Theoretical Analysis of Cu-H2O, Al2O3-H2O, and TiO2-H2O Nanofluid Flow Past a Rotating Disk with Velocity Slip and Convective Conditions

Department of Mathematics, Abdul Wali Khan University, Mardan, Mardan, 23200 Khyber Pakhtunkhwa, Pakistan Department of Mathematics Education, University of Education Winnebakumasi-(Kumasicompus), Kumasi 00233, Ghana Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematical Sciences, University of Lakki Marwat, Lakki Marwat, 28420 Khyber Pakhtunkhwa, Pakistan


Introduction
The suspension of nanosized (between 1 nm and 100 nm) material into conventional fluids such as oil, ethylene glycol, water, and sodium alginate is called nanofluids. Nanofluids with their innovative and advanced ideas have intriguing thermal transfer properties as opposed to traditional heat transfer fluids. There has been a great deal of research into nanofluids' dominant heat transfer properties, especially convective heat transfer and thermal conductivity. With these properties, nanofluid implementations in industries like heat exchange systems look promising. The nanofluids can be used in the subsequent precise areas like chemical nanofluids, environmental nanofluids, heat transfer nanofluids, pharmaceutical nanofluids, drug delivery nanofluids, and process/extraction nanofluids. In short, the number of engineering and industrial applications of nanofluids technologies, as well as their emphasis on particular industrial applications, has been increased recently [1][2][3][4][5][6][7]. The capability of thermal transmission of nanofluids can be quantified by their properties like specific heat, density, viscosity, and thermal conductivity. The thermal properties are contingent on the shape, base fluid, particle size, material, and concentration. To utilize the applications towards engineering and industries, researchers are working on the evaluation and characterization of the thermophysical properties of nanofluids for heat transfer analysis [8]. Sheikholeslami [9] analyzed the different shapes of aluminum oxide using the Darcy porous medium with thermal radiation. Hayat et al. [10] investigated the nanofluid flow with Hall and Ohmic influences. They deliberated the thermal convective and velocity slip boundary conditions. The Hall and Ohmic parameters have reduced the velocity and heat transfer rate. Sheikholeslami [11] presented the analysis different shapes of nanoparticles of copper oxide water with Brownian motion. It has been introduced that the platelet shape nanoparticles has leading impression as associated to other shapes of nanoparticles. Thumma et al. [12] investigated the non-Newtonian nanofluid flow containing water-based CuO and Cu nanoparticles past porous extending sheet with entropy optimization and velocity condition. A non-Fourier has been implemented to analyze the heat transfer rate. Hayat et al. [13] examined the Cu, Fe 2 O 3 , and Au nanoparticles with Hall and Ohmic effects using constant and variable viscosities. Sheikholeslami et al. [14] addressed the Al 2 O 3 -water nanoparticles through a channel with Brownian motion impact. Thumma et al. [15] deliberated the radiative boundary layer nanofluid flow past a nonlinear extending surface with viscous dissipation. Rout et al. [16] analyzed the water-based Cu and kerosene oil-based Cu between two parallel plates with thermal radiation. Further studies related to nanofluids are mentioned in [17][18][19][20][21][22][23][24][25][26].
The flow behavior of a flowing conducting liquid is described by magnetohydrodynamic (MHD), which polarizes it. In industrial activities such as nuclear power plants, crystal manufacture, electric generators, and fuel industry, the impact of magnetic fields is assessed. Tamim et al. [27] addressed the MHD mixed convective flow of nanofluid on a vertical plate. They studied both opposing and assisting flows. The water-based Cu, Al 2 O 3 , and TiO 2 are examined. Ghadikolaei et al. [28] implemented the induced magnetic field on hybrid nanofluid flow through an extending surface. Hayat et al. [29] explore the unsteady MHD viscous fluid flow with Joule heating, thermal radiation, and thermal stratification influences. Ahmad et al. [30] expressed the MHD flow of ferrofluid past an exponentially extending surface. Singh et al. [31] investigated the MHD flow of water-based alumina nanofluid past a flat plate with slip condition. Mliki et al. [32] evaluated the convective nanofluid flow with MHD effect. Upreti et al. [33] presented the CNT nanofluids past an extending surface with nonuniform heat source/sink and Ohmic heating. Pandey et al. [34] presented the MHD water-based copper nanofluid flow inside a convergent/divergent channel. Upreti et al. examined the MHD Ag-kerosene oil nanofluid with suction/injection roles. Turkyilmazoglu [35] presented the viscous fluid flow with magnetic field impact past a spinning disk. The MHD viscous fluid flow considering wall slip conditions has been investigated by Hussain et al. [36]. Dawar et al. [19] presented the highly magnetized and nonmagnetized non-Newtonian fluid flow past an extending cylinder. Further related results can be seen in [18,[37][38][39][40][41][42][43][44][45].
Magnetic nanoparticles pique the researchers' interest in various fields, including homogeneous and heterogeneous catalysis, magnetic fluids, environmental remediation, biomedicine, data storage, and magnetic resonance imaging (MRI) for instance purification of water. The literature proves that the nanoparticles of size less than the critical value (i.e., 10-20 nm) perform best [46]. Nanoparticles' magnetic properties effectively monopolize at such a small scale, rendering them beneficial and helpful in a wide range of applications [46][47][48][49]. In light of the abovementioned applications, we have considered a mathematical model for the flow of nanofluid containing the nanoparticles of Cu-H 2 O, Al 2 O 3 -H 2 O, and TiO 2 -H 2 O, and pure water with a strong magnetic field. According to the authors knowledge, there is no study based on spherical-shaped nanoparticles of the Cu, Al 2 O 3 , and TiO 2 using water as a based fluid past a rotating disk. Furthermore, the velocity slip and convective conditions are considered to analyze the flow behavior in the presence and absence of slip conditions. The mathematical model is solved with the help of the homotopic approach.

Physical Model
We consider the water-based nanomaterials (Cu, Al 2 O 3 , and TiO 2 ) past a rotating disk. The velocity componentsũ 1 ,ũ 2 , and u 3 are taken alongr,ϕ, andz directions, respectively. The disk rotates with an angular velocity Ω atz = 0 (see Figure 1). A magnetic field of strength B 0 is applied normal to the fluid flow. The flow is subjected to velocity slip and thermal convective conditions. The leading equations are defined as follows [35]: with boundary conditions: Journal of Nanomaterials The thermophysical properties of the nanofluids are defined as [50] In the above equations, μ is the dynamic viscosity, ρ is the density, c p is the heat capacitance, L is the wall slip parameter,p is the pressure, k is the thermal conductivity, and φ represents the volume fraction of the nanoparticles. Furthermore, the subscript f indicates the base fluid, nf shows the nanofluids, and np is used for nanoparticles.
The correspondence variables are defined as [53][54][55] The above system is transformed as where Here, The surface drag force C fr and heat transfer rate Nu r are defined as [53,55] where τr, τ ϕ , and q w are defined as The dimensionless form of Equation (7) is: ffiffiffiffiffi where Re = Ωr 2 /ν f is the local Reynolds number.

HAM Solution
The initial guesses and linear operators are defined as with the following properties: where c i ði = 1 − 6Þ are called arbitrary constants.

Results and Discussion
This segment compacts with the impressions of different embedded factors on velocities and temperature, surface drag force and heat transfer rate. We have considered the spherical-shaped three different nanoparticles like Cu, Al 2 O 3 , and TiO 2 with a base fluid H 2 O. Since water is used as a base fluid, therefore, Pr = 6:2. The thermophysical properties of Cu, Al 2 O 3 , TiO 2 , and H 2 O are presented in Table 1. The shape factor and sphericity of the different nanoparticles are presented in Table 2. In Table 3, we have presented the numerical values of skin friction via magnetic parameter for different water-based spherical-shaped nanoparticles and pure water. Both slip and no-slip conditions are considered here. The greater magnetic parameter augments the skin friction coefficient. Actually, the magnetic parameter drops off the velocity function due to Lorentz force. The heightening Lorentz force means the skin friction coefficient augments which has been seen for the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles and pure water for the case of no-slip condition. For the case of slip condition, interesting results have been introduced here. Physically, the presence of slip parameter reduces the velocity of the fluid due augmenting skin friction coefficient as occurs which allow more fluid to past the disk as found for pure water. However, for the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles, the presence of slip and magnetic parameters have diverse impact on surface drag force. In addition, the greater impact of magnetic parameter occurs in the absence of slip effect. Table 4 shows the numerical values of surface drag force via spherical-shaped nanoparticle volume fraction for the different water-based nanoparticles. Physically, the increasing nanoparticle volume fraction means that the nanoparti-cles and the base fluid collide with each other which accelerates the fluid motion; consequently, the momentum boundary layer thickness decreases and upsurges the surface drag force. Also, the impact of spherical-shaped nanoparticles volume fraction is the same for the local Nusselt number as portrayed in Table 5. Additionally, the surface drag force is greater for the case of no-slip condition. The increasing thermal Biot number augments the heat transfer rate. Tables 6-8 show the comparison of analytical and numerical techniques for f ðηÞ, gðηÞ, and θðηÞ. Here, a close agreement between both techniques is found. Figure 3 shows the impact of nanoparticle volume fraction on spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles. Figure 4 shows the variation in radial velocity of the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles and pure water (H 2 O) via a magnetic parameter for the case of no-slip condition. The greater magnetic factor diminishes the radial velocity of the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles and          Figures 4 and 5. However, the impact of slip condition is greater for f ðηÞ as compared to gðηÞ. Figure 10       Journal of Nanomaterials φ reduces the velocity profile gðηÞ of the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles (see Figure 11). Additionally, the reducing impact of φ is greater for α = 0:5 as compared to α = 0:0. Figure 12 shows the variation in temperature profile θðηÞ of the spherical-shaped Cu, Al 2 O 3 , and TiO 2 nanoparticles via φ. The greater φ augments the temperature profile. Physically, the greater φ upsurges the thermal conductivity of the Cu, Al 2 O 3 , and TiO 2 nanoparticles and thermal transfer rate. Therefore, the nanoparticle which has high thermal conductivity has the dominant impact on temperature profile and heat transfer rate as shown in Figure 3 and Table 5. Here, Cu nanoparticle has greater thermal conductivity than Al 2 O 3 nanoparticle, and Al 2 O 3 nanoparticle has greater thermal conductivity than TiO 2 nanoparticle. So, the greatest impact of Cu nanoparticle is found here. Figure 13 shows the variation in temperature profile θðηÞ of the spherical-shaped Cu,

Journal of Nanomaterials
Biot number raises the convection and thermal profile significantly. Additionally, the spherical-shaped Cu nanoparticle has greater impact on thermal profile as compared to Al 2 O 3 and TiO 2 nanoparticles.

Conclusion
In this work, we have examined the water-based sphericalshaped nanoparticles of copper-water, aluminum oxidewater, titanium dioxide-water, and pure water past a rotating disk. Slip and no-slip conditions are considered in order to examine the variations in radial and tangential velocities due to the magnetic field, nanoparticle volume fraction, and thermal Biot number. The final points are mentioned below: Thermal Biot number C f : Skin friction coefficient c i ði = 1 − 6Þ: Arbitrary constants c p : Heat capacitance f 0 , θ 0 , g 0 : Initial guesses k: Thermal conductivity L: Wall slip parameter L f , L g , L θ : Linear operators M: Magnetic parameter Nu: Nusselt number p: Pressure Re: Reynolds number Pr: Prandtl number r,ϕ,z: Coordinates u 1 ,ũ 2 ,ũ 3 : Velocity components Greek Letters Ω: Angular velocity σ: Electrical conductivity ρ: Density μ: Dynamic viscosity α: Dimensionless wall slip parameter φ: Volume fraction of the nanoparticles Subscripts f : Fluid nf : Nanofluids np: Nanoparticles.

Data Availability
All the supporting data are within the manuscript.

Conflicts of Interest
The authors declare that they have no conflict of interest.  Journal of Nanomaterials