Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect

Abstract

The focal interest in this article is to investigate the Stefan blowing and Dufour and Soret effects on hybrid nanofluid (HNF) flow towards a stretching cylinder with thermal radiation. The governing equations are converted into ODE by using suitable transformations. The boundary value problem solver (bvp4c), which is a package in the MATLAB, is used to solve the resulting ODE equations. Results show that rise in the Stefan blowing enhances velocity, temperature, and concentration profiles. Heat transfer rate increases by up to 10% in the presence of 4% nanoparticle/HNF but mass transfer rate diminishes. Additionally, skin friction coefficient, Nusselt number and Sherwood number are examined for many parameters entangled in this article. Additionally, results are deliberatively discussed in detail.

1. Introduction

Recently, many investigators have been drawn in the direction of nano technology because of its significant applications in various industries. Base fluids differ from nanofluids, which have poor heat conductivity in terms of their thermo-physical properties. Choi [1] first introduced nanofluids in 1995 by incorporating nano-sized solid particles into water and claimed that, compared to base fluid, nanofluid has higher thermal conductivity. Such fluids have implications for appliances, which includes refrigerators, processors, cooling systems, hydraulic systems, solar energy machines, biomedical equipment, and microelectronics. Previously, it seems that Crane [2] examined the flow across a linearly stretching surface.

By choosing the proper nanoparticle combination, recent investigators added two different kinds of nanoparticles into the base fluid known as HNF. Specially, nanofluid is well known for having a higher heat transfer rate than regular fluid. Hayat et al. [3] analyzed heat transfer by considering the HNF obtained by the combination of CuO-Ag. Stagnation flow near a stretchy cylinder, along with partial slip condition, was analyzed by Wang [4]. By taking copper and alumina nanoparticles, Maskeen et al. [5] looked into the flow over a stretchy cylinder and the enhancement of heat transfer in HNF. Rehman et al. [6] investigated flow over a stretching sheet with Powell–Eyring fluid model along with joule heating. Salmi et al. [7] examined two-phase chemical reactions HNF flow over a stretchy cylinder. Waini et al. [8] discussed stagnation point HNF flow towards shrinking/stretching cylinder and found that heat transfer rate improved when nanoparticles were present. Waini et al. [9] investigated HNF flow over a shrinking cylinder with prescribed heat flux. Related work is found in Waini et al. [10]. Khashi’ie et al. [11] investigated unsteady squeezing HNF flow over a horizontal channel. Ali et al. [12] analyzed the effect of nonlinear thermal radiation and non-uniform heat flux on hybrid magneto-hydrodynamic (MHD) nanofluid across a stretching cylinder. Rangi et al. [13] examined the impact of boundary layer flow and variable thermal conductivity towards a stretching cylinder. Natural convection flow over a cylindrical annulus with the effect of either axial or radial magnetic field was examined statistically by Sankar et al. [14]. Siddiqui et al. [15] investigated 3D nanofluid flow over a stretching cylinder with entropy generation. The effect of chemical reactions, thermal radiation, and Carreau fluid flow towards a stretching cylinder was discussed by Lim et al. [16].

Several researchers have studied heat transfer phenomena caused by the concentration and temperature gradients. The mechanism of heat transfer that occurs due to the concentration gradient is called the (diffusion-thermo) Dufour effect, whereas the mechanism of heat transfer that happens due to the temperature gradient is called (thermal-diffusion) Soret effect. These effects are encountered in many practical applications, such as in the areas of geosciences, waste disposal, and chemical engineering, etc. Hayat et al. [17] examined Dufour and Soret effects on the MHD flow of Casson nanofluid and found that temperature field upsurges as the Dufour number rises. Jagan et al. [18] explored at the MHD flow of Jeffrey nanofluid with Dufour and Soret effects in the direction of a stretching cylinder and the results showed that width of the solutal boundary layer increases as the Soret number rises, which lowers the mass transfer rate. Most of the related research was performed by Shaheen et al. [19].

There is no doubt that the thermal radiation effect has been involved in various engineering processes, including die forging, gas turbines, thermal engineering storage and nuclear turbines, etc. Hayat et al. [20] examined Jefferey fluid flow over a stretching cylinder with thermal radiation. The non-linear heat radiation on a 3D unsteady MHD nanofluid flow towards a stretchable surface was examined by Jagan et al. [21]. Gholinia et al. [22] examined the impact of thermal radiation in HNF flow over a porous stretched cylinder. Sreedevi et al. [23] analyzed heat and mass transfer through thermal radiation unsteady HNF flow over a stretching sheet. Waqas et al. [24] investigated thermal transport MHD flow of HNF over a vertical stretching cylinder and found that thermal transport increases as magnetic number rises.

On an impermeable surface, a blowing effect arises. The species (concentration) field and velocity field are related by the Stefan blowing effect, which states that the flow field is directly proportional to the concentration of species. Additionally, some of the applications are found in glass blowing, evaporation in paper drying process, etc. Fang et al. [25] investigated heat and species transfer flow over a stretchy sheet with the effect of Stefan blowing and found that rise in the velocity and concentration profiles as Stefan blowing rises. Rana et al. [26] discovered that reducing the Stefan blowing lowers skin friction while considering non-Fourier and non-Fick’s law in their finite element study of bio-convective HNF towards a stretching cylinder. Gowda et al. [27] examined magnetized movement of the Sutterby nanofluid under Stefan blowing conditions and the Cattaneo–Christov concept of heat diffusion.

The current study scrutinized HNF flow with the effect of Stefan blowing, Soret–Dufour, and thermal radiation over a stretchable cylinder, which have yet to be studied. The study of heat and mass transfer in the presence of Stefan blowing and Soret–Dufour effect together is performed in the current work, which shows its novelty. Moreover, the physical quantities of interest are presented for different parameters in the form of tables, 2D graphs, and bar graphs.

2. Mathematical Formulation

Consider an HNF (Al₂O₃–Cu/H₂O) flow over a stretching cylinder with radius ‘a’, as shown in Figure 1. Here, stretching cylinder taken along z-axis and r-axis is perpendicular to it. The free stream velocity and surface velocity are w_e = 2cz and w_w = 2bz, where $b>0$ and $c>0$ are constants. Stefan blowing, Soret–Dufour and thermal radiation effects are considered.

The governing equations (referring to Waini et al. [8]) are described as:

Continuity Equation

$$\frac{\partial }{\partial z}(rw)+\frac{\partial }{\partial r}(ru)=0$$

(1)

Momentum Equation

$$w\frac{\partial w}{\partial z}+u\frac{\partial w}{\partial r}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}\right)+w_e\frac{d{w}_e}{dz}$$

(2)

Temperature Equation

$$\left.\begin{array}{rr}& w\frac{\partial T}{\partial z}+u\frac{\partial T}{\partial r}=\frac{k_{hnf}}{{(\rho C_p)}_{hnf}}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)-\frac{1}{{(\rho C_p)}_{hnf}}\left(\frac{\partial \left(q_r\right)}{\partial r}\right)\\ & +\frac{D{k}_T}{C_s{C}_p}\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)\end{array}\right\}$$

(3)

Concentration Equation

$$w\frac{\partial C}{\partial z}+u\frac{\partial C}{\partial r}=D\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)+\frac{D{k}_T}{C_s{C}_p}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)$$

(4)

The associated boundary conditions are:

$$\left.\begin{array}{c}u=\frac{-D}{\left(1-C_w\right)}\left(\frac{\partial C}{\partial r}\right),w=w_w,T=T_w,C=C_w \mathrm{at} r=a.\\ w\to w_e,T\to T_{\infty },C\to C_{\infty }\mathrm{as}r\to \infty \end{array}\right\}$$

(5)

where the z- and r- axis’ respective velocity components are w and u. T denotes the temperature of HNF. Additionally, the physical features of the HNF are given in

Table 1. Thermophysical attributes of base fluid (H₂O) and nanoparticles (Al₂O₃ and Cu) (referring to Waini et al. [8]).

Properties	Al2O3	Cu	H2O
ρ (kg m−3)	3970	8933	997.1
$k$ (W m−1 K−1)	40	400	0.613
$C_P$ (J kg−1 K−1)	765	385	4179

and the physical attributes of nanoparticles and base fluid are given in

Table 2. Thermophysical attributes of nanofluid and HNF (referring to Waini et al. [8]).

Properties	Nanofluid	HNF
Density	${\rho }_{nf}=\left(1-{\phi }_1\right){\rho }_f+{\phi }_1{\rho }_{n1}$	${\rho }_{hnf}=\left(1-{\phi }_2\right)\left[\left(1-{\phi }_1\right){\rho }_f+{\phi }_1{\rho }_{n1}\right]+{\phi }_2{\rho }_{n2}$
Heat Capacity	${\left(\rho C_p\right)}_{nf}=\left(1-{\phi }_1\right){\left(\rho C_p\right)}_f+{\phi }_1{\left(\rho C_p\right)}_{n1}$	${\left(\rho C_p\right)}_{hnf}=\left(1-{\phi }_2\right)\left[\left(1-{\phi }_1\right){\left(\rho C_p\right)}_f+{\phi }_1{\left(\rho C_p\right)}_{n1}\right]+{\phi }_2{\left(\rho C_p\right)}_{n2}$
Dynamic Viscosity	${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-{\phi }_1\right)}^{2.5}}$	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\phi }_1\right)}^{2.5}{\left(1-{\phi }_2\right)}^{2.5}}$
Thermal Conductivity	$\frac{k_{nf}}{k_f}=\frac{k_{n1}+2k_f-2{\phi }_1\left(k_f-k_{n1}\right)}{k_{n1}+2k_f+{\phi }_1\left(k_f-k_{n1}\right)}$	$\frac{k_{hnf}}{k_{nf}}=\frac{k_{n2}+2k_{nf}-2{\phi }_2\left(k_{nf}-k_{n2}\right)}{k_{nf}+2k_{nf}+{\phi }_2\left(k_{nf}-k_{n2}\right)}$ $\mathrm{where} \frac{k_{nf}}{k_f}=\frac{k_{n1}+2k_f-2{\phi }_1\left(k_f-k_{n1}\right)}{k_{n1}+2k_f+{\phi }_1\left(k_f-k_{n1}\right)}$

. Here, ${\phi }_1$ and ${\phi }_2$ denotes volume fraction of alumina (Al₂O₃) and copper (Cu). The hybrid nanoparticle volume fraction (Al₂O₃–Cu) $\phi$ (referring to Waini et al. [9]) can be written as:

$$\phi ={\phi }_1+{\phi }_2$$

(6)

The suitable transformations are (referring to Waini et al. [8]):

$$\begin{array}{c}u=-\frac{caf\left(\eta \right)}{\sqrt{\eta }},w=2cz{f}^{\prime }\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }} \mathrm{and} \\ \eta ={\left(\frac{r}{a}\right)}^2.\end{array}$$

(7)

Equation (1) is identically satisfied by Equation (7). By Equation (7), Equations (2)–(5) are reduced as follows:

$$\raisebox{1ex}{${\mu }_r$}\!\left/ \!\raisebox{-1ex}{${\rho }_r$}\right.\left[\eta f^{‴}+f^{″}\right]+\mathrm{Re}\left[{ff}^{″}-{f^{\prime }}^2+1\right]=0,$$

(8)

$$\begin{array}{r}\left.\begin{array}{r}\raisebox{1ex}{$k_r$}\!\left/ \!\raisebox{-1ex}{${\left(\rho C_p\right)}_r$}\right.\left[\eta {\theta }^{″}+{\theta }^{\prime }\right]+\raisebox{1ex}{$2Rd$}\!\left/ \!\raisebox{-1ex}{$3{\left(\rho C_p\right)}_r$}\right.\left[2\eta {\theta }^{″}+{\theta }^{\prime }\right]+Du\left[\eta {\varphi }^{″}+{\varphi }^{\prime }\right]\\ +\mathrm{Pr}\mathrm{Re}f{\theta }^{\prime }=0\end{array}\right\},\\ \end{array}$$

(9)

$$\left[\eta {\varphi }^{″}+{\varphi }^{\prime }\right]+ScSr\left[\eta {\theta }^{″}+{\theta }^{\prime }\right]+\mathrm{Re}Scf{\varphi }^{\prime }=0,$$

(10)

subjected to:

$$\left.\begin{array}{c}f\left(1\right)=\frac{Sb}{\mathrm{Re}Sc}{\varphi }^{\prime }\left(1\right),f^{\prime }\left(1\right)=\epsilon ,\theta \left(1\right)=1,\varphi \left(1\right)=1\\ f^{\prime }\left(\infty \right)=1,\theta \left(\infty \right)=0,\varphi \left(\infty \right)=0\end{array}\right\},$$

(11)

where $\mathrm{Re}=\frac{c{a}^2}{2{\nu }_f}$ represents Reynolds number, $Sb=\frac{\left(C_w-C_{\infty }\right)}{\left(1-C_{\infty }\right)}$ represents Stefan blowing parameter, $Rd=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k_f{k}^{*}}$ thermal radiation parameter, $Du=\frac{D{k}_T\left(C_w-C_{\infty }\right)}{{\alpha }_f{C}_s{C}_p\left(T_w-T_{\infty }\right)}$ Dufour number, $Sc=\frac{v_f}{D}$ Schmidt number, $Sr=\frac{D{k}_T\left(T_w-T_{\infty }\right)}{{\nu }_f{C}_s{C}_p\left(C_w-C_{\infty }\right)}$ Soret number, $\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f}$ represents Prandtl number, ${\mu }_r=\raisebox{1ex}{${\mu }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\mu }_f$}\right.$, ${\rho }_r=\raisebox{1ex}{${\rho }_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.$, $k_r=\raisebox{1ex}{$k_{hnf}$}\!\left/ \!\raisebox{-1ex}{$k_f$}\right.$ and ${\left(\rho C_p\right)}_r=\raisebox{1ex}{${\left(\rho C_p\right)}_{hnf}$}\!\left/ \!\raisebox{-1ex}{${\left(\rho C_p\right)}_f$}\right.$. Here, the stretching parameter is denoted by $\epsilon =\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$c$}\right.$.

Equation (12) defines the skin friction coefficient (C_f), Nusselt number (Nu), and Sherwood number (Sh) (referring to Waini et al. [8] and Waqas et al. [24])

$$C_f=\frac{2{\tau }_w}{{\rho }_f{w}_e^2},Nu=\frac{a{q}_w}{k_f\left(T_w-T_{\infty }\right)} \mathrm{and} Sh=\frac{a{q}_m}{D\left(C_w-C_{\infty }\right)}$$

(12)

where shear stress, heat flux, and mass flux are defined as (referring to Waqas et al. [24]):

$${\tau }_w={\mu }_{hnf}{\left(\frac{\partial w}{\partial r}\right)}_{r=a},q_w=-k_{hnf}{\left(\frac{\partial T}{\partial r}\right)}_{r=a}+q_r \mathrm{and} q_m=-D{\left(\frac{\partial C}{\partial r}\right)}_{r=a}$$

(13)

Using Equations (7) and (13) in Equation (12), following Equation (14) is obtained.

$$\left(\frac{\mathrm{Re}z}{a}\right)C_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(1\right),Nu=-2\left(\frac{k_{hnf}}{k_f}+\frac{4Rd}{3}\right){\theta }^{\prime }\left(1\right) \mathrm{and} Sh=-2{\varphi }^{\prime }\left(1\right).$$

(14)

3. Numerical Method

The boundary value problem solver, MATLAB (bvp4c) software, is used to solve Equations (8)–(11) numerically, as described by Waini et al. [8].

Equation (8) is taken as:

$$\left.\begin{array}{c}f=f\left(1\right)\\ f^{\prime }=f^{\prime }(1)=f(2)\end{array}\right\},$$

(15a)

$$f^{″}=f^{\prime }(2)=f(3),$$

(15b)

$$f^{‴}=f^{\prime }(3)=-\frac{1}{\eta }\left\{\frac{{\rho }_r}{{\mu }_r}\mathrm{Re}\left(f(1)f(3)-f{(2)}^2+1\right)+f(3)\right\},$$

(15c)

Equation (9) becomes:

$$\left.\begin{array}{c}\theta =f\left(4\right)\\ {\theta }^{\prime }=f^{\prime }\left(4\right)=f\left(5\right)\end{array}\right\},$$

(16a)

$${\theta }^{″}=f^{\prime }\left(5\right)=-\frac{1}{\eta }\left\{\begin{array}{l}1/\left(k_r+\raisebox{1ex}{$4Rd$}\!\left/ \!\raisebox{-1ex}{$3$}\right./{\left(\rho C_p\right)}_r-DuScSr\right)\\ f\left(5\right)\left(k_r+\raisebox{1ex}{$2Rd$}\!\left/ \!\raisebox{-1ex}{$3$}\right./{\left(\rho C_p\right)}_r-ScDuSr+\mathrm{Pr}\mathrm{Re}f\left(1\right)\right)\\ +f\left(7\right)Du\left(\mathrm{Re}Scf\left(1\right)+1\right)\end{array}\right\},$$

(16b)

Equation (10) becomes:

$$\left.\begin{array}{c}\varphi =f\left(6\right)\\ {\varphi }^{\prime }=f^{\prime }\left(6\right)=f\left(7\right)\end{array}\right\},$$

(17a)

$${\varphi }^{″}=f^{\prime }(7)=-\frac{1}{\eta }\left\{\begin{array}{l}f\left(7\right)-ScSr\left(1/\left(k_r+\raisebox{1ex}{$4Rd$}\!\left/ \!\raisebox{-1ex}{$3$}\right./{\left(\rho C_p\right)}_r-DuScSr\right)\right)\\ \left[\begin{array}{l}f\left(5\right)\left(k_r+2Rd/3/{\left(\rho C_p\right)}_r-ScDuSr+\mathrm{Pr}\mathrm{Re}f\left(1\right)\right)\\ +f\left(7\right)Du\left(\mathrm{Re}Scf\left(1\right)+1\right)\end{array}\right]\\ +ScSrf(5)+\mathrm{Re}Scf(7)f(1)\end{array}\right\},$$

(17b)

with boundary conditions:

$$\left.\begin{array}{c}fa\left(1\right)=\frac{Sb}{\mathrm{Re}Sc}fa(7),fa(2)=\epsilon ,fa(4)=1,fa(6)=1\\ fb(2)=1,fb(4)=0,fb(6)=0\end{array}\right\}.$$

(18)

The necessary solutions are then obtained by solving Equations (15)–(17) using bvp4c MATLAB package.

4. Results and Discussion

In this study, various combinations of important parameters are discussed. The nanoparticle volume fraction of alumina Al₂O₃ (${\phi }_1$) and copper Cu (${\phi }_2$) changes from 0 to 0.02 (2%). In

Table 3. Comparison of $f^{″}\left(1\right)$ and $-2{\theta }^{\prime }\left(1\right)$ when Pr = 6.2, $\epsilon =0$, and Sb = Sr = Du= Rd = 0, ${\phi }_1={\phi }_2=0$.

Re	${\mathit{f}}^{″}\left(1\right)$			$-2{\mathit{\theta }}^{\prime }\left(1\right)$
	Wang [4]	Waini et al. [8]	Present	Present
0.2	0.78604	0.786042	0.786040	1.508638
1	1.48418	1.484183	1.484186	2.793424
10	4.16292	4.162920	4.162921	7.701474

, the present results of $f^{″}\left(1\right)$ and $-2{\theta }^{\prime }\left(1\right)$ are in comparison with those of Wang [4] and Waini [8] for different values of Re, and we found that the results display good agreement.

Comparison of C_f and Nu when ${\phi }_1$ = 0.02, Sc = Sb = Sr = Du = Rd = 0 and Pr = 6.2 for different values of ε, ${\phi }_2$ and Re are given in

Table 4. Comparison values of $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)C_f$ and Nu when Pr = 6.2, and Sb = Sr = Du = Rd = 0.

			Waini et al. [8]		Present Result
${\phi }_2$	Re	$\epsilon$	$\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)C_f$	Nu	$\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)C_f$	Nu
0	0.2	0	0.873892	1.632938	0.873890	1.632940
0.01			-	-	0.946854	1.712795
0.02	0.5	0.2	1.021036	1.792922	1.021036	1.792928
	1		1.457949	2.509315	1.457940	2.509317
	2	0.5	1.092842	4.177081	1.092840	4.177085

. In

Table 5. Numerical values of $\left(\raisebox{1ex}{$\mathrm{Re}z$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)C_f$, Nu and Sh when Pr = 6.2, Re = 1 and ${\phi }_1$ = 0.02.

Sb	Sr	Du	Rd	$\mathit{\epsilon }$	${\mathit{\phi }}_2$	$\left(\raisebox{1ex}{$\mathrm{Re}\mathit{z}$}\!\left/ \!\raisebox{-1ex}{$\mathit{a}$}\right.\right){\mathit{C}}_{\mathit{f}}$	Nu	Sh
0.0	0.1	0.1	0.1	0.3	0.02	1.312229	5.235561	2.147327
0.1	-	-	-	-	-	1.356625	3.852460	2.181614
1.0	-	-	-	-	-	0.899479	0.015974	1.434119
2.0	-	-	-	-	-	0.699197	0.000416	1.031123
-	0.2	-	-	-	-	0.687271	0.000026	1.061832
-	0.3	-	-	-	-	0.676204	−0.000335	1.090929
-	0.4	-	-	-	-	0.637198	−0.000383	1.123820
-	-	0.2	-	-	-	0.658677	−0.001201	1.120188
-	-	0.3	-	-	-	0.658589	−0.001785	1.20365
-	-	0.4	-	-	-	0.658499	−0.002355	1.120542
-	-	-	0.2	-	-	2.083692	21.114771	0.253986
-	-	-	0.4	-	-	1.810421	20.757020	0.505700
-	-	-	0.6	-	-	1.690819	22.816878	0.465192
-	-	-	-	0.1	-	2.124273	22.499878	0.458991
-	-	-	-	0.2	-	0.711615	−0.003096	1.103116
-	-	-	-	0.3	-	1.690819	22.816878	0.465192
-	-	-	-	-	0.01	0.657825	−0.004825	1.122316
-	-	-	-	-	0.015	0.657825	−0.004825	1.122316
-	-	-	-	-	0.02	0.657825	−0.004825	1.122316

, the numerical values of skin friction, Nusselt number, and Sherwood number are presented for different values of Sb, Sr, Du, Rd, ε, and ${\phi }_2$.

Figure 2 depicts the velocity, temperature, and concentration profiles against the Stefan blowing. The boundary layer is growing larger as the blowing parameter rises up to 20%. Physically, the injection of tiny particles (nanoparticles) through the boundary energizes species diffusion as a result the velocity, temperature, and concentration profiles rises.

Increasing the values of ${\phi }_2$ up to 2% when ${\phi }_1$ = 0.02 (2%), Sb = 0.1, ε = 1.5, the velocity and temperature profiles decrease slowly (See Figure 3). Physically, an upsurge in the volume fraction can cause the fluid motion to experience resistance, which reduces the fluid motion.

Figure 4a,b show the profile of velocity and temperature decline as the Soret number rises. In fact, an increase in Soret values reduces the viscosity, which provides less resistance and consequently temperature reduces. From Figure 4c, it is shown that concentration profile enhances when elevating the Soret number. This figure gives the impression that as the Soret number rises, the fluid concentration profile rises as a result of temperature gradients influencing species diffusion. From Figure 5a, it is shown that increasing the Dufour number causes decline in the temperature field. As a result, the fluid receives less heat and its viscosity increases. In Figure 5b, the concentration profile slightly increases due to low friction, which, in turn, enhances the concentration. The thickness of thermal boundary layer decreases as thermal radiation increase. This is because large values of radiation parameter correspond to an increase in dominance of conduction over radiation, thereby decreasing the thickness of thermal boundary layer and increasing the heat loss at the ambient temperature (see Figure 6a). Meanwhile, a similar trend is observed in the concentration boundary layer thickness, with higher values of radiation parameters (see Figure 6b).

In Figure 7a, while increasing the value of ${\phi }_1$ and ${\phi }_2$ (up to 2%), the skin friction coefficient is found to be decreasing. At ε = 1, the C_f is found to be zero because the surface velocity is equal to free stream velocity. For ε < 1, C_f is positive because the surface velocity is greater than the free stream velocity and vice versa is found in case of ε > 1. From Figure 7b, the Nusselt number is increasing while the value of ${\phi }_1$ and ${\phi }_2$ is increasing (up to 2%). Figure 7c, the Sherwood number increases slightly alongside the increase in the nanoparticle volume fraction of ${\phi }_1$ and ${\phi }_2$ (up to 2%).

Figure 8a,b display the different values of Sb when Sc = 0.6, Sr = Rd = Du = 0.2, Pr = 6.2, and ${\phi }_1$ = ${\phi }_2$ = 0.02. The Nusselt number decreases as the blowing parameter increases, which results in a decrease in the heat transfer rate. Additionally, this plot shows that the effect of Sb is less dominant, in comparison to thermal radiation. The Sherwood number increases with increasing of Sb. Physically, an increase in mass blowing at surface results in an increase in the mass flow rate.

Figure 9a displays the different values of Rd when Sc = 0.6, Sr = Du = 0.2, Pr = 6.2, and ${\phi }_1$ = ${\phi }_2$ = 0.02. Nu increases with rising thermal radiation parameter because there will be a rise in temperature within the boundary layer. Additionally, Figure 9b shows that the heat transfer rate increases with an increase in Dufour number. Physically, Du relates to the effect of concentration gradient to the thermal energy flux in the flow.

Increasing the Soret number decreases the mass transfer rate. Further, the Soret effect is the cause of the diffusion of species from higher to lower concentration due to temperature gradient and, as a result, mass transfer rate diminishes, as shown in Figure 10.

5. Conclusions

In this study, the impact of Stefan blowing, Soret, Dufour and thermal radiation on HNF flow towards a stretching cylinder has been accomplished.

This study has the following potential limitations:

• The Schmidt number is fixed in our model as water is taken as the base fluid.
• The Stefan blowing effect only arises at the impermeable surface and perpendicular to the flow direction.
• The value of the Prandtl number is dependent on the base fluid, so it ranges from 1.7 to 13.7.
• The main fallout of the current study is listed below as follows:
• As Stefan blowing intensifies, the thickness of the velocity, thermal, and concentration boundary layers grows. As a result, the hate transfer rate declines while the mass transfer rate rises.
• The convective heat transfer and mass transfer rate is improved by up to 20% with the inclusion of HNF 2%.
• Concentration (temperature) boundary layer enhanced (declines) by evaluating the Dufour and Soret numbers.
• The inclusion of thermal radiation improved the heat transfer rate as the stretching parameter increases.
• Higher values of Soret number reduce the mass transfer rate, while Stefan blowing parameter has a contrary impact on it.