Unsteady Magnetohydrodynamics (MHD) Flow of Hybrid Ferrofluid Due to a Rotating Disk

Abstract

The flow of fluids over the boundaries of a rotating disc has many practical uses, including boundary-layer control and separation. Therefore, the aim of this study is to discuss the impact of unsteady magnetohydrodynamics (MHD) hybrid ferrofluid flow over a stretching/shrinking rotating disk. The time-dependent mathematical model is transformed into a set of ordinary differential equations (ODE’s) by using similarity variables. The bvp4c method in the MATLAB platform is utilised in order to solve the present model. Since the occurrence of more than one solution is presentable, an analysis of solution stabilities is conducted. Both solutions were surprisingly found to be stable. Meanwhile, the skin friction coefficient, heat transfer rate—in cooperation with velocity—and temperature profile distributions are examined for the progressing parameters. The findings reveal that the unsteadiness parameter causes the boundary layer thickness of the velocity and temperature distribution profile to decrease. A higher value of magnetic and mass flux parameter lowers the skin friction coefficient. In contrast, the addition of the unsteadiness parameter yields a supportive effect on the heat transfer rate. An increment of the magnetic parameter up to 30% reduces the skin friction coefficient by 15.98% and enhances the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is rapidly enhanced by improving the mass flux parameter by almost 20%.

1. Introduction

Convective transport has had a noteworthy impact io many real applications such as energy-producing plants, energy distribution systems, and environmental problems [1]. The convective transport can be induced by diffusion and advection where the diffusion refers to the random Brownian motion, whereas the advection refers to the transportation of heat by a larger-scale motion. Due to the substance of its theory and concepts, the ideas of convective transport have been extended to many fluid models that cover Newtonian and non-Newtonian fluid types. Additionally, the model of Eyring Powell fluid has been investigated by Al Jabali et al. [2] and Khalil et al. [3], whereas the Jeffrey model was considered by Shehzad et al. [4] and Kasim et al. [5]. Other interesting models such as viscoelastic can be found in Kasim et al. [6]. Additional previously studied models include the viscous model [7], Casson model [8], and micropolar model [9]. The development of the fluid flow modelling is continued with the discovery of the upgraded thermal fluid. In the late 20th century, nanofluids were introduced in an effort to host the concept of the diffusion of nanoscale particles into fluids. The presence of nanofluids has been shown to increase the capacity for thermal conductivity. A pioneering study on nanofluids was conducted by Choi and Eastman [10]. Since then, this field of research has continued to receive high attention from researchers both experimentally and theoretically. This situation can be seen from the increase in publications every year where the contribution is based on the analysis of the fluid characteristics induced by the different geometries such as rectangular enclosure [11,12,13], stretching/shrinking sheet [14,15,16,17], thin needle [18,19], rotating disk [20], asymmetric wavy channel [21], vertical plate [22], and moving inclined [23]. These investigations also involve several significant effects including chemical reactions, magnetic fields, viscous dissipation circumstance, Dufour and Soret consequence, activation energy, and Joule heating. Simulations investigating the convective transport and heat transfer in cryogenic fuel tanks have been documented in refs. [24,25] where the improved algorithms determining the output were proposed. The analysis of heat transfer characteristics in a thermal energy storage system using single and multi-phase cooled heat sinks is presented in Alireza [26] where the evolution of the experimental, numerical, and computational efforts on energy storage was presented in detail.

The hybrid nanofluid offers a higher heat transfer rate in comparison to conventional nanofluid. This fluid is widely used in several applications and mainly can be found in heat exchangers activities, vehicle brake systems, solar industries, and also in refrigerator production [27]. The earlier studies which applied the hybrid nanoparticles in their investigations were Turcu et al. [28] and Jana et al. [29]. The composition of Cu-Al₂O₃/water was successfully studied by Suresh et al. [30] in discovering fluid with good thermal conductivity. The input stated, that despite having low thermal conductivity, Al₂O₃ offers a good chemical motionlessness in alumina which compromises a stable composition. A further study on combining Al₂O₃ with supplementary nanoparticles was established by Singh and Sarkar [31], Farhana et al. [32], and Takabi and Salehi [33]. Some other interesting publications on the development of the theoretical study of hybrid nanofluid can be viewed in refs. [34,35,36,37,38,39,40,41,42] where the investigations were performed using different particles, approaches, and various surfaces. Furthermore, some other publications reviewing the progress of hybrid nanofluids are documented in refs. [43,44,45,46].

Taking advantage of the progress of nanotechnology, scientists first created ferrofluid to counter the problem of logistics of bringing rocket fuel. The idea of this innovation is based on how to direct the fluid in space, and it is supported by the concept of the magnetic fluid can be well-ordered by a magnetic field. Then, the liquid fuel is mixed with the ferrofluid in the system together with the external magnetic field. The liquid with particles of Fe₃O₄ in nanometres size that happened in conventional base fluids is referred to as ferro-nanofluids or sometimes it is just called ferrofluid. Similar to nanofluids, ferrofluid also has high thermal conductivities and a great heat transfer rate, which is very important in device production. In fact, this technological fluid can be found in the production of hard drives where it is used to seal the interior of devices in order to avoid dust or external source damage to the delicate plate.

Due to the importance and many applications, interest in investigating ferrofluid came into demand for both experimental and simulated procedures. Khan et al. [47] provided a solution for heat transfer analysis of ferrofluid in the presence of viscous dissipation and concluded that kerosene-based ferrofluids developed high skin friction and Nusselt numbers compared to water-based ferrofluids. The investigation of ferrofluid, later on, was extended by other researchers by considering different surfaces. For instance, Qasim et al. [48] consider the stretching cylinder as the surface where the fluid is moving while Mehrez and Cafsi [49] proposed the flow in a rectangular channel. Further research on the topic of ferrofluid has been proposed by Sekar and Raju [50] and Hamid et al. [51] where the ferrofluid is considered together with dust particles and was studied as a two-phase flow. Goshayeshi et al. [52] considered the effect of particle size and types in their investigation while Rashad [53] investigated the anisotropic impact on the flow field. In the same year, the same author documented the analysis of ferrofluid under thermal radiation and MHD circumstances [54]. The demand for investigating ferrofluid led to the development of new ideas and this progress is detailed out in refs. [55,56,57,58,59,60,61].

The interest in innovating the effective fluid has led to the discovery of hybrid ferrofluids where the investigations have involved multiple nanoparticles scattered in ferrofluid. Chu et al. [62] provided a solution for hybrid ferrofluids along with multi-wall carbon nanotubes (MWCNT) in a cavity for natural convection by applying a finite element scheme. It is revealed that the hybridization of ferrite nanoparticles uplifts in a physical thermos to see the studied flow. In the same year, Kumar et al. [63] proposed an investigation under the topic of MHD hybrid ferrofluid together with the effect of irregular heat source/sink towards the flow of the radiative thin film. The output declared from the hybrid ferrofluid intensifies the heat transfer rate in comparison with the conventional ferrofluid. A very recent study on this topic was presented by Anuar et al. [64] where their study of hybrid ferrofluid (CoFe₂O₄–Fe₃O₄/water) was focused on exponentially deformable sheets in stagnation point region. Apart from these mentioned studies, other interesting reports on hybrid ferrofluids can be found in refs. [65,66,67]. One real application that can be highlighted from the uses of hybrid ferrofluids is that it can be used as seeds for acid mine drainage (AMD) treatment. The use of a hybrid ferrofluid will provide a sustainable remedy for wastewater treatment, especially for AMD since the presence of AMD will cause severe destruction to the environment and predominantly disturb human life, aquatic organisms, animals, and also plants. The special properties of hybrid ferrofluids are unique magnetized characteristics with additional low toxicity and strong capacity for contaminant removal, which have led to enhancing the chemical reactivities [68].

As mentioned previously, the study on fluid flow normally deviates from its geometries. The pioneering study on infinite rotating disk was performed by Von Kármán [69] where the flow is taken as a viscous flow in which the disk rotates through uniform angular velocity. The Navier–Stokes equations representing the mathematical model are reduced to ordinary differential equations by adopting the similarity variables. The ideas from Von Kármán’s work have been extended by Fang [70] where the investigation was focused on the flow over the stretchable rotating disk and later Fang and Zhang [71] extend to two infinite stretchable disks. Turkyilmazoglu [72] considered the presence of a magnetic field together with viscous dissipation and joule heating in the investigation after considering the unsteady flow over a rotating disk with outer radial flow [73]. The unsteady flow over a decelerating rotating sphere was then established by Turkyilmazoglu [74]. Another analysis of flow over a rotating disk was documented in refs. [75,76,77,78]. The study of flow over a rotating disk is still in demand due to its available applications in industries and engineering activities such as the viscometers instrument, turbines industries, rotating disk electrodes, mechanical devices, and Brake rotors [79,80].

Motivated by the studies available in the literature and fulfilling the gap in the study of hybrid ferrofluids, this study discusses the investigation on the unsteady flow of hybrid ferrofluid flow over a rotating disk. It is worth mentioning that the non-unique solutions are available but are restricted to pertinent parameters and the stability analysis affirms the physical solution.

2. Mathematical Formulation

Consider the unsteady flow of hybrid ferrofluids with different base fluids (Fe₃O₄-CoFe₂O₄/H₂O-EG and Fe₃O₄-CoFe₂O₄/H₂O) induced by a rotating disk as illustrated in Figure 1 with the imposition of suction/injection effect. The disk rotates with the velocity, $v=v_w$ while being stretched/shrunk with velocity, $u=u_w$. Moreover, it is considered that the suction/injection effect is embedded with the mass flux velocity, $w=w_w$. The constant surface and ambient temperatures are symbolized as $T_w$ and $T_{\infty }$, respectively. A further assumption is that the magnetic field is applied normally to the disk such that $B^{*}=B_0/\sqrt{1-ct}$ with constant magnetic strength $B_0$ (see Appendix A for the derivation of magnetic field).

Thus, the governing equations in cylindrical coordinates $\left(r,\vartheta ,0\right)$ are [77,80,81]:

$$\frac{\partial }{\partial r}\left(ru\right)+\frac{\partial }{\partial z}\left(rw\right)=0,$$

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial r}-\frac{v^2}{r}=-\frac{1}{{\rho }_{hnf}}\frac{\partial p}{\partial r}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{\partial }^{2}u}{\partial z^2}-\frac{u}{r^2}\right)-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}^{\ast }{}^{2}u,$$

(2)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}+\frac{uv}{r}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{\partial }^{2 }v}{\partial z^2}-\frac{v}{r^2} \right)-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}^{\ast }{}^{2}v,$$

(3)

$$\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=-\frac{1}{{\rho }_{hnf}}\frac{\partial p}{\partial z}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{{\partial }^{2}w}{\partial z^2}\right),$$

(4)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2 }T}{\partial z^2}\right),$$

(5)

subject to:

$$u=\lambda u_w, v=v_w, w=w_w, T=T_w \mathrm{at} z=0$$

(6)

$$u\to 0, v\to 0, w\to 0, T\to T_{\infty } \mathrm{as} z\to \infty$$

(7)

where $\left(u,v,w\right)$ are the velocities for $\left(r,\vartheta ,0\right)$ directions while $T$ is the temperature. It is further assumed that $u_w=\mathsf{\Omega }r/\left(1-ct\right), v_w=\mathsf{\Omega }r/\left(1-ct\right)$ where $t$ and $\Omega$ are the respective time and angular velocity (constant), accordingly. Furthermore, $c$ represents the unsteadiness strength (constant) while $w_w=-w_0/\sqrt{1-ct}$ is the mass velocity with constant $w_0$. Moreover, $\lambda =0$ represents a static disk while $\lambda >0,\lambda <0$ stand for the stretching/shrinking disk, respectively.

The thermophysical properties of the base fluids (water and water–ethylene glycol), magnetite, and cobalt ferrite nanoparticles are described in

Table 1. Physical properties of the respective base fluid and nanoparticles.

Properties	Base Fluid		Nanoparticles
	Water	Water–Ethylene Glycol	Magnetite	Cobalt Ferrite
$\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	997.1	1057	5180	4908
$C_p\left(\mathrm{J}/\mathrm{kgK}\right)$	4179	3287	670	700
$k\left(\mathrm{W}/\mathrm{mK}\right)$	0.613	0.424	9.8	3.6
$\sigma \left(\mathrm{S}/\mathrm{m}\right)$	0.05	0.005	0.74 × 106	1.1 × 107
Prandtl number, $\left(\mathrm{Pr}\right)$	6.2	30	-	-

[66]. Meanwhile, the correlations for hybrid nanofluid are shown in

Table 2. Thermophysical properties of hybrid nanofluid.

Thermophysical Properties	Correlations
Thermal conductivity	$\frac{k_{hnf}}{k_f}=\frac{\frac{{\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}}{{\phi }_{hnf}}+2k_f+2\left({\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}\right)-2{\phi }_{hnf}{k}_f}{\frac{{\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}}{{\phi }_{hnf}}+2k_f-\left({\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}\right)+{\phi }_{hnf}{k}_f}$
Electrical conductivity	$\frac{{\sigma }_{hnf}}{{\sigma }_f}=\frac{\frac{{\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}}{{\phi }_{hnf}}+2{\sigma }_f+2\left({\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}\right)-2{\phi }_{hnf}{\sigma }_f}{\frac{{\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}}{{\phi }_{hnf}}+2{\sigma }_f-\left({\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}\right)+{\phi }_{hnf}{\sigma }_f}$
Heat capacity	${(\rho C_p)}_{hnf}=\left(1-{\phi }_{hnf}\right){(\rho C_p)}_f+{\phi }_1{(\rho C_p)}_{n1}+{\phi }_2{(\rho C_p)}_{n2}$
Density	${\rho }_{hnf}=\left(1-{\phi }_{hnf}\right){\rho }_f+{\phi }_1{\rho }_{n1}+{\phi }_2{\rho }_{n2}$
Dynamic viscosity	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\phi }_{hnf}\right)}^{2.5}}$

[33].

Now, consider the following similarity transformations [77,80,81]:

$$u=\frac{\mathsf{\Omega }r}{1-ct}{f}^{\prime }\left(\eta \right),v=\frac{\mathsf{\Omega }r}{ 1-ct}g\left(\eta \right),w=-\frac{2\sqrt{\mathsf{\Omega }{\nu }_f} }{\sqrt{1-ct}}f\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\eta =\sqrt{\frac{\mathsf{\Omega }}{{\nu }_f}}\frac{z}{\sqrt{1-ct}}.$$

(8)

Equation (1) is fully satisfied by considering the similarity variables in Equation (8). Furthermore, the reduced Equations (2)–(5) are:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{f}^{‴}+2f{f}^{″}-{f^{\prime }}^2+g^2-S\left(f^{\prime }+\frac{1}{2}\eta f^{″}\right)-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M{f}^{\prime }=0,$$

(9)

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{g}^{″}+2f{g}^{\prime }-2f^{\prime }g-S\left(g+\frac{1}{2}\eta g^{\prime }\right)-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}Mg=0,$$

(10)

$$\frac{1}{\mathrm{Pr}}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}{\theta }^{″}+2f{\theta }^{\prime }-\frac{1}{2}S\eta {\theta }^{\prime }=0,$$

(11)

subject to:

$$f\left(0\right)=\frac{B}{2}, f^{\prime }\left(0\right)=\lambda , g\left(0\right)=1, \theta \left(0\right)=1,$$

(12)

$$f^{\prime }\left(\eta \right)\to 0, g\left(\eta \right)\to 0, \theta \left(\eta \right)\to 0, \mathrm{as} \eta \to \infty$$

(13)

The parameters relevant to this problem are the unsteadiness parameter $S=c/\mathsf{\Omega }$, Prandtl number $\mathrm{Pr}={\left(\mu C_p\right)}_f/k_f$, suction/injection parameter $B=w_0/\sqrt{\mathsf{\Omega }{\nu }_f}$, and magnetic parameter $M=B_0^2{\sigma }_f/{\rho }_f\Omega$. Meanwhile, the skin friction coefficients are $C_f$ (radial direction) and $C_g$ (azimuthal direction), and the local Nusselt number for evaluating heat transfer performance is $N{u}_r$ (see Waini et al. [81]):

$$C_f=\frac{{\mu }_{hnf}}{{\rho }_f{u}_w^2}{\left(\frac{\partial u}{\partial z}\right)}_{z=0},C_g=\frac{{\mu }_{hnf}}{{\rho }_f{v}_w^2}{\left(\frac{\partial v}{\partial z}\right)}_{z=0},N{u}_r=-\frac{r{k}_{hnf}}{k_f\left(T_w-T_{\infty }\right)}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}$$

(14)

Using Equations (8) and (13), one obtains (see Waini et al. [81]):

$${\mathrm{Re}}_r^{1/2}{C}_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),{ \mathrm{Re}}_r^{1/2}{C}_g=\frac{{\mu }_{hnf}}{{\mu }_f}{g}^{\prime }\left(0\right),{ \mathrm{Re}}_r^{-1/2}N{u}_r=-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right),$$

(15)

where ${\mathrm{Re}}_r=u_{w}r/{\nu }_f$ is the local Reynolds number.

3. Stability Analysis

The dual solutions are examined to test their stability by employing stability analysis [82,83]. Firstly, consider the following variables:

$$\begin{array}{l}u=\frac{\mathsf{\Omega }r}{1-ct}\frac{\partial f}{\partial \eta }\left(\eta ,\tau \right),v=\frac{\mathsf{\Omega }r}{ 1-ct}g\left(\eta ,\tau \right),w=-\frac{2\sqrt{\mathsf{\Omega }{\nu }_f} }{\sqrt{1-ct}}f\left(\eta ,\tau \right),\theta \left(\eta ,\tau \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\\ \eta =\sqrt{\frac{\mathsf{\Omega }}{{\nu }_f}}\frac{z}{\sqrt{1-ct}},\tau =\frac{\mathsf{\Omega }r}{ 1-ct}t\end{array}$$

(16)

where $\tau$ is the time variable. Substituting Equation (16) into Equations (2)–(5),

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\frac{{\partial }^{3}f}{\partial {\eta }^3}+2f\frac{{\partial }^{2}f}{\partial {\eta }^2}-{\left(\frac{\partial f}{\partial \eta }\right)}^2+g^2-S\left(\frac{\partial f}{\partial \eta }+\frac{1}{2}\eta \frac{{\partial }^{2}f}{\partial {\eta }^2}\right)-\left(1+S\tau \right)\frac{{\partial }^{2}f}{\partial \eta \partial \eta }-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M\frac{\partial f}{\partial \eta }=0,$$

(17)

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\frac{{\partial }^{2}g}{\partial {\eta }^2}+2f\frac{\partial g}{\partial \eta }-2\frac{\partial f}{\partial \eta }g-S\left(g+\frac{1}{2}\eta \frac{\partial g}{\partial \eta }\right)-\left(1+S\tau \right)\frac{{\partial }^{2}g}{\partial \eta \partial \eta }-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}Mg=0,$$

(18)

$$\frac{1}{\mathrm{Pr}}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\frac{{\partial }^2\theta }{\partial {\eta }^2}+2f\frac{\partial \theta }{\partial \eta }-\frac{1}{2}S\eta \frac{\partial \theta }{\partial \eta }-\left(1+S\tau \right)\frac{\partial \theta }{\partial \tau }=0,$$

(19)

subject to:

$$f\left(0,\tau \right)=\frac{B}{2}, \frac{\partial f}{\partial \eta }\left(0,\tau \right)=\lambda , g\left(0,\tau \right)=1, \theta \left(0,\tau \right)=1,$$

(20)

$$\frac{\partial f}{\partial \eta }\left(\eta ,\tau \right)\to 0, g\left(\eta ,\tau \right)\to 0, \theta \left(\eta ,\tau \right)\to 0 \mathrm{as} \eta \to \infty$$

(21)

The perturbation function as suggested by early reference is [83]:

$$\begin{array}{l}f\left(\eta ,\tau \right)=f_0\left(\eta \right)+e^{-\alpha \tau }F\left(\eta ,\tau \right),\\ g\left(\eta ,\tau \right)=g_0\left(\eta \right)+e^{-\alpha \tau }G\left(\eta ,\tau \right),\\ \theta \left(\eta ,\tau \right)={\theta }_0\left(\eta \right)+e^{-\alpha \tau }H\left(\eta ,\tau \right)\end{array}$$

(22)

where $F\left(\eta ,\tau \right)$, $G\left(\eta ,\tau \right)$, and $H\left(\eta ,\tau \right)$ are arbitrary functions and relatively smaller than $f_0\left(\eta \right)$, $g_0\left(\eta \right)$, and ${\theta }_0\left(\eta \right)$, while $\alpha$ is the corresponding eigenvalue. Equation (22) is applied into Equations (17)–(21) and by setting $\tau =0$, $F\left(\eta ,\tau \right)=F_0\left(\eta \right)$, $G\left(\eta ,\tau \right)=G_0\left(\eta \right)$, and $H\left(\eta ,\tau \right)=H_0\left(\eta \right)$, the linearized equations are

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{F}_0^{‴}+2\left(f_0{F}_0^{″}+f_0^{″} F_0\right)-2f_0^{\prime }{F}_0^{\prime }+2g_0{G}_0-S\left(F_0^{\prime }+\frac{1}{2}\eta F_0^{″}\right)+\alpha F_0^{\prime }-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M{F}_0^{\prime }=0,$$

(23)

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{G}_0^{″}+2\left(f_0{G}_0^{\prime }+g_0^{\prime }{F}_0\right)-2\left(f_0^{\prime }{G}_0+g_0{F}_0^{\prime }\right)-S\left(G_0+\frac{1}{2}\eta G_0^{\prime }\right)+\alpha G_0-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M{G}_0=0,$$

(24)

$$\frac{1}{\mathrm{Pr}}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}{H}_0^{″}+2\left(f_0{H}_0^{\prime }+{\theta }_0^{\prime }{F}_0\right)-\frac{1}{2}S\eta H^{\prime }+\alpha H_0=0$$

(25)

subject to:

$$F_0(0)=0, F_0^{\prime }(0)=0, G_0(0)=0, H_0(0)=0,$$

(26)

$$F_0^{\prime }(\eta )\to 0, G_0(\eta )\to 0, H_0(\eta )\to 0 \mathrm{as} \eta \to \infty .$$

(27)

The set of eigenvalues $\alpha$ are obtained from Equations (23)–(25) if $F_0^{\prime }(\eta )\to 0 \mathrm{as} \eta \to \infty$ in Equation (27) is replaced by $F^{″}(0)=1$ [84].

4. Results and Discussion

Discussions of the obtained results are provided in this section. The bvp4c solver was used for the computation. This solver occupies a finite difference method that employs the three-stage Lobatto IIIa formula, see Shampine et al. [85,86]. The present results were validated by conducting a comparison with previously published data. In this respect,

Table 3. Validation of numerical values when ${\phi }_{hnf}=B=M=0$ and different $S$.

$\mathit{S}$	Present		Fang and Tao [77]		Waini et al. [81]
	$f^{″}\left(0\right)$	$g^{\prime }\left(0\right)$	$f^{″}\left(0\right)$	$g^{\prime }\left(0\right)$	$f^{″}\left(0\right)$	$g^{\prime }\left(0\right)$
−1	0.719787	−0.236575	0.7198	-0.2366	0.719787	−0.236575
−2	0.931507	0.154981	0.9315	0.1550	0.931507	0.154981
−5	1.562797	1.360850	1.5627	1.3609	1.562797	1.360850
−10	2.600801	3.413860	2.6008	3.4139	2.600801	3.413860

provides the validation of test values when ${\phi }_{hnf}=B=M=0$ with different values of $S$, and an excellent agreement can be observed from the comparison. Therefore, the present results are acceptable and accurate.

Furthermore, the total composition of Al₂O₃ and Cu concentrations in this study are considered as $1\%$ of Fe₃O₄ $\left({\phi }_1=1\%\right)$ and $1\%$ of CoFe₂O₄ $\left({\phi }_2=1\%\right).$ The Prandtl number subject to the water base fluid is $\mathrm{Pr}=6.2$ and $\mathrm{Pr}=30$ for water–ethylene glycol. The dual solution’s availability is possible, as displayed 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 when the parameters are used within the allocated interval; unsteadiness decelerating parameter $-1.2\le S\le -1$, magnetic parameter $0\le M\le 0.2$, suction/injection parameter $-0.2\le B\le 0.2$ while other parameters are fixed such that ${\varphi }_1,{\varphi }_2=1\%$ $\left({\varphi }_{hnf}=2\%\right)$ and $\lambda =0$.

The new numerical results obtained from this present study are presented in

Table 4. Values of ${\mathrm{Re}}_r^{1/2}{C}_f$, ${\mathrm{Re}}_r^{1/2}{C}_g, \mathrm{and} {\mathrm{Re}}_r^{-1/2}N{u}_r$ for Fe₃O₄-CoFe₂O₄/H₂O-EG when ${\phi }_{hnf}=2\%, B=0, S=-1$ and different values of $M$.

$\mathit{M}$	First Solution			Second Solution
	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{g}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{r}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{g}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{r}}$
0	0.765534	−0.251611	3.604480	0.512769	−0.088499	3.452206
0.1	0.722637	−0.302647	3.581163	0.478139	−0.136966	3.430586
0.2	0.681833	−0.354398	3.558560	0.446589	−0.185258	3.410523
0.3	0.643206	−0.406753	3.536762	0.418191	−0.233521	3.392171

and

Table 5. Values of ${\mathrm{Re}}_r^{1/2}{C}_f$, ${\mathrm{Re}}_r^{1/2}{C}_g,\mathrm{and} {\mathrm{Re}}_r^{-1/2}N{u}_r$ for Fe₃O₄-CoFe₂O₄/H₂O when ${\phi }_{hnf}=2\%, B=0, S=-1$ and different values of $M$.

$\mathit{M}$	First Solution			Second Solution
	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{g}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{r}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{1/2}{\mathit{C}}_{\mathit{g}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{r}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{r}}$
0	0.767572	−0.252281	1.755377	0.514134	−0.088735	1.617641
0.1	0.724783	−0.303179	1.737683	0.479588	−0.137073	1.599005
0.2	0.684071	−0.354788	1.720410	0.448097	−0.185238	1.581450
0.3	0.645516	−0.406997	1.703642	0.419732	−0.233372	1.565181

. Here, the values of the physical quantities such as ${\mathrm{Re}}_r^{1/2}{C}_f$, ${\mathrm{Re}}_r^{1/2}{C}_g$, and ${\mathrm{Re}}_r^{-1/2}N{u}_r$ are obtained for varied values of $M$ with different base fluids (H₂O-EG and H₂O). For the first solution, the results show a decrement in these physical quantities as $M$ increases. Quantitatively, a 15.98% decrement is observed for the values of ${\mathrm{Re}}_r^{1/2}{C}_f$ when $M$ increases up to 30% $\left(M=0.3\right)$. Meanwhile, the reduction in ${\mathrm{Re}}_r^{1/2}{C}_g$ is prominent with a 61.66% decrement and the values of ${\mathrm{Re}}_r^{-1/2}N{u}_r$ decrease with 1.88%. However, a significant increment is observed for ${\mathrm{Re}}_r^{-1/2}N{u}_r$ when H₂O-EG is considered with 107.60% compared to H₂O as the base fluid. Furthermore, the numerical results provided could be important to other researchers for future reference. Interestingly, multiple solutions are obtained under certain circumstances. To determine a realistic or stable solution,

Table 6. Smallest eigenvalues when ${\phi }_{hnf}=2\%, B=M=0,$ and various $S$ for Fe₃O₄-CoFe₂O₄/H₂O-EG.

$\mathit{S}$	Smallest Eigenvalues
	First Solution	Second Solution
−1.1	6.0014	0.7442
−1.2	6.7395	0.8547
−1.3	6.9401	1.0462

shows the smallest eigenvalues $\alpha$ that were obtained from the eigenvalue problems (see Section 3). Surprisingly, both solutions give positive values $(\alpha >0)$ that reveal the stability of both solutions and the possibility of another solution with a negative eigenvalue [81].

The effects of magnetic $M$ and unsteadiness $S$ parameters on the physical quantities (${\mathrm{Re}}_r^{1/2}{C}_f$, ${\mathrm{Re}}_r^{1/2}{C}_g$, and ${\mathrm{Re}}_r^{-1/2}N{u}_r$) are elucidated in Figure 2, Figure 3 and Figure 4. The declining behaviour is noticed when a larger $M$ value is considered. However, the opposite trend is shown for stronger deceleration strength $(S<0)$. Physically, the Lorentz force is created when the magnetic field is imposed on the boundary layer, and it is intensified for stronger magnetic field strength. This force possesses the flow and increases the shear stress on the surface, and consequently increases the thermal rate. It should be noted that the outcomes of this study are contradicted by its physical phenomenon due to the interaction between the magnetic field and the unsteadiness of the flow. This observation is supported by a previous study, see Waini et al. [81], which stated that the unsteadiness condition creates an obstacle to the flow and thermal behaviour of the fluid.

Figure 5, Figure 6 and Figure 7 displayed the values of ${\mathrm{Re}}_r^{1/2}{C}_f$, ${\mathrm{Re}}_r^{1/2}{C}_g$, and ${\mathrm{Re}}_r^{-1/2}N{u}_r$ for different values of $B$ (mass flux parameter). The quantities of ${\mathrm{Re}}_r^{1/2}{C}_f$, and ${\mathrm{Re}}_r^{1/2}{C}_g$ are lower for $B=0.2$ (suction case). Physically, the wall suction causes a higher drag force and torque on the revolving disk and thus raising the shear stress. However, the opposite observation was seen in this study (see Figure 5 and Figure 6). This is due to the simultaneous effect of the physical parameter on the flow. However, the behaviour is contradicted for ${\mathrm{Re}}_r^{-1/2}N{u}_r$ where $B=0.2$ (suction case) gives higher values of ${\mathrm{Re}}_r^{-1/2}N{u}_r$.

Figure 8, Figure 9 and Figure 10 are provided in order to investigate the impact of the different base fluids (H₂O and H₂O-EG) on the flow and thermal behaviour. Here, H₂O-EG contributes to enhancing the values of ${\mathrm{Re}}_r^{1/2}{C}_g$, and ${\mathrm{Re}}_r^{-1/2}N{u}_r$. Meanwhile, the values of ${\mathrm{Re}}_r^{1/2}{C}_f$ decline. This observation can be explained from the values of their Prandtl number, Pr. Note that H₂O-EG has larger Pr, i.e., $\mathrm{Pr}=30$ if compared to H₂O $\left(\mathrm{Pr}=6.2\right)$. Larger $\mathrm{Pr}$ means the convection process for the flow is dominant. As a result, fluid momentum, rather than fluid conduction, is the best way to transmit heat. Additionally, the velocity and the temperature profiles of Fe₃O₄-CoFe₂O₄/H₂O-EG for several $S$ are given in Figure 11, Figure 12 and Figure 13. The thickness of the respective boundary layers is lessened as S becomes smaller. This implies that the gradient of these profiles near the wall is increased for stronger deceleration strength $(S<0)$ and gives higher values of the shear stress and heat transfer on the wall, as reported in Figure 4, Figure 5 and Figure 6.

5. Conclusions

In this numerical study, various effects of the controlling parameters in the unsteady MHD hybrid ferrofluid flow and heat transfer over a rotating stretching/shrinking disk are discussed. The physical properties of the fluid are affected by different values of the magnetic parameter, unsteadiness parameter, and the mass flux parameter which generate an impact on the skin friction coefficient, heat transfer rate, velocity profile, and temperature field. It is perceived that velocity profile and temperature field distributions decrease when increasing the unsteadiness parameter. On the other hand, it is found that a highly oscillating magnetic field forces the particles to rotate slower than the ferrofluid; therefore, resulting in an increment of viscosity. Consequently, a deterioration in the flow speed is observed as the value of the magnetic parameter increases up to 30%, thus decreasing the skin friction coefficient by 15.98% and the heat transfer rate approximately up to 1.88%, significantly. In contrast, the heat transfer is enhanced rapidly by improving the mass flux parameter by almost 20%. Furthermore, the use of H₂O-EG improves the thermal efficiency of the system when compared to H₂O as the base fluid. Finally, depending on the analysis of solution stability, both solutions are surprisingly found to be stable.