u=U_{w}f\prime \left(\eta \right),\hfill \\ \multicolumn{1}{c}{v=-\sqrt{\frac{a\nu }{1-\gamma t}}f\left(\eta \right)}\\ \theta \left(\eta \right)=\frac{T-T_h}{T_f-T_0},\varphi \left(\eta \right)=\frac{C-C_h}{C_f-C_0}\hfill \end{array}$$

(8)

Pressure is eliminating from equations (2) and (3) and governing equations (2)–(6) yield

$$\begin{array}{l}{f}^{\left(iv\right)}+f{f}^{‴}-f^{\prime }{f}^{″}-\frac{S_q}{2}\left(3f^{″}+\eta f^{‴}\right)-\beta ({\left(\frac{S_q}{2}\right)}^2\left(15f^{″}+14\eta f^{‴}+{\eta }^2{f}^{\left(iv\right)}\right)\\ +S_q\left(2f^{\prime }{f}^{″}+\eta {\left(f^{″}\right)}^2-\delta \left(3f{f}^{″}+\eta f^{‴}\right)\right)-\left(2f{\left(f^{″}\right)}^2+2{\left(f^{\prime }\right)}^2{f}^{″}-f^2{f}^{\left(iv\right)}\right))\\ -M^2\left(f^{″}+\beta \left(\frac{S_q}{2}\left(3f^{″}+\eta f^{‴}\right)-f^{\prime f^{″}}-f{f}^{‴}\right)\right)=0\end{array}$$

(9)

$$\begin{array}{c}\theta ″+\mathrm{Pr}(f\theta \prime -\theta f\prime )-\mathrm{Pr}{S}_q(s_1+\theta )-\mathrm{Pr}(s_{1}f\prime +\frac{S_q}{2}\eta \theta \prime )\hfill \\ \multicolumn{1}{c}{+\frac{\mathrm{PrNb}}{\mathrm{Re}}(s_1+\theta )(s_2+\varphi )}\\ +\mathrm{PrNb}\theta \prime \varphi \prime +\frac{\mathrm{PrNt}}{\mathrm{Re}}{(s_1+\theta )}^2+\mathrm{PrNt}{(\theta \prime )}^2=0\hfill \end{array}$$

(10)

$${\varphi }^{″}+PrLe\left(f{\varphi }^{\prime }-\varphi f^{\prime }\right)-PrLe{S}_q\left(s_2+\varphi \right)-PrLe\left(s_2{f}^{\prime }+\frac{S_q}{2}\eta {\varphi }^{\prime }\right)+\left(\frac{Nt}{Nb}\right){\theta }^{″}=0$$

(11)

with boundary conditions

$$\begin{array}{c}f\left(0\right)=S,f\left(1\right)=\frac{S_q}{2},f\prime \left(0\right)=1,f\prime \left(1\right)=0\hfill \\ \theta \prime \left(0\right)=-B_1(1-s_1-\theta \left(0\right)),\theta \left(1\right)=0\hfill \\ \varphi \prime \left(0\right)=-B_2(1-s_2-\varphi \left(0\right)),\varphi \left(1\right)=0\hfill \end{array}$$

(12)

where squeezing parameter $S_q$ , Deborah number $\beta$ , length parameter $\delta$ , Hartman number M, Prandtl number Pr, Brownian diffusion parameter Nb, injection/suction parameter S, thermal stratified parameter $s_1$ , Thermal Biot number $B_1$ , solutal stratified parameter $s_2$ , Lewis number Le, thermophoresis parameter Nt, Reynolds number Re, and Solutal Biot number $B_2$ are given by

$$\begin{array}{c}{S}_q=\frac{\gamma }{a},\beta =\frac{a\lambda }{(1-\gamma t)},\delta =\sqrt{\frac{\nu (1-\gamma t)}{\mathrm{ax}}},\hfill \\ \multicolumn{1}{c}{M=\frac{\sigma B_0^2}{a(1-\gamma t)},\mathrm{Pr}=\frac{\nu }{\alpha }}\\ S=\frac{V_0}{\mathrm{ah}\left(t\right)},s_1=\frac{d_1}{d},s_2=\frac{e_1}{e},\mathrm{Nb}=\frac{\tau D_B(C_f-C_0)}{\nu },\hfill \\ \multicolumn{1}{c}{\mathrm{Nt}=\frac{\tau D_T(T_f-T_0)}{\nu T_h}}\\ \mathrm{Re}=\frac{U_{w}x}{\nu },\mathrm{Le}=\frac{\alpha }{D_B},B_1=\frac{h_f}{K}\sqrt{\frac{\nu (1-\gamma t)}{a}},\hfill \\ \multicolumn{1}{c}{B_2=\frac{h_f^{*}}{D_B}\sqrt{\frac{\nu (1-\gamma t)}{a}}}\end{array}$$

(13)

It is witnessed that $S_q<0$ reflects the away movement of plates, while $S_q>0$ depicts that plates are approaching each other. Thermal and solutal stratification effects are vanished when $s_1=s_2=0$ . Furthermore, when $\beta =0$ , Maxwell fluid converts to viscous fluid.

The expressions of the Nusselt and Sherwood number are

$$N{u}_x=\frac{-\mathrm{xK}{\left(\frac{\partial T}{\partial y}\right)|}_{y=h\left(t\right)}}{K(T_f-T_0)},S{h}_x=\frac{-x{D}_B{\left(\frac{\partial C}{\partial y}\right)|}_{y=h\left(t\right)}}{D_B(C_f-C_0)}$$

(14)

In dimensionless scale, equation (14) takes the form

$${\left(R{e}_x\right)}^{-\frac{1}{2}}N{u}_x=-\left(\frac{1}{1-s_1}\right){\theta }^{\prime }\left(1\right),{\left(R{e}_x\right)}^{-\frac{1}{2}}S{h}_x=-\left(\frac{1}{1-s_2}\right){\varphi }^{\prime }\left(1\right)$$

(15)

here, $R{e}_x=U_{w}x/\nu$ represents the local Reynolds number.

Solutions by HAM

For homotopic series solutions, it is required to choose the suitable initial guesses and corresponding linear operators for present flow problem as

$$f_0\left(\eta \right)=\frac{1}{2}\left(2S+2\eta -4{\eta }^2+3S_q{\eta }^2-6S{\eta }^2+2{\eta }^3-2S_q{\eta }^3+4S{\eta }^3\right)$$

(16)

$${\theta }_0\left(\eta \right)=\frac{1}{1+B_1}(B_1-B_1{s}_1-B_1\eta +B_1{s}_1\eta )$$

(17)

$${\varphi }_0\left(\eta \right)=\frac{1}{1+B_2}(B_2-B_2{s}_2-B_2\eta +B_2{s}_2\eta )$$

(18)

$${\mathcal{L}}_f=f^{\left(\mathrm{iv}\right)},{\mathcal{L}}_{\theta }=\theta ″,{\mathcal{L}}_{\varphi }=\varphi ″$$

(19)

with

$${ℒ}_f\left(C_1+C_2\eta +C_3{\eta }^2+C_4{\eta }^3\right)=0,{ℒ}_{\theta }\left(C_5+C_6\eta \right)=0,{ℒ}_{\varphi }\left(C_7+C_8\eta \right)=0$$

(20)

where $C_i(i=1-8)$ are arbitrary constants.

Zeroth order problems

Here

$$\begin{array}{l}\left(1-q\right){ℒ}_f\left[\stackrel{⌣}{f}\left(\eta ;q\right)-f_0\left(\eta \right)\right]=q{\hslash }_f{N}_f\left[\stackrel{⌣}{f}\left(\eta ;q\right)\right]\\ \stackrel{⌣}{f}\left(0;q\right)=S,\stackrel{⌣}{f}\left(1;q\right)=\frac{{\mathrm{S}}_q}{2},{\stackrel{⌣}{f}}^{\prime }\left(0;q\right)=1,{\stackrel{⌣}{f}}^{\prime }\left(1;q\right)=0\end{array}$$

(21)

$$\begin{array}{l}\left(1-q\right){ℒ}_{\theta }\left[\stackrel{⌣}{\theta }\left(\eta ;q\right)-{\theta }_0\left(\eta \right)\right]=q{\hslash }_{\theta }{N}_{\theta }\left[\stackrel{⌣}{\theta }\left(\eta ;q\right)\right]\\ {\stackrel{⌣}{\theta }}^{\prime }\left(0;q\right)-B_1\stackrel{⌣}{\theta }\left(0;q\right)=-B_1\left(1-s_1\right),\stackrel{⌣}{\theta }\left(1;q\right)=0\end{array}$$

(22)

$$\begin{array}{l}\left(1-q\right){ℒ}_{\varphi }\left[\stackrel{⌣}{\varphi }\left(\eta ;q\right)-{\varphi }_0\left(\eta \right)\right]=q{N}_{\varphi }\left[\stackrel{⌣}{\varphi }\left(\eta ;q\right)\right]\\ {\stackrel{⌣}{\varphi }}^{\prime }\left(0;q\right)-B_2\stackrel{⌣}{\varphi }\left(0;q\right)=-B_2\left(1-s_2\right),\stackrel{⌣}{\varphi }\left(1;q\right)=0\end{array}$$

(23)

Defining non-linear operators as

$$\begin{array}{l}{N}_f\left[\stackrel{⌣}{f}\left(\eta ;q\right)\right]=\frac{{\partial }^4\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^4}+\stackrel{⌣}{f}\left(\eta ;q\right)\frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}-\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}\\ -\frac{S_q}{2}\left(3\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+\eta \frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}\right)-\beta ({\left(\frac{S_q}{2}\right)}^2\left(15\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+14\eta \frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}+{\eta }^2\frac{{\partial }^4\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^4}\right)\\ +S_q\left(2\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+\eta \frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}-\delta \left(3\stackrel{⌣}{f}\left(\eta ;q\right)\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+\eta \frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}\right)\right)\\ -\left(2\stackrel{⌣}{f}\left(\eta ;q\right)\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+2\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}-{\left(\stackrel{⌣}{f}\left(\eta ;q\right)\right)}^2\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}\right))\\ -M^2\left(\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+\beta \left(\begin{array}{c}\frac{S_q}{2}\left(3\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}+\eta \frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}\right)-\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\frac{{\partial }^2\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^2}\\ -\stackrel{⌣}{f}\left(\eta ;q\right)\frac{{\partial }^3\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial {\eta }^3}\end{array}\right)\right)\end{array}$$

(24)

$$\begin{array}{l}{N}_{\theta }\left[\stackrel{⌣}{\theta }\left(\eta ;q\right)\right]=\frac{{\partial }^2\stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial {\eta }^2}+Pr\left(\stackrel{⌣}{f}\left(\eta ;q\right)\frac{\partial \stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial \eta }-\stackrel{⌣}{\theta }\left(\eta ;q\right)\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\right)-Pr{S}_q\left(s_1+\stackrel{⌣}{\theta }\left(\eta ;q\right)\right)\\ -Pr\left(s_1\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }+\frac{S_q}{2}\eta \frac{\partial \stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial \eta }\right)+\frac{PrNb}{Re}\left(s_1+\stackrel{⌣}{\theta }\left(\eta ;q\right)\right)\left(s_2+\stackrel{⌣}{\varphi }\left(\eta ;q\right)\right)\\ +PrNb\frac{\partial \stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial \eta }\frac{\partial \stackrel{⌣}{\varphi }\left(\eta ;q\right)}{\partial \eta }+\frac{PrNt}{Re}{\left(s_1+\stackrel{⌣}{\theta }\left(\eta ;q\right)\right)}^2+PrNt{\left(\frac{\partial \stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial \eta }\right)}^2\end{array}$$

(25)

$$\begin{array}{l}{N}_{\varphi }\left[\stackrel{⌣}{\varphi }\left(\eta ;q\right)\right]=\frac{{\partial }^2\stackrel{⌣}{\varphi }\left(\eta ;q\right)}{\partial {\eta }^2}+PrLe\left(\stackrel{⌣}{f}\left(\eta ;q\right)\frac{\partial \stackrel{⌣}{\varphi }\left(\eta ;q\right)}{\partial \eta }-\stackrel{⌣}{\varphi }\left(\eta ;q\right)\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }\right)\\ -PrLe{S}_q\left(s_2+\stackrel{⌣}{\varphi }\left(\eta ;q\right)\right)-PrLe\left(s_2\frac{\partial \stackrel{⌣}{f}\left(\eta ;q\right)}{\partial \eta }+\frac{S_q}{2}\eta \frac{\partial \stackrel{⌣}{\varphi }\left(\eta ;q\right)}{\partial \eta }\right)+\left(\frac{Nt}{Nb}\right)\frac{{\partial }^2\stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial {\eta }^2}\end{array}$$

(26)

where $q\in [0,1]$ is embedding parameter and auxiliary non-zero parameters are ${\hslash }_f,{\hslash }_{\theta },\mathrm{and}{\hslash }_{\Phi }$ .

mth order problems

Here

$$\begin{array}{c}{\mathcal{L}}_f[f_m\left(\eta \right)-{\chi }_m{f}_{m-1}\left(\eta \right)]={\hslash }_f{R}_m^f\left(\eta \right)\hfill \\ f_m\left(0\right)=0,f_m\left(1\right)=0,{f\prime }_m\left(0\right)=0,{f\prime }_m\left(1\right)=0\hfill \end{array}$$

(27)

$$\begin{array}{c}{\mathcal{L}}_{\theta }[{\theta }_m\left(\eta \right)-{\chi }_m{\theta }_{m-1}\left(\eta \right)]={\hslash }_{\theta }{R}_m^{\theta }\left(\eta \right)\hfill \\ {\theta \prime }_m\left(0\right)-B_1{\theta }_m\left(0\right)=0,{\theta }_m\left(1\right)=0\hfill \end{array}$$

(28)

$$\begin{array}{c}{\mathcal{L}}_{\varphi }[{\varphi }_m\left(\eta \right)-{\chi }_m{\varphi }_{m-1}\left(\eta \right)]={\hslash }_f{R}_m^{\varphi }\left(\eta \right)\hfill \\ {\varphi \prime }_m\left(0\right)-B_2{\varphi }_m\left(0\right)=0,{\varphi }_m\left(1\right)=0\hfill \end{array}$$

(29)

Defining non-linear operators as

$$\begin{array}{l}{R}_m^f\left(\eta \right)=f_{m-1}^{\prime \prime \prime \prime }+{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}{f}_k^{\prime \prime \prime }-{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}^{\prime }{f}_k^{\prime \prime }-\frac{S_q}{2}\left(3f_{m-1}^{\prime \prime }+\eta f_{m-1}^{\prime \prime \prime }\right)\\ -\beta ({\left(\frac{S_q}{2}\right)}^2\left(15f_{m-1}^{\prime \prime }+14\eta f_{m-1}^{\prime \prime \prime }+{\eta }^2{f}_{m-1}^{\prime \prime \prime \prime }\right)+S_q(2{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}^{\prime }{f}_k^{\prime \prime }+\eta {\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}^{\prime \prime }{f}_k^{\prime \prime }\\ -\delta \left(3{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}{f}_k^{\prime \prime }+\eta f_{m-1}^{\prime \prime \prime }\right))-(2{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}{\displaystyle \sum }_{p=0}^k{f}_{k-p}^{\prime \prime }{f}_p^{\prime \prime }+2{\displaystyle \sum }_{k=0}^{m-1}{f^{″}}_{m-1-k}{\displaystyle \sum }_{p=0}^k{f}_{k-p}^{\prime }{f}_p^{\prime }\\ -{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}{\displaystyle \sum }_{p=0}^k{f}_{k-p}{f}_p^{\prime \prime \prime \prime }))-M^2(f_{m-1}^{\prime \prime }+\beta \left(\frac{S_q}{2}\left(3f_{m-1}^{\prime \prime }+\eta f_{m-1}^{\prime \prime \prime }\right)\right)-{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}^{\prime }{f}_k^{\prime \prime }\\ -{\displaystyle \sum }_{k=0}^{m-1}{f}_{m-1-k}{f}_k^{\prime \prime \prime }))\end{array}$$

(30)

$$\begin{array}{c}{R}_m^{\theta }\left(\eta \right)={\theta }_{m-1}^{″}+\mathrm{Pr}(\sum _{k=0}^{m-1}{f}_{m-1-k}{\theta }_k^{\prime }-\sum _{k=0}^{m-1}{\theta }_{m-1-k}{f}_k^{\prime })-\mathrm{Pr}(s_1{f}_{m-1}^{\prime }+\frac{S_q}{2}\eta {\theta }_{m-1}^{\prime })\hfill \\ -\mathrm{Pr}{S}_q(s_1+{\theta }_{m-1})+\frac{\mathrm{PrNb}}{\mathrm{Re}}(s_1{s}_2+s_1{\varphi }_{m-1}+s_2{\theta }_{m-1}+\sum _{k=0}^{m-1}{\theta }_{m-1-k}{\varphi }_k)\hfill \\ +\mathrm{PrNb}\sum _{k=0}^{m-1}{\theta }_{m-1-k}^{\prime }{\varphi }_k^{\prime }+\frac{\mathrm{PrNt}}{\mathrm{Re}}({s_1}^2+2s_1{\theta }_{m-1}+\sum _{k=0}^{m-1}{\theta }_{m-1-k}{\theta }_k)+\mathrm{PrNt}\sum _{k=0}^{m-1}{\theta }_{m-1-k}^{\prime }{\theta }_k^{\prime }\hfill \end{array}$$

(31)

$$\begin{array}{c}{R}_m^{\varphi }\left(\eta \right)={\varphi }_{m-1}^{″}+\mathrm{PrLe}(\sum _{k=0}^{m-1}{f}_{m-1-k}{\varphi }_k^{\prime }-\sum _{k=0}^{m-1}{\varphi }_{m-1-k}{f}_k^{\prime })\hfill \\ \multicolumn{1}{c}{-\mathrm{PrLe}{S}_q(s_2+{\varphi }_{m-1})}\\ -\mathrm{PrLe}(s_2{f\prime }_{m-1}+\frac{S_q}{2}\eta {\varphi }_{m-1}^{\prime })+\left(\frac{\mathrm{Nt}}{\mathrm{Nb}}\right){\theta }_{m-1}^{″}\hfill \end{array}$$

(32)

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

(33)

For $q=0\mathrm{and}q=1$ , we can write

$$\begin{array}{l}\stackrel{⌣}{f}\left(\eta ;0\right)=f_0\left(\eta \right),\stackrel{⌣}{f}\left(\eta ;1\right)=f\left(\eta \right)\\ \stackrel{⌣}{\theta }\left(\eta ;0\right)={\theta }_0\left(\eta \right),\stackrel{⌣}{\theta }\left(\eta ;1\right)=\theta \left(\eta \right)\\ \stackrel{⌣}{\varphi }\left(\eta ;0\right)={\varphi }_0\left(\eta \right),\stackrel{⌣}{\varphi }\left(\eta ;1\right)=\varphi \left(\eta \right)\end{array}$$

(34)

and with the variation of $q\mathrm{from}0\mathrm{to}1,\stackrel{⌣}{f}(\eta ;q),\stackrel{⌣}{\theta }(\eta ;q),\mathrm{and}\stackrel{⌣}{\varphi }(\eta ;q)$ vary from the initial solutions $f_0(\eta ),{\theta }_0(\eta ),\mathrm{and}{\varphi }_0(\eta )$ to the final solutions $f(\eta ),\theta (\eta ),\mathrm{and}\varphi (\eta )$ , respectively. By Taylor series, we have

$$\begin{array}{l}\stackrel{⌣}{f}\left(\eta ;q\right)=f_0\left(\eta \right)+{\displaystyle \sum }_{m=1}^{\infty }{f}_m\left(\eta \right)q^m,{f_m\left(\eta \right)=\frac{1}{m!}\frac{{\partial }^m\stackrel{⌣}{f}\left(\eta ;q\right)}{\partial q^m}|}_{q=0}\\ \stackrel{⌣}{\theta }\left(\eta ;q\right)={\theta }_0\left(\eta \right)+{\displaystyle \sum }_{m=1}^{\infty }{\theta }_m\left(\eta \right)q^m,{{\theta }_m\left(\eta \right)=\frac{1}{m!}\frac{{\partial }^m\stackrel{⌣}{\theta }\left(\eta ;q\right)}{\partial q^m}|}_{q=0}\\ \stackrel{⌣}{\varphi }\left(\eta ;q\right)={\varphi }_0\left(\eta \right)+{\displaystyle \sum }_{m=1}^{\infty }{\varphi }_m\left(\eta \right)q^m,{{\varphi }_m\left(\eta \right)=\frac{1}{m!}\frac{{\partial }^m\stackrel{⌣}{\varphi }\left(\eta ;q\right)}{\partial q^m}|}_{q=0}\end{array}$$

(35)

choose suitable value for auxiliary parameter that the series equation (35) converges at $q=1$ , that is

$$\begin{array}{c}f\left(\eta \right)=f_0\left(\eta \right)+\sum _{m=1}^{\infty }{f}_m\left(\eta \right)\hfill \\ \theta \left(\eta \right)={\theta }_0\left(\eta \right)+\sum _{m=1}^{\infty }{\theta }_m\left(\eta \right)\hfill \\ \varphi \left(\eta \right)={\varphi }_0\left(\eta \right)+\sum _{m=1}^{\infty }{\varphi }_m\left(\eta \right)\hfill \end{array}$$

(36)

$f_m,{\theta }_m,\mathrm{and}{\varphi }_m$ represent general solutions for equations (27)–(29) in the form of special solutions $(f_m^{*},{\theta }_m^{*},{\varphi }_m^{*})$ are given by

$$\begin{array}{c}{f}_m\left(\eta \right)=f_m^{*}\left(\eta \right)+C_1+C_2\eta +C_3{\eta }^2+C_4{\eta }^3\hfill \\ {\theta }_m\left(\eta \right)={\theta }_m^{*}\left(\eta \right)+C_5+C_6\eta \hfill \\ {\varphi }_m\left(\eta \right)={\varphi }_m^{*}\left(\eta \right)+C_7+C_8\eta \hfill \end{array}$$

(37)

Convergence of problem

It is noted that $f(\eta ),\theta (\eta ),\mathrm{and}\varphi (\eta )$ have three unknown convergence-control parameters $c_0^f,c_0^{\theta },\mathrm{and}{c}_0^{\varphi }$ which are very helpful for convergent series solutions. It is observed that convergence-control parameters have significant role in method named as HAM. Thus, these convergence-control parameters make HAM different to all other analytic approximation solutions. Here, average residual error is utilized to substantially reduce the CPU time at kth order of approximation, defined by

$${\epsilon }_k^f(c_0^f,c_0^{\theta },c_0^{\varphi })=\frac{1}{N+1}\sum _{j=0}^N{\left[N_f\left(\sum _{i=0}^k{f}_i\right)\right|\eta =j\delta \eta ]}^2$$

(38)

$${\epsilon }_k^{\theta }\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)=\frac{1}{N+1}{\displaystyle \sum }_{j=0}^N{\left[N_{\theta }{\left({\displaystyle \sum }_{i=0}^k{f}_i,{\displaystyle \sum }_{i=0}^k{\theta }_i,{\displaystyle \sum }_{i=0}^k{\varphi }_i\right)|}_{\eta =j\delta \eta }\right]}^2$$

(39)

$${\epsilon }_k^{\varphi }\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)=\frac{1}{N+1}{\displaystyle \sum }_{j=0}^N{\left[N_{\varphi }{\left({\displaystyle \sum }_{i=0}^k{f}_i,{\displaystyle \sum }_{i=0}^k{\theta }_i,{\displaystyle \sum }_{i=0}^k{\varphi }_i\right)|}_{\eta =j\delta \eta }\right]}^2$$

(40)

for the considered constitutive equations, respectively. Where N is an integer and the total error at the kth order of approximation is expressed as

$${\epsilon }_k^t\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)={\epsilon }_k^f\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)+{\epsilon }_k^{\theta }\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)+{\epsilon }_k^{\varphi }\left(c_0^f,c_0^{\theta },c_0^{\varphi }\right)$$

(41)

The analogous maximal parameters that control the convergence are tabulated up to 12th order of approximations (Table 1).

Table 1.

Maximal convergence-control parameters in various orders of approximation.

k (Order of approximation)	${\epsilon }_k^f$	${\epsilon }_k^{\theta }$	${\epsilon }_k^{\varphi }$
2	1.13536 × 10−7	8.71223 × 10−7	3.03922 × 10−6
4	1.65453 × 10−14	5.91291 × 10−10	7.03573 × 10−9
6	3.41298 × 10−21	2.79772 × 10−13	6.24197 × 10−12
8	1.40942 × 10−27	1.84626 × 10−16	3.32404 × 10−15
10	4.58965 × 10−34	3.69838 × 10−20	1.10675 × 10−18
12	2.09866 × 10−40	5.03191 × 10−22	1.34029 × 10−2

Note that total error ${\epsilon }_k^t$ decreases to 4.02398 × 10⁻⁶ in respect of maximal convergence-control parameters $c_0^f=-0.994565$ , $c_0^{\theta }=-0.885284$ , $c_0^{\varphi }=-1.07045$ .

Graphical representation for 12th order of approximation is disclosed in Figure 2. Here ${\epsilon }_k^t$ illustrates the total discrete squared residual error such that

$${\epsilon }_k^t={\epsilon }_k^f+{\epsilon }_k^{\theta }+{\epsilon }_k^{\varphi }$$

(42)

Here, the ${\epsilon }_k^t$ is utilized to determine the optimal convergence-control parameters.

Figure 2.

Graph for 12th order of approximation.

Results and discussion

This segment discloses how the flow parameters affect various quantities of interest such as velocity, temperature, and nanoparticle concentration. To this end, Figures 3–22 are plotted. The graphical results are constructed using the representative parameters involved in the squeezing heat and mass flow phenomenon. Deviation of squeezing parameter $S_q$ on velocity components is observed in Figures 3 and 4. It is noted that an increment in $S_q$ results enhancement in vertical and horizontal velocities. Since pressure is generated by the movement of the top plate toward the lower plate which causes rapid flow of squeezed nanoparticles. Hence, velocity distribution grows up. Figure 5 reflects the behavior of injection/suction parameter S on the vertical velocity field. Velocity field increases with an increment in suction/injection parameter. Physically fluid particles are added to the system which enhances the velocity profile. Figure 6 reflects the behavior of injection/suction parameter S on a horizontal velocity profile. It is analyzed that velocity distribution decreases for enlarge injection/suction parameter. Because that near the upper plate, the velocity field decays more rapidly in comparison to neighborhood of the lower porous stretchable plate. Hence, decline is noticed in the velocity field. Response of Hartman number $(M)$ on velocity field is addressed in Figure 7. Two distinct regions can be identified from the plot. In the region $0⩽\eta ⩽0.5$ , higher magnetic parameter resists the flow while in the region $0.5⩽\eta ⩽1.0$ it assists the flow due to weak resistive force (Lorentz force). The assistance or resistance increases with dominating magnetic parameter. Figure 8 discloses the variation of Deborah number $\beta$ on horizontal velocity. It is noted that velocity profile strengthens with an increment in $\beta$ in the region $0⩽\eta ⩽0.5$ while in the region $0.5⩽\eta ⩽1.0$ , velocity reduces for increasing $\beta$ . In fact, enlarge Deborah number corresponds to greater relaxation time which offers more resistance to the fluid flow. Thus, velocity field decays. Figure 9 shows that the observed temperature reduction is due to the squeezing effect produced by the movement of upper plate. With an increase in $S_q$ , the squeezing force diminishes on the fluid resulting in a decreased flow temperature. Figure 10 shows variation of temperature profile versus thermal stratified parameter $s_1$ . Here temperature distribution is larger for greater thermal stratified parameter. In fact, an increment in thermal stratified parameter means that the fluid density decays in vertical direction which increases the surface temperature. Hence, temperature profile grows up. Impact of Brownian diffusion parameter Nb on temperature distribution is disclosed in Figure 11. It is concluded that stronger Brownian diffusion parameter enhances the fluid temperature. Brownian motion parameter has strong dependence on Brownian diffusion coefficient. The Brownian diffusion coefficient is incremented when we increase the Brownian motion parameter due to which the fluid temperature dominants. Figure 12 illustrates the variation of thermophoresis parameter Nt on temperature distribution. Greater thermophoresis parameter causes higher thermophoresis force which strengthens the fluid temperature. Figure 13 addresses the variation in temperature field for thermal Biot number $B_1$ . An enlargement in Biot number causes sturdy thermal convection which increases the temperature profile. Response of Lewis number Le on particles concentration is disclosed in Figure 14. Decline is observed in concentration distribution for greater Lewis number. It is due to fact that Brownian diffusion coefficient shows inverse behavior to Lewis number. Brownian diffusion coefficient decays for larger Lewis number and consequently concentration profile reduces. Impact of Brownian diffusion parameter Nb on concentration distribution is illustrated in Figure 15. Here concentration profile enhances for dominant Brownian motion parameter. Such increment in concentration distribution is noted because of the motion of the nanoparticles from top plate to the fluid. Figure 16 shows the variation of thermophoresis parameter Nt on concentration field. This demonstrates that greater thermophoresis parameter causes resistance to mass diffusion which leads to decrement in concentration distribution. Influence of solutal stratified parameter $s_2$ on nanoparticle concentration is disclosed in Figure 17. It is found that concentration distribution diminishes with an increment in solutal stratified parameter $s_2$ . Increment in $s_2$ reduces the difference in concentration between fluid and upper plate. Hence, concentration profile decreases. Figure 18 portrays the impact of solutal Biot number $B_2$ on concentration field. It is recognized that as $B_2$ increases, the concentration profile increases. Since mass transfer coefficient revolves around the mass transfer Biot number $B_2$ . With an increment in $B_2$ , the mass transfer coefficient enhances which gives enhancement in concentration field. Figures 19 and 20 represent behavior of Brownian diffusion parameter Nb, thermal Biot number $B_1$ , thermophoresis parameter Nt, and thermal stratified parameter $s_1$ on the Nusselt number. It is revealed that Nusselt number grows up with an increment in $\mathrm{Nb},\mathrm{Nt},B_1$ , and $s_1$ . Variation of $\mathrm{Le},S_q,B_2$ , and $s_2$ on Sherwood number is illustrated in Figures 21 and 22. It is noted that Sherwood number decreases with an increment in Lewis number Le, squeezing parameter $S_q$ , and solutal stratified parameter $s_2$ while it increases with solutal Biot number $B_2$ .

Figure 3.

Effect of $S_q$ on $f(\eta )$ .

Figure 4.

Effect of $S_q$ on $f\prime (\eta )$ .

Figure 5.

Effect of S on $f(\eta )$ .

Figure 6.

Effect of S on $f\prime (\eta )$ .

Figure 7.

Effect of M on $f\prime (\eta )$ .

Figure 8.

Effect of $\beta$ on $f\prime (\eta )$ .

Figure 9.

Effect of $S_q$ on $\theta (\eta )$ .

Figure 10.

Effect of $s_1$ on $\theta (\eta )$ .

Figure 11.

Effect of Nb on $\theta (\eta )$ .

Figure 12.

Effect of Nt on $\theta (\eta )$ .

Figure 13.

Effect of $B_1$ on $\theta (\eta )$ .

Figure 14.

Effect of Le on $\varphi (\eta )$ .

Figure 15.

Effect of Nb on $\varphi (\eta )$ .

Figure 16.

Effect of Nt on $\varphi (\eta )$ .

Figure 17.

Effect of $s_2$ on $\varphi (\eta )$ .

Figure 18.

Effect of $B_2$ on $\varphi (\eta )$ .

Figure 19.

Effect of $\mathrm{Nb}\mathrm{and}\mathrm{Nt}$ on Nu.

Figure 20.

Effect of $B_1$ and $s_1$ on Nu.

Figure 21.

Effect of $S_q\mathrm{and}\mathrm{Le}$ on Sh.

Figure 22.

Effect of $B_2$ and $s_2$ on Sh.

Closing remarks

A theoretical investigation for Maxwell fluid based on MHD squeezing flow of nanoparticles is accounted in the presence of convective heating.

Key points of our inspection are noted as follows:

• Cross flow variation of velocity distribution is noted for greater material parameter $\beta$ .

• The fluid temperature increases for enhanced thermophoresis and Brownian motion parameters.

• Concentration profile follows an opposite trend for themophoresis and Brownian motion parameters.

• The temperature field enhances with an enlargement in the values of thermal stratified parameter and thermal Biot number.

• Decline in concentration field is noted for larger solutal stratified parameter whereas solutal Biot number raises the concentration profile.