Heat transport and magnetohydrodynamic hybrid micropolar ferrofluid flow over a non-linearly stretching sheet

1.   Introduction

In order to meet industrial and technical requirements, a smart fluid with exceptional thermal capabilities is desired. A novel heat transfer fluid created using nanotechnology termed a nanofluid was first described by Choi ^[1] in 1955. A growing proportion of nanotechnology for heat transmission is called nanofluids, which are colloidal mixtures of nanoparticles (1–100 nm) and a base liquid (nanoparticle fluid suspensions) (see Buongiorno et al. ^[2]). Figure 1 shows a comparative sample of different sizes of materials from large scales to nanoscales. Polymers, metals, metal oxides, and other substances may all be found in nanofluids. The thermophysical properties of traditional thermal fluids are improved significantly by the inclusion of nanoparticles, improving the heat transfer coefficient. Nanofluids are currently widely used in a number of industries, including the electronics, automotive, solar energy, medicinal, and oil recovery industries, as a result of such a wider acceptance (see Singh et al. ^[3] and Suvardhan et al. ^[4]).

Researchers are presently drawn to hybrid and dihybrid nanofluid due to the advantages of nanofluid and the growing study into flow, heat, and mass transport phenomena. Hybrid nanofluid is the term used to describe a combination that contains several nanoparticles. It is referred to as a "dihybrid nanofluid" if the combination contains several nanoparticles and fluids ^[5]. According to Lou et al. ^[6], the key benefit of hybrid nanofluids is increased energy efficiency, conservation, and savings. In order to achieve uniform shear flow over a stretched sheet, Waini et al. ^[7] investigated the effects of transpiration on the hybrid nanofluid flow. They discovered that an improvement in copper nanoparticle volume fractions results in an improvement in heat transmission.

The heat conduction study of a stretching surface constantly moving along an ambient fluid has received a lot of attention and has been widely described in the literature. The composition of the ambient liquid and the pace of stretching determines the properties of products created through engineering procedures such as metallurgy and polymer extrusion. Because quick stretching or irregular temperature changes in the extrudate can compromise the intended qualities of the last product, the heat exchange rate must be exactly controlled. Nanofluids are mixtures of a base fluid and nanoparticles that are homogeneous. The particles in these suspensions can be metal or non-metal. Ferrofluid is a class of nanofluid in which the suspension of nanoparticles can be magnetized. Ferrofluids offer a lot of potential in these kinds of applications. Ferrofluids are fluids made artificially from extremely reduced mixture suspensions of tiny magnetic molecules suspended in a carrier fluid that is not conducting. This fluid acts normally except for the fact that it is subjected to a force owing to magnetization. Ferrohydrodynamics is the study of magnetic fluid motion mechanics that is transformed by strong magnetic polarization forces. The existence of an external magnetic source may affect significantly the dynamics of electrically conducting fluids flowing effectively in its spawned field ^[8]. The effects of magnetization on fluid flow with magnets were first described by Rosensweig ^[9]. In the last several years, for Newtonian and non-Newtonian fluids, as well as flat horizontal surfaces, flow and heat transfer have been extensively studied under various hydrodynamic and thermal boundary conditions ^[10,11,12]. Near the infinite plate, heat transport and magnetohydrodynamic (MHD) flow have been investigated. To make a ferrofluid flow, Andersson and Valnes ^[13] were the first to use a stretched surface with a magnetic dipole. Andersson ^[13,14] showed that the impact of an externally provided magnetic field on flow is similar to that of viscoelasticity.

The effects of nonlinear stretching or shrinking on MHD flow are discussed ^[15,16,17]. For flow past a stretching/shrinking sheet, two solutions were discovered in ^[18]. Furthermore, it's crucial to research boundary layer flow and heat transmission across a stretched surface, because it has a wide range of uses in industrial engineering and production processes, such as plastic sheet aerodynamic expulsion and paper manufacturing, manufacture of glass fiber. In the polymer sector, heat-treated materials are transported between feed and wind-up rolls, drawing using wires, whirling metal, and many others ^{[19,20,21,22]}. Hayat et al. ^[23] also came up with a solution based on an analysis of a second-grade fluid's slip flow and heat transfer through a planar stretching sheet. Martin and Boyd ^[24] looked at the steady wall temperature, heat transfer, and slide flow past a flat surface. Their findings show that boundary layer equations may be utilized to examine flow at the microelectromechanical systems (MEMS) scale and that they can be used to investigate rarefaction's impact on shear stress and flow structure. In chemical industries, such as the development of nuclear waste disposal canisters and the cooling systems for nuclear reactors, it's common to employ mixed convectional flow with heat transmission. These physical processes take place when the effects of buoyant forces become even more powerful. Ali ^[25] looked into the impact of viscosity that varies with temperature on heat transmission via laminar mixed convection along a surface that is constantly shifting. According to the literature study, there are just a few research on heat transmission from ferrofluids. Li and Xuan ^[26] explored the effects of an external magnetic field on convective heat transport using a ferrofluid flow through a thin wire. They discovered that the magnetic field from the outside affects the ferrofluids' heat transmission via convection performance. When Yamaguchi et al. ^[27] analyzed the convection naturally of the fluid with magnetic properties in a square hollow with partitions having a magnetic field imposed from the outside computationally, they discovered that a substantial magnetic field influences the convection flow. Motozawa et al. ^[28] explored the influence of heat transfer in a magnetic fluid flow with a magnetic field in a duct that is rectangular. They observed that the heat transfer coefficient rises locally in the presence of a magnetic field. The coefficient of heat transfer increases as the magnetic field increases. Characteristics of ferrofluid flow over a linearly stretching have already been discussed in previous work.

The thorough review of the literature demonstrates that, the study of heat transfer and MHD hybrid nano-fluid with micropolar ferrofluid flow over a non-linearly stretching sheet does not exist in literature. In this article, we considered this problem with two different nanoparticles ($A{l}_{2}{O}_{3}$ -Cu) suspended in the base fluid ethylene glycol (EG). Using the suitable similarity transformations, the governing equations are transformed into a set of nonlinear ordinary differential equations, which are then solved using the built-in MATLAB module BVP-4c.

2.   Materials and methods

The topic of micropolar and classical nanofluid across a stretched sheet has been studied by several researchers. The flow of a micropolar/classical ferrofluid on a stretched sheet in the y > 0 regions was presumed steady-state in this research. At x = 0, the surface is stretched in the x-direction and normal in the y-direction. The stretched sheet's surface velocity is u_w(x) = a ${x}^{m}$, where $a$ is an optional constant that is positive. A magnetic field perpendicular to the surface is also present (See Figure 2). Here are the micropolar ferrofluid governing equations based on the preceding assumptions:

The continuity equation and momentum equations are ^[29],

The thermophysical properties ${\mu }_{hnf}\left(\beta \right), {\rho }_{hnf}, {K}_{hnf}, {\left(\rm{ \mathsf{ ρ} }{\beta }_{T}\right)}_{hnf}, {\left(\rm{ \mathsf{ ρ} }{\beta }_{C}\right)}_{hnf}, {\left(\rm{ \mathsf{ ρ} }{C}_{p}\right)}_{hnf}~~and~~{\sigma }_{hnf}$ for hybrid nanofluid (Al₂O₃-Cu/EG) are defined as,

where,

where,

where ${\rm{ \mathsf{ ϕ} }}_{1}, {\rm{ \mathsf{ ϕ} }}_{2}$ are solid volume fraction of Alumina/Aluminium oxide, Cuprum/Copper, respectively, C_p is the specific heat, k_f denotes the thermal conductivity of the regular fluid. The values for the thermophysical parameters for hybrid nanofluid (Al₂O₃-Cu/EG) is shown in Table 1.

Also, flow is incompressible with no external forces, and the governing equations are:

The body force proposed by Hussanan et al. is:

The governing equations (momentum equations) take the form ^[30,31],

The energy equation is

and, according to Rosseland's guess, it is

The problem's boundary conditions are

For a better understanding of these formulas, the velocity components in the x and y directions are assumed to be u and v, respectively. The angular velocity is denoted by N. $\mu$ is viscosity in a dynamic state, $\delta$ is a constant in the $0\le \delta \le 1$ region. When $\delta = 0$, microelements close to the wall surface are not allowed to spin. If $\delta = \frac{1}{2}$ demonstrated the insufficiency for the concentration of microelements. Furthermore, $\delta = 1$ when $\delta$ involve in the turbulent boundary layer flow. Micro inertia per unit mass is j, $\gamma$ is the viscosity of a spinning gradient and k is the viscosity of the vortex. To solve these nonlinear differential equations quickly and easily, we made use of similarity transformations such as,

By replacing the variables into the Eqs (16), (17), (19)–(21), we have

And the boundary conditions are

The non-dimensional variables are subsequently presented

and,

where K represents the Micro-rotation parameter, ${K}_{l}$ is the local micro-rotation parameter, j is micro-inertia density, R is radiative parameter stretching and Pr is Prandtl number.

3.   Engineering quantities of interest

For this modeled flow problem, the skin friction coefficient and Nusselt, Sherwood number are defined as

where ${q_w}$ represents surface heat flux, ${\tau _z}$ denotes surface shear stress, and ${q_m}$ defines surface mass flux for ferrofluid:

Now apply the similarity transformation Eq (22) to both Eqs (30) and (31), and we get the results:

4.   Numerical results and discussion

In this article, a numerical approach is used to investigate and assess some of the physical parameters of the ferrofluid flow over a stretching sheet, such as stream function $f\left(\eta \right)$, velocity ${f}^{\text{'}}\left(\eta \right)$, micro-rotation velocity $G\left(\eta \right)$, and temperature $\theta \left(\eta \right)$. The numerical solution of nonlinear coupled differential equations is obtained by using the MATLAB software BVP4c method. Figures 3 and 4 depict that with the increase in the weakness of concentration of microelement $\delta$, the stream function $f\left(\eta \right)$ and the velocity ${f}^{\text{'}}\left(\eta \right)$ decreases. Whereas the effect of the weakness of concentration of micro-element $\delta$ on the micro-rotating velocity $G\left(\eta \right)$ and temperature $\theta \left(\eta \right)$ is opposite as can be seen from Figures 5 and 6. Figures 7, 8, and 10 exhibit that with the increase in micro-rotation parameter K, the stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$ and micro-rotation velocity $G\left(\eta \right)$ increase. While the effect of micro-rotation parameter K on temperature is converse as is visible in Figure 9. Figure 11 shows that the stream function $f\left(\eta \right)$ increases with the increase of the local micro-rotation parameter ${K}_{1}$. On the other hand, micro-rotation velocity $G\left(\eta \right)$ decreases whenever the local micro-rotation parameter k1 increases as depicted in Figures 12–14 represent the effect of m on stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$. As m increases the stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$ increases. Even so with the increase of m temperature $\theta \left(\eta \right)$ and micro-rotation velocity $G\left(\eta \right)$ decreases as disposed of in Figures 15 and 16.

Figures 17–20 are created to represent the effects of magnetic parameters on stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$, temperature $\theta \left(\eta \right)$ and micro-rotation velocity $G\left(\eta \right)$ respectively, under the presumption that the other parameters have default values. When M increases then stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$ decrease display in Figures 17 and 18. The velocity field decreases as greater resistance to the flow phenomena is created by the application of the applied magnetic field. So, it is seen that the stream function $f\left(\eta \right)$ and velocity curve ${f}^{\text{'}}\left(\eta \right)$ are reduced as a result of the Magnetic improvement. In contrast increases for temperature profile $\theta \left(\eta \right)$ and micro-rotation velocity $G\left(\eta \right)$ as present in Figures 19 and 20. Magnetic materials experience a change in flux density as their temperature increases and decreases from an ambient temperature. Figure 21 illustrates the effect of shape factor n on temperature $\theta \left(\eta \right)$. When n = 3, 6.3698 and 16.1576 then its sphere, cylinder, and laminar respectively. Figures 22–24 exhibit the operation of ${\phi _1}$ on-stream function $f\left(\eta \right),$ velocity ${f}^{\text{'}}\left(\eta \right)$ temperature $\theta \left(\eta \right)$ increase with the value ${\phi _1}$ increases. Unlike for micro-rotation velocity $G\left(\eta \right)$ in Figure 25. Stream function $f\left(\eta \right)$ and velocity ${f}^{\text{'}}\left(\eta \right)$ are decrease as well as increase the value of ${\phi }_{2}$ as present in Figures 26 and 27. Whereas in Figures 28 and 29 illustrate temperature $\theta \left(\eta \right)$ and micro-rotation velocity $G\left(\eta \right)$ increase when increasing the value of ${\phi }_{2}$. Figures 22–29 show, the effects of the solid volume fractions of Alumina/Aluminum oxide and Copper/Copper on the thermal field. The volume fraction of Alumina/Aluminium oxide and Cuprum/Copper are boosting the thermal phenomena. However, as compared to ${\phi _1}$, the thermal profiles in case of ${\phi }_{2}$ are more obvious. Due to the nanoparticle volume fractions, the behaviour of these figures is consistent with the physical behaviour of the nanofluid. The thermal conductivity of the nanoparticles is greater than that of the base fluid, which increases the total thermal conductivity of the nanofluid and contributes to the rise in boundary layer temperature. Figure 30 display that as the Prandtl number increases the temperature $\theta \left(\eta \right)$ decrease. It is evident that the temperature distribution is a decreasing function of Prandtl number. This is because the thermal diffusivity of the fluid decreases due to higher values of Pr which further led the reduction in the thermal boundary layer thickness. The impact of radiation factors on temperature $\theta \left(\eta \right)$ are examined in Figure 31. More heat is introduced to the thermal phenomena as a result of changing the radiation parameter. More heat is introduced to the thermal phenomena as a result of changing the radiation parameter. As a result, the temperature curves are increased by the increasing radiation parameter value. Physically, by raising the value of the parameter R, we may enhance the radiative heat transfer.

The numerical outcomes of the Nusselt number, skin coefficient friction and Sherwood number are shown. The heat transfer factor is shown in Table 2 to see if the values of $\delta$, Pr, ${\phi }_{1}$, ${\phi }_{2}$, Alpha, and M are inclined then heat transfer is also inclined. The effect of a different number of stretching parameters and other numbers on the skin coefficient friction is numerically shown in Table 2. When the quantities of $\delta$, Pr, and M increase then skin coefficient friction expands, and if ${K}_{1}$ expands, and the skin coefficient friction declines. The effect of M, Pr, ${K}_{1}$, and $\delta$, ${\phi }_{1}$, ${\phi }_{2}$, on the Sherwood number is shown in Table 3. If the value of ${K}_{1}$ increases, then the mass transfer rate decreases. When the values M, m, K, and $\delta$ increase then the mass flow rate increase. The effect of various parameters $\delta$, Pr, R, n, ${K}_{1}$ and M on Nusselt number has been depicted in Table 4. It has been observed that when the values of parameters $\delta$, M, R and N increase then the heat transfer expands, and if k, Pr and ${K}_{1}$ expand, the Nusselt number decline.

Table 5 provides the comparative study of hybrid nanofluid and nanofluid under the effect of magnetic parameter $M$ on Nusselt number, Skin friction, and Sherwood number. It is observed that the Nusselt number or heat transfer rate and shear stress or skin friction and Sherwood number or surface mass flux of hybrid nanofluid are better as compared to nanofluid flow.

Table 6 offers the proportional study of hybrid nanofluid and nanofluid under the effect of boundary parameter $\delta$ on Nusselt number, Skin friction, and Sherwood number. It is detected that the Nusselt number or heat transfer rate and shear stress or skin friction and Sherwood number or surface mass flux of hybrid nanofluid are better as compared to nanofluid flow. The values of other parameters in Tables 5 and 6 are taken as $Rd = 1.2$, $R = 3$, $n = 3$, $K = 6$, $Pr = 0.8$, $m = 3$, ${K}_{1} = 2.5$, $\delta = 0.2$, $\alpha = 5$, $M = 2$.

In the comparison Tables 5 and 6, we have considered (a) ${\phi }_{1} = 0.2$ and ${\phi }_{2} = 0$ in the case of nanofluid (b) ${\phi }_{1} = 0.1$ and ${\phi }_{2} = 0.1$ in the case of hybrid nanofluid, where $A{l}_{2}{O}_{3}\to {\phi }_{1}$ and $Cu\to {\phi }_{2}$.

5.   Conclusions

In this study, the microstructure and inertial characteristics of a magnetite ferrofluid over a non-linearly stretching sheet using the BVP4c method are examined. The effect of related parameters on stream function, velocity, micro-rotation velocity, and temperature are demonstrated graphically. Results revealed that:

(1) On the stream function subject, there is an inverse relation between magnetic and micro-rotation parameters.

(2) On the velocity subject, there is a reverse relation between boundary and micro-rotation parameters.

(3) On the micro-rotation velocity subject, in the deficiency of magnetic parameter, micro-rotation velocity is in the lowest form.

(4) On the temperature subject, the highest temperature is attained for the lowest value of the micro-rotation parameter.

The work done in this article can be extended for other non-Newtonian fluids, like, Maxwell fluid, Second grade fluid, and Oldroyd-B fluid, etc.

Acknowledgments

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018).

Conflict of interest

The authors declare no conflict of interest.