Investigation of three-dimensional hybrid nanofluid flow affected by nonuniform MHD over exponential stretching/shrinking plate

Nomenclature

1 Introduction

Recently, the utilization of heat transfer enhancement in various engineering fields and industries has attracted the interest of many researchers. The functioning and compactness of engineering electronic devices suffer from the poor thermal conductivity of traditional heat-transfer fluids (for instance, oil, water, ethylene glycol mixture, etc.). It is, however, possible to enhance the thermal conductivity of these materials. One such process, which has first been introduced by Choi and Eastman [1], involves mixing single-type nanoscale (with sizes below 100 nm) particles with the initial fluid. The new mixture, which now has a higher thermal conductivity, is called nanofluid and is more efficient in improving heat transfer.

Maiga et al. [2] conducted a numerical simulation of the nanofluid behavior. They estimated the heat transfer enhancement of the nanofluid in an evenly heated tube under turbulent and laminar flow conditions, and employed approximated correlations for experimental data. Their findings indicated that the heat transfer enhancement is more significant in the turbulent flow conditions due to the higher Reynolds number and its effect on nanoparticles.

Waqas et al. [3] investigated the nanomaterial flow via stretchable impermeable rotating disk subjected to partial slip and carbon nanotubes (CNTs). Velocity, Nusselt numbers, temperature, and skin friction have been computed and addressed.

In another study, Wasqas conducted numerical simulation for stagnation point flow of cross nanofluid in frames of hydromagnetics. His results indicated that the radius of curvature and temperature-dependent heat sink-source significantly affect the heat-mass transport mechanisms for cylindrical surface [4]. Some numerical and experimental studies have been reported on the nanofluid characteristics [5,6,7,8,9,10,11].

Replacing single-type nanoparticles with hybrid nanoparticles is also a recent subject of interest among researchers. Hybrid nanofluids are novel heat-transfer fluids that contain more than one type of nanoscale solids in their initial fluid (water, oils such as engine oil, kerosene, paraffin, and vegetable oil, water ethylene or ethylene glycol, and glycol mixture).

The advantages of “hybrid nanofluids” include stability, superior thermal conductivity, and better heat transfer, and advantages in individual suspension that can be the result of the good aspect ratio and the synergistic effect present in nanomaterials. The desirable results of better thermal conductivity of hybrid nanofluids are higher efficiency in energy consumption, improved performance, and decreased operating expenses. The first study on the thermal conductivity of hybrid nanofluids was conducted by Jana et al. in 2007 [12] in which they investigated the thermal conductivity enhancement by mixing the fluid with single and hybrid nano-additives. In their experiment, Madhesh et al. [13] examined the heat transfer potential and the rheological characteristics of copper–titanium hybrid nanofluids (HyNF). They added copper–titania hybrid nanocomposites (HyNC) to the base fluids in tubular heat exchangers with the concentrations varying from 0.1 to 2.0 vol%. HyNC had an average size of 55 nm. The results of the experiment indicated a 49% larger Nusselt number, a 52% larger convective heat transfer coefficient, and a 68% larger overall heat transfer coefficient up to 1.0% volume concentration of HyNC. Ho et al. (2010) investigated the thermal conductivity of the water-based hybrid suspension containing microencapsulated phase change material (MEPCM) particles and Al₂O₃ nanoparticles. They showed that compared to the PCM suspension, hybrid nanofluids have a better thermal conductivity in higher mass concentrations of nanoparticles. Suresh et al. [14] examined a method to synthesize Al₂O₃–Cu/water hybrid nanofluid and studied the thermal conductivity of hybrid nanofluids at the room temperature with varying concentrations of nanoparticles. Labib et al. [15] also investigated how the concentration of nanoparticles and the Reynolds number affect heat transfer properties of Al₂O₃–CNT/water hybrid nanofluids. They found that adding two different types of nanoparticles (CNTs and Al₂O₃) to water results in a considerable increase in the convective heat transfer. They also showed that increasing the nanoparticles concentration enhances the heat transfer coefficient and adds to the friction factor of hybrid nanofluids. Moghadassi et al. conducted a simulation in single- and two-phase SIMPLE algorithms to investigate how Al₂O₃–Cu/water hybrid nanofluids affect forced convective heat transfer [16]. They indicated that the Nusselt number for Al₂O₃–Cu/water hybrid nanofluids in the single-phase model showed a 4.73% increase compared to Al₂O₃/water nanofluids and 13.46% compared to water. In the two-phase model, the Nusselt number showed a 4% increase compared to Al₂O₃/water nanofluids. Hamida and Hatami used Galerkin Finite Element Method (GFEM) for modeling the heat transfer in a channel filled with hybrid nanofluids under the electric field. They used electric field and heated fins to increase the heat transfer in a channel filled with different hybrid nanofluids [17]. Considerable investigations have concentrated on the hybrid nanofluid thermophysical characteristics [18,19,20,21].

Recently, several scientific, industrial, and engineering utilizations have been suggested for magnetohydrodynamics (MHD) boundary layer flow and heat transfer of electrically conductive fluids [22]. The transportation of MoS₂ nanoparticles with an existing MHD slip flow was studied by Khan [23]. Chamkha and Ismael [24] investigated how magnetic field affects the mixed convection in lid-driven trapezoidal cavities that are filled with Cu-water nanofluid. Alsabery et al. studied the magnetic mixed convection of Al₂O₃-water nanofluid with a two-phased Buongiorno’s model in a lid-driven enclosure subjected to a square cavity [25]. Ellahi studied MHD non-Newtonian fluid flow in a pipe. The findings showed that the decrease in flow motion is associated with the MHD parameter [26]. Sheikholeslami and Ganji examined MHD nanofluid flow inside a permeable channel and showed that the velocity boundary layer becomes thicker along with the Hartmann number (Ha). On the other hand, higher Reynolds number and nanoparticle volume fraction result in a lower thickness of the velocity boundary layer [27]. Seyyedi et al. simulated the natural convection by CVFEM and ANSYS fluent in a semi-annulus enclosure to study how MHD affects the entropy generation [28]. Rashidi et al. investigated the second law of thermodynamics and entropy generation on a porous rotating disc with a MHD nanofluid flow. Magnetic rotating disc can be useful when the heat transfer enhancement is the main objective in renewable energy systems [29]. In a study by Sheikholeslami and Ellahi, the Lattice Boltzmann method (LBM) method was used to investigate three-dimensional (3D) MHD free convective heat transfer and found that Lorentz forces decrease the temperature gradient [30]. Ma et al. presented a two-dimensional (2D) numerical simulation to study the effect of magnetic field on Ag-MgO nanofluid forced convection and heat transfer in a channel with active heaters and coolers. They used Fortran code according to LBM. They investigated the effect of thermal arrangement, block side length, Reynolds number, Hartmann number, and volume fraction of nanoparticles on flow pattern and heat transfer characteristics [31]. Mahabaleshwar et al. conducted a numerical simulation to investigate the steady MHD incompressible hybrid nanofluid flow and mass transfer due to porous stretching surface with quadratic velocity in the presence of mass transpiration and chemical reaction. They presented the asymptotic solution of concentration filed for large Schmidt number using Wentzel–Kramer–Brillouin (WKB) method [32].

The practical capabilities of the boundary layer flow across a stretching sheet have drawn the attention of scientists in recent years. It has several applications e.g., in production of glass-fiber, metallurgy, wire drawing, melt-spinning, annealing, tinning of copper wires, glass blowing, plastic and rubber sheet manufacturing, and crystal growing. Sakiadis [33,34] pioneered the concept of boundary layer flow on a continuous solid surface at a constant moving speed. Expanding on Sakiadis’ ideas, Crane [35] found similar solutions for steady, 2D stretching when the velocity on the boundary is proportional to a fixed point at the distance. Gupta and Gupta, Chen and Char, and Pavlov extended the works of Crane to other physical conditions [36,37,38]. Magyari and Keller [39] studied the heat and mass transfer inside the boundary layers on a continuous surface exponential stretching that were subjected to suction/blowing. Recently, shrinking of sheets have caused problems in industrial applications where the fluid flow shrinks towards/back to the origin/original surface. The unusual form of flow as a result of shrinking was first indicated by Wang [40] during their investigation on how a liquid film behaves on an unsteady stretching sheet. Miklavcic and Wang [41] conducted a study on the flow over a shrinking sheet and demonstrated the exact solution of the Navier–Stokes equations. They also showed that maintaining the flow over a shrinking sheet requires mass suction. Bhattacharyya [42] investigated fluid flow and heat transfer across a sheet undergoing exponential shrinkage. Their findings showed that the feasibility of the steady flow depends on the mass suction parameter exceeding a specific critical value. In another study, Bhattacharyya and Vajravelu [43] examined the stagnation point flow and heat transfer across a sheet subjected to exponential shrinkage. Bachok et al. [44] examined the steady 2D stagnation-point flow of an aqueous nanofluid over a sheet undergoing exponential stretching/shrinkage. In the recent developments, a substantial amount of research has been dedicated to analysis of fluid flow across the stretching/shrinking plate but we refer few recent studies [45,46,47,48,49,50,51].

A review on the existing literature shed light on research gaps and inadequate studies. As far as the authors know, there has been no study on 3D hybrid nanofluid flow past a bidirectional exponential stretching/shrinking plate while taking the heat generation/absorption and MHD into account. The current study aims to fill in this gap by numerically analyzing the effect of heat absorption/generation over a bidirectional, exponentially stretching/shrinking plate on the MHD flow in a hybrid nanofluid containing Al₂O₃–Cu/H₂O.

2 Mathematical formulations

It is supposed that the steady 3D incompressible MHD fluid flow and heat transfer in the hybrid Al₂O₃–Cu/H₂O nanofluid over an elastic sheet (bidirectional, exponentially stretching/shrinking) is situated at x, y = 0 and z = 0. This sheet is affected by heat transfer in/on the plane and the origin O (as shown in Figure 1). The wall’s surface temperature is assumed to exponentially vary according to the following equation: T w ( x , y ) = T ∞ + T 0 e a ( x + y ) / 2 L in which T ₀ and T _∞ are the reference temperature and the ambient fluid temperature. a represents the exponent temperature parameter. Also, the velocity of the stretching sheet is considered bidirectional exponentially with u w = c e x + y L in x-axis and also with v w = d e x + y L in y-axis, while a and b are velocity references and L represents the characteristic length.

Figure 1

Sketch of physical configuration.

Space-dependent magnetic field of strength is written as: B ( x , y ) = B 0 e x + y 2 L , in which B ₀ represents the magnetic field strength exerted in the perpendicular direction on the plane of the stretching/shrinking sheet. This variable is considered while the fluid is electrically conducted. It is possible to overlook the induced magnetic field as the magnetic Reynolds number supposedly has a negligible quantity. The Lorentz force produced by the magnetic field in the boundary layer impedes the fluid flow. Based on the Bernoulli’s principle and Boussinesq approximations (buoyancy) in the free stream, the hybrid Al₂O₃–Cu/H₂O nanofluid governing equations can be obtained as following [52]:

In which u represents the velocity component along the – x axis, v the velocity component along the – y axis, and w the velocity component in the – z direction. T represents the local temperature of the nanofluid. The boundary conditions can be expressed as following:

where T _∞ expresses the constant temperature of the fluid distant to the sheet, T ₀ is the reference temperature, a denotes the temperature exponent parameter, and w ₀ represents the constant corresponding to the mass flux velocity.

In the above equations, the density of hybrid Al₂O₃–Cu/H₂O nanofluid is given by ρ h n f , α h n f is the thermal diffusivity, ( ρ C p ) h n f denotes the effective heat capacity of hybrid Al₂O₃–Cu/H₂O nanofluid, σ expresses the Al₂O₃–Cu/H₂O electrical conductivity, and Q represents the heat generation/absorption coefficient. Also, μ h n f denotes the dynamic viscosity of the hybrid nanofluid.

Table 1 displays the thermophysical characteristics of the nanoparticles Al₂O₃ and Cu and the base fluid H₂O [53].

In this study, the nanoparticle volume fraction is given by ϕ ; ρ _f and ρ _s express the base fluid and hybrid nanoparticles densities; (ρC _p ) _s and (ρC _p ) _f represent the hybrid nanoparticles heat capacity along with base fluid; C _p represents the constant pressure of heat capacity; and k _f and k _s are the thermal conductivity of the base fluid and hybrid nanoparticles, respectively.

The shape of nanoparticles can substantially modify the thermal conductivity and the amount of heat transfer. Therefore, the thermophysical characteristics of the nanofluids and hybrid nanofluids while considering the shape factor are displayed in Table 2 [54].

The following similarity transformation can be defined:

In the above expressions, prime represents the differentiation related to η.

Also, it can be showed that:

in which S represents the non-dimensional mass flux parameter for suction and injection.

According to similarity transformation, the continuity (Eq. (1)) is fulfilled. (Eqs. (2)–(4)) together with boundary conditions (Eq. (5)) turn into:

in which Pr represents the Prandtl number of the base fluid, M and H denote the magnetic parameter and heat generation/absorption, respectively, S stands for the suction/injection parameter and λ represents the stretching/shrinking parameter, as follows: λ > 0 shows the stretching sheet, λ < 0 shows the shrinking sheet, and λ = 0 indicates the stagnation sheet. These parameters are expressed as following:

The surface friction coefficients represented by C _fx and C _fy and the Nusselt number Nu _x can be expressed:

where τ w x and τ w y are the shear stress of the surface along the x- and y-axes, respectively. q _w denotes the surface heat flux. These parameters can be obtained as follows:

By applying Eq. (6) and Eq. (13), Eq. (14) is turned to:

provided that Re x = u w L ν f and Re y = v w L ν f , where Re _x and Re _y are the Reynolds numbers along the – x and – y axes, respectively.

3 Numerical method

To obtain numerical results, it is necessary to solve ordinary differential Eqs. (8)–(10) in the boundary conditions (11) using a fifth-order Runge–Kutta–Fehlberg technique.

The mentioned technique is used for estimating the ordinary differential equations. It also incorporates a trial step at the middle point of an integral to cancel errors of lower orders. Here first-order nonlinear differential equations are converted to first-order linear equations. The following variables are required:

After inserting these variables in the Eqs. (10–12) and boundary conditions (13) and using fifth-order Runge–Kutta method, the ultimate equations can be expressed:

It is noteworthy that to solve the equations using fifth-order Runge–Kutta technique, the following preconditions are necessary:

• The appropriate value for ( η → ∞ ) is selected as 7.

• In corresponding boundary conditions, b ₁, b ₂, and b ₃ are, respectively, equivalent to f ′ ( η ) → 0 , g ′ ( η ) → 0 , and θ ( η ) → 0 when η → 0 .

• Convergence criterion has been considered as 10⁻⁶

• The step size is considered Δ η = 0.001 .

Figure 2 presents a flowchart to have a better understanding of the numerical solution approach of the fifth-order Runge–Kutta technique [55]. Several noteworthy methods of numerical solution for the ordinary nonlinear differential equations are also provided in refs. [56,57,58,59,60,61,62].

Figure 2

Problem flowchart [55].

4 Validation

In this section, the results of the presented approach are compared with previous studies for the purpose of validation of the presented numerical method. The results regarding the wall temperature gradient are confirmed by Zainal et al. [52] and Jusoh et al. [63] who used the bvp4c function. Ahmad et al. [64] utilized a standard fifth-order Runge–Kutta technique in combination with the shooting method (as shown by Table 3). The results of the current study are consistent with two other studies. Therefore, the accuracy and practicality of the numerical model are confirmed in addressing the problems of heat transfer and boundary layer flow.

5 Results and discussion

After validating the computations, the effect of various parameters and nanoparticle shapes on the velocity and temperature profiles and on the skin friction coefficient and also the local Nusselt number will be further explained in this section. Table 4 displays the following shapes of nanoparticles: brick, cylinder, platelet, and blade, each with different shape factors [65].

Table 4

Nanoparticles shape with their shape factor. Ref [65]

Nanoparticle type	Shape	Shape factor
Bricks		3.7
Cylinders		4.9
Platelets		5.7
Blades		8.6

Figure 3 shows the temperature distribution for different shape factors. As can be seen, with increasing shape factor, the temperature over the surface is also increased. This is because increasing the shape factor results in the boundary layer becoming less thick.

Figure 3

Effect of shape factor (sf) on temperature profile (θ).

The temperature profiles for varying values of exponent parameter a are displayed in Figure 4. When this parameter is decreased, the temperature profiles go near the surface. Therefore, the temperature penetration depth is proportional to a.

Figure 4

Effect of exponent parameter (a) on temperature profile (θ).

Variation in temperature with heat generation/absorption parameter (H) is displayed in Figure 5. It is evident that enhancing the parameter (H), which represents the decrease in heat absorption/increase in heat generation, leads to a sharp increase in the temperature distribution. This is because increasing the heat generation/absorption parameter leads to a thicker thermal boundary layer.

Figure 5

Effect of heat generation/absorption parameter (H) on temperature profile (θ).

The temperature gradient for various shape factor is displayed in Figure 6. Near the surface, increasing the shape factor lowers the absolute of temperature gradient (heat transfer rate). In contrast, farther from the surface and with a larger η, the absolute of temperature gradient is increased with increase in the shape factor.

Figure 6

Effect of shape factor (sf) on temperature gradient profile (θ′).

The effect of exponent parameter a on the heat transfer rate is displayed in Figure 7.

Figure 7

Effect of exponent parameter (a) on temperature gradient profile (θ′).

In the hybrid nanofluid, a larger temperature exponent parameter means decreased heat transfer rate closer to the surface. The opposite is true for a higher η where larger values of a increase heat transfer rate.

Figure 8 indicates that increase in the heat generation/absorption value near the surface has a negative effect on the heat transfer rate. Table 5 estimates the results of skin friction coefficient for different values of magnetic parameter M and suction/injection parameter S. As can be seen, a higher skin friction coefficient is achieved by increasing the magnetic parameter M and suction/injection parameter S. Table 6 shows how the local Nusselt number changes with the exponent parameter a and heat generation/absorption parameter H. As illustrated, local Nusselt number is decreased with larger exponent parameter a and heat absorption/generation parameter H.

Figure 8

Effect of heat generation/absorption parameter (H) on temperature gradient profile (θ′).

6 Conclusion

This study analyzed the changes in temperature, temperature gradient, Nusselt number and friction coefficient, and MHD hybrid nanofluid (hybrid Al₂O₃–Cu/H₂O) across bidirectional exponential stretching/shrinking plate under the influence of heat generation/absorption. By implementing an efficient similarity transformation, the dominant partial differential equations were simplified to obtain the ordinary differential equations system. The findings indicated that increasing the shape factor and generation/absorption parameters adds to the temperature over the surface. However, decreasing this parameter leads the temperature profiles to move towards the surface and the temperature is increased. Also, regarding the impact of shape factor, heat generation/absorption parameter, and exponent parameters on temperature gradient, it can be observed that the absolute of temperature gradient (heat transfer rate) near the surface decreases with increase in these parameters. But farther from the surface, the absolute of temperature gradient rises along with a larger η value. The skin friction coefficient has a direct relationship with magnetic M and suction/injection S parameters, and the local Nusselt number has a negative relationship with exponent a and heat generation/absorption H parameters.