Magnetohydrodynamic hybrid nanofluid flow through moving thin needle considering variable viscosity and thermal conductivity

Abstract

In modern science and technology, industrial applications that deal with the problem of continuously moving thin needle, surrounded with fluid in sectors like hot rolling, crystal growing, heat extrusion, glass fiber drawing, etc., are rapidly increasing. Such processes involve high temperatures which may affect the fluid properties that is, viscosity and thermal conductivity. So, it’s crucial to understand temperature-dependent fluid properties. Focused on these assumptions, the main objective of the current research work is to investigate how temperature-dependent fluid properties might improve the heat transfer efficiency and performance evolution of hybrid nanofluid in the presence of transverse magnetic field over a moving thin needle. Variable Prandtl number is also introduced to observe flow fluctuation, the effect of adding nanoparticles, and enhancement in heat transmission. The results are obtained for different needle thicknesses, temperature-dependent viscosity, temperature-dependent thermal conductivity, and heat generation. Moreover, Fe₃O₄/Graphene nanoparticles are considered to be dispersed in water. The governing partial differential equations of flow and heat transfer are transformed into a system of coupled nonlinear ordinary differential equations using analysis of similarity conversion. Subsequently, the numerical solution of the problem is attained by employing the MAPLE software. The fourth-fifth-order Runge-Kutta-Fehlberg (RFK45) approach is used by default in this MAPLE program to address the numerical problem of boundary value. The velocity and temperature field are pictured for different values of the parameters as well as physical quantities of interest such as skin friction coefficient and rate of heat transfer are visually depicted in graphs and tables. It is found that fluid motion and energy transport are highly regulated by the variation of magnetic field strength. As the volume fraction of Fe₃O₄ is increased, the heat generation, and thermal conductivity parameter vehemently enhance the temperature profile which leads to a rise in thermal boundary layer. A strong augmentation in the heat transfer rate has been found with the increment in the variable Prandtl number.

Introduction

Hybrid nanofluids, the upgraded version of nanofluids ¹ are more reliable fluids with exceptional heat transfer properties. The heat and mass transfer features have paramount significance in the field of modern technology. This new class of fluid promises to provide efficient thermal management as a coolant. As a result, the application of hybrid nanofluids has increased rapidly as a safe coolant for nuclear reactors ² and electronic devices. Especially in the presence of the magnetic field,^3–5 the applications of hybrid nanofluids are much promising for heat transfer enhancement. Nanoparticles with magnetic properties are significant in loudspeaker construction, medicine, separation of sink float, tumor analysis, ⁶ and cancer therapy. Hybrid nanofluids are created by suspending two nanoparticles with sizes ranging from 1 to 100 mm in a base fluid (water/engine oil/ethylene glycol). The mixture of metal oxide nanoparticles (ZnO, Fe₃O₂, CuO, Al₂O₃, etc.) to the traditional fluid like water/engine oil/ethylene glycol are proven to increase the thermal conductivity of hybrid nanofluids with improved thermal properties.^7–11 In molten salt, ¹² heat transfer capacity and enhancement in thermal conductivity are investigated for graphene nanoparticles.

Over the past few decades, the boundary layer flow over a thin needle has been vastly investigated because of the paramount applications in industrial areas such as power generation of geothermal devices, anemometer of hot wire, and delicate electronic devices. The revolutionary findings about thin needle state that the thickness of thin needle is smaller than the thickness of boundary layer. An initial study of thin needle examined for viscous fluid flow. ¹³ It is also found that the heat transfer characteristics affected by sloid volume fraction on thin needle. ¹⁴ Dual solutions were found in continuously moving thin needle. ¹⁵ Rosseland radiation ¹⁶ and entropy generation of moving thin needle on self-similar surface were examined in the presence of viscous dissipation. The needle sizes influenced ¹⁷ the thermal conductivity, velocity, and temperature profiles of the fluid. The forced convection on non-isothermal flow was also examined over thin needle to observe heat transition properties for power-law thermal fluctuation. ¹⁸ The investigation of temporal stability on moving thin needle ¹⁹ revealed that just one of the solutions remained stable and physically trustworthy over time. For moving thin needle, research carried out to check the flow behavior under joule heating, thermal radiation, and viscous dissipation. ²⁰ With the prescribed heat flux of moving thin needle, investigation showed that thinner needle incremented the rate of heat transfer as well as skin friction coefficient of the surface than that of thicker needle. ²¹ To improve the thermophysical properties of base fluid a study was carried out for nanofluid, which revealed that suction parameter fastens the cooling process by escalating the skin coefficient. ²² Through FEM approach, ²³ three-dimensional spinning flow of nanofluid observed a great impact on fluid velocity due to Lorentz force. An investigation took place to examine the thermal conductivity and viscosity of a nanofluid consisting of graphene nanoparticles on water over a magnetized stretching sheet. ²⁴

Moving surfaces cause velocity and temperature variations to the heated or cooled surfaces, which are related to many industrial engines, space crafts, electronic devices, and many other practical applications. A heat source is something that can heat up a spacecraft. The findings of heat sources and their impact on heat transport are critical in terms of various physical problems. The heat line technique was applied to investigate the heat source effect of hybrid nanofluid. ²⁵ Different conditions of heating and cooling were observed in a dual heat source system of ejector-compression. ²⁶ A hydromagnetic flow over a semi-infinite flat plate, was investigated with the presence of heat generation/absorption. ²⁷ As heat generation or absorption affects the fluid flow in many ways, researchers examined this effect on several surfaces by using different methods.^28–30

Viscosity and thermal conductivity are the basic physical properties of any fluid, which fluctuate with temperature in hybrid nanofluid. For viscosity, internal friction generates heat, which raises the temperature and hence affects the stickiness of the fluid. As a result, fluid viscosity cannot be regarded as constant. So, it is necessary to assume that viscosity will vary with temperature. There are some investigations to understand the temperature dependent viscosity in different aspects and surfaces.^31–35 For thin needle ³⁶ findings suggested that the viscosity parameter undermines the fluid motion. The study revealed that the variable viscosity parameter enhances the temperature distribution of hybrid nanofluid. ³⁷ With variations in temperature, the thermal conductivity of the fluid fluctuates linearly. Focusing on this particular point, temperature-dependent thermal conductivity was introduced. A study took place for maxwell fluid, ³⁸ but for hybrid nanofluid ³⁹ it was found that the fluid temperature of fluid increased with higher increments in thermal conductivity parameter. Some other investigations have been done on this topic.^40–42

Our study is motivated by the above-mentioned literature where research works showed interest in scrutinizing the variable fluid properties without considering the variable Prandtl number. An unrealistic outcome may affect the flow behavior due to constant Prandtl number, as the Prandtl number fluctuates for variable viscosity and variable thermal conductivity. Thus, to our best knowledge, we witnessed that no other effort has been carried out to scrutinize the variable thermophysical properties of hybrid nanofluid over moving thin needle influenced by strong magnetic field with variable Prandtl number. The originality of this study is to explore the performance evolution of magnetic ferrite (Fe₃O₄) and graphene nanoparticles on the base fluid water. This problem has significant implications for enhancing the heat transmission capability through iron-based nanoparticles to the conventional base fluid. Moreover, heat transfer is a key element of widely used application sectors of thermal usage which are connected with the temperature-dependent fluid properties. The computational physical problem with a strong magnetic effect may play a vital role in heat transfer characteristics by escalating the internal heat generation. In geothermal power generation, hybrid nanofluid can be employed as a working fluid to cool down the exposed area regulated by high temperature (500°C–1000°C) due to energy extraction from the crust of earth. The evaluation of the proposed model can be used for textile production, hot rolling, paper manufacturing, wire drawing procedure, and plastic film drawing procedure. The impact of the variation of the physical parameters on the dimensionless velocity and temperature profile along with shear stress and the rate of heat transfer is studied. For validation purposes, we performed comparisons with previously published results and excellent agreement was found.

The differences in our analysis of the physical problem related to the physical problems already investigated are: (i) In this present study two-dimensional laminar flow of Fe₃O₄/Graphene-water hybrid nanofluid flow is considered with nanoparticle influenced by strong magnetic fluid while the published research ³⁶ was limited to nanofluid. (ii) Both variable viscosity and variable thermal conductivity have been taken into account for hybrid nanofluid simultaneously while in the published researches^37,38 these variations are investigated separately. (iii) Variable Prandtl number has been introduced for hybrid nanofluid. (iv) Different needle thicknesses are examined and discussed to understand the flow and thermal behavior throughout the whole study. We’ve attempted to provide the answer to the following questions,

1. How does needle thickness affect the flow behavior and heat transfer rate?

2. How does the magnetic field strength affect the fluid motion, shear stress, and heat transfer rate due to variable fluid properties?

3. How does adding nanoparticles improve the heat transmission?

4. What kind of impact variable Prandtl number have on thermal distribution?

Mathematical formulation

We consider the steady laminar fluid flow of an electrically conducting hybrid nanofluid past a continuously moving thin needle. The flow is subject to a magnetic field of flux B₀ which is applied in the r direction. The flow configuration is illustrated at Figure 1. The needle velocity, $U_w$ is parallel to the free stream velocity, $U_{\infty }$ . The size of the needle is c, x, and r stand for the axial and radial coordinates and u and v are the corresponding velocity components. The needle is considered to be thin, and the needle thickness is smaller than that of the momentum and thermal boundary layer thickness. The surface $T_w$ and ambient temperature $T_{\infty }$ are constants, where $T_w-T_{\infty }>0$ . Based on the aforementioned assumptions, the fundamental equations^36,37,39 of the magnetic hybrid nanofluid flow are:

$$\frac{\partial }{\partial x}(\mathrm{ru})+\frac{\partial }{\partial r}(\mathrm{rv})=0,$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=U_{\infty }\frac{d{U}_{\infty }}{\mathrm{dx}}+\frac{1}{{\rho }_{\mathrm{hnf}}}\frac{1}{r}\frac{\partial }{\partial r}(r{\mu }_{\mathrm{hnf}}(T)\frac{\partial u}{\partial r})}\\ -\frac{{\sigma }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}{B}_0^2(u-U_{\infty }),\hfill \end{array}$$

(2)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{1}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}\frac{1}{r}\frac{\partial }{\partial r}(r{\kappa }_{\mathrm{hnf}}(T)\frac{\partial T}{\partial r})}\\ +\frac{Q_0}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}(T-T_{\infty })+\frac{{\sigma }_{\mathrm{hnf}}}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}{B}_0^2{(u-U_{\infty })}^2.\hfill \end{array}$$

(3)

with associated boundary conditions,

$$u=U_w,v=0,T=T_w\mathrm{at}r=R\left(x\right),$$

$$u\to U_{\infty },T\to T_{\infty }\mathrm{as}r\to \infty .$$

(4)

Figure 1.

Configuration of the flow field.

In this, T, ${\mu }_{\mathrm{hnf}}$ , ${\rho }_{\mathrm{hnf}}$ , ${\sigma }_{\mathrm{hnf}}$ , ${\left(\rho C_p\right)}_{\mathrm{hnf}}$ , ${\kappa }_{\mathrm{hnf}}$ characterize temperature, dynamic viscosity, density, electrical conductivity, effective heat capacity, and thermal conductivity of hybrid nanofluid. The thermophysical properties of hybrid nanofluid are given as follows,^19,31

$$A={\mu }_{\mathrm{hnf}}=\frac{{\mu }_f}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}},$$

$$B=\frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}=\frac{(1-{\phi }_2)[(1-{\phi }_1){\rho }_f+{\phi }_1{\rho }_{n1}]+{\phi }_2{\rho }_{n2}}{{\rho }_f},$$

$$\begin{array}{c}\multicolumn{1}{c}{C=\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_f}}\\ =[1+\frac{3({\sigma }_{n1}{\phi }_{n1}-\phi {\sigma }_f)+{\phi }_{n2}{\sigma }_{n2}}{{\sigma }_{n1}(1-{\phi }_{n1})+{\sigma }_{n2}(1-{\phi }_{n2})+(2+\phi ){\sigma }_f}],\hfill \\ \mathrm{where},\phi ={\phi }_{n1}+{\phi }_{n2},\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{D=\frac{{(\rho C_p)}_{\mathrm{hnf}}}{{(\rho C_p)}_f}}\\ =\frac{(1-{\phi }_2)[(1-{\phi }_1){(\rho C_p)}_f+{\phi }_1{(\rho C_p)}_{n1}]+{\phi }_2{(\rho C_p)}_{n2}}{{(\rho C_p)}_f},\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{E={\kappa }_{\mathrm{hnf}}}\\ =\left[\frac{{\phi }_1{\kappa }_{n1}+{\phi }_2{\kappa }_{n2}+2\phi {\kappa }_f+2\phi ({\phi }_1{\kappa }_{n1}+{\phi }_2{\kappa }_{n2})-2{\phi }^2{\kappa }_f}{{\phi }_1{\kappa }_{n1}+{\phi }_2{\kappa }_{n2}+2\phi {\kappa }_f+\phi ({\phi }_1{\kappa }_{n1}+{\phi }_2{\kappa }_{n2})+{\phi }^2{\kappa }_f}\right]\hfill \\ \times {\kappa }_f,\mathrm{where},\phi ={\phi }_{n1}+{\phi }_{n2}.\hfill \end{array}$$

The thermophysical characteristics of nanoparticles and water are defined in Table 1. The volume fraction of Graphene and Fe₂O₃ are denoted with $\phi 1$ and $\phi 2$ , and the notations $n1,n2$ are stand for the solid components, respectively. Here, $C_p,$ ρ, κ, σ are denoted as specific heat capacity at constant pressure, density, thermal conductivity, and electrical conductivity, respectively.

Table 1.

Physiothermal properties of relevant nanoparticles. ⁷

Thermophysical properties	Water	Graphene	Fe3O4
$C_p$ (J/kgK)	4180	2100	670
ρ (kg/m3)	997	2250	5180
κ (W/mK)	0.6071	2500	9.7
σ $({\Omega }^1{\mathrm{m}}^{-1}$ )	0.005	107	25,000

The simplification of the basic governing equations is done by introducing the following stream function and similarity transformations,^19,36

$$u=2U{f}^{\prime }\left(\eta \right),v=-\frac{{\nu }_f}{r}(f\left(\eta \right)-f^{\prime }\left(\eta \right),\lambda =\frac{U_{\infty }}{U},$$

$$\Psi =\nu \mathrm{xf}\left(\eta \right),\eta =\frac{U{r}^2}{{\nu }_{f}x},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}.$$

(5)

Here we consider the composite velocity, $U=U_w+U_{\infty }$ . Also, ${\nu }_f,{\mu }_f$ , ${\rho }_f$ , ${\sigma }_f$ , $\lambda ,{\kappa }_f$ stands for the kinematic viscosity, dynamic viscosity, density, electric conductivity, moving ratio parameter, and thermal conductivity of the fluid. We set up $\eta =c$ (refer to the wall of the needle). So, the surface of the needle can be expressed as,

$$R\left(x\right)={\left(\frac{{\nu }_f\mathrm{cx}}{U}\right)}^{1/2}.$$

(6)

The variable viscosity appearing at equation (2) is given by the relation,

$${\mu }_{\mathrm{hnf}}=\frac{{\mu }_f}{[1+\gamma (T-T_{\infty })]},$$

(7)

So that the viscosity established a relation of inverse linear function of temperature T,

$$\frac{1}{{\mu }_{\mathrm{hnf}}}=\mathrm{\Gamma }(T-T_r),$$

(8)

Where $\Gamma =\frac{\gamma }{{\mu }_f}$ and $T_r=T_{\infty }-\frac{1}{\gamma }$ in this above Γ and $T_r$ are constants, their values depend on the reference state and γ is a thermal property of the fluid. The dimensionless temperature is:

$$\theta =\frac{T-T_r}{T_w-T_{\infty }}+\Omega ,$$

(9)

Here,

$$\Omega =\frac{T_r-T_{\infty }}{T_w-T_{\infty }}=-\frac{1}{\gamma \left(T_w-T_{\infty }\right)}=constant,$$

(10)

The value of $\Omega$ can be defined by the viscosity and temperature characteristic of the considered fluid where $\mathrm{\Delta }T=T_w-T_{\infty }$ . So, equation (8) become, ³⁷

$${\mu }_{\mathrm{hnf}}\left(T\right)={\mu }_{\mathrm{hnf}}\left(\frac{\mathrm{\Omega }}{\mathrm{\Omega }-\theta }\right)$$

(11)

The variation of the thermal conductivity κ in this study is expressed as a function of temperature, ³⁹

$${\kappa }_{\mathrm{hnf}}\left(T\right)={\kappa }_{\mathrm{hnf}}(1+\epsilon \theta \left(\eta \right)).$$

(12)

Where, ε stands for thermal conductivity parameter.

After introducing all the above relations to the system of equations (1)–(3) we have the following system of nonlinear ordinary differential equations:

$$\mathrm{ff}″+\frac{2A}{B}\left(\frac{\mathrm{\Omega }}{\mathrm{\Omega }-\theta }\right)[(\eta f″)\prime +\eta \left(\frac{\theta \prime f″}{\mathrm{\Omega }-\theta }\right)]-\frac{C}{4B}M(2f^{\prime }-\lambda )=0,$$

(13)

$$\begin{array}{c}\frac{E}{D}(1+\epsilon \theta )[(\eta \theta \prime )\prime +\frac{\eta \theta {\prime }^2\epsilon }{(1+\epsilon \theta )}]+\frac{1}{2}P{r}_{f}f\theta \prime +\frac{1}{4D}P{r}_{f}Q\theta \hfill \\ +\frac{C}{4D}\mathrm{MEc}(2f\prime -\lambda {)}^2=0.\hfill \end{array}$$

(14)

Variable Prandtl number

Due to the fluctuation of viscosity over the boundary layer, the variable Prandtl Number is defined as following, ³⁶

$$P{r}_f\left(T\right)=\frac{{\mu }_{\mathrm{hnf}}\left(T\right)C_p}{k_{\mathrm{hnf}}\left(T\right)}=\frac{A\left(\frac{\mathrm{\Omega }}{\mathrm{\Omega }-\theta }\right)C_p}{E(1+\epsilon \theta )},$$

(15)

So, we get,

$$\frac{E}{A}(1-\frac{\theta }{\mathrm{\Omega }})(1+\epsilon \theta )P{r}_f\left(T\right)=P{r}_f.$$

By employing (15), equation (14) became,

$$\frac{E}{D}(1+\epsilon \theta )[(\eta \theta \prime )\prime +\frac{\eta {\theta \prime }^2\epsilon }{(1+\epsilon \theta )}]+\frac{1}{2}\frac{E}{A}(1-\frac{\theta }{\mathrm{\Omega }})(1+\epsilon \theta )P{r}_f\left(T\right)f\theta \prime$$

$$\begin{array}{c}+\frac{1}{4D}\frac{E}{A}(1-\frac{\theta }{\mathrm{\Omega }})(1+\epsilon \theta )P{r}_f\left(T\right)Q\theta +\frac{C}{4D}\frac{E}{A}(1-\frac{\theta }{\mathrm{\Omega }})\hfill \\ (1+\epsilon \theta )P{r}_f\left(T\right)\mathrm{MEc}{(2f\prime -\lambda )}^2=0.\hfill \end{array}$$

(16)

Subject to the corresponding boundary conditions,

$$f\prime \left(c\right)=\frac{1}{2}(1-\lambda ),f\left(c\right)=\frac{c}{2}(1-\lambda ),\theta \left(c\right)=1,$$

$$f\prime \left(\infty \right)=\frac{\lambda }{2},\theta \left(\infty \right)=0.$$

(17)

Here, prime’ indicate the differentiation with respect to η, the symbolic meaning of $\mathrm{Ec},M,Q,P{r}_f,\lambda ,R{e}_x$ denotes Eckert number, Magnetic field parameter (Hartmann number), Heat generation parameter, Prandtl number, velocity ratio parameter, and Reynolds number, respectively, defined as:

$$\begin{array}{c}\mathrm{Ec}=\frac{U^2}{{\left(C_p\right)}_f(T_w-T_{\infty })},M=\frac{\sigma B_0^{2}x}{{\rho }_{f}U},Q=\frac{Q_{0}x}{{\left(\rho C_p\right)}_{f}U},\hfill \\ \lambda =\frac{U_{\infty }}{U},R{e}_x=\frac{\mathrm{Ux}}{{\nu }_f}.\hfill \end{array}$$

(18)

Though the parameters are identified as local similarity solutions, for a very small value of x, the transformed ODE equations are valid. However, if we want to consider the global similarity solutions then the magnetic field parameter, M and heat generation parameter, Q should be x independent. To achieve that,

we have, $M=\frac{\sigma B_0^{2}x}{{\rho }_{f}U}$ . Thus, we can consider $x^{-\frac{1}{2}}$ as a form of $B_0$ ,

so that we have, $M=\frac{\sigma B^2}{{\rho }_{f}U}$ where, $B_0~\left(x^{-\frac{1}{2}}\right)$ .

Again, $Q=\frac{Q_{0}x}{{\left(\rho C_p\right)}_{f}U}$ . Thus, we can consider $x^{-1}$ as a form of $Q_0$ , so that we have $Q=\frac{q}{{\left(\rho C_p\right)}_{f}U}$ , where $Q_0~(x^{-1})$ .

Finally, we have, $M=\frac{\sigma B^2}{{\rho }_{f}U}$ and $Q=\frac{q}{{\left(\rho C_p\right)}_{f}U}$ .

The engineering interest parameters are the local coefficient of skin friction $C_f$ and the local Nusselt Number $N{u}_x$ defined as given below,

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{Re}}_x^{\frac{1}{2}}(1-\frac{\theta (\eta )}{\mathrm{\Omega }})C_f=8c^{\frac{1}{2}}\frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}f″(c),{\mathrm{Re}}_x^{-\frac{1}{2}}N{u}_x}\\ =-2c^{\frac{1}{2}}\frac{k_{\mathrm{hnf}}}{k_f}(1+\epsilon \theta \left(\eta \right))\theta \prime \left(c\right).\hfill \end{array}$$

(19)

Numerical method

The solution of the non-linear coupled system of equations (13) and (16) subject to the boundary conditions (17) is attained using MAPLE 13 worksheet software. To solve numerically the boundary value problem, this particular software uses fourth fifth order Runge-Kutta-Fehlberg (RFK45) method. For all parameter’s values, (101–100λ) is replaced the unity determined by the pertinent parameter in order to obtain the results. In the dsolve command abserr   =   0.04 , continuation = λ. Without this adjustment, MAPLE provides results which don’t correspond to the asymptotic values specified by equation (17). Additional information can be found in MAPLE’s help section on overcoming convergence challenges under the section of Numerical Solution of Difficult ODE Boundary Value Problems. Using the value of similarity variable, ${\eta }_{\max }$ = 6 in the asymptotic boundary condition equation (17) is replaced as given below,

$${\eta }_{\max }=6,f\prime \left(6\right)=\frac{\lambda }{2},\theta \left(6\right)=0\mathrm{when}\lambda =0.6.$$

(20)

The values used here guaranteed that all numerical solutions appropriately approached. Heat transfer related several papers have repeatedly proven the robustness and accuracy of this program.^43–45 The classic Runge-Kutta (RK4) method is a widely used and effective numerical method for solving nonlinear ODEs. To achieve sufficient reasonable accuracy and overcome the challenges of solving the initial-value problems, this method is constructed with multiple small steps. Later on, another adaptive Runge-Kutta method has been proposed and named as fourth-fifth-order Runge-Kutta-Fehlberg (RKF45) method which employs embedded integration formulas. Tracking the truncation error in every step of integration, the RKF45 method is proven to reduce the error within the permitted range by adjusting the size of steps. An investigation has been carried out to look at the former’s computational advantage by comparing both classic RK4 and RKF45 methods ⁴⁶ which found that the RKF45 method is more efficient to get the precise accuracy than that of the RK4 method. Through the strategy of adaptive time stepping, RKF method upgrades the code robustness, integrity of results, as well as overall efficiency.

Results and discussion

This current study interprets the attributes of distinguished physical parameters that is, viscosity parameter Ω, magnetic field parameter M, thermal conductivity parameter ε, heat generation parameter Q, Eckert number Ec, Prandtl number Pr, moving ratio parameter over the common profiles of fluid flow like velocity, temperature, skin friction coefficient, and heat transfer rate. For the computation, the volume fraction of Fe₂O₃ ( ${\phi }_2$ ) and Graphene ( ${\phi }_1$ ) were at 0.1.

In Table 2, the numerical results of the present study are compared with former published data^14,15,17,18 to validate the authenticity of the existing computation. The comparison of $f″(c)$ is performed for viscous fluid ( ${\phi }_1={\phi }_2=0$ ) when λ = 0 and Pr = 0.733 in the absence of other parameters with different thin needle thickness (c). Depilation in the numerical values of $f″(c)$ have observed for the thicker needle size. Furthermore, different needle sizes have been taken under consideration to observe the flow behavior.

Table 2.

Comparison of numerical results of $f″(c)$ when λ = 0.

c	Present study	Ishak et al. 15	Chen and Smith 18	Grosan and Pop 14	Soid et al. 17
0.01	8.496486	8.4924	8.49244	8.492173	8.491454
0.1	1.288634	1.28881	1.28881	1.289074	1.288778
0.2	0.778743				0.751665

The flow behavior concerning the velocity and temperature profile of hybrid nanofluid for incremental values of magnetic field is depicted at Figures 2 and 3, respectively. By imposing higher values of M, the magnetic field increases. For these phenomena the Lorentz force induced, elevates the internal friction and hence the fluid motion is increased. So, the momentum boundary layer thickness is decreased. While the temperature distribution of the fluid shows lessen impact as stronger magnetic field is applied. The thinner the needle, the thicker the momentum boundary layer thickness is. However, reverse trend has been evident in thermal boundary layer. The hydromagnetic flow etiquette of hybrid nanofluid are represented with the nonzero (M $>0$ ) values of M.

Figure 2.

Variation of M with needle size on $\mathrm{f}\prime (\eta )$ .

Figure 3.

Variation of M with needle size on $\theta (\eta )$ .

The performance evolution of escalated volume fraction of Fe₂O₃ ( ${\phi }_2$ ) is illustrated at Figures 4 and 5 on $f\prime (\eta )$ and $\theta (\eta )$ with different thickness of needle. Physically, the larger values of ${\phi }_2$ result to vehemently subdue fluid motion. The rationality behind this deceleration is due to the incremented viscidness which slow down the fluid motion. Though for thinner needle the fluid motion is quicker. However, from Figure 5 it is clear that the temperature of fluid rises for higher values of ${\phi }_2$ due to the augmentation in thermal conductivity. In comparison with other needle, thicker needle c = 0.1 is more efficient for temperature distribution and greater thermal boundary layer thickness has been observed for it.

Figure 4.

Variation of ${\phi }_2$ with needle size on $\mathrm{f}\prime (\eta )$ .

Figure 5.

Variation of ${\phi }_2$ with needle size on $\theta (\eta )$ .

The impact of advancing values of Eckert number, Ec is demonstrated at Figure 6 for the temperature distribution. It reveals that larger Eckert number enhance the temperature in fluid region where thermal energy is created by converting mechanical energy. The logic for this construction of upsurge is that the viscous force is produced by the heat dissipation which induces the friction between fluid particles. As a result, the drag force of particles facilitates the production of heat so internal heat generation developed which augments the temperature.

Figure 6.

Variation of Ec with needle size on $\theta (\eta )$ .

Figures 7 and 8 demonstrate the upshot of variation of the diverse variable viscosity parameter on fluid velocity and temperature regimes. The fluid velocity upgrades with the enlarged magnitude of Ω. Whereas, a diminishing trend of fluid motion has been found with incrementing needle size. This significant formation indicates that the contact surface decreases between fluid particles and the moving thin needle. As a result, drag force recedes down, thereby velocity enhances. So, the momentum boundary layer decreased with the augment of needle thickness. Moreover, inertial force overpasses the viscous force due to belittle fluid velocity, which escalates the fluid motion. However, a reverse trend has been found in temperature distribution.

Figure 7.

Variation of Ω with needle size on $\mathrm{f}\prime (\eta )$ .

Figure 8.

Variation of Ω with needle size on $\theta (\eta )$ .

Figures 9 and 10 demonstrate the fluctuation of distinct values of heat generation parameter Q, on the velocity and temperature profile. Reduction of velocity $f\prime (\eta )$ has been noticed for the rising values of Q. Momentum boundary layer decreases with escalating needle size. The deceleration indicates that heated fluid restrains the fluid motion. While Q = 0 represents that the fluid absorbs no heat, Q $>0$ represents the presence of heat source. For higher input of Q, fluid absorb more heat which enhance the temperature of the fluid. The rise in temperature is due to the upsurge in internal source of fluid energy. The thermal boundary layer is escalated for thicker needle size.

Figure 9.

Variation of Q with needle size on $\mathrm{f}\prime (\eta )$ .

Figure 10.

Variation of Q with needle size on $\theta (\eta )$ .

Figure 11 elucidates the influence of dissimilar thermal conductivity parameter ε, on the dimensionless temperature field. Temperature of the fluid is getting accelerated by larger input in thermal conductivity parameter. According to physical theory, the augmentation in fluid temperature fully depends on the higher thermal conductivity which is one of the fundamental physical properties of the fluid. So, it is clarified that, for constant thermal conductivity (ε = 0), the fluid temperature is minimum throughout the boundary layer while incremented thermal conductivity (ε > 0) results in a significant rise of the temperature. Moreover, the thermal boundary layer thickness is enhanced with thicker (c = 0.1) needle.

Figure 11.

Variation of ε with needle size on $\theta (\eta )$ .

The velocity and temperature curve for the fluctuation of the Prandtl number, Pr, are respectively shown in Figures 12 and 13. While fluid motion is uplifted, fluid tempeture decreases as Pr (Pr $>>1$ . ) values rise, indicating a smaller thermal boundary layer. According to the definition, the ratio of momentum diffusivity to thermal diffusivity is defined as Pr, implying that the Prandtl number is reciprocal to thermal conductivity and proportional to momentum diffusivity. As a result, velocity acceleration and temperature declination are noticed in the area of fluid movement for higher input in Pr. The noteworthy fact is that for thinner needle, lowest thermal boundary layer and highest momentum boundary layer are formed.

Figure 12.

Variation of Pr with needle size on $\mathrm{f}\prime (\eta )$ .

Figure 13.

Variation of Pr with needle size on $\theta (\eta )$ .

Figure 14 demonstrates the reaction of wall shear stress $f″(c)$ for dissimilar needle sizes c, and the fluid viscosity parameter Ω, with the variation of the magnetic field parameter M. Incremented values of c and Ω uplift the wall shear stress of the surface at stronger magnetic field. Figure 15 portrays the fluctuation of heat transfer rate for distinct thermal conductivity parameter, ε with different needle size in response to magnetic field. For the increasing magnitude of thermal conductivity parameter and needle size heat transfer rate diminish in the presence of augmented magnetic field. The dissimilarity of Prandtl number, Pr for different needle has been depicted on $-\theta \prime (c)$ in the influence of magnetic field in Figure 16. Escalated Pr and c show a noteworthy rise on heat transfer rate distribution of moving thin needle over enhanced magnetic field. Figure 17 exhibits the flow formation for larger numerical values of Pr and thermal conductivity parameter ε, on $-\theta \prime (c)$ for distinct size of needle which demonstrate significant upsurge in heat transfer rate. For thinner needle the heat transfer rate results to higher rate of heat transfer rather than the thicker one.

Figure 14.

Impact of Ω and M on $f″(c)$ .

Figure 15.

Impact of ε and M on $-\theta \prime (c)$ .

Figure 16.

Impact of Pr and M on -θ′(c).

Figure 17.

Impact of Pr and ε on $-\theta \prime (c)$ .

In this following section, the numerical computation of skin friction coefficient $f″(c)$ and heat transfer rate $-\theta \prime (c)$ will be discussed which has been shown in Table 3. To observe the characteristics of the distinct inflicted relevant parameter which are Ec, M, ${\phi }_2$ , Ω, Q, ε, and Pr on heat flux and drag force when c = 0.01, Ec = 0.3, M = 1, ${\phi }_2$  = 0.1, Ω = 2, Q = 0.2, ε = 0.6, Pr = 6.2. Incremented amount of both magnetic field M and fluid viscosity parameter Ω enhance the wall shear stress and heat transfer rate of moving thin needle surface. Increasing needle thickness c, results to reduction in both wall shear stress and heat transfer. For higher input in Ec, ${\phi }_2$ , Q, ε heat transfer rate is escalated but the skin friction coefficient is receded down. With the larger Prandtl number Pr, the heat transfer rate strongly shows an improvement which indicates that Prandtl number is proportional to heat transfer rate. The physical explanation for this trend is because of the lower thermal diffusivity which is implied by the higher input in Pr. For this reason, t thickness of temperature boundary layer couldn’t rise relatively fast as that for a smaller Prandtl number. The higher wall temperature gradient implied by the thinner boundary layer results in an incremented value of the local Nusselt number. Reverse tendency is noticed for the skin friction coefficient.

Table 3.

Numerical values of shear stress and heat transfer rate.

c	Ec	M	${\mathit{\phi }}_2$	Ω	Q	ε	Pr	$\mathit{f}″(\mathit{c})$	$-\mathit{\theta }\prime (\mathit{c})$ .
0.010.030.1	0.3	1	0.1	2	0.2	0.6	6.2	1.44480.60910.2619	15.65646.39752.8085
	012							1.44371.447981.4521	15.763915.401315.0417
		013						1.12871.45011.8159	15.551315.592215.6184
			0.010.050.1					1.42671.43671.4448	16.249215.978415.6564
				2510				1.44481.91972.0480	15.656415.709515.7241
					00.20.4			1.42001.45021.5010	17.761615.592212.3484
						00.30.7		1.43031.45021.4683	17.793815.592213.8505
							36.210	1.46611.44481.4237	14.689815.656416.7133

It is concluded that thinner needle size results to higher heat transfer rate of needle surface. The boundary layer is proportional to several parameters like magnetic parameter, viscosity parameter, and Prandtl number and inversely proportional to heat generation parameter and volume fraction of Fe₃O₄. With a strong magnetic effect, incremented viscosity parameter and Prandtl number enhance shear stress and heat transfer rate of surface, respectively. Increasing values of thermal conductivity parameter and Prandtl number augment the rate of heat transfer to the moving needle surface.

Conclusions

The incompressible laminar steady hybrid nano-coating fluid (Fe₃O₄/Graphene-water) flow through moving thin needle in the presence of magnetic field has been studied numerically using the MAPLE software. The findings reveal a significant impact on the temperature adopting variable viscosity and thermal conductivity for different Prandtl numbers which have significant implications in industrial sectors of manufacturing. The flow energy and mass transport are systematically explored for various values of the parameters along with the skin friction coefficient and heat transfer rate for different needle sizes. The final outcomes are given below:

• For thinner needle, the momentum boundary layer thickness is higher but opposite trend is found for the temperature boundary layer. In comparison with thicker needle, thermal boundary layer shows reduction for thin needle. This indicates that thin needle (c = 0.01) increases the velocity but decreases the fluid temperature.

• With the greater values in magnetic field strength M, momentum boundary layer increases as $f\prime (\eta )$ is uplifted. Declination is found for temperature distribution. For stronger magnetic influence both the wall shear stress and heat transfer rate of surface escalates.

• Augment in volume fraction of Fe₂O₃ ( ${\phi }_2$ ) and heat generation parameter undermine fluid motion while significant upsurge has been found for fluid temperature. The rate of heat transfer decreases for added nanoparticles.

• Larger input in temperature dependent fluid viscosity parameter Ω, results to subdue the temperature of fluid but greater performance in fluid motion has been observed.

• Thermal distribution shows a notable acceleration which is influenced by both temperature dependent thermal conductivity parameter ε, and Eckert number Ec, which represents enlarged thermal boundary layer.

• Rise in the Prandtl number Pr, elevate the velocity and reduce the temperature.

This present work can be extended for three-dimensional flow of different fluids like Casson fluid, power-law fluid, and non-newtonian viscous fluid. Other surfaces like cylindrical, rotated cone, vertical plate, etc could be taken under consideration. We investigate the flow subject to variable thermophysical aspects to achieve thermal efficiency. Though we consider low viscous fluid in our study, it has been found that highly viscous fluid elevates skin friction as well as wall heat transfer rate. However, the peak enhancement in wall heat transfer rate was computed when the fluid absorbs no heat.

Limitations of the defined problems are:

1. The hybrid nanofluid (Fe₃O₄/Graphene-water) is flowing over a moving thin needle along the x-axis.

2. The heat convections are taken at the boundary of the needle surface for the purpose of temperature regulation.

3. Temperature of the fluid is controlled by considering the effect like heat generation where the exponential temperature dependent on heat source/sink.

4. The hybrid nanofluid is scrutinized to be stable without any accumulations phenomenon of selected magnetic ferrite (Fe₃O₄) and graphene nanoparticles.