Heat transfer optimisation through viscous ternary nanofluid flow over a stretching/shrinking thin needle

Abstract

The current investigation interprets the flow and the thermal characteristics of the ternary nanofluid composed of MoS₂, ZnO, and SiO₂ spherical nanoparticles and water. The resulting nanofluid is $Mo{S}_2-\mathit{ZnO}-Si{O}_2-(H_{2}O+EG)$ where $(H_{2}O+EG)$ act as the base fluid which help in the flow and the nanoparticles contribute to enhancing the heat conductivity. The flow is assumed to occur across a thin needle whose surface is maintained at a higher temperature than the surroundings. The mathematical model is framed by incorporating radiation introduced by Rosseland in terms of partial differential equations (PDE). This system of equations governs the flow and thermal properties of fluid which are converted to a system of ordinary differential equations (ODE). The major outcomes of the study indicated that the increase in the amount of molybdenum disulfide enhanced the heat conducted by the nanofluid whereas it reduced the flow speed. The positive values of the heat source/sink parameter caused the heat conduction of the nanofluid to go high.

Keywords:

• Heat transfer
• nanoparticles
• radiation
• slip
• ternary nanofluid

1. Introduction

Nanofluids, a fascinating and rapidly evolving field of research, represent a promising class of engineered fluids that have gained substantial attention in recent years. Comprised of a base fluid combined with nanoparticles at the nanoscale, nanofluids deliver enhanced mechanical, thermal, and optical properties than their base counterparts. This amalgamation of nanotechnology and fluid dynamics has opened up new avenues for various industrial applications, ranging from advanced heat transfer systems to innovative cooling technologies. Some of the instances where nanofluids are highly influential include heat transfer and thermal management, electronics and microsystems, energy conversion and storage, oil and gas industries, automotive and aerospace, textiles and Fabrics, construction and building materials, environmental applications etc. In this regard, the analysis performed by Waqas et al. [1] on the nanofluid highlighted that the fluid flows faster with a higher Deborah number. The experiment designed by Javadpour et al. [2] concluded that the nanofluid having $0.069wt\%$ of MWCNT with a flow rate of $2.092kg/\min$ is an optimum setup for achieving effective cooling. Mohammadpour et al. [3] discussed the momentum of the fluid flow in the microchannel having protrusions in the double synthetic jets. The symmetrical analysis performed by Yaseen et al. [4] concluded that the $Mo{S}_2-Si{O}_2/H_{2}O-C_2{H}_6{O}_2$ hybrid nanofluid had an enhanced Nusselt number than $Mo{S}_2-H_{2}O$ nanofluid at the lower plate of the horizontal channel. Ali et al. [5, 6] discussed the impact of gravity modulation and magnetic dipole effects on the dynamics of the micropolar fluid. Anandika et al. [7] designed a multilayer model to optimize the heat conducted by the nanofluid. Xin et al. [8] discussed the impact of Thompson and Troian slips for the velocity of the nanofluid at the boundary layer. Alharbi et al. [9] incorporated the Cattaneo–Christov heat flux model to analyze the heat transfer characteristics of water having the suspensions of Cu nanoparticles. Khan et al. [10] analyzed the significance of suspending alumina nanoparticles for enhancing the thermal properties of water flowing in a vertical tube. Puneeth et al. [11] observed that the blade shaped nanoparticles helped in conducting more heat than any other shape of the nanoparticles in Casson hybrid nanofluid.

The enhanced thermal properties of nanofluids have increased the use of nanofluids in many industries. Hence, with the constant growth rate in the industries, the demand for nanofluids or other heat carriers increased rapidly. In this regard, the concept of ternary nanofluid was introduced which contains the suspension of 3 different nanoparticles. Each of these nanoparticles has a prominent role in acting on the heat conduction capability of the base fluid and its chemical stability. Apart from various experimental works [12–15], the theoretical model of ternary nanofluid was introduced by Manjunatha et al. [16, 17]. Arif et al. [18] compared the performance of ternary nanofluids having different shapes of particles in effectively cooling a heated system. Shahzad et al. [19] performed second–order convergence analysis to analyze the motion of ternary nanofluid across a spinning disk under the influence of Hall current. It was observed that the heat conducted by $A{l}_2{O}_3-Ti{O}_2-\mathit{CuO}/H_{2}O$ nanofluid increased for higher ranges of radiation parameters in the study conducted by Puneeth et al. [20]. Wang et al. [21] examined the feasibility of ternary nanofluid in effective heat transfer in cooling electronic devices. Ahmed et al. [22] studied the $\mathit{ZnO}+A{l}_2{O}_3+Ti{O}_2/DW$ ternary nanofluid which is a sonochemically synthesized metal oxide to analyze its heat transfer properties in a square flow. A numerical study was made by Hasnain et al. [23] to observe the growth in the heat conducted due to the suspension of three differently shaped nanoparticles. Puneeth et al. [24] concluded that the temperature detected in the ternary nanofluid was higher for the high thermophoresis. Ali et al. [25] further analyzed the importance of bioconvection in the heat transfer analysis of nanofluid by designing the Falkner–Skan model for the flow of Maxwell nanofluid.

The thin needle is a geometric surface whose body is identified as a long and narrow cylinder with a sharp end point. A needle whose surface/body thickness becomes smaller than the boundary layer is identified as a thinner needle. The flow over such a surface (thin needle) has grabbed the interest of many scholars worldwide due to its continuous availability in electrical devices, geothermal power generation, hot wire anemometers etc. For instance, by implementing the cross–diffusion model, Rehman et al. [26] examined the Soret and Dufour impact on the transfer of energy and transfer mass transfer in the nanofluid flowing over the needle. Further, this study was continued by Shafiq et al. [27] to implement the artificial neural network. Nazar et al. [28] performed stability analysis to check the stability of gold–suspended nanofluid flowing past a moving needle. Meanwhile, Yasir et al. [29] also obtained dual solutions to study the stability of the nanofluid suspended with carbon nanotubes. Khan et al. [30] discussed the bioconvection activated flow of water–based hybrid nanofluid considering the chemical reaction parameter which actively influences the mass transfer profile. Puneeth et al. [31] studied the fourth–degree of autocatalysis that occurred in the nanofluid set to motion past a thin needle. Akinshilo et al. [32] observed a decrease in the velocity of the flow for the higher porosity parameter of the needle. The prominent factors such as the radiation and viscous dissipation were analyzed by Jahan et al. [33] on the non–isothermal flow of nanofluid across a thin needle.

$Mo{S}_2-\mathit{ZnO}-Si{O}_2$ refers to a composite material that is composed of Molybdenum disulfide $(Mo{S}_2),$ Zinc oxide (ZnO), and Silicon dioxide $(Si{O}_2).$ This combination offers synergistic properties and functionalities that are not present in the individual components. The specific applications and the properties of this composition are dependent on the ratios of the components, the fabrication process, and the intended use. A thorough literature survey was performed to arrive at the research gap and it was noticed that most of the scholarly contributions to the field of nanofluids were regarding the analysis of the thermal properties of nanofluids. These contributions of scholars motivated many researchers to extend the study of nanofluids to a new concept of hybrid and ternary nanofluids having multiple suspensions of nanoparticles. Also, many articles available in the field of study discuss the heat conduction performance of hybrid nanofluids. But the volumetric analysis of the ternary nanofluid flowing past a thin needle is rarely available. Thus, this article tries to implement the Newtonian viscosity model and the Maxwell thermal conductivity model to analyze the flow and energy transfer properties of $Mo{S}_2-\mathit{ZnO}-Si{O}_2-(H_{2}O+EG)$ ternary nanofluid. Finally, the solutions to the system of equations were obtained through the RKF–45 method, and the obtained graphical results are interpreted rigorously in this article.

2. Mathematical description and model

The laminar flow of $Mo{S}_2-\mathit{ZnO}-Si{O}_2-H_{2}O-EG$ ternary nanofluid in two–dimensions is considered to flow over a thin needle of varying radius as shown in Figure 1. The $H_{2}O-EG$ mixture is in the ratio $80:20$ whose viscosity upon suspending the nanoparticles is designed by considering the Newtonian viscosity model. The temperature of the medium is set to $T_{\infty }$ while on the surface, the temperature is assumed to be T_w such that $T_w>T_{\infty }.$ The uniform–sized spherical MoS₂, ZnO, and SiO₂ nanoparticles are suspended in $H_{2}O$ to form the ternary nanofluid flowing at a constant speed of $U_{\infty }.$ The agglomeration impact is not considered since a stable nanoparticle suspension is assumed in the formation of the $Mo{S}_2-\mathit{ZnO}-Si{O}_2-(H_{2}O+EG)$ ternary nanofluid. The thermophysical constants of the constituents of this combination are mentioned in Table 1. The energy equation is designed by using Maxwell’s thermal conductivity model and heat source/sink parameter. Taking into account, the optical dense property of the mixture of water and ethylene glycol, the Rosseland approximation is implemented for modeling the thermal radiation as follows: (1) $$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial r}$$(1)

Figure 1. Flow configuration.

Table 1. The thermophysical constants of the constituents [4, 34].

Nanoparticle	ρ	Cp	κ
ZnO	5600	495.2	13
MoS 2	5060	397.21	34.5
SiO2	2650	730	1.5
$H_{2}O+EG$	1056	3288	0.425

These assumptions lead to the formation of the following system of partial differential equations [28]: (2) $$\frac{\partial (ru)}{\partial x}+\frac{\partial (rv)}{\partial r}=0$$(2) (3) $$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}={\nu }_{tf}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)$$(3) (4) $$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}={\alpha }_{tf}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)-\frac{1}{{(\rho Cp)}_{tf}}\frac{\partial q_r}{\partial r}+\frac{Q_0}{{(\rho Cp)}_{tf}}(T-T_{\infty })$$(4)

The energy equation reduces to the following expression after substituting the expression of q_r in (4)(4) $$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}={\alpha }_{tf}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)-\frac{1}{{(\rho Cp)}_{tf}}\frac{\partial q_r}{\partial r}+\frac{Q_0}{{(\rho Cp)}_{tf}}(T-T_{\infty })$$(4) (5) $$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}={\alpha }_{tf}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{(\rho Cp)}_{tf}}\frac{{\partial }^{2}T}{\partial r^2}+\frac{Q_0}{{(\rho Cp)}_{tf}}(T-T_{\infty })$$(5)

The flow is subjected to the following boundary conditions: (6) $$\begin{array}{cc}\hfill & u=U_{\infty }\lambda ,\hspace{1em}v=v_w,\hspace{1em}T=T_w\hspace{1em}\text{at}\hspace{1em}r=R(x)\hfill \\ \hfill & u\to U_{\infty },\hspace{1em}T\to T_{\infty }\hspace{1em}\text{as}\hspace{1em}r\to \infty \hfill \end{array}$$(6)

The physical quantities of interest, especially the skin friction coefficient and the Nusselt numbers are defined as: (7) $$C{f}_x=\frac{{\mu }_{tf}{\left(\frac{\partial u}{\partial r}\right)}_{r=R(x)}}{{\rho }_f{U}_{\infty }^2},\hspace{1em}N{u}_x=-x\frac{{\left({\kappa }_{tf}\frac{\partial T}{\partial z}+q_r\right)}_{r\hspace{0.17em}=\hspace{0.17em}R(x)}}{{\kappa }_f(T_w-T_{\infty })}$$(7)

The system (2)(2) $$\frac{\partial (ru)}{\partial x}+\frac{\partial (rv)}{\partial r}=0$$(2) –(5)(5) $$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}={\alpha }_{tf}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}{(\rho Cp)}_{tf}}\frac{{\partial }^{2}T}{\partial r^2}+\frac{Q_0}{{(\rho Cp)}_{tf}}(T-T_{\infty })$$(5) and the surface constraints (6)(6) $$\begin{array}{cc}\hfill & u=U_{\infty }\lambda ,\hspace{1em}v=v_w,\hspace{1em}T=T_w\hspace{1em}\text{at}\hspace{1em}r=R(x)\hfill \\ \hfill & u\to U_{\infty },\hspace{1em}T\to T_{\infty }\hspace{1em}\text{as}\hspace{1em}r\to \infty \hfill \end{array}$$(6) are converted into a system of ODEs by implementing the following relations: (8) $$\eta =\frac{U_{\infty }{r}^2}{{\nu }_{f}x},\hspace{1em}u=2U_{\infty }f\prime ,\hspace{1em}v=-\frac{{\nu }_f}{r}\left(f-\eta f\prime \right),\hspace{1em}{v}_w=-\frac{{\nu }_f}{r}\frac{\lambda }{2}\delta ,\hspace{1em}\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}$$(8)

Upon setting the value of $\eta =\delta$ in Eq. (6)(6) $$\begin{array}{cc}\hfill & u=U_{\infty }\lambda ,\hspace{1em}v=v_w,\hspace{1em}T=T_w\hspace{1em}\text{at}\hspace{1em}r=R(x)\hfill \\ \hfill & u\to U_{\infty },\hspace{1em}T\to T_{\infty }\hspace{1em}\text{as}\hspace{1em}r\to \infty \hfill \end{array}$$(6) which signifies the wall of the needle, the surface of the needle shall be defined as (9) $$R(x)=\sqrt{\frac{{\nu }_f\delta x}{U_{\infty }}}$$(9)

 The continuity equation (Eq. (2)(2) $$\frac{\partial (ru)}{\partial x}+\frac{\partial (rv)}{\partial r}=0$$(2) ) is satisfied for the similarity transformation defined above in (8)(8) $$\eta =\frac{U_{\infty }{r}^2}{{\nu }_{f}x},\hspace{1em}u=2U_{\infty }f\prime ,\hspace{1em}v=-\frac{{\nu }_f}{r}\left(f-\eta f\prime \right),\hspace{1em}{v}_w=-\frac{{\nu }_f}{r}\frac{\lambda }{2}\delta ,\hspace{1em}\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }}$$(8) and the resulting system equations are of the form: (10) $$\frac{{\mu }_{tf}}{{\mu }_f}\left(\eta f^{″\prime }+f^{″}\right)+0.5\frac{{\rho }_{tf}}{{\rho }_f}f{f}^{″}=0$$(10) (11) $$\left(\frac{{\kappa }_{tf}}{{\kappa }_f}+\frac{4}{3}R\right)\eta {\theta }^{″}+\frac{{\kappa }_{tf}}{{\kappa }_f}\theta \prime +Pr\left[0.25Q\theta +\frac{{(\rho Cp)}_{tf}}{{(\rho Cp)}_f}f\theta \prime \right]=0$$(11)

The boundary conditions corresponding to (6)(6) $$\begin{array}{cc}\hfill & u=U_{\infty }\lambda ,\hspace{1em}v=v_w,\hspace{1em}T=T_w\hspace{1em}\text{at}\hspace{1em}r=R(x)\hfill \\ \hfill & u\to U_{\infty },\hspace{1em}T\to T_{\infty }\hspace{1em}\text{as}\hspace{1em}r\to \infty \hfill \end{array}$$(6) take the following form: (12) $$f\prime (\delta )=\frac{\lambda }{2},\hspace{1em}f(\delta )=\lambda \delta ,\hspace{1em}\theta (\delta )=1,\hspace{1em}{f}^{\prime }(\infty )\to 0.5,\hspace{1em}{\theta }^{\prime }(\infty )\to 0$$(12)

The nondimensionalized expression for Cf_x (skin friction coefficient) and Nu_x (Nusselt number) takes the following form: (13) $$\frac{C{f}_x\sqrt{R{e}_x}}{4\sqrt{\delta }}=\frac{{\mu }_{tf}}{{\mu }_f}{f}^{″}(0),\hspace{1em}\frac{N{u}_x}{2\sqrt{\delta Re}}=-\left(\frac{{\kappa }_{tf}}{{\kappa }_f}+\frac{4}{3}R\right)\theta \prime (0)$$(13)

The nondimensional parameters appearing in this model are expressed as follows: $$Q=\frac{Q_{0}x}{U_{\infty }{(\rho Cp)}_f},\hspace{1em}Pr=\frac{{\nu }_f}{{\alpha }_f},\hspace{1em}R=\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}{\alpha }_f},\hspace{1em}R{e}_x=\frac{x{U}_{\infty }}{{\nu }_f}$$

3. Thermophysical and rheological properties

The rheological and thermophysical properties of the ternary nanofluids are stated below [16]:

1. Density: $${\rho }_{tf}=(1-{\varphi }_3)\{(1-{\varphi }_2)[(1-{\varphi }_1){\rho }_f+{\varphi }_1{\rho }_1]+{\varphi }_2{\rho }_2\}+{\varphi }_3{\rho }_3.$$

2. Viscosity $${\mu }_{tf}=\frac{{\mu }_f}{{(1-{\varphi }_1)}^{2.5}{(1-{\varphi }_2)}^{2.5}{(1-{\varphi }_3)}^{2.5}}.$$

3. Specific heat capacity: $${(\rho Cp)}_{tf}=(1-{\varphi }_3)\{(1-{\varphi }_2)[(1-{\varphi }_1){(\rho Cp)}_f+{\varphi }_1{(\rho Cp)}_1]+{\varphi }_2{(\rho Cp)}_2\}+{\varphi }_3{(\rho Cp)}_3.$$

4. Thermal conductivity $$\frac{{\kappa }_{tf}}{{\kappa }_{\mathit{hnf}}}=\frac{{\kappa }_3+2{\kappa }_{\mathit{hnf}}-2{\varphi }_3({\kappa }_{\mathit{hnf}}-{\kappa }_3)}{{\kappa }_3+2{\kappa }_{\mathit{hnf}}+{\varphi }_3({\kappa }_{\mathit{hnf}}-{\kappa }_3)},$$

where $\frac{{\kappa }_{\mathit{hnf}}}{{\kappa }_{nf}}=\frac{{\kappa }_2+2{\kappa }_{nf}-2{\varphi }_2({\kappa }_{nf}-{\kappa }_2)}{{\kappa }_2+2{\kappa }_{nf}+{\varphi }_2({\kappa }_{nf}-{\kappa }_2)}$ and $\frac{{\kappa }_{nf}}{{\kappa }_f}=\frac{{\kappa }_1+2{\kappa }_f-2{\varphi }_1({\kappa }_f-{\kappa }_1)}{{\kappa }_1+2{\kappa }_f+{\varphi }_1({\kappa }_f-{\kappa }_1)}.$

4. Solution methodology

The RKF-45 method, also known as the Runge-Kutta-Fehlberg method, is a numerical integration technique used to solve ordinary differential equations (ODEs). It is an adaptive step-size method that provides a compromise between accuracy and computational efficiency. It combines two different Runge-Kutta methods of different orders to estimate the solution and an error estimate. The error estimate is then used to adaptively adjust the step size for improved accuracy. The convergence criteria of RKF-45 involve comparing the solutions obtained using the fourth-order and fifth-order methods and using an error estimate to control the step size. This method is widely used in scientific and engineering applications where numerical integration of ODEs is required. Its adaptive nature makes it suitable for solving ODEs with varying behaviors, such as stiff systems or systems with rapid changes in dynamics. Thus, the altered governing equations (10)(10) $$\frac{{\mu }_{tf}}{{\mu }_f}\left(\eta f^{″\prime }+f^{″}\right)+0.5\frac{{\rho }_{tf}}{{\rho }_f}f{f}^{″}=0$$(10) –(12)(12) $$f\prime (\delta )=\frac{\lambda }{2},\hspace{1em}f(\delta )=\lambda \delta ,\hspace{1em}\theta (\delta )=1,\hspace{1em}{f}^{\prime }(\infty )\to 0.5,\hspace{1em}{\theta }^{\prime }(\infty )\to 0$$(12) are converted into an initial value problem and the RKF–45 method is implemented through maple to obtain the solution with an accuracy of ${10}^{-5}.$ Due to these advantages, many authors [35–38] have used RKF-45 method to perform the analysis. In order to apply the RKF-45 method to the system of Eqs. (10)–(12)(12) $$f\prime (\delta )=\frac{\lambda }{2},\hspace{1em}f(\delta )=\lambda \delta ,\hspace{1em}\theta (\delta )=1,\hspace{1em}{f}^{\prime }(\infty )\to 0.5,\hspace{1em}{\theta }^{\prime }(\infty )\to 0$$(12) , it must be converted to initial value problem that can be done by using the following transformation: (14) $$\left(\begin{array}{cccc}f& f\prime & f^{″}& f^{‴}\\ \theta & \theta \prime & {\theta }^{″}& 0\end{array}\right)=\left(\begin{array}{cccc}{D}_1& D_2& D_3& D_3^{\prime }\\ D_4& D_5& D_5^{\prime }& 0\end{array}\right)$$(14)

Upon using the above transformations, the system of equations reduces to the following form: (15) $$\left(\begin{array}{c}{D}_1^{\prime }\\ D_2^{\prime }\\ D_3^{\prime }\\ D_4^{\prime }\\ D_5^{\prime }\end{array}\right)=\left(\begin{array}{c}{D}_2\\ D_3\\ \frac{1}{\eta }\left[\frac{{\mu }_f}{{\mu }_{tf}}\left(-0.5\frac{{\rho }_{tf}}{{\rho }_f}{D}_1{D}_2\right)-D_3\right]\\ D_5\\ -\frac{1}{\eta }{\left[\frac{{\kappa }_{tf}}{{\kappa }_f}+\frac{4}{3}R\right]}^{-1}\left[\frac{{\kappa }_{tf}}{{\kappa }_f}{D}_5+Pr\left(0.25Q{D}_4+\frac{{(\rho Cp)}_{tf}}{{(\rho Cp)}_f}{D}_1{D}_5\right)\right]\end{array}\right)$$(15)

Subject to the constraints: (16) $$D_1(\delta )=\lambda \delta ,\hspace{1em}{D}_2(\delta )=\frac{\lambda }{2},\hspace{1em}{D}_3(\delta )=a_1,\hspace{1em}{D}_4(\delta )=1,\hspace{1em}{D}_5(\delta )=a_2$$(16)

In the above problem, $a_1,\hspace{1em}{a}_2,$ and a₃ are positive constants. The obtained results are verified as shown in Table 2 by referring to the existing literature.

Table 2. Validating the obtained results through comparison.

	Qasim et al. [39]	Suleman et al. [40]	Present
Pr	$-\theta \prime (0.1)$
0.72	1.23664	1.23665	1.23666
1	1.0000	1.00000	0.99999
6.7	0.3333	0.3331	0.3336
10	0.26876	0.26877	0.26877

5. Result analysis

Due to the advantages listed above, the RKF-45 approach and the shooting method are used to solve the dimensionless system of Eqs. (10)–(12)(12) $$f\prime (\delta )=\frac{\lambda }{2},\hspace{1em}f(\delta )=\lambda \delta ,\hspace{1em}\theta (\delta )=1,\hspace{1em}{f}^{\prime }(\infty )\to 0.5,\hspace{1em}{\theta }^{\prime }(\infty )\to 0$$(12) . Graphs and tables are used to discuss the analysis of the flow’s parameters. The Prandtl number (Pr) is kept constant at 6.7 for this study since it is assumed that the base fluid is a Newtonian fluid, and the overall nanoparticle volume fraction $(\varphi )$ is maintained at 0.03 to prevent the potential of sedimentation.

The graphical analysis of the impact of the thickness of the needle is shown in Figure 2 according to which an increase in the velocity profile is observed. The increase in the δ corresponds to the increase in the width/opening of the needle as it is directly linked with the radius. As the value of this parameter is increased, it widens the gap and allows more space for the flow of fluid. As a result, a huge amount of nanofluid flows in a shorter duration of time. Similarly, the impact of the same parameter on the temperature of the nanofluid is seen in Figure 3. Since the fluid flows faster and the quantity of fluid within the region of the needle increases with the increase in δ, the heat conducted by the nanofluid gets distributed between the nanofluid layers and thus a lower temperature is recorded. The existence of the heat sink/source has played a significant impact on the temperature of the nanofluid and this impact is shown in Figures 4 and 5. The parameter Q is responsible for analyzing this effect and it acts as a heat source when Q > 0 and acts as a heat sink when Q < 0. The increase in the positive values of Q signifies that more heat is being emitted from the source which is absorbed by the nanofluid and thus an enhancement in the temperature is recorded. Similarly, an action of a strong heat sink is obtained for lower values of Q which the excess heat and thus reduces the temperature of the nanofluid.

Figure 2. δ influence on $f\prime (\eta ).$

Figure 3. δ influence on $\theta (\eta ).$

Figure 4. Q influence on $\theta (\eta ).$

Figure 5. Q influence on $\theta (\eta ).$

It is important to note that the effect of radiation on the temperature of a nanofluid can be complex and depends on several factors, including the properties of the nanoparticles, the concentration of nanoparticles, the temperature of the fluid, and the radiation source. Detailed analysis and modeling are often required to understand and quantify the specific impact of radiation on the temperature behavior of a nanofluid in a given application. Figure 6 depicts the influence of thermal radiation on the temperature of the nanofluid. An increase in the nanofluid’s temperature is noted for higher values of R. This enhancement in the temperature is observed due to the absorption of heat emitted from the surface by the nanofluid. Overall, radiation can contribute to the heat transfer and temperature of a nanofluid, and its influence should be considered alongside other modes of heat transfer, such as conduction and convection when studying or utilizing nanofluids in various thermal systems.

Figure 6. R influence on $\theta (\eta ).$

Nanofluids with different nanoparticle volume fractions exhibit varying flow behaviors. At low–volume fractions, nanofluids may flow similarly to their base fluids. However, as the volume fraction increases, the presence of nanoparticles can alter the flow characteristics, leading to changes in viscosity, rheology, and pressure drop. It’s worth noting that the optimal nanoparticle volume fraction varies depending on the specific application and the desired property enhancement. Hence, significant analysis is made to understand the impact of the nanoparticle volume fraction of MoS₂ $({\varphi }_3)$ on the velocity and temperature. The volume fraction parameter ${\varphi }_3$ is directly linked to the number of nanoparticles that are being suspended in the nanofluid. Thus, the analysis of this parameter has a significant role in the model. Hence, the examination of the impact of ${\varphi }_3$ on the flow of the nanofluid is performed by the values of ${\varphi }_1={\varphi }_2=0.0025$ so that the overall volume fraction does not exceed 0.03 and the same is depicted in Figure 7. It is observed that as the volume proportion of nanoparticles grows, so does the density of the nanofluid, which lowers the velocity of the nanofluid. The thermal conductivity of the nanofluid will increase with the volume percentage of nanoparticles. Consequently, the temperature of the nanofluid rises as the volume percentage of nanoparticles increases as depicted in Figure 8. Different studies and applications may have different recommended ranges of volume fraction based on the target performance and stability requirements. Therefore, it is important to consider the specific needs and constraints of the application when determining the appropriate nanoparticle volume fraction for a given nanofluid.

Figure 7. ${\varphi }_3$ influence on $f\prime (\eta ).$

Figure 8. ${\varphi }_3$ influence on $\theta (\eta ).$

The role of the stretching parameter λ on the velocity profile is depicted in Figures 9 and 10. It is observed that the velocity of the flow increases for higher values of λ because of the fact that the positive values indicate stretching. The increase in the positive values of λ indicates faster stretching and due to the adhesive force between the layer of fluid and the surface, the nanofluid gets pulled along the surface, and hence an increase in the velocity is noted. In the case of the negative values of λ, the needle is assumed to shrink which creates a drag that opposes the fluid flow. Thus, a reduction in the velocity is seen as the values of λ take lower values. Further, due to the high–speed flow that occurs because of the fast stretching, the heat absorption capacity of the nanofluid cannot reach its maximum capacity thus, a drop in the temperature is seen for higher values of λ as depicted in Figures 11 and 12.

Figure 9. λ influence on $f\prime (\eta ).$

Figure 10. λ influence on $f\prime (\eta ).$

Figure 11. λ influence on $\theta (\eta ).$

Figure 12. λ influence on $\theta (\eta ).$

The influence of the fluid parameters on the skin friction coefficient $(C{f}_x)$ and the Nusselt number $(N{u}_x)$ is tabulated in Table 3. Skin friction is found to increase with the rise in the volume fraction parameter whereas the Nusselt number is showing a decreasing behavior. Meanwhile, it is noticed that the variations in the radiation parameter have no significant impact on the skin friction coefficient but it decreases the Nusselt number effectively. Furthermore, the higher values of δ reduce Cf_x whereas the Nusselt number increases significantly. A similar impact of the heat source/sink parameter is recorded where it is observed that the skin friction coefficient remains unaltered but the Nusselt number decreases.

Table 3. Variations of Cf_x and Nu_x for changes in the fluid flow parameters.

		$-C{f}_x$	Nux			$-C{f}_x$	Nux
$\varphi$	0.005	0.7243	0.9173	δ	0.1	0.7243	–0.0154
	0.01	0.6812	0.7679		0.2	0.8616	0.2569
	0.015	0.6274	6632		0.3	1.0345	0.5040
	0.02	0.5797	0.5048		0.4	1.2536	0.7095
	0.025	0.5547	0.3040		0.5	1.5329	0.8861
R	0.5	0.7243	1.3579	Q	–0.8	0.7243	2.318
	1	0.7243	1.0446		–0.4	0.7243	2.0266
	1.5	0.7243	0.8479		0	0.7243	1.6799
	2	0.7243	0.7121		0.4	0.7243	1.2355
	2.5	0.7243	0.6121		0.8	0.7243	0.5672

6. Conclusion

The analysis of the flow of $Mo{S}_2-\mathit{ZnO}-Si{O}_2-(H_{2}O+EG)$ ternary nanofluid over a stretching/shrinking thin needle was analyzed using a suitable mathematical model. The volumetric analysis was performed by implementing the RKF-45 numerical process. The obtained solutions were observed to be in good agreement with the existing sources and the outcomes were recorded in terms of graphs and tables. The designed mathematical model explains the thermal features of ternary nanofluid considering the effects of radiation and heat source/sink. Furthermore, the analysis was performed for both stretching and shrinking needle of radius R(x). The hybrid nanofluid $\mathit{ZnO}-Si{O}_2-(H_{2}O+EG)$ was assumed to be the base fluid which upon suspension of MoS₂ nanoparticles constituted the ternary nanofluid. Some of the major outcomes of the study are as follows:

• The velocity of the nanofluid decreases as the volume fraction of the nanoparticle is increased.

• The presence of more nanoparticles having higher thermal conductivity helps the nanofluid conduct more heat.

• The radiation causes the heat absorbed by the nanofluid to enhance significantly.

• As the heat source continuously emits the heat that is absorbed by the nanofluid, its temperature increases.

• Similar to the heat source, as the heat sink is strong the heat gets absorbed by it, and the overall temperature of the nanofluid is reduced.

• As the size of the needle is increased, the velocity at which the nanofluid flows is increased.

• As the needle is stretching the flow speed increases whereas the speed decreases as the needle is shrinking.

Ternary nanofluids have shown remarkable improvements in thermal conductivity compared to traditional heat transfer fluids. The presence of multiple types of nanoparticles in the fluid enhances the interfacial interaction and facilitates efficient heat transfer. While ternary nanofluids show great promise, it is important to note that further research and development are still needed to fully understand their behavior, optimize their properties, and address any potential challenges or limitations. Nonetheless, the importance of ternary nanofluids lies in their potential to revolutionize heat transfer and offer innovative solutions in various fields, leading to improved efficiency, performance, and sustainability.

Nomenclature
α	=	thermal diffusivity $(m^2{s}^{-1})$
δ	=	dimensionless thickness parameter
κ	=	thermal conductivity $(W{m}^{-1}{K}^{-1})$
λ	=	stretching parameter
μ	=	dynamic viscosity $(kg{m}^{-1}{s}^{-1})$
ν	=	kinematic viscosity $(m^2{s}^{-1})$
ρ	=	density $(kg{m}^{-3})$
(u, v)	=	velocity components $(m{s}^{-1})$
${\sigma }^{*}$	=	Stefan Boltzmann constant
Cfx	=	skin friction coefficient
Cp	=	specific heat capacity $(Jk{g}^{-1}{K}^{-1})$
$k^{*}$	=	mean absorption coefficient
Nux	=	Nusselt number
Pr	=	Prandtl number
Q	=	dimensionless heat source/sink
Q0	=	heat source/sink
qr	=	thermal radiation
R	=	dimensionless radiation
T	=	temperature (K)
T0	=	temperature at y = 0 (K)

Subscripts
f	=	fluid
nf	=	nanofluid
s	=	nanoparticle

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project Project under the grant number RGP2/107/44.