An Optimal Investigation of Convective Fluid Flow Suspended by Carbon Nanotubes and Thermal Radiation Impact

Abstract

This study is focused towards analyzing the heat and flow movement among two stretching rotating disks inside water-based carbon nanotubes. The idea of thermal boundary conditions and heat convection is used and the system is expressed in partial differential equations. Using the similarity techniques, the model is successfully converted to a nonlinear ordinary differential equation. A familiar collocation method is used to simulate the outcomes of the governed system while the method is validated through a set of tables and assessed with existing literature. The physical aspects of the proposed model have been studied in detail and assisted via graphical diagrams against the variation of different parameters. It is found that the multiple-wall carbon nanotubes intensify the system quickly and improve the rate of heat transmission. It is also noted that the proposed method is in excellent in agreement with already published studies and can be extended for other physical problems. Moreover, when values of Re parameter increase, a drop is noted in the magnitude of radial velocity near the faces of the disks. It is very clear from the tabular comparison that collocation scheme is in good agreement with already published studies and homotopic solutions.

1. Introduction

The area of nanofluids and their extensive real-life application gained a realistic devotion among science and engineering scholars. The area of nanofluids was initiated by Choi and Eastman by introducing a new sort of thermo-fluid tag for “nanofluids” in which solid-liquid combination takes base fluid (mostly bad conductors) and solid nanometer dimensions particles—“nanoparticles” [1]. If we look at the engineering and industrial aspects then operation of nanoparticle arises in various arena, such as biological, chemotherapy, medicine, therapy, surgery, environmental, chemical, material, physical and other interdisciplinary sciences. Later on, various nanofluid models were proposed by different scholars, including Buongiorno, Xuan and Li, Tiwari and Das and Xue and Xu [2,3,4,5]. Numerous investigations have been shown with the similar ideology of studying nanofluids using different nanoparticles including aluminum oxide, copper, copper-oxide, molybdenum disulfide, silver, carbon and other shapes of carbon, such as CNTs [6,7,8,9]. Literature discloses that CNTs display outstanding motorized, ocular, electrical and thermal physiognomies [10]. CNTs are testified and noted as having great elasticity (500 by steel), higher current capacity (1K times) and huge density (1/2 by aluminum), compared with copper (Cu) and ample advanced conductivity (15 times). Moreover, the presence of carbon chain CNTs are not risky factors and it could be useful to the scientific community and environment.

Many studies are available in the literature that deal with thermal analysis using carbon nanotubes. Hayat et al. analyzed the mixed convection flow of blood containing carbon nanotubes (CNTs; both single and multi-wall) over a curved stretching sheet [11]. The nanofluids-based modeling is performed through the concept of Xu idea, while dissipation and Joule heating impacts are incorporated into energy expression. Haq et al. presented viscosity and thermal conductivity of carbon nanotubes (CNTs) including single and multi-walls inside the base fluid of comparable volume over a fluid stream on shallow stretching [12]. Conclusions have been established in support of the entire examination and it is was discovered that engine oil-based CNTs have advanced skin friction and heat transmission rate compared to ethylene glycol and water-based CNTs. Mohyud-Din et al. analyzed the impacts of thermal radiation and carbon nanotubes (CNTs), both single-wall and multi-wall, in the Marangoni convection boundary-layer viscous liquid flow [13]. A numerical study using least squares methods and comparative analysis with existing method is made. Raza et al. presented an arithmetical study to pursue the influences of chemical reaction and radiation features in MHD flow of nanofluids induced by outwardly elastic diskette taking the effects of non-uniform heat sink and source [14]. The modeled equations have been efficiently solved through MATLAB-based numerical command and outcomes analysis is reported via graphical plots. Haq et al. explored a study to seek the impacts of volume fraction of carbon nanotubes (CNTs) on magneto-hydrodynamics stream and heat transmission in dual cross directions over a stretching sheet with convective boundary conditions [15]. Three sorts of base fluids including water, ethylene glycol and engine oil are used and compared while impacts of single and multi-wall carbon nanotubes were studied. The readers can obtain more information on the topic in [16,17,18,19,20,21,22,23,24].

Hamid et al. explored the hydromagnetic flow and heat control inside a partly heated rectangular fin-shaped cavity containing carbon nanotubes (CNTs) incorporated in water [25]. A tiny intense (heated) rod is located in the interior of the cavity to generate a heat transfer source or a resistance. The right wall of the horizontal tip is tested for three dissimilar temperatures (adiabatic, cold and heated). Experimental validation of the study was made and results are explained physically using graphs. It is found that the velocity components are maximum and minimum at the vertical and horizontal corners, respectively. The local Nusselt numbers are improved by incorporating both radiations impacts and solid volume fraction of CNTs, whereas, peak of the Nusselt number is noted at the corners. Khan et al. examined the properties of water-based single-walled carbon nanotubes on free-convection in a partially heated right trapezoidal cavity where the lower boundary is heated while the side and top walls are kept cold and adiabatic, respectively [26]. The proposed dimensionless partial differential equations are explained with designated dimensionless boundary conditions via a finite element scheme and outcomes are illustrated graphically. The readers can see the references to understand the study of nanofluids inside the complex enclosures [27,28,29,30,31,32,33,34,35,36,37].

The motivation of current research work is to examine the heat and flow movement among two stretching rotating disks inside the water-based carbon nanotubes. The idea of thermal boundary conditions and heat convection is used and the system is expressed in partial differential equations. The considered model has significance in real life, such as the governing model used in ground water flows, geothermal extraction, storage of nuclear waste material, oil recovery processes, impurity dispersion in aquifers, etc. A comprehensive assessment of outcomes gained by collocation and RK-4 methods is made to display the competence of recommended procedures. The physical aspects of the proposed model were studied in detail and asserted via graphical diagrams by variation in different parameters. It is found that the multiple-wall carbon nanotubes intensify the system quickly and improve the rate of heat transmission. It is also noted that the proposed method is in excellent agreement with already published studies and homotopic solutions and can be extended for other physical problems [38].

2. Mathematical Formulation

Assuming the flow of nanofluid among double nonequivalent immeasurable disks that are axisymmetric and incompressible, both plates disconnected by $h$ (constant distance), one is at a point $z=0$ and $z=h$ is the location of the other. Assuming the disks are revolving with the angular velocities towards the axial direction, for the first one it is ${\Omega }_1$ and the velocity of second is ${\Omega }_2$. Figure 1 is shown to understand the geometrical aspect of the problem. It can be seen that stretching rates of the disks are, respectively, $a_1$ and $a_2$ in the radial direction. Two sorts of carbon nanotubes (single and multi-wall) were employed as nanoparticles inside water-based fluid. In addition, the surface validates the convective type boundary conditions. Moreover, it is further assumed that that temperature of the upper and lower disks is $T_1$ and $T_0$. Taking all assumptions into consideration, the governing equations of the mechanism in cylindrical coordinates are [20]:

$$\frac{\partial u}{\partial r}+\frac{\partial w}{\partial z}+\frac{u}{r}=0,$$

(1)

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

(2)

$$u\frac{\partial v}{\partial r}+\frac{uv}{r}+w\frac{\partial v}{\partial z}={\nu }_{nf}\left(\frac{{\partial }^{2}v}{\partial r^2}+\frac{{\partial }^{2}v}{\partial z^2}+\frac{1}{r}\frac{\partial v}{\partial r}-\frac{v}{r^2}\right),$$

(3)

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

(4)

$${\left(\rho c_p\right)}_{nf}\left[w\frac{\partial T}{\partial z}+u\frac{\partial T}{\partial r}\right]=k_{nf}\left(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+\frac{16{\sigma }^{*}{T}_1^3}{3k^{*}}\left(\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}\right),$$

(5)

associated with the following boundary conditions

$$z\to 0 then v=r_1{\Omega }_1,k_{nf}\frac{\partial T}{\partial z}=\left(T_0-T\right)\left(-h_1\right),u=r_1{a}_1 w=0,$$

(6)

$$z\to h then v=r_1{\Omega }_2,u=r_1{a}_2, k_{nf}\frac{\partial T}{\partial z}=\left(T-T_1\right)\left(-h_2\right), w=0,$$

(7)

where some parameters are indicated, such as mean absorption coefficient ($k^{*}$), fluid temperature ($T$), pressure ($p$) and Stefan Boltzmann constant ($r_1$). The $h_2$ and $h_1$ are, respectively, represented as convective heat transfer coefficients of upper and at lower walls.

$$\begin{gathered} {\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}},{\rho }_{nf}={\rho }_{CNT}\left(\varphi \right)+{\rho }_f\left(1-\varphi \right),{\left(\rho c_p\right)}_{nf}={\left(\rho c_p\right)}_{CNT}\left(\varphi \right) \\ +{\left(\rho c_p\right)}_f\left(1-\varphi \right),\frac{k_{nf}}{k_f}=\frac{\left(-\varphi +1\right)+2\varphi lnln \frac{k_{CNT}+k_f}{2k_f} \frac{k_{CNT}}{k_{CNT}-k_f}}{\left(-\varphi +1\right)+2\varphi lnln \frac{k_{CNT}+k_f}{2k_f} \frac{k_f}{k_{CNT}-k_f}}. \end{gathered}$$

(8)

In the equation above the parameters $\rho , \mu ,k \mathrm{and} c_p$ are density, nanofluid effective dynamic viscosity, thermal conductivity and heat capacitance. Moreover, the solid volume fraction of nanoparticles, thermo-based physical possessions of base fluid, nanofluid and carbon nanotubes are correspondingly designated as $\varphi , f,nf$ and $CNT$. To change the directly-above structure of the partial differential equations (PDEs) to a nonlinear ordinary differential equations set we have considered the subsequent alteration schemes:

$$\begin{gathered} v=r{\Omega }_{1}g\left(\eta \right), u=r{\Omega }_1{f}^{\prime }\left(\eta \right), w=-2h{\Omega }_{1}f\left(\eta \right), \\ \frac{T-T_1}{T_0-T_1}=\theta \left(\eta \right), \frac{z}{h}=\eta , {\rho }_f{\Omega }_1{\nu }_f\left(\frac{1}{2}\frac{r^2}{h^2}ϵ+P\left(\eta \right)\right)=p. \end{gathered}$$

(9)

Assisted by the similarity viable equations in ($9$), we have gained ($1$)–($7$):

$$\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}{f}^{‴}+Re\left(2f{f}^{″}-{f^{\prime }}^2+g^2\right)-\frac{ϵ}{1-\varphi +\frac{{\rho }_s}{{\rho }_f}\varphi }=0,$$

(10)

$$\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}{g}^{″}+2Re\left(f{g}^{\prime }-f^{\prime }g\right)=0,$$

(11)

$$\frac{1}{\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}{P}^{\prime }=-4Ref{f}^{\prime }-\frac{2}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}{f}^{″}=0,$$

(12)

$$\frac{1}{Pr}\left(\frac{k_{nf}}{k_f}+Rd\right){\theta }^{″}+2Re\left(1-\varphi +\frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\varphi \right)f{\theta }^{\prime }=0.$$

(13)

The boundary conditions obtained as:

$$\begin{gathered} f\left(\eta \right)=0, P\left(\eta \right)=0,g\left(\eta \right)=1, f^{\prime }\left(\eta \right)=A_1,{\theta }^{\prime }\left(\eta \right)=-\frac{k_f}{k_{nf}}{\gamma }_1\left[1-\theta \left(\eta \right)\right],\eta \to 0, \\ {\theta }^{\prime }\left(\eta \right)=-\frac{k_f}{k_{nf}}{\gamma }_2\theta \left(\eta \right), f^{\prime }\left(\eta \right)=A_2,f\left(\eta \right)=0, g\left(\eta \right)=\Omega ,\eta \to 0, \end{gathered}$$

(14)

In the above-described reduced system, involved parameters are, Reynold number ($Re$), Prandtl number ($Pr$), scaled stretching ($A_{1}and A_2$), Biot (Bi), rotation parameter ($\Omega$) and radiation parameter ($Rd$).

$$\frac{{\left(\rho c_p\right)}_f{\nu }_f}{k_f}=Pr,\frac{{\Omega }_1{h}^2}{{\nu }_f}=Re, \frac{a_1}{{\Omega }_1}=A_1,\frac{a_2}{{\Omega }_1}=A_2,\frac{16{\sigma }^{*}{T}_1^3}{3k_f{k}^{*}}=Rd,\frac{{\Omega }_2}{{\Omega }_1}=\Omega , \frac{h{h}_1}{k_f}={\gamma }_1, \frac{h{h}_2}{k_f}={\gamma }_2.$$

Now taking the derivative of the expression, ($10$) with respect to $\eta$ and then eliminating $ϵ$, we found the simple model as follows:

$$\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}{f}^{iv}+2Re\left(f{f}^{‴}+g{g}^{\prime }\right)=0,$$

(15)

and $ϵ$ can be obtained by considering the subsequent expression as

$$ϵ=\frac{1}{{\left(1-\varphi \right)}^{2.5}}{f}^{‴}\left(0\right)-Re\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)\left[{\left(f^{\prime }\left(0\right)\right)}^2-\left(g\left(0\right)\right)+\frac{M}{\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}\frac{{\sigma }_{nf}}{\sigma }{f}^{\prime }\left(0\right)\right],$$

Now, integration of Equation ($12$) will give us the solution for the involved pressure as given below:

$$-2\left[Re\left(f^2\right)+\left(f^{\prime }-f^{\prime }\left(0\right)\right)\frac{1}{{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}\right]=\frac{1}{\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)}P.$$

The entire shear-stress at the bottom disk can be calculated using the expression:

$${\tau }_w=\sqrt{{\tau }_{zr}^2+{\tau }_{z\theta }^2},$$

where ${\tau }_{zr}$ and ${\tau }_{z\theta }$ indicate the radial and tangential directions of shear stress and are defined as:

$${\tau }_{zr}={\mu }_{nf} \frac{\partial u}{\partial z}{|}_{z=0}=\frac{{\mu }_{f}r{\Omega }_1{f}^{″}\left(0\right)}{{\left(1-\varphi \right)}^{2.5}h}, {\tau }_{z\theta }={\mu }_{nf} \frac{\partial v}{\partial z}{|}_{z=0}=\frac{{\mu }_{f}r{\Omega }_1{g}^{\prime }\left(0\right)}{{\left(1-\varphi \right)}^{2.5}h}.$$

At lower and upper disks the skin friction coefficients $C_1$ and $C_2$ are given as:

$$\frac{1}{R{e}_r{\left(1-\varphi \right)}^{2.5}}{\left[{\left(g^{\prime }\left(0\right)\right)}^2+{\left(f^{″}\left(0\right)\right)}^2\right]}^{\frac{1}{2}}=\frac{{\tau }_w{|}_{z=0}}{{\rho }_f{\left(r{\Omega }_1\right)}^2}=C_1,$$

$$\frac{1}{R{e}_r{\left(1-\varphi \right)}^{2.5}}{\left[{\left(g^{\prime }\left(1\right)\right)}^2+{\left(f^{″}\left(1\right)\right)}^2\right]}^{\frac{1}{2}}=\frac{{\tau }_w{|}_{z=h}}{{\rho }_f{\left(r{\Omega }_2\right)}^2}=C_2.$$

Local Reynolds numbers are calculated as $\frac{r{\Omega }_{1}h}{{\nu }_f}=R{e}_r$. The upper disk and lower disk have heat transmission as follows:

$$\frac{h{q}_r}{k_f\left(T_0-T_1\right)}{|}_{z=h}=N{u}_{x2},\frac{h{q}_w}{k_f\left(T_0-T_1\right)}{|}_{z=0}=N{u}_{x1}$$

Here $q_r$ and $q_w$ are the radiative and wall heat flux specified under:

$$q_w{|}_{z=h}=q_r{|}_{z=h}-k_{nf}\frac{\partial T}{\partial z}{|}_{z=h},q_w{|}_{z=0}=q_r{|}_{z=0}-k_{nf}\frac{\partial T}{\partial z}{|}_{z=0},$$

$$\frac{16{\sigma }^{*}{T}_1^3}{3k^{*}}\frac{\partial T}{\partial z}{|}_{z=0},=q_r{|}_{z=0}, \frac{16{\sigma }^{*}{T}_1^3}{3k^{*}}\frac{\partial T}{\partial z}{|}_{z=h}=q_r{|}_{z=h}.$$

The Nusselt numbers at lower and upper disks in dimensionless form are given as:

$$N{u}_2=-{\theta }^{\prime }\left(1\right)\left(\frac{k_{nf}}{k_f}+Rd\right),-\left(Rd+\frac{k_{nf}}{k_f}\right){\theta }^{\prime }\left(0\right)=N{u}_1.$$

3. Methodology and Solution Procedure

Numerical analysis of the considered convective fluid flow suspended by the carbon nanotubes by means of the well-known and efficient collocation approach; this scheme is very simple to apply and has the following procedure for the obtained set of dimensionless equations:

Step 1. Let start with considering the following dimensionless form

$$f^{iv}\left(\eta \right)+2Re{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)\left(f\left(\eta \right)f^{‴}\left(\eta \right)+g\left(\eta \right)g^{\prime }\left(\eta \right)\right)=0,$$

(16)

$$g^{″}\left(\eta \right)+2Re{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)\left(f\left(\eta \right)g^{\prime }\left(\eta \right)-f^{\prime }\left(\eta \right)g\left(\eta \right)\right)=0,$$

(17)

$$\frac{1}{Pr}\left(\frac{k_{nf}}{k_f}+Rd\right){\theta }^{″}\left(\eta \right)+2Re\left(1-\varphi +\frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\varphi \right)f\left(\eta \right){\theta }^{\prime }\left(\eta \right)=0.$$

(18)

Step 2. This scheme allows us to write the following approximate solutions to find the solutions of the problems defined in step 1 as below:

$$\tilde{f}=a_0+a_1\eta +a_2{\eta }^2+\dots +a_M{\eta }^M={\displaystyle \sum }_{k=0}^M a_k{\eta }^k,$$

(19)

$$\tilde{g}=b_1+b_2\eta +b_3{\eta }^2+\dots +b_M{\eta }^M={\displaystyle \sum }_{k=0}^M b_k{\eta }^k,$$

(20)

$$\tilde{\theta }=c_1+c_2\eta +c_3{\eta }^2+\dots +c_M{\eta }^M={\displaystyle \sum }_{k=0}^M c_k{\eta }^k.$$

(21)

Here the parameter $M$ denotes the order of approximation via collocation method, and it is well known that as we enhance the value of $M$, the accuracy improves gradually. After imposing the obtained dimensionless boundary conditions on the above trial solution, it converts to the following modified form and we call it modified trial solutions:

$$\tilde{f}=A_1\eta -\left(A_2+2A_1\right){\eta }^2+\left(A_2+A_1\right){\eta }^3+{\displaystyle \sum }_{k=1}^M a_k\left(k-\left(k+1\right)\eta +{\eta }^{k+1}\right){\eta }^2,$$

(22)

$$\tilde{g}=1-\left(1-\Omega \right)\eta -{\displaystyle \sum }_{k=1}^M b_k\left(1-{\eta }^k\right)\eta ,$$

(23)

$$\begin{gathered} \tilde{\theta }=\frac{{\gamma }_1}{R}\left(k_{nf}+{\gamma }_2{k}_f\right)-\frac{k_f{\gamma }_2{\gamma }_1}{R}\eta \\ +{\displaystyle \sum }_{k=1}^M c_k\left({\eta }^{k+1}-\frac{1}{R}\left({\gamma }_1\eta +\frac{k_{nf}}{k_f}\right)\left(\left(k+1 \right)k_{nf}+{\gamma }_2{k}_f\right)\right). \end{gathered}$$

(24)

where $R=\left({\gamma }_1+{\gamma }_2\right)k_{nf}+{\gamma }_1{\gamma }_2{k}_f.$

Step 3. We can generate the residual functions $R_f,R_g$ and $R_{\theta }$ by incorporating the modified trial solutions into the problem give in step 1, obtaining the following form as:

$$R_f=f^{iv}\left(\eta \right)+2Re{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)\left(f\left(\eta \right)f^{‴}\left(\eta \right)+g\left(\eta \right)g^{\prime }\left(\eta \right)\right)\ne 0,$$

(25)

$$R_g=g^{″}\left(\eta \right)+2Re{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{{\rho }_{CNT}}{{\rho }_f}\varphi \right)\left(f\left(\eta \right)g^{\prime }\left(\eta \right)-f^{\prime }\left(\eta \right)g\left(\eta \right)\right)\ne 0,$$

(26)

$$R_{\theta }=\frac{1}{Pr}\left(\frac{k_{nf}}{k_f}+Rd\right){\theta }^{″}\left(\eta \right)+2Re\left(1-\varphi +\frac{{\left(\rho c_p\right)}_{CNT}}{{\left(\rho c_p\right)}_f}\varphi \right)f\left(\eta \right){\theta }^{\prime }\left(\eta \right)\ne 0.$$

(27)

Step 4. In order to investigate the unknowns present in the set of residual functions given above, it is necessary to generate the set of algebraic equations which are equal in the number the unknowns present in $R_f,R_g$ and $R_{\theta }$, therefore we use the following equal spaced collocation points:

$${\eta }_l=a+\frac{1}{M}l, l=01,2,3,\dots ,M.$$

Step 5. After solving the system of algebraic equations obtained in step 4, we achieved the values of unknowns, and inserting these constants into the above modified trial solution yields the numerical solution of nondimensional form of the governing model.

In order to see the impact of numerous parameters on the velocities and temperature profiles, we simulate the collocation method Maple code against $M=7$ and $\varphi =0.2, Re=0.9, {\gamma }_1=0.4, {\gamma }_2=0.5, A_1=0.7, Pr=1, \Omega =A_2=0.8$.

4. Results and Discussion

The model is numerically examined using the efficient collocation method and the behavior of the profiles including velocity (tangential and axial) and temperature and is shown graphically against the variation of various prominent parameters including the dimensionless form [41,42,43,44,45]. The numerical simulations are made for $M=7$ and $\varphi =0.2, Re=0.9, {\gamma }_1=0.4, {\gamma }_2=0.5, A_1=0.7, Pr=1, \Omega =A_2=0.8$. Figure 2 illustrates the velocities attitude against Re parameters. Figure 2a,c has been respectively asserted to show the behavior of axial $f\left(\eta \right)$ and tangential velocities $g\left(\eta \right)$ with SWCNTs while the axial $f\left(\eta \right)$ and tangential velocities behavior in the presence of MWCNTs is displayed in Figure 2b,d, respectively. The temperature profile is analyzed and plots for SWCNTs and MWCNTs are displayed in Figure 3a and Figure 2b, respectively. The impact of nanoparticles volume friction ϕ on the velocities is illustrated in Figure 4a for the axial case and Figure 4c for the tangential case in the presence of SWCNTs while Figure 4b,d respectively display the behavior for axial and tangential velocities with the incorporation of MWCNTs. Figure 5a,b is plotted to seek the temperature analysis against the nanoparticles volume friction ϕ. It can be seen that the ϕ impact with SWCNTs is depicted in Figure 5a, while the ϕ impact with SWCNTs is presented in Figure 5b. Consequences of parameter $A_1$ on radial velocity are offered in Figure 6a,b for SWCNTs and MWCNTs, correspondingly. We detected that the dual influence of parameter $A_1$ on $f^{\prime }\left(\eta \right)$. The radial velocity shape surges upwards as parameter $A_1$ is improved while $0\le \eta <3.35$, when $3.35\le \eta \le 1$ the radial velocity profile drops when parameter $A_1$ increases. Analogous profile of $f^{\prime }\left(\eta \right)$ is attained for SWCNTs and MWCNTs. Figure 7a,b is plotted to depict the effect of parameter $A_2$ of $f\left(\eta \right)$ for the SWCNTs and MWCNTs, separately. As parameter $A_2$ is boosted the axial velocity drops near the surface of the lower disk. A similar profile for $f^{\prime }\left(\eta \right)$ is achieved for SWCNTs and MWCNTs. Figure 8a,b demonstrate that the tangential velocity profile is a growing function of parameter $A_2$ for SWCNTs and MWCNTs, correspondingly. The magnitude of $g\left(\eta \right)$ has greater values of SWCNTs compared to MWCNTs.

Table 1. Thermo-physical properties of water and carbon nanotube.

Thermophysical Properties	$\mathit{k} \left(\mathit{W}/\mathit{m}\mathit{K}\right)$	$\mathit{\rho } \left(\mathit{k}\mathit{g}/{\mathit{m}}^3\right)$	${\mathit{c}}_{\mathit{p}}\left(\mathit{J}/\mathit{k}\mathit{g}\mathit{K}\right)$
Water	0.613	997.1	4179
SWCNTs	6600	2600	425
MWCNTs	3000	1600	796

is construct to show the thermo-physical properties. Comparative study is made for various values of involved parameters with other methods and existing physical outcomes while a detailed sketch is available in

Table 2. Assessment of the projected outcomes for $f^{″}\left(0\right)$ and $g^{\prime }\left(0\right)$ with existing results [39] and homotopic analysis method (HAM) [40] for $\varphi =A_1=A_2=0, Re=1$ and for different values of $\Omega$.

${\Omega }$	[40]	[39]	CM	[40]	[39]	CM
−	0.06666	0.06666	0.06666314	2.00095	2.00095	2.00095204
−0.8	0.08394	0.08394	0.08394207	1.80259	1.80259	1.80258842
−0.3	0.10395	0.10395	0.10395088	1.30442	1.30442	1.30442358
0	0.09997	0.09997	0.09997221	1.00428	1.00428	1.00427759
0.5	0.06663	0.06663	0.06663420	0.50261	0.50261	0.50261352

. The table is particularly made for $\varphi =A_1=A_2=0, Re=1$ and for different values of $\Omega$. It is very clear from the tabular comparison given in Table 2 that the collocation scheme is in good agreement with already published studies and homotopic solutions. In

Table 3. Error investigation of axil and tangential velocities, and temperature profiles for SWCNTs for dissimilar values of when $\varphi =0.2, A_1=0.7 {\gamma }_2=0.5, {\gamma }_1=0.4, \Omega =A_2=0.8, Rd=0.3,$ and $Re=0.9$.

$\mathit{k}$	$\mathit{M}$	$\mathit{E}\mathit{r}\mathit{r}\mathit{o}{\mathit{r}}_{\mathit{f}}$	$\mathit{E}\mathit{r}\mathit{r}\mathit{o}{\mathit{r}}_{\mathit{g}}$	$\mathit{E}\mathit{r}\mathit{r}\mathit{o}{\mathit{r}}_{\mathit{\theta }}$
1	08	1.71536 × 10−10	6.93642 × 10−11	3.84831 × 10−26
	13	5.81908 × 10−14	1.09435 × 10−16	9.29935 × 10−31
	18	4.23987 × 10−19	4.41625 × 10−22	1.38211 × 10−36
	22	2.13039 × 10−24	1.28969 × 10−27	1.93124 × 10−42
	26	1.26154 × 10−31	1.66696 × 10−35	2.17492 × 10−50
	30	2.26778 × 10−35	4.89294 × 10−39	3.27934 × 10−53

, by taking various values for $M,$ when $\varphi =0.2, A_1=0.7, {\gamma }_2=0.5, {\gamma }_1=0.4, \Omega =A_2=0.8, Rd=0.3$ and $Re=0.9$. an extensive study related to error investigation for axil and tangential velocities, and temperature profile with SWCNTs, was made. It can be seen in Table 3 that higher values of M allow for a better analysis of the problem and accuracy level is improved at tangible level.

Table 4. Convergence of the proposed method when $\varphi =0.2,Re=0.9, {\gamma }_1=0.4, {\gamma }_2=0.5, A_1=0.7,\Omega =A_2=0.8,Pr=6.2$ and $Rd=0.3$.

SWCNTs				MWCNTs
$\mathit{M}$	$-{\mathit{f}}^{″}\left(0\right)$	$-{\mathit{g}}^{\prime }\left(0\right)$	$-{\mathit{\theta }}^{\prime }\left(0\right)$	$-{\mathit{f}}^{″}\left(0\right)$	$-{\mathit{g}}^{\prime }\left(0\right)$	$-{\mathit{\theta }}^{\prime }\left(0\right)$
1	4.4213991	0.6357312	0.0000206393047	4.4130586	0.5695951	0.0000454053457
4	4.3936648	0.1928291	0.0000206392404	4.3942372	0.1945842	0.0000454050291
7	4.3937727	0.1938640	0.0000206392413	4.3943129	0.1953145	0.0000454050328
10	4.3938406	0.1938180	0.0000206392413	4.3943621	0.1952809	0.0000454050330
15	4.3938407	0.1938198	0.0000206392413	4.3943621	0.1952819	0.0000454050330
21	4.3938407	0.1938198	0.0000206392413	4.3943621	0.1952819	0.0000454050330
23	4.3938407	0.1938198	0.0000206392413	4.3943621	0.1952819	0.0000454050330
28	4.3938407	0.1938198	0.0000206392413	4.3943621	0.1952819	0.0000454050330
31	4.3938407	0.1938198	0.0000206392413	4.3943621	0.1952819	0.0000454050330

is given to display the convergence analysis of the proposed scheme when $\varphi =0.2, Re=0.9, {\gamma }_1=0.4, {\gamma }_2=0.5, A_1=0.7, \Omega =A_2=0.8, Pr=6.2$ and $Rd=0.3$.

Table 5. Error analysis of axil velocity, tangential velocity and temperature for SWCNTs for different values of $M$.

$\mathit{k}$	$\mathit{M}$	${\mathit{ϵ}}_{\mathit{f}}$	${\mathit{ϵ}}_{\mathit{g}}$	${\mathit{ϵ}}_{\mathit{\theta }}$
1	10	2.10474 × 10−10	6.10471 × 10−11	7.36521 × 10−26
	15	1.50471 × 10−14	3.66540 × 10−16	1.54891 × 10−31
	19	5.00474 × 10−19	4.45874 × 10−22	1.05478 × 10−36
	23	1.33691 × 10−24	1.75325 × 10−27	2.69801 × 10−42
	28	1.00081 × 10−31	9.35682 × 10−35	2.45701 × 10−50
	31	2.12453 × 10−35	4.00147 × 10−39	3.00689 × 10−53

is constructed to show the effectiveness of the proposed method. In this table, residual error of velocity, temperature and concentration profile is shown against the variation of $M$. It is important to note that as the value of $M$ is enhanced the residual error decreases gradually, this shows the convergence of the proposed method.

5. Concluding Remarks

This study is carried out to examine the heat and flow measure among two stretching rotating disks containing the water-based carbon nanotubes. The concept of thermal boundary conditions and heat convection is utilized and the system is expressed in partial differential equations. By means of the similarity methods, the model is fruitfully changed to a nonlinear ordinary differential equation. A familiar collocation method is used to simulate the outcomes of the governed system while the method is validated through a set of tables and assessment with existing literature.

Hence, significant observations are given below:

• When the values of Re parameter are increased, a drop is noted in the magnitude of radial velocity near the faces of the disks.
• It is very clear from the tabular comparison given in Table 2 that the collocation scheme is in good agreement with already published studies and homotopic solutions.
• It is found that the multiple-wall carbon nanotubes intensify the system quickly and improve the rate of heat transmission.
• It is also noted that the proposed method is in excellent agreement with already published studies and homotopic solutions and can be extended for other physical problems.