Go-MoS₂/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation

Abstract

The impacts of Stefan blowing along with slip and Joule heating on hybrid nanofluid (HNF) flow past a shrinking cylinder are investigated in the presence of thermal radiation. Using the suitable transformations, the governing equations are converted into ODEs, and the MATLAB tool bvp4c is used to solve the resulting equations. As Stefan blowing increases, temperature and concentration profiles are accelerated but the velocity profile diminishes and also the heat transfer rate improves up to 25% as thermal radiation upsurges. The mass transfer rate diminishes as increasing Stefan blowing. The Sherwood number, the Nusselt number, and the skin friction coefficient are numerically tabulated and graphs are also plotted. The outcomes are conscientiously and thoroughly discussed.

1. Introduction

Most scientists, engineers, and researchers are very much interested in the boundary layer flow, heat, and mass transfer towards a stretching/shrinking cylinder because of the numerous applications which include the extraction of metals, annealing, extrusion process, pipe industry, copper wire thinning, etc. Nanofluid comes into existence when we add a little amount of nano-sized particles into base fluids. The term nanofluid was first instituted by Choi and Eastman [1], to enhance the heat transfer rate. Furthermore, studies have shown that a significant increment in the heat transfer rate of nanofluid is attained when two different nanoparticles (HNF) are used. The earlier experimental works utilized hybrid nanoparticles that were considered by Turcu et al. [2] and Jana et al. [3]. They stated that, although Al₂O₃ has low thermal conductivity, there is a good chemical inertness in alumina that could maintain the stability of HNF. Additionally, Takabi et al. [4] researched HNF flow containing Al₂O₃-Cu nanoparticles in a sinusoidal corrugated enclosure.

Early on, Crane [5] emphasized flow towards a stretching plat. Waini et al. [6] inspected HNF flow past a stretching (shrinking) cylinder and concluded that the inclusion of nanoparticles heat transfer rate improved. Wang [7] researched stagnation flow past a shrinking sheet. Waini et al. [8] researched HNF flow past a shrinking cylinder with surface heat flux. Similar research was carried out by Awaludin et al. [9]. Work on viscid flow due to shrinking cylinders with non-uniform radius was conducted by Ali et al. [10]. Jagan et al. [11] investigated thermal radiative Jeffrey nano liquid flow past a stretching cylinder. Natural convection in a linearly heated vertical porous annulus was examined by Sankar et al. [12]. Heat transfer and HNF flow towards stretching/shrinking horizontal cylinder work carried out by Rashid et al. [13]. Girish et al. [14] Developed buoyant convection in vertical porous annuli with unheated entry and exit.

After claiming the improved thermal upshot of nanoparticles, this contribution reflects the thermal enhancement of graphene oxide (GO) and molybdenum disulphide (MoS2) nanoparticles over a vertically moving plate with help of a fractional approach. Chu et al. [15] examined MHD mixed convection in HNF flow over a cylinder with shape factor and scrutinized that the blade-shaped nanoparticles have a maximum temperature and brick-shaped nanoparticles have a low temperature. Khan et al. [16] researched stagnation point flow impinging on a radially permeable moving rotating disk with Go-MoS₂ nanoparticles.

According to theory, Joule heating is the development of heat as a result of resistive loss during the change in electric to thermal state. This method is mainly used in cartridge heaters, MHD (magnetohydrodynamics) thrusters, electrical and electronic devices, and so on. Wahid et al. [17] researched by using Joule heating flow towards a shrinking sheet. Alarabi et al. [18] researched HNF flow towards a shrinking (stretching) cylinder with joule heating. Khashi et al. [19] examined HNF flow toward a shrinking cylinder with Joule heating. To direct the MHD flow toward a stretched cylinder, Jagan et al. [20] considered velocity slip.

A careful review of the literature disclosed that the thermal radiation effect tangled in various engineering processes including thermal engineering storage, nuclear turbines, spectroscopy, and so on. Yashkun et al. [21] observed the HNF flow towards a shrinking /stretching sheet with thermal radiative. Pal et al. [22] investigated radiative heat and mass transfer of nanofluid flow over a stretching/shrinking sheet. Waini et al. [23] inspected thermal radiative flow toward a shrinking cylinder by using two different nanoparticles. Numerical simulations of HNF flow towards a shrinking cylinder with the effect of thermal radiative were inspected by Aladdin et al. [24]. Eswaramoorthi et al. [25] researched thermal radiative bioconvective nanofluid in a stratified medium by using gyrostatic microorganisms.

Stefan blowing effect relates the velocity and species (concentration) field. A blowing effect develops on an impervious surface. Uddin et al. [26] evaluated the numerical outcomes for nano liquid flow with Stefan blowing. Casson fluid moving toward a shrinking sheet with effects of Stefan blowing and velocity slip analyzed by Lund et al. [27]. Cattaneo-Christov model on nanofluid flow with Stefan blowing analyzed by Ali et al. [28]. Rana et al. [29] examined HNF flow towards a stretched cylinder with MHD and Stefan blowing effects.

The current work examines the effects of previously unstudied phenomena on HNF flow, including Stefan blowing, thermal radiation, Joule heating, and velocity slip. By examining heat and mass transmission while stagnation point, thermal radiation, Stefan blowing, and joule heating effects are present, this current work shows its novelty. The main study is in industrial systems such as drying and purifying processes. The pertinent physical quantities for various parameters are also shown via tables and graphs.

2. Mathematical Formulation

The HNF flow over a shrinking cylinder with radius a = 1 as shown in Figure 1. Here, the surface and free stream velocity are respectively $u_w\left(x\right)=\frac{c_{2}x}{L}$ and $u_e\left(x\right)=\frac{c_{1}x}{L}$, where c₁ and c₂ are constants and L is the characteristic length. $B_0$ is the magnetic field applied in the opposite flow direction and the radiative heat flux is defined as $q_r=\frac{-4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial r}$. The impacts of Stefan blowing, stagnation point, thermal radiative, Joule heating, and velocity slip are contemplated. The following assumptions are taken

• The flow is steady, laminar, and 2D-dimensional.
• The flow is incompressible.
• The cylinder is shrinking with uniform velocity along the x-direction.

The equations that govern the HNF flow are described (refer to Waini et al. [6,23]):

Continuity Equation

$$\frac{\partial \left(r u\right)}{\partial x}+\frac{\partial \left(r v\right)}{\partial r}=0$$

(1)

Momentum Equation

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right)+u_e\frac{d{u}_e}{dx}-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^2\left(u-u_e\right)$$

(2)

Temperature Equation

$$\begin{array}{c}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{k_{hnf}}{{(\rho C_P)}_{hnf}}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)-\frac{1}{{(\rho C_P)}_{hnf}}\left(\frac{\partial \left(q_r\right)}{\partial r}\right)\\ +\frac{{\sigma }_{hnf}}{{(\rho C_P)}_{hnf}}{B}_0^2{(u-u_e)}^2\end{array}\}$$

(3)

Concentration Equation

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial r}=D\left(\frac{{\partial }^{2}C}{\partial r^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)$$

(4)

associated boundary conditions are (refer to Waini et al. [6] Rana et al. [29]):

$$\begin{array}{c}v=\frac{-D}{\left(1-c_w\right)}\left(\frac{\partial C}{\partial r}\right),u=u_w+k_1\left(\frac{\partial u}{\partial r}\right),T=T_w,C=C_w at r\to a\\ u\to u_e,T\to T_{\infty },C\to C_{\infty } as r\to \infty \end{array}\}$$

(5)

where u and v are the corresponding x- and r-axis velocity components and T denotes fluid temperature. In addition,

Table 1. The thermophysical properties. (refer to Waini et al. [6] and Raza et al. [30]).

Properties	Go	MoS2	H2O
ρ (kg m−3)	1800	5060	997.1
$C_P$ (J kg−1 K−1)	717	397.21	4179
$k$ (W m−1 K−1)	5000	904.4	0.63
σ (S m−1)	6.30 × 107	2.09 × 104	0.05

presents the physical properties of base fluid (H₂O) and nanoparticles, while

Table 2. Correlations of HNF (referring to Waini et al. [6,23]).

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

provides the physical relations of the HNF. Here, the nanoparticle volume fraction of Graphene oxide (Go) and Molybdenum disulfide (MoS₂) are symbolized by ${\phi }_1$ and ${\phi }_2$ respectively as follows

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

(6)

Similarity transformation (referring to Awaludin et al. [9]) are introduced:

$$u=\frac{c_{1}x{f}^{\prime }\left(\eta \right)}{L},v=-\frac{a}{r}\sqrt{\frac{c_1{\nu }_f}{L}}f\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi \left(\eta \right)=\frac{C-C_w}{C_w-C_{\infty }} \mathrm{and} \eta =\sqrt{\frac{u_e\left(x\right)}{x{\nu }_f}}\frac{r^2-a^2}{2a}$$

(7)

By using similarity transformation (7), one obtains:

$${\mu }_r\left[\left(1+2\lambda \eta \right)f^{‴}+2\lambda f^{″}\right]+{\rho }_r\left(f{f}^{″}-{f^{\prime }}^2+1\right)-{\sigma }_{r}M\left(f^{\prime }-1\right)=0,$$

(8)

$$\begin{array}{r}{k}_r\left[\left(1+2\lambda \eta \right){\theta }^{″}+2\lambda {\theta }^{\prime }\right]+\frac{4}{3}Rd\left[\left(1+2\lambda \eta \right){\theta }^{″}+\lambda {\theta }^{\prime }\right]+{\sigma }_{r}Pr MEc{(f^{\prime }-1)}^2\\ +{\left(\rho C_p\right)}_{r}Pr f {\theta }^{\prime }=0\end{array}\},$$

(9)

$$\left(1+2\lambda \eta \right){\varphi }^{″}+2\lambda {\varphi }^{\prime }+Scf{\varphi }^{\prime }=0,$$

(10)

subjected to:

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

(11)

where $Sb=\frac{\left(C_w-C_{\infty }\right)}{\left(1-C_{\infty }\right)}$ Stefan blowing parameter, ${Re}_x=\frac{u_{e}x}{{\nu }_f}$ local Reynolds number, $A=k_1\sqrt{\frac{c_1}{{\nu }_{f}L}}$ velocity slip, $M=\frac{{\sigma }_{f}L{B}_0{}^2}{{\rho }_f{c}_1}$ magnetic parameter, $Ec=\frac{c_1{}^2}{{(C_p)}_f\left(T_w-T_{\infty }\right)}$ Eckert number, $Rd=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k_f{k}^{*}}$ thermal radiation, $Pr=\frac{{(\mu C_p)}_f}{k_f}$ Prandtl number, $Sc=\frac{v_f}{D_f}$ Schmidt number, $\lambda =\sqrt{\frac{v_{f}L}{c_1{a}^2}}$ curvature parameter, ${\rho }_r=\frac{{\rho }_{hnf}}{{\rho }_f}$, ${\mu }_r=\frac{{\mu }_{hnf}}{{\mu }_f}$, ${\left(\rho C_p\right)}_r=\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}$, $k_r=\frac{k_{hnf}}{k_f}, {\sigma }_r=\frac{{\sigma }_{hnf}}{{\sigma }_f}$ and $\epsilon =\frac{c_2}{c_1}$.

The physical quantities are

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

(12)

with the skin friction coefficient C_f, Nusselt number Nu and Sherwood number Sh (referring to Waini et al. [23]).

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

(13)

Inserting (7) and (13) into Equation (12), one obtains.

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

(14)

3. Numerical Method

Equations (8) to (11) are numerically solved using the boundary value problem solver MATLAB (bvp4c) package which is a finite difference code that implements the 3-stage Lobatto IIIa formula as explained by Waini et al. [6].

Equation (8) is translated into:

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

(15)

$$f^{″}=f^{\prime }\left(2\right)=f\left(3\right)$$

(16)

$$f^{‴}=f^{\prime }\left(3\right)=-\frac{1}{\left(1+2\lambda \eta \right)}\left[\begin{array}{l}\frac{{\rho }_r}{{\mu }_r}Re\left(f\left(3\right)f\left(1\right)-f{(2)}^2+1\right)+2\lambda f\left(3\right)\\ -\frac{{\sigma }_r}{{\rho }_r}\left(f\left(2\right)-1\right)\end{array}\right]$$

(17)

Equation (9) converts:

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

(18)

$${\theta }^{″}=f^{\prime }\left(5\right)=\frac{-1}{\left(1+2\lambda \eta \right)\left(k_r+\frac{4Rd}{3{\left(\rho C_p\right)}_r}\right)}\left[\begin{array}{c}\lambda f\left(5\right)\left(2k_r+\frac{4Rd}{3{\left(\rho C_p\right)}_r}+Pr f \left(1\right)\right)\\ +\frac{{\sigma }_r}{{\left(\rho C_p\right)}_r}{\left(f\left(2\right)-1\right)}^{2}M Ec Pr\end{array}\right]$$

(19)

Equation (10) is written as:

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

(20)

$${\varphi }^{″}=f^{\prime }\left(7\right)=-\frac{1}{\left(1+2 \lambda \eta \right)}\left(2 \lambda {\varphi }^{\prime }+Sc f {\varphi }^{\prime }\right)$$

(21)

with the boundary conditions:

$$\begin{array}{c}fa\left(1\right)=\frac{Sb}{Sc}fa\left(7\right),fa\left(2\right)=\epsilon +Afa\left(3\right),fa\left(4\right)=1,fa\left(6\right)=1\\ fb\left(2\right)=1,fb\left(4\right)=0,fb\left(6\right)=0\end{array}\}$$

(22)

The bvp4c MATLAB package is then used to solve Equations (15)–(22), yielding the required solutions.

4. Results and Discussion

In this present study, flow behavior is examined for various parameters by taking ${\phi }_1$ = ${\phi }_2$ = 0.02. In

Table 3. Comparison results where Sb = λ = M = Ec = Rd = Sc = K = 0 and ${\phi }_1={\phi }_2=0$ when Pr = 6.2.

ε	${\mathit{f}}^{″}\left(\mathit{\eta }\right)$
	Wang [7]	Waini et al. [8]	Present Result
−1	1.328820	1.328817	1.328820
−0.5	1.495670	1.495670	1.495670
0	1.232588	1.232588	1.232588

, the current findings were compared with Wang [7] and Waini et al. [8] and are consistent with the literature stated above.

Table 4. Numerical results of C_f, Nu, and Sh when ${\phi }_1$ = 0.02 and Pr = 6.2.

Sb	M	Ec	Rd	ε	${\mathit{\phi }}_2$	${\mathit{R}\mathit{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$	${\mathit{R}\mathit{e}}_{\mathit{x}}^{-1/2}\mathit{N}\mathit{u}$	${\mathit{R}\mathit{e}}_{\mathit{x}}^{-1/2}\mathit{S}\mathit{h}$
0	0.1	0.1	0.1	−0.5	0.02	1.578609	1.269266	0.471888
0.1	-	-	-	-	-	1.369317	1.098009	0.438235
1.0	-	-	-	-	-	1.180348	0.291620	0.308434
2.0	-	-	-	-	-	1.067028	0.072320	0.237721
-	0.1	-	-	-	-	1.067028	0.072320	0.237721
-	0.2	-	-	-	-	1.161345	0.032563	0.234640
-	0.3	-	-	-	-	1.205204	0.016897	0.237441
-	-	0.1	-	-	-	1.221111	0.008869	0.238595
-	-	0.2	-	-	-	1.172795	−0.022768	0.240634
-	-	0.3	-	-	-	1.197174	−0.101285	0.238227
-	-	-	0.2	-	-	1.139701	−0.095667	0.24396
-	-	-	0.4	-	-	1.139701	−0.078478	0.243946
-	-	-	0.6	-	-	1.139700	−0.049477	0.243946
-	-	-	-	−0.1	-	0.970726	0.054617	0.273412
-	-	-	-	−0.2	-	1.014529	0.036342	0.267537
-	-	-	-	−0.3	-	1.098488	−0.022513	0.256132
-	-	-	-	-	0.01	1.115239	−0.36832	0.254321
-	-	-	-	-	0.015	1.115239	−0.36832	0.254321
-	-	-	-	-	0.02	1.115239	−0.036832	0.254321

provides the numerical results for skin friction, the Nusselt number, and the Sherwood number by fixing ${\phi }_1$ = 0.02 and Pr = 6.2.

From Figure 2a, rising Stefan blowing decreases the velocity profile. Infer that an upsurge in Stefan blowing from the surface to free stream which slows down the speed of HNF as it moves towards the surface due to mass diffusion. With the upsurge of Stefan blowing, the temperature and concentration profiles accelerated and are blown further away from the surface to ambient temperature and ambient concentration. Physically, species diffusion is energized by the inclusion of nanoparticles (tiny particles) in base fluid (water). In turn, the temperature and concentration profiles increase (see Figure 2b,c).

Figure 3a demonstrates that by rising values of the curvature parameter (λ), the velocity profile intensifies slightly. The radius of the cylinder is inversely proportional to the curvature parameter by definition, a rise in the λ implies a decrease in the cylinder radius. The profile of temperature declines due to increasing λ. As the increase in λ, the flatness of the shrinking surface rises, consequently, the flow velocity accelerates, and the profile of temperature decreases because the resistance between the fluid layer reduces (see Figure 3b).

According to Figure 4a, the profile of velocity is enhanced with larger values of M because the magnetic property has a substantial effect on nanoparticles interaction with the electromagnetic field. Additionally, nanoparticles respond strongly to the magnetic field since they are in sync with the electromagnetic field. Since the magnetic field assists the fluid flow, it drives the nanoparticles toward the cylinder surface. This lively process reduces the temperature as well as the concentration field (see Figure 4b,c).

Figure 5a observed that as thermal radiation Rd rises, the temperature profile initially declines, but after moving far from the surface, the temperature profile exhibits the opposite trend. Because thermal radiation more is dominant than thermal conduction. The temperature profile rises due to an increase in Eckert number as depicted in Figure 5b. Physically, an increase in Ec converts kinetic energy into internal energy by work that is conducted against viscous fluid stress, which rises the temperature field.

Figure 6a shows that the rising slip parameter increases the velocity profile. Moreover, a rise in the velocity slip allows fluid to slip over the cylinder surface, which consequently speeds up fluid flow at the boundary. Figure 6b, demonstrates the temperature profile decreases as the slip parameter upsurges, which reduces the thickness of the thermal boundary layer.

Figure 7a witnessed that by raising the nanoparticle volume fraction of ${\phi }_2$ up to 2%, the velocity profile diminishes. In general, an increase in the nanoparticle volume fraction causes resistance in the fluid flow, which slows down the velocity field. The temperature profile upsurges as increasing ${\phi }_2$ up to 2% (see Figure 7b). In turn, this is due to the thermal conductivity of HNF elevates, and as a consequence fluid temperature rises.

Figure 8a,b, display the increasing nanoparticle volume fraction of ${\phi }_1$ and ${\phi }_2$ up to 2% versus Stefan blowing and shrinking state, the skin friction coefficient upsurges. In general, nanoparticle volume fractions produce less resistance at the cylinder surface due to the random motion of HNF/nanoparticles.

Heat transfer rate increases against shrinking parameter and Stefan blowing when a rise in the nanoparticle volume fraction of ${\phi }_1$ and ${\phi }_2$ up to 2% as compared to regular fluid (${\phi }_1$ = ${\phi }_2$ = 0). Physically, an upsurge in the ${\phi }_1$ and ${\phi }_2$ against shrinking parameter and Stefan blowing leads to the thermal conductivity accelerated, which causes a rise in the heat transfer rate. This is due to the synergistic effects of HNF (see Figure 9a,b).

Figure 10a shows Nusselt number Nu upsurges as thermal radiation increases. Physically, thermal radiation increases the thermal conductivity of the fluid, which boosts the heat transfer rate. With an increase in Stefan blowing Sb, the mass transfer rate decreases. Physically, due to intermolecular force between nanoparticles the diffusion causes from ambient concentration to surface concentration (see Figure 10b).

5. Conclusions

The present study addressed the impacts of Stefan blowing, Joule heating, and thermal radiation on HNF flow over a shrinking cylinder has been expertise.

The following potential limitations are accomplished:

• Since water is used as the base fluid in our model, the Schmidt number is fixed.
• Surface velocity is always less than zero.
• The Prandtl number has a range from 1.7 to 13.7 which is based on the base fluid.

The consequences of the current study are recorded as follows:

• There is an augmentation in temperature as well as concentration profiles due to the presence of Stefan blowing, but the velocity profile falls.
• The magnetic and curvature parameters cause a reduction in the temperature and concentration profiles and a rise in the velocity profile.
• The temperature profile rises due to an increase in Eckert number and thermal radiation parameter.
• The velocity profile rises as the magnetic field increases but decreases the thermal and concentration boundary layer.
• The heat transfer rate is improved by the thermal radiation parameter and shrinking case when ε < 0.
• With higher Stefan blowing, heat transmission increases but the mass transfer rate decreases in the presence of 2% of nanoparticles/HNF.