1. Introduction
Hybrid nanofluids are the latest category of heat transfer fluids with great potential for industrial applications derived from the distribution of nanoparticles in conventional fluids. To create the appropriate combination of hybrid nanofluids, these fluids are made up of metallic or non-metallic particles. In a variety of applications, including heat transmission, hybrid nanofluids have been used in mechanical heat sinks, plate heat exchangers, and helical heat exchangers [
1,
2]. Previous research has shown that when nanoparticles are suspended in ordinary fluids, the heat transfer properties improve, increasing thermal conductivity [
3]. However, selecting appropriate nanoparticles is one of the most important aspects of maintaining a stable hybrid nanofluid proportion. In relation to the aim of this study, some related references on nanofluids and hybrid nanofluids in various applications can be found in Rabiei et al. [
4], Renuka et al. [
5], Ur Rehman et al. [
6], and Connolly et al. [
7]. We also mention some recently published papers by Kamis et al. [
8], Nadeem et al. [
9], Elsaid et al. [
10], Hassan et al. [
11], and Sheikholeslami [
12], who presented a numerical simulation of boundary layer flow in a hybrid nanofluid considering various physical phenomena.
The inclusion of magnetohydrodynamics (MHD) has piqued the interest of scholars due to its importance in numerous fields, including geology, astrophysics, drug industries, cosmology, MHD generators, and seismology. According to Khalili et al. [
13], the role of MHD is to restrict fluid flow by aligning it to magnetic fields. Devi and Devi [
14] studied the impacts of MHD boundary layer flow in a hybrid nanofluid over a stretching sheet with a suction effect. They discovered that the hybrid nanofluid has a higher heat transfer rate than nanofluid in a magnetic environment. Meanwhile, Zainal et al. [
15] noticed that the occurrence of a suction parameter and MHD tends to slow fluid motion due to the synchronism of the magnetic and electric fields caused by the Lorentz force formation. Recently, Khashi’ie et al. [
16] investigated the unsteady squeezing flow in a horizontal channel with the influence of a magnetic field in a hybrid nanofluid. Up to now, many scientific researchers have investigated the effect of magnetic field parameters in Newtonian or non-Newtonian fluid flows over stretching/shrinking surfaces by considering various impacts, for example, Yashkun et al. [
17], Zainal et al. [
18], Aly et al. [
19], Shafee et al. [
20], and Dinarvand [
21].
The Joule heating mechanism demonstrates an evolving appeal in massive engineering and manufacturing processes, including the electrical and electronic device configurations. The main benefit of Joule heating is that it transports electrical strength to reduce damage by lowering the current. According to Reddy and Reddy [
22], the nanofluid temperature in the boundary layer flow is raised using this control parameter. Sheikholeslami and Ganji [
23] investigated the nanofluid behaviour with the appearance of magnetic effect and Joule heating. A comparison study has been conducted by Khashi’ie et al. [
24,
25] in hybrid nanofluid and heat transfers towards a permeable shrinking surface by including the Joule heating. Naseem et al. [
26] discovered that the temperature profile increases as the Eckert number rises. Many other researchers have contributed to the study of fluid flow and heat transfer by taking Joule heating phenomena into account, and thus, more detail can be found in the references mentioned therein, as seen in Daniel et al. [
27], Khan et al. [
28], Yan et al. [
29], and Mahanthesh et al. [
30].
Nevertheless, the study mentioned above is about constant flows. In some cases, a change in the free stream velocity or the surface temperature might cause the flow to become unstable. Hayat et al. [
31] observed the unsteady three-dimensional flow over an exponential surface, considering viscous dissipation effects and Joule heating using boundary layer approximations. In another investigation, Chaudhary and Choudhary [
32] discovered that as the unsteadiness parameter improves, the thermal boundary layer thickness and the heat transfer rate decline. Meanwhile, according to Ahmed et al. [
33], higher amounts of the unsteadiness parameter improve the distribution of temperature, while increasing the Eckert number substantially improves the temperature distribution in the Maxwell fluid. Mahanthesh et al. [
30] concluded that the viscous dissipation and thermal radiation influences are critical in the cooling and heating processes; hence, they should be preserved to a reasonable level in cooling systems. As for references, Malekian et al. [
34], Zainal et al. [
35,
36], Rehman and Salleh [
37], and Waini et al. [
38] have scrutinised various analytical and numerical investigations to explore the unsteady-state behaviour in nanofluid and hybrid nanofluid flow.
A previous study by Dholey [
39] only considers the viscous flow without observing the heat transfer in his mathematical model. Today, experimental and numerical research have shown that nanofluid reacts as a better heat transfer fluid when compared to viscous fluid. Motivated by the outstanding work of Dholey [
39], the main objective of this study is to broaden his research by employing hybrid nanofluid flow in a magnetic environment of the boundary layer and heat transfer. Hence, a new mathematical hybrid nanofluid model is introduced. The current study also aims to fill a research gap in the existing literature, particularly in studying the unsteady separated stagnation-point flow by including viscous dissipation and the Joule heating impact. In this particular instance, another objective of this study is to explore the effect of designated physical parameters in the context of hydrodynamic flow and heat transfer of a hybrid nanofluid. Thus, to solve the stated problem, the bvp4c scheme in the MATLAB package is used. Comparative results for a specific case have been obtained, revealing a strong correlation between previous work and the existing outcomes. Since the appearance of dual solutions is observed, the third objective of this study is to perform a stability analysis to evaluate the solution’s dependability. Overall, we believe that an extensive study on the unsteady separated stagnation flows combined with the application of mathematical knowledge should be intensified to gain a better understanding due to its major significance in many industrial applications such as start-up processes and periodic fluid motion. This will lead to progressive improvements in the efficiency, durability, and cost of many fluid dynamic devices in order to establish new and advanced heat transfer technology.
2. Mathematical Formulations
Let us consider the unsteady MHD separated stagnation-point in two-dimensional flow of a hybrid nanofluid, as shown in
Figure 1, where
are Cartesian coordinates with the
-axis measured along the shrinking surface,
is in the direction normal to the surface, and the flow is at
. It is assumed that the velocity of the shrinking surface is
and that of the far-field (inviscid hybrid nanofluid flow) is
, where
denotes time. Next, the magnetic field
is
where
is the applied magnetic field strength,
is a parameter showing the unsteadiness of the problem, and
and
are the dimensionless forms of the constant reference value of time
and general time
. From the definition of
it can be concluded that
which gives
The surface temperature
of the sheet is
where
is the sheet characteristic temperature,
is the sheet length characteristic, and
represents the free stream temperature. The working fluid contains two types of nanoparticles, namely, alumina (Al
2O
3) and copper (Cu), hence forming a hybrid nanofluid (Al
2O
3-Cu/H
2O) with water (H
2O) as the base fluid. In this study, we also consider the effects of viscous dissipation and Joule heating, which are included in the energy equation below.
Based on the above proposed assumptions, the following set of governing equations can be expressed in Cartesian coordinates
, as follows (see Dholey [
39]; Devi and Devi [
14]):
along with the boundary conditions
From the above condition,
is for a static sheet;
and
denote the shrinking/stretching parameter, respectively;
is the temperature of the hybrid nanofluid;
is the characteristic temperature of the hybrid nanofluid; and
is the length characteristic of the sheet. In addition,
is the dynamic viscosity;
is the heat/thermal conductivity; and
and
are the density and heat capacity, respectively. Following that,
Table 1 and
Table 2 demonstrate the hybrid nanofluid correlations and the nanoparticle’s characteristics, respectively, where
is Al
2O
3 (alumina) nanoparticle and
is Cu (copper) nanoparticle. Referring to the experimental work, the characteristic temperature of the hybrid nanofluid
in the solution process is in the range of
–
[
3,
40].
We now introduce the appropriate similarity variables pursuing Dholey [
39], as follows:
where
is a constant, known as the acceleration parameter;
denotes displacement of the sheet, so that
; and the prime denote differentiation with respect to
.
Substituting (5) into Equations (2) and (3), we obtain the following ordinary (similarity) differential equations and boundary conditions:
Here,
is the Prandtl number,
is the magnetic field parameter, and
is the Eckert number, which are given by
As for the quantities of physical interest, now, we have
where
and
present the shear stress and the heat flux at a point on the surface of the sheet, respectively. By employing (5) and (10), we have
Here, and measure the coefficient of skin friction and heat transfer from the sheet’s surface, respectively.
4. Analysis of Findings
This section discusses the effect of various physical parameters, where the results are shown graphically in
Figure 2,
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13. The conclusions are drawn for the flow field and other physical quantities of interest. The bvp4c solver in MATLAB software was used to solve Equations (6) and (7) with respect to boundary conditions (8), numerically. The reliability of the current results is confirmed by comparing the numerical data with Lok and Pop [
45], Ishak et al. [
46], and Wang [
47], as reported in
Table 3. Since the present outcomes are consistent, we believe the obtained results are valid and trustworthy. This also proves that the recommended mathematical model for this problem is appropriate. The influence of specific physical characteristics is then investigated, and the results are presented in graphical form.
The numerical values of
and
as
with various values of
are presented in
Table 4 and
Table 5, respectively. According to
Table 4, the values of skin friction on the shrinking sheet increase as the magnetic effects are enlarged. Based on the given values, the positive sign of
indicates that a drag force is employed in the working fluid, while the negative sign of
shows that the sheet exerts a dragging force on the flow field. Similarly, the values of heat transfer rate given by
demonstrate an increment trend as the magnetic effects are improved in hybrid nanofluid flow as accessible in
Table 5. This appears to prove that the existence of a magnetic field in the flow environment may improve thermal efficiency.
Moreover, it is noticed that there are two possible solutions in this study; therefore, an analysis of solution stability is necessary to obtain a consistent solution. The stability analysis technique reveals the properties of the dual solutions by identifying the smallest eigenvalue. As seen in
Table 6, the first solution is reliable since
is positive, whereas the second solution is not since
is negative, which highlights the dissatisfying stabilising feature.
In this study, the values of the governing parameters such as
, and
are selected based on the availability of the dual solutions. Particularly, any values can be used in order to generate the results, as long as the profiles are asymptotically converged. However, the results may vary, and a unique solution is expected since a different range of the governing parameter is employed. On the other hand, the Pr value is fixed to 6.2 to indicate the base fluid state as water at
. The reduced skin friction coefficients
of viscous fluid
nanofluid
, and hybrid nanofluid
are available in
Figure 2, while the heat transfer rate
is portrayed in
Figure 3. It is observed that the similarity solutions for viscous flow
are available when
. The range of solution becomes wider when the nanoparticle volume fraction is added, where
and
, which denote the Al
2O
3/H
2O nanofluid and Al
2O
3-Cu/H
2O hybrid nanofluid, respectively. However, for
, the full partial differential Equations (1) to (3) should be numerically solved.
Figure 2 also indicates that the growth of
particularly enhances the behaviour of
. Furthermore, in these three different types of fluids, the hybrid nanofluids show the highest trend of
compared to conventional viscous fluid and nanofluid, particularly when 1% of
(Al
2O
3) and 1% of
(Cu) volume concentration are presented.
Figure 3 depicts a positive expansion in
, which represents the first solution’s heat transfer rate. Overall, when the viscous flow transforms into Al
2O
3/H
2O and Al
2O
3-Cu/H
2O as it passes through the stretching/shrinking plate, the heat transfer capability improves.
The effect of a magnetic field
on this particular case is also worth exploring. The characteristics of skin friction coefficient
and heat transfer rate
are presented in
Figure 4 and
Figure 5, respectively. Based on the generated outcomes, the addition of
shows a response where both
and
are increased when
improved. This is because of the Lorentz force, which is caused by the engagement of the induced electric currents and the applied magnetic field in the flow field. This accelerated flow increased the velocity gradient near the plate surface and intensified the velocity inside the boundary layer, escalating the trend of
and
. It is interesting to note that the presence of
when
is proven to improve the efficiency of heat transfer performance compared to
since the first solution exhibits an upward trend, as displayed in
Figure 5.
Figure 6 and
Figure 7 illustrate the unsteadiness parameter
influences towards the stretching/shrinking plate
. The hybrid nanofluid Al
2O
3-Cu/H
2O characteristics are portrayed in
Figure 6 concerning
when
. It is concluded that as
improved, the first solution has increased in
while the response of the second solution was in the reverse direction. Additionally,
is currently improved when
escalates in the first solution, as observed in
Figure 7.
Figure 8,
Figure 9,
Figure 10 and
Figure 11 show the effect of the acceleration parameter
in regard to
on the stretching/shrinking plate. Focusing on the first solution,
Figure 8 illustrates that the increment of
spontaneously intensifies
. This finding also implies that a greater amount of
broadens the flow behaviour, causing the flow velocity to increase and decrease the boundary layer thickness, as displayed in
Figure 10. In addition,
Figure 8 also consequently observes the pattern of
as
. This is due to the no frictional drag force occurrences on the stretching sheet. Sequentially,
Figure 9 depicts the thermal efficiency, with
intensifying in the first solution as the value of
increases in the hybrid nanofluid. The results demonstrate that increasing the acceleration parameter flow promotes thermal conductivity effectively. The temperature profiles
in
Figure 11 back up the trend seen in
Figure 9, which shows the reduction in temperature distributions as
increases. The deterioration in Al
2O
3-Cu/H
2O temperature improves the thermal transmission and gradually upsurges the heat transfer performance. Based on
Figure 10 and
Figure 11, it is noted that all profiles asymptotically satisfy the free stream conditions (8), which then authorises the validity of the numerical solutions.
Figure 12 and
Figure 13 demonstrate the impact of
on heat transfer rate
and temperature profile distributions
. An increase in
contributes to the decrement of
in both solutions, as depicted in
Figure 12. Furthermore, the development of
values does not delay the boundary layer separation of Al
2O
3-Cu/H
2O. In general, the Eckert number is the potential ratio of the advective transport and heat dissipation. Therefore, higher Eckert numbers generate more heat due to friction forces between fluid particles. This argument confirms the trend of
shown in
Figure 13. Evidently, we can deduce that the Eckert number has a tendency to degrade heat conveyance performance in this particular case. To summarise, the generated results may be advantageous to various researchers, allowing the mathematical simulation outcomes to be as close to the actual situation as possible. In addition, the results discussed earlier may also provide better theoretical guidance for engineering applications and scientific research, especially in nanotechnology and thermal systems.