Soret & Dufour and Triple Stratification Effect on MHD Flow with Velocity Slip towards a Stretching Cylinder

Abstract

The phenomenon of convective flow with heat and mass transfer has been studied extensively due to its applications in various fields. The effects of nonlinear thermal radiation (NLTR), slip, thermal-diffusion (Soret) and diffusion-thermo (Dufour) on magenoto-hydrodynamic (MHD) flow towards a stretching cylinder in the presence of triple stratification (TSF) are investigated in this paper. The governing equations are transformed into an ODE by suitable transformations. The homotopy analysis method (HAM) is used to solve the ODE. The revamping of fluid flow, and heat transfer due to the presence of the Soret and Dufour effect, concentration slip and concentration stratification are analyzed. The temperature and local Sherwood number increases as the Dufour number rises, whereas the local Nusselt number decreases. While elevating the Soret number, the Sherwood number diminishes, whereas the concentration profile rises. The thermal boundary layer thickness enhances when thermal radiation increases. The rate of solute transport reduces while the concentration slip increases.

1. Introduction

The importance of nanofluids has been discussed before. The heat transfer in water-based nanofluids (TiO${}_2$–H${}_2$O, Al${}_2$O${}_3$–H${}_2$O and Cu–H${}_2$O) and thermal radiation’s influence on nanofluid flow over a stretchy cylinder were studied by Rahman and Aziz [1], Rasekh et al. [2] and Pandey and Kumar [3]. The effect of thermal radiation on stagnation-point nanofluid flow towards a stretchy sheet, 3D Jeffrey fluid flow, a nanofluid flow due to a contracting cylinder, finite element simulation of MHD convective nanofluid flow in porous media and MHD flow of a casson fluid induced by a semi-infinite stretchy surface with Cattaneo–Christov heat flux were investigated by Das et al. [4], Raju et al. [5], Abbas et al. [6], Uddin et al. [7] and Sandeep et al. [8]. More recent related research was performed by Veera Krishna and Chamkha [9], Veera Krishna [10], Veera Krishna and Chamkha [11] and Animasaun et al. [12].

Concentration stratification occurs due to concentration differences, such as the inclusion of different heterogeneous mixtures. The effects of thermal stratification on flow and heat transfer and mixed convection flow due to a stretching cylinder were discussed by Chamkha et al. [13] and Mukhopadhyay and Ishak [14]. Karthikeyan et al. [15] and Eswaramoorthi et al. [16] analyzed the Soret and Dufour effect on MHD mixed convection of a stagnation point flow over a vertical plate in a porous medium with chemical reaction and viscoelastic flow, over a stretchy sheet with convective boundaries.

Hayat et al. [17] analyzed MHD nanofluid flow with double stratification and slip conditions, and concluded that temperature profile diminishes when thermal and slip stratification parameters are increased. The main aim of this paper is to analyze how the NLTR and Soret and Dufour effect play out for the mathematical model of Hayat et al. [17]. More related research was done by Faraz et al. [18], Ali et al. [19] and Abdal et al. [20]. Thus, in this paper, investigation was done on MHD flow of nanofluid towards a stretchy cylinder with thermal, nanparticle volume fraction and concentration stratification in the presence of NLTR, Soret and Dufour and slip effects. The present investigation is useful to improve the mass and heat transfer processes in thermal systems.

2. Mathematical Formulation

We consider the two-dimensional and steady Newtonian nanofluid flow towards a stretching cylinder, where the stretching cylinder is considered along z-axis and r-axis is perpendicular to it. The velocity of the stretchy cylinder is assumed as $W_w\left(z\right)=\frac{W_{0}z}{l}$ where $W_0>0$ are constants and l is the characteristic length as shown in Figure 1. Fluid flow is considered along z-axis with slip effect and constant magnetic field. The ambient temperature, nanoparticle volume fraction and concentration are taken as $T{e}_{\infty }=T{e}_0+\frac{dz}{l}$, $C{n}_{\infty }=C{n}_0+\frac{ez}{l}$ and $S_{\infty }=S_0+\frac{nz}{l}$. The fluid temperature, nanoparticle volume fraction and concentration are assumed as $T{e}_w=T{e}_0+\frac{az}{l}$, $C{n}_w=C{n}_0+\frac{bz}{l}$ and $S_w=S_0+\frac{mz}{l}$. The following assumptions are taken:

• The flow is steady, laminar and $2D$-dimensional.
• The fluid is incompressible.
• The uniform external magnetic field is applied. The induced magnetic field is neglected.
• The cylinder is stretching with uniform velocity along z-direction.
• The stratification effect for temperature, nanoparticle volume fraction and concentration is considered.

The thermal energy, nanoparticle volume fraction and concentration stratification effects are also considered. The governing equations for the present flow analysis (refer to Hayat et al. [17]) can be written as

Contunity Equation

$$\begin{array}{c}\frac{\partial \left(ru\right)}{\partial r}+\frac{\partial \left(rw\right)}{\partial z}=0,\hfill \end{array}$$

(1)

Momentum Equation

$$\begin{array}{c}u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=\nu \left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}\right)-\frac{\sigma B_0^2}{\rho }w,\hfill \end{array}$$

(2)

Temperature Equation

$$\begin{array}{c}u\frac{\partial \left(Te\right)}{\partial r}+w\frac{\partial \left(Te\right)}{\partial z}=\alpha \left(\frac{{\partial }^2\left(Te\right)}{\partial r^2}+\frac{1}{r}\frac{\partial \left(Te\right)}{\partial r}\right)+\tau \left(D_B\frac{\partial \left(Te\right)}{\partial r}\frac{\partial \left(Cn\right)}{\partial r}+\frac{D_{Te}}{T{e}_{\infty }}{\left(\frac{\partial \left(Te\right)}{\partial r}\right)}^2\right)\hfill \\ -\frac{1}{\left(\rho c_p\right)}\frac{\partial q_r}{\partial r}+\frac{D_s{k}_T}{c_s{c}_p}\left(\frac{{\partial }^2\left(Cn\right)}{\partial r^2}+\frac{1}{r}\frac{\partial \left(Te\right)}{\partial r}\right),\hfill \end{array}$$

(3)

Nano-Particle Volume Fraction Equation

$$\begin{array}{c}u\frac{\partial \left(Cn\right)}{\partial r}+w\frac{\partial \left(Cn\right)}{\partial z}=D_B\left(\frac{{\partial }^2\left(Cn\right)}{\partial r^2}+\frac{1}{r}\frac{\partial \left(Cn\right)}{\partial r}\right)+\frac{D_{Te}}{T{e}_{\infty }}\left(\frac{{\partial }^2\left(Te\right)}{\partial r^2}+\frac{1}{r}\frac{\partial \left(Te\right)}{\partial r}\right),\hfill \end{array}$$

(4)

Concentration Equation

$$\begin{array}{c}u\frac{\partial S}{\partial r}+w\frac{\partial S}{\partial z}=D_s\left(\frac{{\partial }^{2}S}{\partial r^2}+\frac{1}{r}\frac{\partial S}{\partial r}\right)+\frac{D_s{k}_T}{T_m}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)-k_1\left(S-S_{\infty }\right).\hfill \end{array}$$

(5)

where the boundary conditions are

$$\begin{array}{c}w=W_w\left(z\right)+L\frac{\partial w}{\partial r},Te=T{e}_w\left(z\right)=k_1\frac{\partial \left(Te\right)}{\partial r},Cn=C{n}_w\left(z\right)=k_2\frac{\partial \left(Cn\right)}{\partial r},\hfill \\ S=S_w\left(z\right)=k_3\frac{\partial S}{\partial r}atr=R_1.\hfill \\ w\to 0,Te\to T{e}_{\infty }\left(z\right),Cn\to C{n}_{\infty }\left(z\right),S\to S_{\infty }\left(z\right)asr\to \infty .\hfill \end{array}$$

(6)

The transformations are

$$\begin{array}{c}\zeta =\sqrt{\frac{W_w\left(z\right)}{\nu z}}\left(\frac{r^2-R_1^2}{2R_1}\right),w=W_w\left(z\right)f^{\prime }\left(\zeta \right),u=-\sqrt{\frac{\nu W_w\left(z\right)}{z}}\frac{R_1}{r}f\left(\zeta \right),\hfill \\ {\varphi }_1\left(\zeta \right)=\frac{Te-T{e}_{\infty }}{T{e}_w-T{e}_0},{\varphi }_2\left(\zeta \right)=\frac{Cn-C{n}_{\infty }}{C{n}_w-C{n}_0},s\left(\zeta \right)=\frac{S-S_{\infty }}{S_w-S_0}.\hfill \end{array}$$

(7)

Equation (1) is satisfied identically, and Equations (2)–(6) are reduced to Equations (8)–(12) using Equation (7).

$$\left(1+2\gamma \zeta \right){\theta }_1^{‴}+{\theta }_1{\theta }_1^{″}-{{\theta }_1^{\prime }}^2+2\gamma {\theta }_1^{″}-M^2{\theta }_1^{\prime }=0,$$

(8)

$$\begin{array}{c}\left(1+2\gamma \zeta \right){\varphi }_1^{″}+2\gamma {\varphi }_1^{\prime }+\left(1+2\gamma \zeta \right)PrNb{\varphi }_1^{\prime }{\varphi }_2^{\prime }+\left(1+2\gamma \zeta \right)PrNt{\left({\varphi }_1^{\prime }\right)}^2\hfill \\ +Pr\left(f{\varphi }_1^{\prime }-f^{\prime }{\varphi }_1-G_1{f}^{\prime }\right)+\frac{4}{3}Rd\left(1+2\gamma \zeta \right)\left[{\left({\theta }_w\right)}^3\left(3{\theta }^2{\left({\varphi }_1^{\prime }\right)}^2+{\varphi }_1^3{\varphi }_1^{″}\right)\right.\hfill \\ \left.+3{\left({\theta }_w\right)}^2\left(2{\varphi }_1{\left({\varphi }_1^{\prime }\right)}^2+{\varphi }_1^2{\varphi }_1^{″}\right)+3{\theta }_w\left({\left({\varphi }_1^{\prime }\right)}^2+{\varphi }_1{\varphi }_1^{″}\right)+{\varphi }_1^{″}\right]\hfill \\ +\frac{4}{3}Rd\gamma \left[{\theta }_w^3{\varphi }_1^3{\varphi }_1^{\prime }+3{\theta }_w^2{\varphi }_1^2{\varphi }_1^{\prime }+3{\theta }_w{\varphi }_1{\varphi }_1^{\prime }+{\varphi }_1^{\prime }\right]+D_f\left[\left(1+2\gamma \zeta \right){\varphi }_2^{″}\right.\hfill \\ \left.+2\gamma {\varphi }_2^{\prime }\right]=0,\hfill \end{array}$$

(9)

$$\begin{array}{c}\left(1+2\gamma \zeta \right){\varphi }_2^{″}+2\gamma {\varphi }_2^{\prime }+S{c}_n\left({\theta }_1{\varphi }_2^{\prime }-{\theta }_1^{\prime }{\varphi }_2-G_2{\theta }_1^{\prime }\right)+\frac{Nt}{Nb}\left[\left(1+2\gamma \zeta \right){\varphi }_1^{″}\right.\hfill \\ \left.+2\gamma {\varphi }_1^{\prime }\right]=0,\hfill \end{array}$$

(10)

$$\begin{array}{c}\left(1+2\gamma \zeta \right)s^{″}+2\gamma s^{\prime }+Sc\left({\theta }_1{s}^{\prime }-{\theta }_1^{\prime }s-G_3{\theta }_1^{\prime }\right)+Sc{S}_r\left[\left(1+2\gamma \zeta \right){\varphi }_1^{″}\right.\hfill \\ \left.+2\gamma {\varphi }_1^{\prime }\right]-C_{r}Scs=0.\hfill \end{array}$$

(11)

$$\begin{array}{c}{\theta }_1\left(0\right)=0,{\theta }_1^{\prime }\left(0\right)=1+A{\theta }_1^{″}\left(0\right),{\varphi }_1\left(0\right)=1-G_1-B_1{\varphi }_1^{\prime }\left(0\right),\hfill \\ {\varphi }_2\left(0\right)=1-G_2-B_2{\varphi }_2^{\prime }\left(0\right),s\left(0\right)=1-G_3-B_3{s}^{\prime }\left(0\right),\hfill \\ {\theta }_1^{\prime }\left(\zeta \right)\to 0,{\varphi }_1\left(\zeta \right)\to 0,{\varphi }_2\left(\zeta \right)\to 0ands\left(\zeta \right)\to 0as\zeta \to \infty .\hfill \end{array}$$

(12)

where

$$\begin{array}{c}A=L\sqrt{\frac{W_0}{\nu l}},B_1=K_1\sqrt{\frac{W_0}{\nu l}},B_2=K_2\sqrt{\frac{W_0}{\nu l}},B_3=K_3\sqrt{\frac{W_0}{\nu l}},C_r=\frac{k_{1}l}{W_0},\hfill \\ \gamma =\sqrt{\frac{\nu l}{W_0{R}_1^2}},D_f=\frac{D_s{k}_T\left(C{n}_w-C{n}_0\right)}{\alpha c_s{c}_p\left(T{e}_w-T{e}_0\right)},G_1=\frac{d}{a},G_2=\frac{e}{b},G_3=\frac{n}{m},\hfill \\ M^2=\frac{\sigma l{B}_0^2}{\rho W_0},Nb=\frac{{\left(\rho c_p\right)}_p{D}_B\left(C{n}_w-C{n}_{\infty }\right)}{{\left(\rho c_p\right)}_f\nu },Nt=\frac{{\left(\rho c_p\right)}_p{D}_{Te}\left(T{e}_w-T{e}_{\infty }\right)}{{\left(\rho c_p\right)}_f\nu T{e}_{\infty }},\hfill \\ Rd=\frac{4{\sigma }^{\ast }T{e}_{\infty }^3}{k{k}^{\ast }},R{e}_z=\frac{W_0{z}^2}{\nu l},Sc=\frac{\nu }{D_s},S{c}_n=\frac{\nu }{D_B},S_r=\frac{D_s{k}_T\left(T{e}_w-T{e}_0\right)}{T{e}_m\nu \left(S_w-S_0\right)},\hfill \\ Pr=\frac{\mu c_p}{k},{\theta }_w=\frac{T{e}_w-T{e}_0}{T{e}_{\infty }}.\hfill \end{array}$$

(13)

The skin-friction coefficient, local Nusselt and Sherwood number are given by

$$\begin{array}{c}R{e}_z^{1/2}{C}_f={\theta }_1^{″}\left(0\right),\hfill \end{array}$$

(14)

$$\begin{array}{c}R{e}_z^{-1/2}Nu=-\left(1+\frac{4}{3}Rd{\theta }_w^3\right){\varphi }_1^{\prime }\left(0\right)\hfill \end{array}$$

(15)

$$\begin{array}{c}R{e}_z^{-1/2}Sh=-s^{\prime }\left(0\right).\hfill \end{array}$$

(16)

3. Convergence of the Solution

Equations (8)–(11) subject to Equation (12) are solved using HAM by choosing the initial boundary approximations, auxillary function and auxillary linear operators (refer to Noor et al. [21,22]) as

$$\begin{array}{c}{\theta }_{1,0}\left(\zeta \right)=\frac{1}{1+A}\left(1-e^{-\zeta }\right),{\varphi }_{1,0}\left(\zeta \right)=\left(\frac{1-G_1}{1+B_1}\right)e^{-\zeta },\hfill \\ {\varphi }_{1,0}\left(\zeta \right)=\left(\frac{1-G_2}{1+B_2}\right)e^{-\zeta },s_0\left(\zeta \right)=\left(\frac{1-G_3}{1+B_3}\right)e^{-\zeta }.\hfill \end{array}$$

(17)

$$\begin{array}{c}H\left(\zeta \right)=1.\hfill \end{array}$$

(18)

$$\begin{array}{c}{\mathit{L}}_{{\theta }_1}=\frac{d^3{\theta }_1}{d{\zeta }^3}-\frac{d{\theta }_1}{d\zeta },{\mathit{L}}_{{\varphi }_1}=\frac{d^2{\varphi }_1}{d{\zeta }^2}-{\varphi }_1,\hfill \\ {\mathit{L}}_{{\varphi }_2}=\frac{d^2{\varphi }_2}{d{\zeta }^2}-{\varphi }_2,{\mathit{L}}_s=\frac{d^{2}s}{d{\zeta }^2}-s,\hfill \end{array}$$

(19)

which satisfies the property

$$\begin{array}{c}{\mathit{L}}_{{\theta }_1}[A_1+A_2\ast Exp\left(-\zeta \right)+A_3\ast Exp\left(\zeta \right)]=0,\hfill \end{array}$$

(20)

$$\begin{array}{c}{\mathit{L}}_{{\varphi }_1}[A_4\ast Exp\left(-\zeta \right)+A_5\ast Exp\left(\zeta \right)]=0,\hfill \end{array}$$

(21)

$$\begin{array}{c}{\mathit{L}}_{{\varphi }_2}[A_6\ast Exp\left(-\zeta \right)+A_7\ast Exp\left(\zeta \right)]=0.\hfill \end{array}$$

(22)

$$\begin{array}{c}{\mathit{L}}_s[A_8\ast Exp\left(-\zeta \right)+A_9\ast Exp\left(\zeta \right)]=0.\hfill \end{array}$$

(23)

where $A_1$ to $A_9$ are arbitrary constants. Mathematica software is used to solve the above HAM equations. The equations obtained contain the parameters $h_{{\theta }_1}$, $h_{{\varphi }_1}$, $h_{{\varphi }_2}$ and $h_s$. The h-curve is plotted for $M=0.2$, $Rd=0.1$, $Nt=0.1$, ${\theta }_w=0.1$, $Nb=0.1$, $Pr=1.4$, $D_f=0.2$, $S{c}_n=1.9$, $Sc=0.4$, $S_r=0.2$, $\gamma =0.1$, $C_r=0.1$, $A=0.1$, $G_1=G_2=G_3=0.1$ and $B_1=B_2=B_3=0.1$. The admissible ranges of $h_{{\theta }_1}$, $h_{{\varphi }_1}$, $h_{{\varphi }_2}$ and $h_s$ are $-1.8\le h_{{\theta }_1}\le -0.1$, $-1.3\le h_{{\varphi }_1}\le -0.1$, $-1.3\le h_{{\varphi }_2}\le -0.1$ and $-1.5\le h_s\le -0.4$ (see Figure 2).

The convergence value is tabulated in

Table 1. Convergence of the series $-{\theta }_1^{″}\left(0\right)$, $-{\varphi }_1^{\prime }\left(0\right)$, $-{\varphi }_2^{\prime }\left(0\right)$ and $-s^{\prime }\left(0\right)$ for $\gamma =0.1$, $M=0.2$, $Rd=0.1$, ${\theta }_w=0.1$, $Pr=1.4$, $Nt=0.1$, $Nb=0.1$, $D_f=0.2$, $S{c}_n=1.9$, $Sc=0.4$, $C_r=0.1$, $A=0.1$, $S_r=0.2$, $G_1=G_2=G_3=0.1$ and $B_1=B_2=B_3=0.1$.

m-th Order Approximation	$-{\mathit{\theta }}_1^{\prime \prime }\left(0\right)$	$-{\mathit{\varphi }}_1^{\prime }\left(0\right)$	$-{\mathit{\varphi }}_2^{\prime }\left(0\right)$	$-{\mathit{s}}^{\prime }\left(0\right)$
1	$0.9213$	$0.8190$	$0.8491$	$0.7013$
4	$0.9212$	$0.8090$	$0.8546$	$0.5695$
8	$0.9190$	$0.8101$	$0.8598$	$0.5281$
12	$0.9185$	$0.8107$	$0.8617$	$0.5147$
16	$0.9184$	$0.8110$	$0.8622$	$0.5085$
18	$0.9184$	$0.8111$	$0.8623$	$0.5066$
19	$0.9184$	$0.8112$	$0.8624$	$0.5058$
20	$0.9184$	$0.8112$	$0.8624$	$0.5051$

for different orders of approximations.

The square residual errors are defined as follows.

$$\begin{array}{c}{\Delta }_m^{{\theta }_1}={\int }_0^1{\left[R_m^{{\theta }_1}\left(\zeta ,h_{{\theta }_1}\right)\right]}^{2}d\zeta ,\hfill \end{array}$$

(24)

$$\begin{array}{c}{\Delta }_m^{{\varphi }_1}={\int }_0^1{\left[R_m^{{\varphi }_1}\left(\zeta ,h_{{\varphi }_1}\right)\right]}^{2}d\zeta ,\hfill \end{array}$$

(25)

$$\begin{array}{c}{\Delta }_m^{{\varphi }_2}={\int }_0^1{\left[R_m^{{\varphi }_2}\left(\zeta ,h_{{\varphi }_2}\right)\right]}^{2}d\zeta ,\hfill \end{array}$$

(26)

$$\begin{array}{c}{\Delta }_m^s={\int }_0^1{\left[R_m^s\left(\zeta ,h_s\right)\right]}^{2}d\zeta .\hfill \end{array}$$

(27)

Equations (24)–(27) are plotted in Figure 3.

4. Computational Results and Discussion

Different combinations of apt parameters involved in this study are discussed. The ranges of the parameters are: $M\sim 0\mathrm{to}200$, $Rd\sim 0\mathrm{to}0.5$, $Nt\sim 0.1\mathrm{to}0.5$, ${\theta }_w\sim 0\mathrm{to}1.5$, $Nb\sim 0.1\mathrm{to}0.5$, $Pr\sim 1\mathrm{to}13.7$, $D_f\sim 0\mathrm{to}1$, $S_r\sim 0\mathrm{to}1$, $\gamma \sim 0\mathrm{to}0.5$, $C_r\sim 0\mathrm{to}1.5$, $A\sim 0\mathrm{to}2$, $G_1\sim 0\mathrm{to}4$, $G_2\sim 0\mathrm{to}2$, $G_3\sim 0\mathrm{to}1$ and $B_1=B_2=B_3\sim 0\mathrm{to}1$. In

Table 2. Comparison of ${\theta }_1^{\prime }\left(0\right)$ for various values of A when $\gamma =M=0$.

A	Andersson [23]	Mahmoud [24]	Mahmoud & Waheed [25]	Hayat et al. [17]	Present
0	$1.0$	$1.0$	$1.0$	$1.0000$	$1.0000$
$0.1$	$0.9218$	$0.91279$	$0.91279$	$0.9127$	$0.9127$
$0.2$	$0.8447$	$0.84473$	$0.84472$	$0.84473$	$0.84473$
$0.5$	$0.7044$	$0.70440$	$0.70440$	$0.70440$	$0.70440$
$1.0$	$0.5698$	$0.56984$	$0.56982$	$0.56984$	$0.56984$
$2.0$	$0.4320$	$0.43204$	$0.43199$	$0.43204$	$0.43204$
$5.0$	$0.2758$	$0.27579$	$0.27579$	$0.27580$	$0.27580$
$10.0$	$0.1876$	$0.18758$	$0.18759$	$0.18781$	$0.18781$
$20.0$	$0.1242$	$0.12423$	$0.12420$	$0.12456$	$0.12456$

and

Table 3. Comparison of $-{\theta }_1^{″}\left(0\right)$ for various values of A when $\gamma =M=0$.

A	Andersson [23]	Mahmoud [24]	Mahmoud & Waheed [25]	Hayat et al. [17]	Present
0	$1.0$	$1.0$	$1.0$	$1.0000$	$1.0000$
$0.1$	$0.8721$	$0.87208$	$0.87209$	$0.872082$	$0.872082$
$0.2$	$0.7764$	$0.77637$	$0.77639$	$0.77677$	$0.77677$
$0.5$	$0.5912$	$0.59119$	$0.59121$	$0.591195$	$0.591159$
$1.0$	$0.4302$	$0.43016$	$0.43018$	$0.430159$	$0.430159$
$2.0$	$0.2840$	$0.28398$	$0.28400$	$0.283978$	$0.283978$
$5.0$	$0.1448$	$0.14484$	$0.14481$	$0.144841$	$0.144841$
$10.0$	$0.0812$	$0.08124$	$0.08123$	$0.081242$	$0.081242$
$20.0$	$0.0438$	$0.04378$	$0.04381$	$0.043772$	$0.043772$

, the present results of ${\theta }_1^{\prime }\left(0\right)$ and ${\theta }_1^{″}\left(0\right)$ are compared with those of Andersson [23], Mahmoud [24], Mahmoud and Waheed [25] and Hayat et al. [17] for different positive values of A. The results show good agreement.

From

Table 4. Numerical values of skin friction coefficient ${\theta }_1^{″}\left(0\right)$ for different parameters.

$\mathit{\gamma }$	M	A	Hayat et al. [17]	Present
$0.0$	$0.2$	$0.1$	$0.9511$	$0.9511$
$0.5$			$1.1007$	$1.1007$
$1.0$			$1.2337$	$1.2337$
$0.1$	$0.0$		$0.9021$	$0.9021$
	$0.5$		$1.0889$	$1.0889$
	$1.0$		$1.2391$	$1.2391$
	$0.2$	$0.0$	$1.1352$	$1.1352$
		$0.5$	$0.6554$	$0.6554$
		$1.0$	$0.4724$	$0.4724$

, it is clear that the skin friction coefficient ${\theta }_1^{″}\left(0\right)$ increases while elevating $\gamma$ (>0) and M but it decreases while elevating A (>0).

The velocity profile is plotted for magnetic $\left(M\right)$, curvature $\left(\gamma \right)$ and velocity slip $\left(A\right)$ parameters. From Figure 4a, it is clear that the thickness of momentum boundary layer diminishes when elevating the magnetic parameter. The physical magnetic field, along with velocity slip, develop a retarding force which controls the fluid velocity. A similar result is shown in Figure 4c because the stretching sheet cannot transmit its drag force completely to the fluid. In Figure 4b, it is shown that the thickness of momentum boundary layer rises while the curvature parameter is boosted.

Thermal boundary layer thickness increases while thermal radiation increases, as more heat is generated in the fluid (see Figure 5a). The same result is shown in Figure 5d because temperature inside the boundary layer rises. When magnetic, curvature and velocity slip parameters are boosted, thermal boundary layer thickness grows (see Figure 5b,c and Figure 6a), but it diminishes with increases in thermal stratification, nanoparticle volume fraction stratification and thermal slip parameters (see Figure 6b–d). A physical rise in thermal stratification decreases the local bouyance level and a rise in thermal slip decreases the molecular movement of the fluid, which results in a decrease in temperature profile.

It can be observed from Figure 7 that when magnetic parameter, curvature parameter, Dufour number and velocity slip factor are increased, the concentration profile increases. From Figure 8, it is clear that concentration profile diminishes if thermal stratification, concentration stratification parameter and concentration slip are increased.

It is clear from Figure 9 that the skin friction coefficient rises with enhancement in velocity slip factor. In Figure 10 and Figure 11, the local Nusselt number decreases while increasing thermal radiation, Dufour number, thermal stratification and velocity slip factors. However, when the thermal slip factor is increased, it decreases for $0\le G_1<2.13$ and increases for $2.13<G_1\le 4$. It remains unchanged (0.24) for different values of thermal slip factor when $G_1=2.13$. While increasing the NLTR and Dufour number, the local Sherwood number increases (see Figure 12), whereas it decreases while increasing the parameters Soret number, concentration stratification, concentration slip and velocity slip factor (see Figure 13).

The regression equation for skin friction with A, $\gamma$ and M for 0.1, 0.2, 0.3, 0.4 and 0.5 are

$$\begin{array}{c}R{e}_x^{1/2}{C}_f=-0.9149+0.7983\ast A-0.2942\ast \gamma -0.2598\ast M.\hfill \end{array}$$

(28)

The regression equation for local Nusselt number with A, $B_1$, $B_2$, $D_f$, $G_1$, $G_2$, $\gamma$, M and $Rd$ for 0.1, 0.2, 0.3, 0.4 and 0.5 is

$$\begin{array}{c}R{e}_x^{-1/2}Nu=1.0397-0.2440\ast A-0.5730\ast B_1+0.1084\ast B_2-0.4534\ast D_f\hfill \\ -0.2748\ast G_1+0.0467\ast G_2+0.1838\ast \gamma -0.0666\ast M-0.5210\ast Rd.\hfill \end{array}$$

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The regression equation for a local Sherwood number with A, $B_1$, $B_2$, $B_3$, $C_r$, $D_f$, $G_1$, $G_2$, $G_3$, $\gamma$, M, $Rd$ and $S_r$ for 0.1, 0.2, 0.3, 0.4 and 0.5 is

$$\begin{array}{c}R{e}_x^{-1/2}Sh=0.5653-0.1131\ast A+0.0274\ast B_1-0.0082\ast B_2-0.2566\ast B_3\hfill \\ +0.3382\ast C_r+0.0288\ast D_f+0.0032\ast G_1-0.0036\ast G_2\hfill \\ -0.3505\ast G_3+0.3146\ast \gamma -0.0407\ast M+0.0325\ast Rd-0.1982\ast S_r.\hfill \end{array}$$

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5. Conclusions

The study of the impacts of slip, thermal radiation and Soret and Dufour on MHD nanofluid flow with TSF towards a stretching cylinder is offered. The main findings of this analysis are listed below as follows:

• While increasing the concentration slip, the solutal concentration boundary layer thickness shrinks, which results in a reduction in the rate of mass transfer.
• When the NLTR parameter value is increased, the local heat transfer rate reduces, whereas the local mass transfer rate increases.
• The thermal (concentration) boundary layer is thickened by increasing the Dufour (Soret) number.
• While raising thermal radiation, the thermal boundary layer thickness enhances.
• When the thermal energy, nanoparticle volume fraction and solutal stratification are increased, the thickness of thermal and solutal concentration boundary layers diminishes.
• The thermal and concentration boundary layer thickness and skin friction raise when the velocity slip is increased, whereas the momentum boundary layer thickness, heat and mass transfer rate diminish.
• The present results are very useful to the thermal science community to improve the cooling processes in heat transfer systems.