Three-Dimensional Casson Nanofluid Thin Film Flow over an Inclined Rotating Disk with the Impact of Heat Generation/Consumption and Thermal Radiation

Abstract

In this research, the three-dimensional nanofluid thin-film flow of Casson fluid over an inclined steady rotating plane is examined. A thermal radiated nanofluid thin film flow is considered with suction/injection effects. With the help of similarity variables, the partial differential equations (PDEs) are converted into a system of ordinary differential equations (ODEs). The obtained ODEs are solved by the homotopy analysis method (HAM) with the association of MATHEMATICA software. The boundary-layer over an inclined steady rotating plane is plotted and explored in detail for the velocity, temperature, and concentration profiles. Also, the surface rate of heat transfer and shear stress are described in detail. The impact of numerous embedded parameters, such as the Schmidt number, Brownian motion parameter, thermophoretic parameter, and Casson parameter (Sc, Nb, Nt, γ), etc., were examined on the velocity, temperature, and concentration profiles, respectively. The essential terms of the Nusselt number and Sherwood number were also examined numerically and physically for the temperature and concentration profiles. It was observed that the radiation source improves the energy transport to enhance the flow motion. The smaller values of the Prandtl number, Pr, augmented the thermal boundary-layer and decreased the flow field. The increasing values of the rotation parameter decreased the thermal boundary layer thickness. These outputs are examined physically and numerically and are also discussed.

1. Introduction

Energy is a requirement of production for every industry and is used in every engineering field. Important sources of energy are gas turbines, exchange membrane, and fuel cells [1], hydraulic-fracturing [2,3], etc. Suspensions of nanoparticles in fluids show a vital enrichment of their possessions at modest nanoparticle concentrations. Numerous researchers have worked on nanofluids and studied their role in heat transfer analysis, like nuclear reactors and other transportations. Nanofluids are smart fluids, where heat transfer can be decreased or increased in the base fluids. This research work focuses on investigating the vast range of uses that involve nanofluids, emphasizing their enriched heat transfer possessions, which are governable, and the defining features that these nanofluids preserve that make them suitable for such uses. Moreover, nanofluids are a new kind of energy transference fluid that are the suspension of base fluids and nanoparticles. For cooling requirements, usual heat transfer liquids cannot be used, due to their lesser thermal conductivity. By implanting nanoparticles into normal fluids, their thermal enactment can be enriched considerably. Choi [4] is widely accepted as the first publication that introduces the concept of nanofluids. He clarifies nano liquids as a liquid containing smaller scale particles known as nanoparticles about 1 to 100 nm in measure.

Bhatti et al. [5] explored the simultaneous impacts of the varying magnetic field of Jeffrey nanofluid. They examined the impact of physical parameters over the flow field. Xiao et al. [3] examined the relative permeability of nanofibers with the capillary pressure effect using the Fractal-Monte Carlo technique. They observed the impact of the embedding parameters with applications. Ellahi et al. [6] investigated the MHD non-Newtonian nanofluid with a temperature dependent viscosity flow through a pipe. The microchannel heat sink flow exploration cooled by a Cu water nanofluid by applying the least square method and the porous media approach was observed by Hatami et al. [7]. Hatami et al. [8] explored nanofluid laminar flow between rotating disks with heat transfer. Srinivas Acharya et al. [9] investigated nanofluid mixed convection flow with ion slip and Hall effects between two concentric cylinders. Khan et al. [10] investigated boundary-layer nanofluid flow through a stretching surface. Khanafer et al. [11] described two-dimensional Buoyancy driven flow with enhanced heat transfer enclosure utilizing nanofluids. Mahanthesh et al. [12] investigated unsteady MHD three-dimensional Eyring-Powell nanofluid flow with thermal radiation through a stretching sheet. Rashidi et al. [13] explored nanofluid with entropy generation and MHD flow on a steady porous rotating disk. Rashidi et al. [14] investigated 3-D film condensation on a steady inclined rotating disk.

Gul et al. [15] studied the heat and mass transfer analysis of a liquid film over an inclined plane. They compared integer and non-integer order results under the influence of embedded parameters. Saleh et al. [16] studied carbon-nanotubes suspended nanofluid flow with convective conditions using the Laplace transform. Sheikholeslami et al. [17] examined nanofluid flow in a semi-annulus enclosure with heat transfer and MHD effects. Sheikholeslami et al. [18] investigated flow in a semi-porous channel of MHD nanofluid with an analytical investigation. Later, these investigators [19] deliberated unsteady nanofluid flow through a stretching surface. Hayat et al. [20] explored the boundary layer flow of Maxwell nanofluid. Malik et al. [21] explored MHD flow through a stretching Erying-Powell nanofluid. Nadeem et al. [22] examined the flow of Maxwell liquid with nanoparticles through a vertical stretching surface. Raju et al. [23] examined flow with free convective heat transfer through a cone of MHD nano liquid. Rokni et al. [24] explored flow with the heat transfer of nanofluids through plates. Nadeem et al. [25] investigated flow on a stretching sheet of nano non-Newtonian liquid. Shehzad et al. [26] investigated the convective boundary conditions of Jaffrey nanoliquid flow with an MHD effect. Sheiholeslami et al. [27] explored flow with a magnetic field and heat transfer of nano liquid. Mahmoodi et al. [28] examined flow for cooling applications of nanonfluid with heat transfer. Recently, Shah et al. [29,30,31,32] investigated a rotating system in the effects of hall current and thermal radiations of nanofluid flow. Further theoretical investigations were examined by Sheikholeslami using different phenomena for nanofluids, with present usages and possessions with applications of numerous methods, can be found in [33,34,35,36,37]. Pour and Nassab [38] examined the convectional flow of nanofluids using the numerical technique. The influence of the physical parameters was observed in their study.

The exploration of thin film has achieved substantial presentation due to its frequent usages in the field of technology, industry, and engineering in a short interval of time. The investigation of thin liquid flow is necessary, due to its practical uses, such as cable and fibber undercoat. Several well-known uses of thin film are the fluidization of devices, elastic sheet drawing, and constant formation. Regarding their uses, it is vital that scientists develop research on the stretching sheet of liquid films. Sandeep et al. [39] studied non-Newtonian nanoliquids’ thin films’ fluid flow with heat transfer. Wang [40] detected an unsteady flow of thin film fluid through a stretching sheet. Usha et al. [41] investigated unsteadily finite thin liquid past a stretching sheet. Liu et al. [42] investigated thin film flow with heat transfer on a stretching surface. Aziz et al. [43] perceived the flow on a stretching sheet of a thin fluid film for the production of heat inside. Tawade et al. [44] examined fluid flow with thermal radiation and heat transmission of a thin film. Fluid film flow on a stretching sheet with heat transfer was investigated by Andersson et al. [45]. Also, investigators [46,47,48,49,50,51] examined the flow of liquid film on a stretching surface for further dissimilar cases. Hatami et al. [52] examined 3-D nanofluid flow on a steady rotating disk. A similar related study about nanofluid can be seen in [52,53,54,55,56]. Jawad et al. [57] examined Darcy-Forchheimer nanofluid thin film flow with Joule dissipation and Navier’s partial slip of the MHD effect. Jawad et al. [58] studied 3-D single-wall carbon nanotubes rotating flow with the impact of nonlinear thermal radiation and viscous dissipation in the presence of aqueous suspensions. Other related work can be seen in [59,60,61,62,63].

In view of the above important discussion, the aim of the current study is to investigate liquid film flow over an inclined plane. The momentum, thermal, and concentration boundary-layers under the influence of physical constraints for heat and mass transfer analysis will be examined physically and numerically.

2. Problem Formulation

Consider a steady three-dimensional Casson nanofluid thin-film flow over a rotating disk. The rotation of the disk is due to the angular velocity (Ω) in its own plane as displayed in Figure 1. An angle, β, is made by the inclined disk with the horizontal axis. Also, h denotes the film thickness of the nanofluid, and W represents the spraying velocity. The radius of the disk is very large as compared to the liquid film thickness and hence the termination influence is unnoticed. $\overline{g}$ is gravitational acceleration, T₀ is the temperature at the film surface, while T_w represents the surface temperature of the disk. Likewise, C₀ and C_h are the concentration on the film and on the disk surfaces, respectively. Pressure is a function of the z-axis only and the ambient pressure (P₀) at the sheet of the film is kept constant. The equations of continuity, momentum, concentration, and energy for a steady state are shown in Equations (1) to (6) [8,9,10]:

$$u_x+u_y+u_z=0$$

(1)

$${\rho }_{nf}\left(u{u}_x+v{u}_y+w{u}_z\right)=\left(1+\frac{1}{\gamma }\right){\mu }_{nf}\left(u_{xx}+u_{yy}+u_{zz}\right)+\overline{g}\sin \beta$$

(2)

$$u{v}_x+v{v}_y+w{v}_z=\left(1+\frac{1}{\gamma }\right)\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(v_{xx}+v_{yy}+v_{zz}\right)$$

(3)

$$u{w}_x+v{w}_y+w{w}_z=\left(1+\frac{1}{\gamma }\right)\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(w_{xx}+w_{yy}+w_{zz}\right)-\overline{g}\cos \frac{\beta }{{\Omega }^{\prime }}-\frac{P_z}{{\rho }_{nf}}$$

(4)

$$u{T}_x+v{T}_y+w{T}_z=\frac{k_{nf}}{{\left({\rho }_{cp}\right)}_{nf}}\left(T_{xx}+T_{yy}+T_{zz}\right)$$

(5)

$$u{C}_x+v{C}_y+w{C}_z=D_{\beta }\left(C_{xx}+C_{yy}+C_{zz}\right)+\left(\frac{D_T}{T_0}\right)\left(T_{xx}+T_{yy}+T_{zz}\right)$$

(6)

In the above equations, u, v, and w represent the velocity components in the x, y, and z axis, respectively.

The boundary conditions are as follows:

$$\begin{array}{l}u=-\Omega y,v=\Omega x,w=0,T=T_w,C=C_h\\ u_z=v_z=0,w=0,T=T_w,C=C_0,P=P_0\end{array}\begin{array}{l}atz=0\\ atz=h\end{array}$$

(7)

Consider the similarity transformations of the form:

$$\begin{array}{l}u=-\Omega yg\left(\eta \right)+\Omega x{f}^{\prime }\left(\eta \right)+\overline{g}k\left(\eta \right)\sin \frac{\beta }{{\Omega }^{\prime }}\\ v=\Omega xg\left(\eta \right)+\Omega y{f}^{\prime }\left(\eta \right)+\overline{g}s\left(\eta \right)\sin \frac{\beta }{{\Omega }^{\prime }}\\ w=-2\sqrt{\Omega v_{nf}}f\left(\eta \right),T=\left(T_0-T_w\right)\theta \left(\eta \right)+T_w\\ \eta \varphi \left(\eta \right)=\frac{C-C_w}{C_0-C_w},\eta =z\sqrt{\frac{\Omega }{v_{nf}}}\end{array}$$

(8)

The transformations introduced in Equation (8) are implemented in Equations (2) to (7). Equation (1) is proved identically and Equations (2) to (6) are obtained in the forms:

$$\left(1+\frac{1}{\gamma }\right)f^{‴}-{f^{\prime }}^2+g^2+2f{f}^{″}=0$$

(9)

$$\left(1+\frac{1}{\gamma }\right)k^{″}+gs-k{f}^{\prime }+2kf=0$$

(10)

$$\left(1+\frac{1}{\gamma }\right)g^{″}-2g{f}^{\prime }+2g^{\prime }f=0$$

(11)

$$\left(1+\frac{1}{\gamma }\right)s^{″}-kg-s{f}^{\prime }+2s^{\prime }f=0$$

(12)

If θ(η) and ϕ(η) are a function of z only, Equations (5) and (6) take the forms:

$$\left(1-\frac{4}{3}R\right){\theta }^{″}+2\mathrm{Pr}\frac{A_2{A}_3}{A_1{A}_4}\left(I_1{f}^{\prime }+j_1\right)\theta =0$$

(13)

$${\varphi }^{″}+2Scf{\varphi }^{\prime }+\frac{Nt}{Nb}{\theta }^{″}=0$$

(14)

$$\begin{array}{l}f\left(0\right)=0,f^{\prime }\left(0\right)=0,f^{″}\left(\delta \right)=0,\varphi \left(0\right)=0,\varphi \left(\delta \right)=1\\ g\left(0\right)=0,g^{\prime }\left(\delta \right)=0,k\left(0\right)=0,k^{\prime }\left(\delta \right)=0\\ s\left(0\right)=0,s^{\prime }\left(\delta \right)=0,\theta \left(0\right)=0,{\theta }^{\prime }\left(\delta \right)=1.\end{array}$$

(15)

Physical parameters and other dimensionless numbers of interest are defined as:

$$\begin{array}{l}\mathrm{Pr}=\frac{v_f}{{\alpha }_f},Sc=\frac{\mu }{\rho fD},Nb=\frac{{\left(\rho c\right)}_p{D}_b\left(C_h\right)}{{\left(\rho c\right)}_f\alpha }\\ Nt=\frac{{\left(\rho c\right)}_p{D}_T\left(T_H\right)}{{\left(\rho c\right)}_f\alpha T_c},S=\frac{\alpha }{\Omega }\end{array}$$

(16)

Here, Pr is the Prandtl number, Sc is the Schmidt number, Nb is the Brownian motion parameter, and Nt is the thermophoretic parameter.

Where the normalized thickness constant is presented as:

$$\delta =h\sqrt{\frac{\Omega }{v_{nf}}}$$

(17)

The condensation velocity is defined as:

$$f\left(\delta \right)=\frac{W}{2\sqrt{\Omega \nu }}=\alpha$$

(18)

The pressure can be attained by the integration of Equation (4).

For the exact solution, let Pr = 0 and using θ(δ) = 1, the exact solution is:

$${\theta }^{\prime }\left(0\right)=\frac{1}{\delta }$$

(19)

An asymptotic limit for small, δ, is defined in Equation (17). The reduction of θ’(0) for rising δ is not monotonic. So, Nu is defined as:

$$Nu=\frac{k_{nf}}{k_f}\frac{{\left(T_z\right)}_w}{\left(T_0-T_w\right)}=A_4\delta {\theta }^{\prime }\left(0\right)$$

(20)

The Sherwood number is defined as:

$$Sh=\frac{{\left(C_z\right)}_w}{C_0-C_w}=\delta {\varphi }^{\prime }\left(0\right)$$

(21)

3. Solution by Homotopy Analysis Method

The optimal approach is used for the solution process. Equations (9) to (14) with boundary conditions (15) are solved by HAM. Mathematica software is used for this aim. The basic derivation of the model equation through HAM is given in detail below.

Linear operators are denoted as $L_{\widehat{f}},L_{\widehat{\theta }}$ and $L_{\widehat{\varphi }}$ is represented as

$$\begin{array}{l}{L}_{\widehat{f}}(\widehat{f})={\widehat{f}}^{‴},L_{\widehat{k}}(\widehat{k})={\mathrm{k}}^{″},L_{\widehat{g}}(\widehat{g})={\mathrm{g}}^{″},\\ L_{\widehat{s}}(\widehat{s})={\mathrm{s}}^{″},L_{\widehat{\theta }}(\widehat{\theta })={\widehat{\theta }}^{″},L_{\widehat{\varphi }}(\widehat{\varphi })={\varphi }^{″}\end{array}$$

(22)

The modelled Equations (9) to (14) with boundary conditions (15) are solved analytically as well as numerically. The comparison between the analytical and numerical solution is shown graphically as well as numerically in

Table 1. Comparison of HAM and numerical solution for f(η).

f(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	0.000000	−2.812710 × 10−10	2.812710 × 10−10
0.1	0.003258	0.003258	5.703240 × 10−8
0.2	0.012408	0.012408	2.280140 × 10−7
0.3	0.026573	0.026573	5.104210 × 10−7
0.4	0.044948	0.044948	8.995600 × 10−7
0.5	0.066795	0.066793	1.388520 × 10−6
0.6	0.091431	0.091429	1.965710 × 10−6
0.7	0.118224	0.118221	2.615740 × 10−6
0.8	0.146586	0.146583	3.327870 × 10−6
0.9	0.175966	0.175962	4.078740 × 10−6
1.0	0.205842	0.205837	4.847430 × 10−6

,

Table 2. Comparison of HAM and numerical solution for k(η).

k(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	0.000000	3.431260 × 10−8	3.431260 × 10−8
0.1	0.082367	0.082366	2.022930 × 10−7
0.2	0.155134	0.155134	4.388100 × 10−7
0.3	0.218705	0.218705	6.667190 × 10−7
0.4	0.273439	0.273438	8.749510 × 10−7
0.5	0.319620	0.319619	1.071030 × 10−6
0.6	0.357437	0.357436	1.231030 × 10−6
0.7	0.386981	0.386980	1.356500 × 10−6
0.8	0.408247	0.408246	1.432420 × 10−6
0.9	0.421138	0.421137	1.471570 × 10−6
1.0	0.425484	0.425483	1.467400 × 10−6

,

Table 3. Comparison of HAM and numerical solution for g(η).

g(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	1.000000	1.000000	1.286450 × 10−8
0.1	0.950141	0.950139	2.377450 × 10−6
0.2	0.903441	0.903436	4.705070 × 10−6
0.3	0.860810	0.860803	6.954740 × 10−6
0.4	0.822957	0.822948	9.080750 × 10−6
0.5	0.790417	0.790406	0.000011
0.6	0.763566	0.763553	0.000013
0.7	0.742641	0.742627	0.000014
0.8	0.727756	0.727740	0.000015
0.9	0.718908	0.718892	0.000016
1.0	0.715997	0.715981	0.000016

,

Table 4. Comparison of HAM and numerical solution for s(η).

s(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	0.000000	−7.170820 × 10−9	7.170820 × 10−9
0.1	−0.019540	−0.019539	1.154300 × 10−6
0.2	−0.038351	−0.038349	2.304890 × 10−6
0.3	−0.055850	−0.055846	3.427710 × 10−6
0.4	−0.071593	−0.071588	4.501730 × 10−6
0.5	−0.085251	−0.085246	5.509380 × 10−6
0.6	−0.096596	−0.096590	6.416000 × 10−6
0.7	−0.105479	−0.105471	7.187270 × 10−6
0.8	−0.111819	−0.111811	7.792270 × 10−6
0.9	−0.115595	−0.115587	8.183170 × 10−6
1.0	−0.116839	−0.116831	8.326040 × 10−6

,

Table 5. Comparison of HAM and numerical solution for θ(η).

θ(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	0.000000	−1.777730 × 10−9	1.777730 × 10−9
0.1	0.116628	0.116628	1.151200 × 10−7
0.2	0.232964	0.232964	2.309850 × 10−7
0.3	0.348690	0.348690	3.439950 × 10−7
0.4	0.463468	0.463469	4.517690 × 10−7
0.5	0.576946	0.576946	5.526490 × 10−7
0.6	0.688763	0.688763	6.424670 × 10−7
0.7	0.798558	0.798559	7.163620 × 10−7
0.8	0.905975	0.905976	7.751320 × 10−7
0.9	1.010670	1.010670	8.130730 × 10−7
1.0	1.112300	1.112300	8.304690 × 10−7

and

Table 6. Comparison of HAM and numerical solution for ϕ(η).

ϕ(η)	HAM Solution	Numerical Solution	Absolute Error
0.0	0.000000	−2.935480 × 10−9	2.935480 × 10−9
0.1	0.109309	0.109309	8.676730 × 10−8
0.2	0.218448	0.218448	1.752770 × 10−7
0.3	0.327225	0.327226	2.613220 × 10−7
0.4	0.435439	0.435440	3.434820 × 10−7
0.5	0.542882	0.542883	4.241530 × 10−7
0.6	0.649351	0.649352	4.969910 × 10−7
0.7	0.754655	0.754656	5.581250 × 10−7
0.8	0.858619	0.858620	6.055710 × 10−7
0.9	0.961090	0.961091	6.356540 × 10−7
1.0	1.061940	1.061940	6.465200 × 10−7

for the velocities, temperature, and concentration profiles. From these tables, an excellent agreement between the HAM and numerical (ND-Solve Techniques) methods is obtained.

4. Results and Discussion

The three-dimensional flow of the liquid film through a steady rotating inclined surface with mass and heat transmission was examined. The influence of the embedded parameters, magnetic field, M, Casson parameter, γ, Schmidt number, Sc, Brownian motion parameter, Nb, and thermophoretic parameter, Nt, was investigated for the axial velocity, f(η), radial velocity, k(η), drainage flow, g(η), and induced flow, s(η), temperature field, θ(η), and concentration profile, ϕ(η), respectively. Figure 2, Figure 3, Figure 4 and Figure 5 display the influence of the Casson fluid parameter, γ, on f(η), k(η), g(η), and s(η). Rising γ generates resistance in the flow path and decreases the flow motion of nanoparticles. It is observed that an increase of the Casson fluid parameter, γ, leads to a decrease of f(η), k(η), g(η), and s(η). The opposite trend is found in case of the z-direction, that is the enormous value of γ decreases the f(η), k(η), g(η), and s(η). The influence of Pr on θ(η) is displayed in Figure 6. It is interesting to note that θ(η) decreases with large values of Pr and increases with smaller values. In fact, the thermal diffusivity of nanofluids has greater values by reducing Pr, and this effect is inconsistent for larger Pr. Hence, the greater values of Pr drop the thermal boundary layer. The influence of the radiation parameter, R, on θ(η) is presented in Figure 7. It is observed that if R increases, then the boundary layer area θ(η) is augmented. The effect of Nb on θ(η) is displayed in Figure 8. The converse influence was created for ϕ(η) and θ(η), which means augmented Nb decreases the concentration profile, ϕ(η). The concentration boundary layer thickness decreased due to the rising values of Nb and as a result, the concentration field, ϕ(η), declined. The features of the thermophoretic parameter, Nt, on the concentration profile, ϕ(η), are presented in Figure 9. The enhancement of Nt increases ϕ(η). Thus, Nt depends on the temperature gradient of the nanofluids. The kinetic energy of the nanofluids rises with the increasing value of Nt, and as a result, ϕ(η) increases. Figure 10 identifies the influence of Sc. The dimensionless number, Sc, is stated as the ratio of momentum and mass diffusivity. It is obvious that the amassed Sc reduces the ϕ(η) and as a result, the boundary layer thickness is decreased.

Figure 11 and Figure 12 demonstrate the effects of Pr and R. It can be seen that rising values of Pr and R increase Nu. In fact, the coaling phenomenon is enhanced with increased values of these parameters. Figure 13 identifies that Nu reduces for the amassed values of k.

5. Conclusions

In this article, the three-dimensional thin-film Casson fluid flow over an inclined steady rotating plane was examined. The thin film flow was thermally radiated and the suction/injection effect was also considered. By the similarity variables, the PDEs were converted into ODES. The obtained ODEs were solved by the HAM with association of the MATHEMATICA program. The main features of the study are highlighted as:

• Smaller values of the Prandtl number enhance the thermal boundary layer.
• An increasing value of the magnetic field stops the fluid motion.
• Larger amounts of the thermal radiation parameter and thermophoretic parameter enhances the thermal boundary layer.
• The Casson fluid parameter produces a resistance force and its increasing value decreases the fluid motion.