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

Authors: Abdullah Dawar; Ebenezer Bonyah; Saeed Islam; Ahmed Alshehri; Zahir Shah
Journal: Journal of Nanomaterials
 (2021)

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## Abstract

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 nanofluid technologies, as well as their emphasis on particular industrial applications, has been increased recently. Therefore, this exploration is carried out to analyze the nanofluid flow past a rotating disk with velocity slip and convective conditions. The water-based spherical-shaped nanoparticles of copper, alumina, and titanium have been considered in this analysis. The modeled problem has been solved with the help of homotopic technique. Convergence of the homotopic technique is shown with the help of the figure. The role of the physical factors on radial and tangential velocities, temperature, surface drag force, and heat transfer rate are displayed through figures and tables. The outcomes demonstrate that the surface drag force of the water-based spherical-shaped nanoparticles of Cu, Al2O3, and TiO2 has been reduced with a greater magnetic field. The radial and tangential velocities of the water-based spherical-shaped nanoparticles of Cu, Al2O3, and TiO2, and pure water have been augmented via magnetic parameter. The radial velocity of the water-based spherical-shaped nanoparticle of Cu has been augmented via nanoparticle volume fraction, whereas reduced for the Al2O3 and TiO2 nanoparticles. The tangential velocity of the water-based spherical-shaped nanoparticles of Cu, Al2O3, and TiO2 has reduced via nanoparticle volume fraction. Also, the variations in radial and tangential velocities are greater for slip conditions as compared to no-slip conditions.


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## Body

## 1. 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 [1–7]. 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, Fe2O3, and Au nanoparticles with Hall and Ohmic effects using constant and variable viscosities. Sheikholeslami et al. [14] addressed the Al2O3-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 [17–26].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, Al2O3, and TiO2 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, 37–45].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 [46–49]. In light of the abovementioned applications, we have considered a mathematical model for the flow of nanofluid containing the nanoparticles of Cu-H2O, Al2O3-H2O, and TiO2-H2O, 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, Al2O3, and TiO2 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.


## 2. Physical Model

We consider the water-based nanomaterials (Cu, Al2O3, and TiO2) past a rotating disk. The velocity components u~1,u~2, and u~3 are taken along r~,ϕ, and z~ directions, respectively. The disk rotates with an angular velocity Ω at z~=0 (see Figure 1). A magnetic field of strength B0 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]:
(1)∂u~1∂r~+u~1r~+∂u~3∂z~=0,u~1∂u~1∂r−u~22r+u~3∂u~1∂z=−1ρnf∂p~∂r~+μnfρnf∂2u~1∂r~2+1r~∂u~1∂r~−u~1r~2+∂2u~1∂z~2−σnfρnfB02u~1,u~1∂u~2∂r~+u~1u~2r~+u~3∂u~2∂z~=μnfρnf∂2u~2∂r~2+1r~∂u~2∂r~−u~2r~2+∂2u~2∂z~2−σnfρnfB02u~2,u~1∂u~3∂r~+u~3∂u~3∂z~=−1ρnf∂p~∂z~+μnfρnf∂2u~3∂r~2+1r~∂u~3∂r~+∂2u~3∂z~2,u~1∂T~∂r~+u~3∂T~∂z~=knfρcpnf∂2T~∂r~2+1r~∂T~∂r~+∂2T~∂z~2,with boundary conditions:
(2)u~1=L∂u~1∂z~,u~2=L∂u~2∂z~+Ω⋅r~,u~3=0,−knf∂T~∂z~=hfT~f−T~atz~=0u~1⟶0,u~2⟶0,T~⟶T~∞asz~⟶∞.Figure 1
Geometrical representation of the flow problem.The thermophysical properties of the nanofluids are defined as [50]
(3)μnfμf=11−φ2.5,ρnfρf=1−φ+φρnpρf,ρcpnfρcpf=1−φ+φρcpnpρcpfσnfσf=1+3σnp/σf−1φσnp/σf+2−σnp/σf−1φ,knfkf=knp+n−1kf−n−1kf−knpφknp+n−1kf+kf−knpφ.In the above equations,μ is the dynamic viscosity, ρ is the density, cp 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 [53–55]
(4)u~1=r~Ωfη,u~2=r~Ωgη,u~3=Ωνfhηp~=p~∞+2ΩμfP~η,T~=T~∞+T~f−T~∞θη,η=Ωνfz~.The above system is transformed as(5)h′+2f=0,f″+M¯2M¯1hf′−f2+g2−M¯3M¯1Mf=0,g″−M¯2M¯12gf+g′h−M¯3M¯1Mg=0,θ″−M¯5M¯4Prθ′h=0,where
(6)M¯1=μnfμf,M¯2=ρnfρf,M¯3=σnfσf,M¯4=knfkf,M¯5=ρcpnfρcpf.with
(7)f0=αf′0,g0=1+αg′0,h0=0,knfkfθ′0=Biθ0−1fη⟶∞⟶0,gη⟶∞⟶0,θη⟶∞⟶0.Here,M=σfB02/ρfΩ is the magnetic parameter, Pr=νf/αf is the Prandtl number, α=Lr~Ω/νf is the wall slip parameter, and Bi=hf/kfΩ/νf is the thermal Biot number.The surface drag forceCfr~ and heat transfer rate Nur are defined as [53, 55]
(8)Cfr~=1ρfr~2Ω2τr~2+τϕ2,Nur~=r~qwkfTf−Tw,where τr~, τϕ, and qw are defined as
(9)τr=μnf∂u~1∂z~+∂u~3∂r~z~=0,τϕ=μnf∂u~2∂z~+1r~∂u~3∂ϕz~=0,qw=−knf∂T~∂z~z~=0.The dimensionless form of Equation (7) is:
(10)ReCfr~=M¯1f′20+g′20,NurRe=−M¯4θ′0,where Re=Ωr~2/νf is the local Reynolds number.


## 3. HAM Solution

The initial guesses and linear operators are defined as(11)f0η=0,g0η=11+αe−η,θ0η=kfknfBi1+Bie−η,Lf=f″−f,Lg=g″−g,Lθ=θ″−θ,with the following properties:
(12)Lfc1e−η+c2eη=0,Lgc3e−η+c4eη=0,Lθc5e−η+c6eη=0,where cii=1−6 are called arbitrary constants.


## 4. HAM Convergence

Figure2 is displayed for the convergence of series solutions. The auxiliary parameters ℏf, ℏg, and ℏθ are responsible for the adjustment and controlling of the series solutions. Therefore, the acceptable values for velocities and thermal profiles are −2.1≤ℏf≤−0.1, −1.9≤ℏg≤0.0, and −2.5≤ℏθ≤0.2.Figure 2
ℏ-curves for f′0, g′0, and θ′0.


## 5. 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, Al2O3, and TiO2 with a base fluid H2O. Since water is used as a base fluid, therefore, Pr=6.2. The thermophysical properties of Cu, Al2O3, TiO2, and H2O 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, Al2O3, and TiO2 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, Al2O3, and TiO2 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 nanoparticles 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 fη, gη, and θη. Here, a close agreement between both techniques is found. Figure 3 shows the impact of nanoparticle volume fraction on spherical-shaped Cu, Al2O3, and TiO2 nanoparticles. Figure 4 shows the variation in radial velocity of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles and pure water (H2O) via a magnetic parameter for the case of no-slip condition. The greater magnetic factor diminishes the radial velocity of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles and pure water (H2O). Physically, the applied magnetic field creates Lorentz force during the fluid flow which opposes the motion of the flow nanoparticles; consequently, a reducing impact is observed. For the case of slip condition, a similar impact of magnetic parameter is observed for radial velocity of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles and pure water (H2O) as displayed in Figure 5. Furthermore, the presence of a slip parameter reduces the velocity of the fluid due to augmenting skin friction coefficient occuring which allows more fluid to past the disk. So, the combination of magnetic and slip parameters has greater impact on velocity profile of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles and pure water (H2O) as compared to no-slip condition. Figure 6 displays the variation in radial velocity fη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles via φ for the case of no-slip condition. The greater φ augments the radial velocity of the spherical-shaped Cu nanoparticle, while it reduces the radial velocity of the spherical-shaped Al2O3 and TiO2 nanoparticles. Physically, the greater φ opposes the motion of the spherical-shaped Al2O3 and TiO2 nanoparticles which augments the boundary layer thickness and slows down the velocity profile, while this impact is opposite for Cu nanoparticle. For the case of slip condition, the greater φ augments the radial velocity of the spherical-shaped Cu nanoparticle, while it reduces the velocity profile for Al2O3 and TiO2 nanoparticles next to the surface of the rotating disk and moderates the increasing effect as η⟶∞ (see Figure 7). Figures 8 and 9 portray the variation in gη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles via a magnetic parameter for the case of no-slip and slip conditions, respectively. For both no-slip and slip conditions, similar impacts are found here as seen in Figures 4 and 5. However, the impact of slip condition is greater for fη as compared to gη. Figure 10 shows the variation in velocity profile gη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles via φ for the case when α=0.0. The greater φ augments the velocity profile gη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles. Physically, the greater φ opposes the motion of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles which augments the boundary layer thickness and slows down the velocity profile. For the case when α=0.5, the greater φ reduces the velocity profile gη of the spherical-shaped Cu, Al2O3, and TiO2 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 θη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles via φ. The greater φ augments the temperature profile. Physically, the greater φ upsurges the thermal conductivity of the Cu, Al2O3, and TiO2 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 Al2O3 nanoparticle, and Al2O3 nanoparticle has greater thermal conductivity than TiO2 nanoparticle. So, the greatest impact of Cu nanoparticle is found here. Figure 13 shows the variation in temperature profile θη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles via the thermal Biot number. The greater Biot number augments the thermal profile θη of the spherical-shaped Cu, Al2O3, and TiO2 nanoparticles. Physically, the heat transfer coefficient caused by the hot fluid is directly related to the Biot number. Therefore, the greater Biot number raises the convection and thermal profile significantly. Additionally, the spherical-shaped Cu nanoparticle has greater impact on thermal profile as compared to Al2O3 and TiO2 nanoparticles.Table 1
Numerical values of the thermophysical properties of H2O, Cu, Al2O3, and TiO2 [13, 51, 52].
Base fluid and nanoparticlesρkg/m3cpJ/kgKkW/mKσ1/ΩmH2O997.141790.6130.05Al2O33970765401×10−10Cu89333854015.96×107TiO24250685.28.95392.6×106Table 2
Shape factor and sphericity of different particle shapes [56, 57].
Shape of the nanoparticleSphericityShape factorSphere1.03.00Cylinder0.624.84Blade0.368.33Platelet0.525.77Brick0.813.70Table 3
Numerical values of the skin friction via magnetic parameter for different water-based spherical-shaped nanoparticles and pure water.
Magnetic parameterValuesNo-slip conditionCuAl2O3TiO2Pure waterM1.0α=0.00.974550.959010.974840.949042.01.339311.308451.340761.304323.01.709711.662921.711841.66454Slip condition1.0α=0.50.698130.693690.697400.821722.00.781740.773650.781040.578233.00.865420.853680.864750.33615Table 4
Numerical values of the skin friction via spherical-shaped nanoparticle volume fraction for different water-based nanoparticles.
Nanoparticles volume fractionValuesNo-slip conditionCuAl2O3TiO2φ0.1α=0.00.974550.959010.974840.21.001050.969601.001420.31.028540.980861.02879Slip condition0.1α=0.50.698130.693690.697400.20.717270.708320.715780.30.737020.723520.73397Table 5
Numerical values of the local Nusselt number via the Biot number and spherical-shaped nanoparticle volume fraction for different water-based nanoparticles and pure water.
ParametersValuesCuAl2O3TiO2Pure waterBi0.10.084330.084490.085040.087870.20.150210.150500.151460.156480.30.208820.203210.204500.21124φ0.10.161240.157820.14667—0.20.295200.282920.24451—0.30.549980.515120.41199—Table 6
Analytical and numerical solutions forfη.
ηHAMShootingAbsolute error0.01.277200×10−170.0484580.0484580.50.0799260.1578460.0779211.00.0929980.1996630.1066661.50.0837150.2033720.1196572.00.0685440.1864090.1178662.50.0535230.1597240.1062023.00.0405350.1298660.0893303.50.0297380.1000120.0702744.00.0204720.0704700.0499984.50.0113940.0384150.0270215.08.673620×10−19-0.0040140.004014Table 7
Analytical and numerical solutions forgη.
ηHAMShootingAbsolute error0.01.0000000.9188800.0811200.50.6225650.7302150.1076501.00.4015480.5889910.1874431.50.2638760.4742050.2103292.00.1753980.3776810.2022822.50.1175300.2961260.1785963.00.0791690.2275440.1483753.50.0532750.1695570.1162824.00.0350730.1183680.0832964.50.0208450.0672990.0464545.00.0067380.0033950.003343Table 8
Analytical and numerical solutions forθη.
ηHAMShootingAbsolute error0.00.5207280.5149730.0057550.50.4072580.3037780.1034771.00.3169870.1709610.1460291.50.2488840.0985080.1503762.00.1962180.0597820.1364362.50.1535660.0380720.1154943.00.1173790.0248780.0925013.50.0854130.0160850.0693284.00.0562150.0096520.0465634.50.0287720.0044870.0242855.00.002246-0.0001050.002351Figure 3
Impact of nanoparticle volume fraction on spherical-shaped nanoparticles of water-based Cu, Al2O3, and TiO2.Figure 4
Variation infη via M when α=0.0.Figure 5
Variation infη via M when α=0.5.Figure 6
Variation infη via φ when α=0.0.Figure 7
Variation infη via φ when α=0.5.Figure 8
Variation ingη via M when α=0.0.Figure 9
Variation ingη via M when α=0.5.Figure 10
Variation ingη via φ when α=0.0.Figure 11
Variation ingη via φ when α=0.5.Figure 12
Variation inθη via φ.Figure 13
Variation inθη via Bi.


## 6. Conclusion

In this work, we have examined the water-based spherical-shaped nanoparticles of copper-water, aluminum oxide-water, 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:(a)
Forα=0.5, the surface drag force of Cu, Al2O3, and TiO2 have reduced with the increasing magnetic parameter, while for α=0.0, the surface drag force of the Cu, Al2O3, and TiO2 nanoparticles have augmented with the increasing magnetic parameter. Additionally, the greater impact of magnetic parameter occurs when α=0.5(b)
The surface drag force and heat transfer rate of spherical-shaped nanoparticles of Cu, Al2O3, and TiO2 is augmented via nanoparticle volume fraction(c)
Forα=0.5 and α=0.0, the radial and tangential velocities of the spherical-shaped nanoparticles of Cu, Al2O3, and TiO2 and pure water have augmented via a magnetic parameter. Additionally, the impact of magnetic field is greater for radial velocity as compared to tangential velocity(d)
Forα=0.5 and α=0.0, the tangential velocity of the spherical-shaped nanoparticle of Cu, Al2O3, and TiO2 has reduced via nanoparticle volume fraction. Additionally, the reducing impact of nanoparticle volume fraction is greater for α=0.5 as compared to α=0.0(e)
The greater nanoparticle volume fraction and thermal Biot number have increased the temperature profile of the spherical-shaped nanoparticles of Cu, Al2O3, and TiO2



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