A comparative analysis of Newtonian heating and convective conditions for double stratified magnetized second grade nanofluid

Abstract

A detailed examination of the effects of double stratification on the magnetized second grade nanofluid created by a stretching sheet within the porous medium has been carried out in this study. The CBC (convective boundary conditions) and NH (Newtonian heating) conditions are utilized for the analysis. The relevant similarity transformations are defined in order to transform the governing nonlinear PDEs into dimensionless ODEs. The transformed equations are numerically treated with bvp4c in Matlab software package. The computations are expressed in tabular and graphical forms. The flow parameters involved in the modeling are demonstrated on temperature, velocity, and concentration of nanofluid. The calculations of Nusselt number $- θ' (0)$ , Sherwood number $- ϕ' (0)$ and skin-friction coefficient $C_{fx} (0)$ against the distinct values ofconstraints for general, CBC and NH are expressed to examine the flow situation. The numeric computations are matched with the literature published earlier for particular cases which shows the validity of current computations. It is noted that the skin friction is decreasing with an increment in second grade parameter. Flow velocity is increased by rising the second grade parameter.

Keywords

Second grade fluid nanofluid thermal stratification stretching sheet Newtonian conditions

Introduction

The rheological behavior of non-Newtonian fluids has engaged the researchers due to its involvement in industrial and technology processes such as chemical engineering and petroleum production. The researchers had proposed the various models of non-Newtonian materials because an individual fluid model is insufficient to demonstrate the complete characteristics of all these materials. These models are mainly categorized as rate, differential and integral type fluids. A paramount subclass of differential type fluids is the Rivlin Ericksen fluid of grade two or second grade fluid proposed by the Rivlin and Ericksen¹ to address the normal stress effects. Tauseef Saeed et al.² studied the exact analysis of second grade fluid with generalized boundary conditions. Abbas et al.³ presented the chemical reactive species features of second-grade nanofluid flow past an exponential curved stretching surface with the help of numerical approach. Siddique et al.⁴ explored the bioconvection of MHD second-grade fluid with nanoparticles over an exponentially stretched sheet.

Several practical applications of natural and engineering include dynamics of nanofluid to control heat transfer rate. Nanofluid has suitable thermal characteristics to enhance thermal conductivity. Nanofluid is combination of tiny solid particles in a parent fluid as presented by Choi.⁵ Nanofluids are extensively utilized for thermal procedure where declining and growing temperatures involve. In these kind of procedures, heat transmission is assumed to be largely essential factor. Nanofluid proves best candidate as compared to other fluids (oil and water etc.) in this situation. In view of reliabilty, various flow models are formulated to explore the nanofluids. Tiwari and Das⁶ is proposed to discuss the volumetric fraction of the nano-particles. In addition, Buongiorno⁷ model mainly addresses the thermophoresis influences and Brownian motion in convective state in comparison with thermophysical properties and highlighted that both impacts improve the thermal conductivity. Hussain et al.⁸ discussed the numerical investigation of three-dimensional water-based magneto-hydrodynamic rotating nanofluid flow produced by a linearly extended sheet. Jain et al.⁹ investigated the solutal and thermal transportation of Williamson hybrid nanofluid past a permeable stretching/shrinking surface with mixed convected boundary condition. Salawu et al.¹⁰ discussed the Maxwell Arrhenius kinetic nanofluid flow over a stretchable surface with nonlinear variable properties. Cui et al.¹¹ studied the impact of non-similar modeling of forced convection nanofluid flow over stretching sheet with chemical reaction and heat generation.

Thermal stratification occurs when two fluids at different temperatures come together. The heavier/colder fluid freezes at the bottom of the surface because of their differences in temperature whereas letting the warm/ light fluid to floating above the cooler fluid. Temperature or density variation causes the continuous outflow of the TBL (thermal boundary layer) within the medium. Fluids of variably warm sidewalls have thermal stratification properties. Thermal stratification in various flows is applicable to significant engineering applications such as polymer extrusion. During the production of polymer extrusion, the heat transfer is important for cooling the objects as it passes via a die to achieve the desired properties of the obtained product. Besthapu et al.¹² discussed the mixed convected thermal stratified nanofluid flow produced by an exponential stretchable sheet. Anjali Devi and Kandasamy¹³ addressed the mass species and thermal influences on steady-state magnetized flow over an accelerating thermally stratified sheet under suction/blowing situation. The behavior of radiated thermally stratifying Maxwell fluid flow was presented by Hayat et al.¹⁴ Nakayama and Koyama¹⁵ discussed the thermal stratification behavior on free convected transportation through porous space.

The stretching surface has become the important part and extensive range of benefits is found where heat transfer relate with final product. The manufacturing of paper, glass blowing, drawing wires, crystal growth, heat decline of continuous filaments or strips and extrusion of plastic sheets are a few instances of importance. It is commonly found feature that the velocity in numerous cases is not linear and it may be in nonlinear or exponential form. The concept of stretching surface continuously moving with constant speed was initiated by Sakiadis¹⁶ to observe the flow characteristics. Further, Crane¹⁷ initiated the steady flow caused by linear stretching plate. The extension of this work was adopted by Gupta and Gupta¹⁸ by involving the suction and blowing cases. The flow on a continuously stretchable surface with temperature variation was illustrated by Carragher and Crane.¹⁹ Magyari and Keller²⁰ considerably discussed the boundary-driven flow engendered by exponential stretchy sheet by taking temperature’s exponential variation. Subhani and Nadeem²¹ adopted a numerical analysis to address the 3D micropolar nanofluid flow induced by an exponentially stretching surface embedded in porous medium. Ayub et al.²² inspected the slip effects on electromagnetohydrodynamics (EMHD) nanofluid flow through a horizontal Riga plate. Saleem et al.²³ demonstrated the convected thermal and solutal transmission in magneto Walter’s B nanofluid flow induced by a rotating cone. Ur Rehman et al.²⁴ explored the thermophysical analysis for three-dimensional MHD stagnation-point flow of nano-material influenced by an exponential stretching surface.

There are a number of heating mechanisms discussed by Merkin²⁵ mentioning the temperature division starting from the wall to the ambient situation such as CHF (constant heat flux), NH, CBC, and CWT (constant wall temperature). The heat movement from boundary surface related to heat’s finite capacity in NH conditions commonly referred as the conjugate convected flow. CBC is the flow of heat via finite thickness bounding surface signifying the heat’s finite capacity. The impact of heated processes on fluid motion is substantial and investigated by many researchers. Mohd Kasim et al.²⁶ investigated the constant heat flux solution for mixed convection boundary layer viscoelastic fluid. Hayat et al.²⁷ delibrated the boundary-layer flow of Walters’ B fluid with Newtonian heating. Rehman et al.²⁸ delibrated the Newtonian heating convective flow by way of two different surfaces. RamReddy et al.²⁹ delibrated the similarity solution for free convection flow of a micropolar fluid under convective boundary condition.

The problem under consideration is investigated by assuming the NH and CBC for the effects of double stratification on the magnetized second grade nanofluid created by a stretching sheet. A detailed analysis of temperature distributions, velocity variation and concentration change is computed and interpreted. These cases are also taken to evaluate the physical quantities of interest related to flow equations. The calculated results are new and useful for various industrial and engineering applicable processes where heat transfer involved.

Problem formulation

The nanofluid dynamic across a two-dimensional (2-D) stretchable sheet of length $l$ is assumed in a manner that the sheet stretched in the $x$ -axis direction (see Figure 1). The stretching and free stream velocities are expressed as $u_{w} (x) = a e^{x / ℓ}$ and $u_{\infty} (x) = 0 .$ In the temperature stratification layer, the fluid temeprture $T$ at the wall and the variable ambient state are assumed as $T_{w} (x) = T_{o} + a e^{x / 2 ℓ}$ and $T_{\infty} (x) = T_{o} + b e^{x / 2 ℓ}$ accordingly where $T_{w} (x) > T_{\infty} (x)$ and $T_{o}$ is reference temperature. The nano-particles fraction $C$ at the wall and variable ambient state are considered in the form of $C_{w} (x) = C_{o} + c e^{x / 2 ℓ}$ and $C_{\infty} (x) = C_{o} + d e^{x / 2 ℓ}$ accordingly where $C_{o}$ signifies the reference concentration and $a, b, c$ and $d$ are positive contants. The ambient conditions of $T$ and $C$ are attained when $y \to \infty$ accordingly. The flow geomentry is demonstrated by Figure 1.

Figure 1.

Flow geometry with coordinates.

The governing equations are^30–32:

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

(1)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{μ}{ρ_{f}} \frac{\partial^{2} u}{\partial y^{2}} + \frac{α_{1}}{ρ_{f}} [\frac{\partial}{\partial x} (u \frac{\partial^{2} u}{\partial y^{2}}) - \frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial x \partial y} + v \frac{\partial^{3} u}{\partial y^{3}}],

(2)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α \frac{\partial^{2} T}{\partial y^{2}} + τ [D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial y})}^{2}] \\ + \frac{D_{T} K_{T}}{C_{s} C_{p^{*}}} \frac{\partial^{2} C}{\partial y^{2}}, \end{matrix}

(3)

u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{T} K_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}} .

(4)

Here, the symbols are denoted as: $x$ and $y$ indicates the Cartesian co-ordinates measured directed toward stretching surface and normal to it, $u$ and $v$ the velocity components, $α_{1}$ the second-grade constant, $μ$ the vicsocity, $ρ_{f}$ the density, $ν$ the fluid’s kinematic viscosity, $σ$ the electric conductivity, $α = \frac{k}{{(ρ c)}_{f}}$ the thermal diffusivity, $k$ the thermal coductivity, $(ρ c)_{f}$ the basefluid heat capacitance, $τ = \frac{{(ρ c)}_{p}}{{(ρ c)}_{f}}$ the ratio of nanoparticles heat capacitance to basefluid heat capacitance, $D_{B}$ the Brownian diffusion constant, $D_{T}$ the thermophoretic $diffusion constant, K_{T}$ the thermal $diffusion ratio, C_{p^{*}}$ the specific heat capacity and $C_{S}$ is concentration susceptibility.

The subjected BCs (boundary conditions) are:

u = u_{0} e^{\frac{x}{l}}, v = 0, T = T_{w} (x), C = C_{w} (x),

\frac{\partial T}{\partial y} = - h_{s} T (NH), - k_{f} \frac{\partial T}{\partial y} = h_{f} (T_{f} - T) (CBC) at y = 0,

u \to 0, v \to 0, T \to T_{\infty} (x), C \to C_{\infty} (x) .

(5)

Here $h_{s}$ and $h_{f}$ are the coefficients of heat transmission for CBC and NH, respectively. Moreover, in $CBC$ and $NH$ cases, it is assumed that:

θ (η) = \frac{T - T_{f}}{T_{f} - T_{\infty}}, θ (η) = \frac{T - T_{\infty}}{T_{\infty}} .

(6)

To derive the non-linear ordinary form of flow equations, a suitable stream function $ψ = ψ (x, y)$ satisfying the continuity equation (1) along similarity variables are taken as:

\begin{matrix} ψ = \sqrt{2 ℓ υ u_{0}} e^{\frac{x}{2 ℓ}} f (η), u = \frac{\partial ψ}{\partial y} = u_{0} e^{\frac{x}{l}} f' (η), \\ v = - \frac{\partial ψ}{\partial x} = - \sqrt{\frac{v u_{0}}{2 l}} e^{\frac{x}{2 l}} {f (η) + η f' (η)}, \\ η = y \sqrt{\frac{u_{0}}{2 lv}} e^{\frac{x}{2 l}}, θ (η) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, ϕ (η) = \frac{C - C_{\infty}}{C_{w} - C_{\infty}} . \end{matrix}

(7)

By using equation $(7)$ , equations $(2), (4)$ , and $(5)$ are transformed into the following form:

f ″' + f f ″ - 2 f'^{2} + A (3 f' f ″' + 2 η f ″ f ″' + 3 f'^{2} - f f^{iv}) = 0,

(8)

\frac{1}{\Pr} θ ″ + f θ' - f' θ + Nb θ' ϕ' + Nt θ'^{2} + D_{f} ϕ'' - S t_{1} f' = 0,

(9)

ϕ ″ + Lef ϕ' - Le f' ϕ + S_{r} Le θ ″ - LeS t_{2} f' = 0 .

(10)

The superscript ‘ denotes the derivative of variables w.r.t. $η$ . Also $A = \frac{α_{1} u_{0} e^{\frac{x}{l}}}{ρ_{f} vl}$ is second-grade parameter, $\Pr = \frac{ν}{α} is the$ Prandtl $number, Le = \frac{ν}{D_{B}}$ is Lewis number, $S_{r} = \frac{D_{T} K_{T} (T_{w} - T_{\infty})}{ν T_{\infty} (C_{w} - C_{\infty})} is the Soret number, D_{f} = \frac{D_{T} K_{T} (C_{w} - C_{\infty})}{ν C_{s} C_{p^{*}} (T_{w} - T_{\infty})}$ is the Dufour $number, Nt = \frac{τ D_{T} (T_{w} - T_{\infty})}{ν T_{\infty}}$ is thermophoresis $parameter and Nb = \frac{τ D_{B} (C_{w} - C_{\infty})}{ν}$ is Brownian motion parameter respectively.

The corresponding BCs (8) become:

\begin{matrix} f (η) = 0, f' (η) = 1, f ″ (η) = 0, θ (η) = 1 - S t_{1}, \\ ϕ (η) = 1 - S t_{2}, \end{matrix}

\begin{matrix} θ' (η) = - γ (1 + θ (η)) (CBC), θ' (η) = - β (1 - θ (η)) \\ (NH), at η = 0, \end{matrix}

f' (η) \to 0, θ (η) \to 0, ϕ (η) \to 0 at η \to \infty .

(11)

Here, $S t_{1} = \frac{b}{a}$ is the thermal stratification parameter, $S t_{2} = \frac{c}{d}$ is solutal stratification parameter, $γ = \frac{h_{f}}{k_{f} \sqrt{\frac{u_{0}}{2 lv}} e^{\frac{x}{2 l}}}$ thermal slip parameter and $β = \frac{h_{s} v}{\sqrt{\frac{u_{0}}{2 lv}} e^{\frac{x}{2 l}}}$ the Newtonian heating parameter respectively.

The physical quantities generated by the current flow problem are:

C_{f} = \frac{τ_{w}}{\frac{1}{2} ρ u_{w}^{2}}, N u_{x} = \frac{x q_{w}}{k_{1} (T_{w} - T_{\infty})}, S h_{x} = \frac{x q_{m}}{D_{B} (C_{w} - C_{\infty})} .

Where the expressions of the wall shear stress $τ_{w}$ , heat flux $q_{w}$ and mass flux $q_{m}$ are defined by:

τ_{w} = [μ \frac{\partial u}{\partial y} + ρ_{f} α_{1} (v \frac{\partial^{2} u}{\partial y^{2}} + 2 \frac{\partial u}{\partial x} \frac{\partial u}{\partial y} + u \frac{\partial^{2} u}{\partial x \partial y})] |_{y = 0},

q_{w} = - k_{1} (\frac{\partial T}{\partial y}) |_{y = 0},

q_{m} = - D_{B} (\frac{\partial C}{\partial y}) |_{y = 0} .

The coefficients like $C_{f}, Nu$ , and $Sh$ in dimensionless expressions are given as:

\begin{matrix} C_{fx} (0) = [f ″ (0) + A (7 f' (0) f ″ (0) - f (0) f ″' (0))] \\ \sqrt{(\frac{2 l}{x}) R e_{x}}, - θ' (0) = \frac{Nu}{\sqrt{(\frac{x}{2 l}) R e_{x}}}, \\ - ϕ' (0) = \frac{Sh}{\sqrt{(\frac{x}{2 l}) R e_{x}}} at η = 0 . \end{matrix}

(12)

$Where R e_{x} = \frac{U_{w} (x) x}{ν}$ is the local Reynolds number.

Results and discussion

This section is prepared to illustrate the numerical solution of relevant equations with bvp4c in MATLAB technique. The procedure of numerical technique is illustrated in Figure 2. The conversion of differential equations to ordinary form is taken with the defined similarity variables and then the solution related to thermophoresis parameter $(Nt)$ , Brownian motion $(Nb)$ , Prandlt number $(\Pr),$ Lewis numbe $(Le)$ , Dufour number $(Df)$ , Soret number $(Sr)$ , thermal stratification parameter $(S t_{1})$ , solutal stratification parameter $(S t_{2})$ , and thermal slip parameter $(γ)$ , second grade parameter $(A)$ , and the Newtonian heating parameter $(β)$ in relation with velocity profile, temperature (T) profile, and the concentration (C) profile is obtained. The graphs and variation of skin friction, Nusselt number, and Sherwood number are shown via tables for several cases of interest like (GC) general case, NH (Newtonian heating) case and CBC (convective boundary condition) case accordingly. A comparison with the literature has been conducted in order to confirm the numerical accuracy and establish a beautiful agreement (see Table 1).

Figure 2.

Numerical procedure.

Table 1.

The values of $- θ' (0)$ for $Nt = Nb$ = $A$ = $D_{f}$ = $S_{r}$ = $Le$ = $S t_{1}$ = $S t_{2}$ = $γ$ = $β$ = 0.

$\Pr$	$- θ' (0)$	$- θ' (0)$
$\Pr$	Bidin and Nazar³³	Present results
1.0	0.9548	0.9548
2.0	1.4714	1.4714
3.0	1.8691	1.8691

Table 1 gives the accuracy of current flow problem as one can see that there is a good agreement for the Nusselt number $- θ' (0)$ with literature work. Table 2 addressed the numeric data of $C_{fx} (0)$ against the ditsinct values of $A .$ The values of $C_{fx} (0)$ are rising by enhancing $A .$ Table 3 presents that the variations in $- θ' (0)$ and $- ϕ' (0)$ gainst the variation of distinct parameters under several situations. It is noted that for general case, $- θ' (0)$ exhibits an increase in case of increasing values of $A, \Pr, S t_{1}$ , and $S_{r}$ . But for greater values of $Le, Nb, Nt, S t_{2}$ and $D_{f}, - θ' (0)$ declines. It is observed that the concentration $- ϕ' (0)$ profile enhances when $A, Le, Nb, S t_{2}, D_{f}$ , and $β$ increased but when $\Pr, Nt, S t_{1}$ , and $D_{f}$ enhance, $- ϕ' (0)$ depicts decline. For CBC case, temperature $- θ' (0)$ profile displays an increase in case of increasing values for $Le, Nb, Nt, S t_{2}$ , $D_{f}$ , and $γ$ but for greater values of $A, \Pr, S t_{1}$ , and $S_{r}, - θ' (0)$ reduces. It is also seen that the concentration $- ϕ' (0)$ profile rises when $A, Le, Nb, Nt, S_{r}, S t_{2}$ , and $γ$ enhance but when $\Pr, S t_{1}$ , and $D_{f}$ enhances, $- ϕ' (0)$ depicts decline. For NH case, the temperature $- θ' (0)$ profile dipicts an increase in case of increasing $\Pr, S t_{1}, S_{r}$ , and $β$ values but for greater values of $A, Le, Nb, Nt, S t_{2}$ , and $D_{f}$ , the $- θ' (0)$ profile diminish. The concentration $- ϕ' (0)$ profile boost up when $A, Le, Nb, Nt, S t_{2}$ , and $D_{f}$ enhance but when $\Pr, S t_{1}, S_{r}$ , and $β$ enhances, $- ϕ' (0)$ depicts decline.

Table 2.

Numeric values of $C_{fx} (0)$ for distinct values of $A .$

$A$	$C_{fx} (0)$
0.1	−1.7218
0.5	−2.7350

Table 3.

The calculations of $- θ' (0)$ and $- ϕ' (0)$ with variations for various cases.

$A$	$\Pr$	$Le$	$Nb$	$Nt$	$S t_{1}$	$S t_{2}$	$D_{f}$	$S_{r}$	$γ$	$β$	$- θ' (0)$			$- ϕ' (0)$
$A$	$\Pr$	$Le$	$Nb$	$Nt$	$S t_{1}$	$S t_{2}$	$D_{f}$	$S_{r}$	$γ$	$β$	GC	CBC	NH	GC	CBC	NH
0.1	6.5	0.3	0.1	0.1	0.2	0.2	0.1	0.1	0.2	0.2	2.3095	0.1727	0.2005	0.2965	0.3539	0.3531
0.5	6.5	0.3	0.1	0.1	0.2	0.2	0.1	0.1	0.2	0.2	2.3700	0.1710	0.2020	0.3397	0.3976	0.3968
0.1	10.0	0.3	0.1	0.1	0.2	0.2	0.1	0.1	0.2	0.2	2.8038	0.1688	0.2044	0.2821	0.3538	0.3529
0.1	6.5	0.6	0.1	0.1	0.2	0.2	0.1	0.1	0.2	0.2	2.2351	0.1776	0.1962	0.4847	0.5883	0.5873
0.1	6.5	0.3	2.5	0.1	0.2	0.2	0.1	0.1	0.2	0.2	1.008	0.1853	0.1890	0.3342	0.3540	0.3540
0.1	6.5	0.3	0.1	6.0	0.2	0.2	0.1	0.1	0.2	0.2	0.9193	0.1892	0.1849	0.2682	0.3540	0.3541
0.1	6.5	0.3	0.1	0.1	0.5	0.2	0.1	0.1	0.2	0.2	2.8811	0.1222	0.2451	0.1624	0.2363	0.2329
0.1	6.5	0.3	0.1	0.1	0.2	0.5	0.1	0.1	0.2	0.2	1.6352	0.1742	0.1992	0.3812	0.4205	0.4199
0.1	6.5	0.3	0.1	0.1	0.2	0.2	0.4	0.1	0.2	0.2	2.1974	0.1882	0.1868	0.2996	0.3537	0.3537
0.1	6.5	0.3	0.1	0.1	0.2	0.2	0.1	0.2	0.2	0.2	2.3405	0.1725	0.2007	0.2322	0.3486	0.3471
0.1	6.5	0.3	0.1	0.1	0.2	0.2	0.1	0.1	0.4	0.2	2.3095	0.3485	0.2005	0.2965	0.3716	0.3531
0.1	6.5	0.3	0.1	0.1	0.2	0.2	0.1	0.1	0.2	0.4	2.3095	0.1727	0.3777	2.3095	0.3539	0.3483

Figure 3 expressed the velocity profile for numerous values of second grade parameter $A .$ The velocity profiles increased in relation with the increment of $A$ (0.5, 1.0, 1.5, 2.0). The material constraint $α_{1}$ is higher for incrementing $A$ values which resulted an improvement in the velocity profiles $f' (η) .$ Figure 4 elucidates the nature of temperature $θ (η)$ against the distinct Dufour number $D_{f}$ values for thermally stratified condition. The temperature is augmented by improving the $D_{f}$ (0.1, 0.2, 0.3, 0.4) values. The difference between the wall temperature and ambient temperature is minimized due to higher $D_{f}$ values which resulted an enhancing behavior of $θ (η)$ curves. Figures 5 and 6 evaluated the nature of $D_{f}$ on $θ (η)$ for CBC ( $D_{f} =$ 0.1, 0.3, 0.5, 0.7) and NH ( $D_{f} =$ 0.4, 0.5, 0.6, 0.7) cases. For both CBC and NH cases, temperature $θ (η)$ profiles are risen. Temparature $θ (η)$ at wall is higher for CBC case as comparative to NH case. The solutions in CBC case converge earlier as comparative to NH case.

Figure 3.

The impacts of $A$ on $f' (η)$ .

Figure 4.

The impacts of $D_{f}$ on $θ (η)$ .

Figure 5.

The impacts of $D_{f}$ on $θ (η)$ (CBC).

Figure 6.

The impacts of $D_{f}$ on $θ (η)$ (NH).

Figures 7 –9 represent the influences of dissimilar values of Brownian movement constraint $Nb$ on temparature $θ (η)$ profiles for thermally stratified, CBC and NH cases. The profiles of temparature $θ (η)$ are augemnted for higher $Nb$ = 0.5, 1.0, 1.5, 2.0 (thermally stratified), $Nb$ = 0.1, 0.2, 0.3, 0.5 (CBC) and $Nb$ = 0.5, 1.0, 1.5, 2.0 (NH) values. The Brownian diffusion coefficient is stronger due to higher $Nb$ values which resulted an improvement in $θ (η)$ profiles for thermally stratified, CBC and Newtonian heating cases. The nature of thermophresis constraint $Nt$ on temperature $θ (η)$ for thermally stratified ( $Nt$ = 0.5, 1.0, 1.5, 2.0), CBC ( $Nt$ = 0.2, 0.3, 0.4, 0.5), and NH ( $Nt$ = 0.5, 1.0, 1.5, 2.0) situations is expressed in the Figures 10 –12. The temperature $θ (η)$ profiles are risen against the incermenting $Nt$ values. The thermal condutance is improved due to inclusion of nanoparticles which augment the temperature $θ (η)$ profiles. Tempearture $θ (η)$ profiles at the wall are in case of CBC as comapartive to NH and thermal stratification case. The temperature profile displays reduction by rising $\Pr$ values in case of thermally stratified ( $\Pr$ = 1.5, 3.0, 4.5, 7.0), CBC ( $\Pr$ = 2.5, 3.0, 3.5, 4.0), and NH ( $\Pr$ = 2.5, 3.3, 4.5, 5.0) cases as illustrated in Figures 13 –15. The inclusion of such parameter in a decline temperature profile makes sense because the ratio of thermal diffusion rate to viscous diffusion rate illustrates the fundamental meaning of Prandlt number which is enhanced by an increase in parent fluid viscosity and nano-particle size. In this way, the flow system experiences the conduction phenomenon and leading to a decrease in thermal diffusivity.

Figure 7.

The impacts of $Nb$ on $θ (η)$ .

Figure 8.

The impacts of $Nb$ on $θ (η)$ (CBC).

Figure 9.

The impacts of $Nb$ on $θ (η)$ (NH).

Figure 10.

The impacts of $Nt$ on $θ (η)$ .

Figure 11.

The impacts of $Nt$ on $θ (η)$ (CBC).

Figure 12.

The impacts of $Nt$ on $θ (η)$ (NH).

Figure 13.

The impacts of $\Pr$ on $θ (η)$ .

Figure 14.

The impacts of $\Pr$ on $θ (η)$ (CBC).

Figure 15.

The impacts of $\Pr$ on $θ (η)$ (NH).

Figures 16 –18 revealed the temperature $θ (η)$ profiles against the multiple values of thermal stratification constraint $S t_{1}$ for thermally stratified ( $S t_{1}$ = 0.1, 0.2, 0.3, 0.4), CBC ( $S t_{1}$ = 0.1, 0.2, 0.3, 0.4), and NH ( $S t_{1}$ = 0.2, 0.3, 0.4, 0.5) cases. Temperature profiles are decaying against the boosted $S t_{1}$ values. The decrement in tempearture profiles is significant in NH situation as comparative to CBC and thermally startified condition. Figure 19 reflected the behavior of Lewis number $Le$ on concentration $ϕ (η)$ profiles. An enlaregment in $Le$ (0.1, 0.2, 0.3, 0.4) values resulted the lower $ϕ (η)$ profiles. The Lewis number strongly dependent on Brownian diffusion coefficient. This coefficient is smaller against the higher $Le$ values which corresponds to a decrement in $ϕ (η)$ profiles. The inreasing trend of Soret number $S_{r}$ (0.1, 0.3, 0.5, 0.7) values lead to an augmentation in $ϕ (η)$ profiles (see Figure 20). Figure 21 displayed an enhancement in the values of $S t_{2}$ (0.1, 0.2, 0.3, 0.4) resulted a decrement in concentration profile.

Figure 16.

The impacts of $S t_{1}$ on $θ (η)$ .

Figure 17.

The impacts of $S t_{1}$ on $θ (η)$ (CBC).

Figure 18.

The impacts of $S t_{1}$ on $θ (η)$ (NH).

Figure 19.

The impacts of $Le$ on $ϕ (η)$ .

Figure 20.

The impacts of $S_{r}$ on $ϕ (η)$ .

Figure 21.

The impacts of $S t_{2}$ on $ϕ (η)$ .

Conclusions

The present study is conducted to analyze the double stratification impacts on a second grade nanofluid flow produced by a stretchable sheet with the help of a numerical approach in MATLAB called bvp4c.The CBC and NH cases are discussed in detail. The numeric computations and comparison with the literature are presented and evaluated. This research work can also be extended for Riga surface, power law surface, curved surface, micropolar and hybrid nanofluid. Moreover, the results obtained from this present analysis can be summed up as:

(1) The enlargement of the second grade constraint caused an enhancement in the velocity profile.

(2) Temperature profile is reduced by rising the Prandlt number and thermal stratification parameter for NH and CBC cases, respectively.

(3) Temperature profile decreases as a result of an increase in Soret parameter. On contrary, greater the solutal stratification parameter and Lewis number resulted a decrement in concentration profile.

(4) The skin friction increased by enhancing the second grade constraint values.

(5) For general case, the reduced Nusselt number displays an increase in case of increasing values of second grade parameter, thermal stratification parameter, and Prandlt number.

(6) For CBC case, the reduced Nusselt number resulted an augmentation against higher Lewis number, Brownian movement and solutal stratification constraint.

(7) For NH case, the reduced Nusselt number displays an increase in case of increasing values of Soret number, thermal stratification and Newtonian heating constraints.

Footnotes

Appendix

Handling Editor: Dharmendra Tripathi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sabir Ali Shehzad

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