Radiative flow of magneto hydrodynamics single-walled carbon nanotube over a convectively heated stretchable rotating disk with velocity slip effect

Abstract

In this article, we have examined the three-dimensional flow of heat and mass transport of carbon nanotube–based nanoliquid over a rotating stretchable disk. A uniform magnetic field $B_0$ is applied in a transverse direction to the flow of nanofluid. Moreover, we have considered carbon nanotube nanoparticles termed as single-walled carbon nanotubes within the base liquid (water). In addition, at the boundaries of current problem, the effect of velocity slip and thermal convection is deliberated. The heat transport mechanism is also incorporated thermal radiation. The pertinent strong nonlinear ordinary differential system after utilizing the appropriate variables is intended. Homotopy analysis method technique is employed to estimate the analytical results for velocities and thermal fields. For the sake of comparison, the numerical method ND-Solve solution is also obtained. The results are found to be in an excellent agreement. Various graphs have been plotted in order to study the effect of different model variables on the velocities and thermal fields. The main features of physical quantities of flow like $C_F{\mathrm{Re}}_{\alpha }^{0.5},C_G{\mathrm{Re}}_{\alpha }^{0.5}$ (local skin friction coefficient), and $\mathrm{Nu}{\mathrm{Re}}_{\alpha }^{-0.5}$ (heat transfer rate) have been formulated and deliberated graphically. It is found that velocity is reduced under the influence of the exterior magnetic field. Concluding remarks are pinched under the analysis of complete investigation.

Introduction

In recent years, nanoliquids have gained tremendous concentration of scientists and engineers due to their fascinating thermal transport characteristics in different practical fields. Various classical heat transport liquids such as oil, water, and ethylene glycol have stumpy heat transport features. Nanofluids which are the homogeneous mixture of solid nanoparticles of dimension 1–100 nm in classical fluid are utilized to enhance the thermal conductivity of conventional heat transfer base liquids. Carbon nanotubes (CNTs) are the most favorable nano-elements in terms of their wonderful and high capability to conduct heat. There are numerous types of CNTs; however, they are commonly classified as single-walled carbon nanotubes (SWCNTs)/multi-walled carbon nanotubes (MWCNTs). SWCNTs are similar to a usual straw with just single wall having a diameter of 0.4–3 nm, while MWCNTs are the group of nested tubes with diameter ranges from 0.4 to 30 nm. In the present analysis, the thermal enhancement properties of water-based nanoliquid in the presence of SWCNTs are presented. The dispersion of SWCNTs in a based liquid forms a wide network of SWCNTs that accelerates thermal conductivity. The emerging problems in heat and mass transport enhancement depend on the use of nanoliquids.^1–4 Nadeem and ijaz ⁵ investigated the analysis of blood flow of variable nanoliquid viscosity in the presence of SWCNTs nanoparticles through a multiple stenosed arteries. Kamali et al. ⁶ examined the heat transport enhancement by using CNT-based non-Newtonian nanoliquids. The laminar flow of CNT-based nanoliquid is discussed by Wang et al., ⁷ and the squeezing motion of CNTs water-based nanoliquid between similar disks is illustrated by Haq et al. ⁸ Ellahi et al. ⁹ considered the magneto hydrodynamics (MHD) streaming of CNT-based nanoliquid. Arani et al. ¹⁰ computed the flow and heat transport water-based SWCNTs in a new proposal of double-layered micro-channel. Mahanthesh et al. ¹¹ examined the influence of nonlinear thermal radiation on CNT-based nanoliquid flow with Marangoni convection. Recently, Hayat et al. ¹² scrutinized the importance of CNTs considering chemical reaction, melting heat, and Darcy–Forchheimer.

Currently, the flow due to rotating disk has been an advanced area of investigation for the researchers and engineers because of their wide-ranging applications in several fields of science and modern technologies. In aeronautical discipline, disk-shaped bodies are usually encountered. In various real-life areas and scientific uses, such studies consist in rotating viscometers, computer storage appliances, rotors of gas turbines, centrifugal pumps, crystal growing developments, medicinal equipments, thermal-power producing system, air vacuuming apparatuses, chemical industries, and many other rotating mechanisms. The flow of liquid over a rotating disk was first defined by Karman. ¹³ He obtained the approximate solution of partial differential equations for rotating disk by using the well-known similarity transformation. Later, several researchers of the world studied the Karman famous work in different aspects for several physical problems. Cochran ¹⁴ used the appropriate variables proposed by Karman ¹³ to describe the rotating disk streaming by using numerical integration method. Mellor et al. ¹⁵ investigated and formulated the flow between the rotating disks. The characteristics of heat exchange between rotating disks are explained by Arora and Stokes. ¹⁶ Turkyilmazoglu and Senel ¹⁷ examined and computed numerically the effect of mass and heat transport in a rotating flow of liquid due to permeable disk. The characteristics of flow and heat transfer over a rotating disk immersed in five distinct nanofluids have been investigated by Turkyilmazoglu. ¹⁸ Sheikholeslami et al. ¹⁹ computed the mathematical results of the nanoliquid flow problems due to the inclined rotating disk. Turkyilmazoglu ²⁰ examined the heat transfer and liquid flow produced by the unsteady motion of the rotating disk, moving upright downward and upward direction. Moreover, Qayyum et al. ²¹ and Aziz et al. ²² explained the effect of MHD and slip aspects in different nanoparticles flow between rotating stretchable disks. The recent study on CNTs with rotating system and their application can be seen in literature.^23–25 Furthermore, recent investigations and theoretical research on nanofluids using dissimilar phenomena, with modern applications, possessions, and properties with the use of diverse approaches, can be demonstrated in previous works.^26–32 The prime objective of this endeavor is to explore water-functionalized CNT three-dimensional flow of nanoliquid over a stretchable disk in the presence of slip velocity and uniform magnetic field. The innovative idea of this investigation is considered as using CNTs nanoparticles called SWCNTs within the classical liquids. Heat transport mechanism is explored via convective condition and thermal radiation. The governing nonlinear expression of velocity and heat is computed through homotopy analysis method (HAM) approach.^33–35 The recent investigation of nanofluid can be studied in Sheikholeslami et al.^37–39 Solution with this method is important because it involves all the physical parameters of the problem, and we can easily discuss its behavior. Recently, HAM is mostly used due to its fast convergence.^40–45

Moreover, the behavior of some sundry parameters appeared in mathematical modeling is elaborated through some tabular aids and graphs. The dependency of the $C_F,C_G$ (local skin friction coefficient) and $\mathrm{Nu}$ (heat transfer rate) on the sundry variables is mathematically presented via plots.

Mathematical modeling

Consider the incompressible three-dimensional flow of water-based SWCNTs nanoliquid over a stretchable rotating disk. The characteristics of energy transport mechanism are inspected, such as thermal radiation and thermal convection. From the physical interpretation of flow model in Figure 1, we assume that disk rotates at $z=0$ with constant angular velocity $\Omega$ . Furthermore, the derivatives of $\phi$ are ignored due to the axial symmetry of motion. $T_f\mathrm{and}{T}_{\infty }$ refer to the disk surface and ambient liquid temperature, respectively. A uniform magnetic field of strength $B_0$ acts in z-direction. Also, $(u,v,w)$ describes the velocity components in the form of cylindrical coordinate $(r,\phi ,z)$ . In view of above-mentioned assumptions, the resulting momentum and energy equation for three-dimensional flows take the form ²²

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

(1)

$${\rho }_{\mathrm{nf}}[u\frac{\partial u}{\partial r}-\frac{v^2}{r}+w\frac{\partial u}{\partial z}]={\mu }_{\mathrm{nf}}[\frac{{\partial }^{2}u}{\partial r^2}-\frac{u}{r^2}+\frac{1}{r}\left(\frac{\partial u}{\partial r}\right)+\frac{{\partial }^{2}u}{\partial z^2}]-{\sigma }_{\mathrm{nf}}{B}_0^2\left(u\right)$$

(2)

$${\rho }_{\mathrm{nf}}[u\frac{\partial v}{\partial r}-\frac{\mathrm{uv}}{r}+w\frac{\partial v}{\partial z}]={\mu }_{\mathrm{nf}}[\frac{{\partial }^{2}v}{\partial r^2}+\frac{1}{r}\left(\frac{\partial v}{\partial r}\right)-\frac{v}{r^2}+\frac{{\partial }^{2}v}{\partial z^2}]-{\sigma }_{\mathrm{nf}}{B}_0^2\left(v\right)$$

(3)

$${\rho }_{\mathrm{nf}}[u\frac{\partial v}{\partial r}+w\frac{\partial v}{\partial z}]={\mu }_{\mathrm{nf}}[\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\left(\frac{\partial w}{\partial r}\right)+\frac{{\partial }^{2}w}{\partial z^2}]$$

(4)

$${\left(\rho c_p\right)}_{\mathrm{nf}}(u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z})=(k_{\mathrm{nf}}+\frac{16}{3}\frac{{\sigma }^{\otimes }{T}_{\infty }^3}{k^{\otimes }})(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial }^{2}T}{\partial z^2})$$

(5)

Here, equation (1) describes the law of conservation of mass (continuity equation), which confirmed automatically while equations (2)–(5) define the momentum expression and thermal field. Furthermore, the various model physical quantities embedded in equations (2)–(5) are the effective density $({\rho }_{\mathrm{nf}})$ of nanoliquid, dynamic viscosity $({\mu }_{\mathrm{nf}})$ , electrical conductivity $({\sigma }_{\mathrm{nf}})$ , thermal conductivity $k_{\mathrm{nf}}$ coefficient, effective heat capacity $(\rho c_p{)}_{\mathrm{nf}}$ of nanofluid, Stefan Boltzmann $({\sigma }^{\otimes })$ constant, mean absorption $(k^{\otimes })$ coefficient, and temperature $(T_f-T)$ (fluid, ambient).

Figure 1.

Physical interpretation of flow and coordinate axis.

Some proposed thermophysical properties for CNTs nanoliquid are^5,8

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\rho }_{\mathrm{nf}}}{{\rho }_f}=(1-\varphi )+\varphi \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_f},\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}={(1-\varphi )}^{-2.5},}\\ \multicolumn{1}{c}{\frac{{\left(\rho \mathrm{cp}\right)}_{\mathrm{nf}}}{{\left(\rho \mathrm{cp}\right)}_f}=(1-\varphi )+\left(\frac{{\left(\rho \mathrm{cp}\right)}_{\mathrm{CNT}}}{{\left(\rho \mathrm{cp}\right)}_f}\right)\varphi }\\ \multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=1+\frac{3(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}-1)\varphi }{(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}+2)-(\frac{{\sigma }_{\mathrm{CNT}}}{{\sigma }_f}-1)\varphi },}\\ \multicolumn{1}{c}{\frac{k_{\mathrm{nf}}}{k_f}=\frac{1-\varphi +2\varphi \left(\frac{k_{\mathrm{CNT}}}{k_{\mathrm{CNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{CNT}}+k_f}{2k_f}\right)}{1-\varphi +2\varphi \left(\frac{k_f}{k_{\mathrm{CNT}}-k_f}\right)\ln \left(\frac{k_{\mathrm{CNT}}+k_f}{2k_f}\right)}}\end{array}$$

(6)

here, $\varphi$ indicates nanoparticle solid volume fraction; ${\rho }_f\mathrm{and}{\rho }_{\mathrm{CNT}}$ signifies (base fluids, CNTs) density; ${\sigma }_f\mathrm{and}{\sigma }_{\mathrm{CNT}}$ signifies (base fluids, CNTs) electrical conductivity, $(\rho c_p{)}_f\mathrm{and}{(\rho c_p)}_{\mathrm{CNT}}$ represents (base fluids, CNTs) specific heat capacities, and $k_f\mathrm{and}{k}_{\mathrm{CNT}}$ shows (base fluids, CNTs) thermal conductivity.

The set of supporting boundary conditions for problems that are under discussion is

$$\begin{array}{c}\multicolumn{1}{c}{u=r\epsilon +L_1\left(\frac{\partial u}{\partial z}\right){\mu }_{\mathrm{nf}},v=r\Omega +L_1\left(\frac{\partial v}{\partial z}\right){\mu }_{\mathrm{nf}},w=0,}\\ \multicolumn{1}{c}{-k_{\mathrm{nf}}\left(\frac{\partial T}{\partial z}\right)=h_f(T_f-T),\mathrm{at}z=0}\\ \multicolumn{1}{c}{u\to 0,v\to 0,T\to T_{\infty },z=\infty }\end{array}$$

(7)

Here, $L_1$ denotes the coefficient of wall slip and $\epsilon$ identifies the stretching variable.

The following set of self-similar transformation is introduced for dimensionless functions $F,G$ , and $\beta$ with similarity parameter $\eta$ as

$$u=r\Omega F\prime \left(\eta \right),v=r\Omega G\left(\eta \right),w=-\sqrt{2\Omega {\upsilon }_f}F\left(\eta \right),\beta \left(\eta \right)=\frac{T-T_{\infty }}{T_f-T_{\infty }},\eta =\sqrt{\frac{2\Omega }{{\upsilon }_f}}z$$

(8)

After certain essential calculations, the dimensional flow problem equations (2)–(7) are converted into a non-dimensional flow model as follows

$$\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\{2F″\prime \left(\eta \right)\}+G^2\left(\eta \right)-F\prime \left(\eta \right)-2F″\left(\eta \right)·F\left(\eta \right)-\mathrm{MF}\prime \left(\eta \right)=0$$

(9)

$$\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\{2G″\left(\eta \right)\}-2F\prime \left(\eta \right)G\left(\eta \right)+2F\left(\eta \right)G\prime \left(\eta \right)-\mathrm{MG}\left(\eta \right)=0$$

(10)

$$\frac{1}{\mathrm{Pr}}(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4R_d}{3})\beta ″\left(\eta \right)+(1-\varphi +\frac{{\left(\rho c_p\right)}_{\mathrm{CNT}}\varphi }{{\rho }_f})F\left(\eta \right)\beta \prime \left(\eta \right)=0$$

(11)

The relevant transform non-dimensional boundary conditions of flow model are

$$\begin{array}{c}\multicolumn{1}{c}{F\left(0\right)=0,F\prime \left(0\right)=S+\chi {(1-\varphi )}^{-2.5}F″\left(0\right),}\\ \multicolumn{1}{c}{G\left(0\right)=1+\chi {(1-\varphi )}^{-2.5}G\prime \left(0\right)}\\ \multicolumn{1}{c}{\beta \prime \left(0\right)=-\frac{k_f}{k_{\mathrm{nf}}}\gamma (1-\beta \left(0\right)),\mathrm{at}\eta \to 0}\\ \multicolumn{1}{c}{F\prime \left(\infty \right)\to 0,G\left(\infty \right)\to 0,\beta \left(\infty \right)\to 0,\mathrm{at}\eta \to \infty }\end{array}$$

(12)

Here, $S=\epsilon /\Omega$ (stretching strength parameter), $\chi =L_1{\mu }_f\sqrt{2\Omega /{\upsilon }_f}$ (velocity slip parameter), $\gamma =h_f/k_f\sqrt{{\upsilon }_f/2\Omega }$ (Biot number), $M=4{\sigma }_f{B}_0^2/\Omega {\rho }_f$ (magnetic parameter), $\mathrm{Pr}={\upsilon }_f(\rho c_p{)}_f/k_f$ (Prandtl number), and $R_d=4{\sigma }^{\otimes }{T}_{\infty }^3/k^{\otimes }{k}_f$ (radiation parameter).

From the engineering point of view, the factors $C_F\sqrt{R{e}_{\alpha }}$ , $C_G\sqrt{R{e}_{\alpha }}$ (skin friction), and $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ (Nusselt number) have played a dynamic role. These parameters are defined by the following formulae in non-dimensional form

$$C_F\sqrt{R{e}_{\alpha }}=\frac{F″\left(0\right)}{{(1-\varphi )}^{2.5}},C_G\sqrt{R{e}_{\alpha }}=\frac{G\prime \left(0\right)}{{(1-\varphi )}^{2.5}},\mathrm{Nu}{\left(R{e}_{\alpha }\right)}^{-1/2}=-\left(\frac{k_{\mathrm{nf}}}{k_f}\right)\beta \prime \left(0\right)$$

(13)

Here, in the above expression, $R{e}_{\alpha }$ depicts local Reynolds number and $R{e}_{\alpha }=2r^2\Omega /{\upsilon }_f$ .

Solution by HAM

In this section, the nonlinear ordinary differential equations (9)–(11) with boundary condition (equation (12)) have been solved by using HAM. Liao in 1992 suggested HAM scheme, which is applied to compute the series solution of highly complex nonlinear problems. Furthermore, HAM techniques have some benefits, such as it confirms the convergence of results and delivers full independence to choose the base function and primary gauss. To continue with HAM, it is very essential to state the initial guess $(F_0,G_0,{\beta }_0)$ and linear operator like $(L_F,L_G,L_{\beta })$ .^23,24

To control and improve the convergence of problem, we sued auxiliary constant $h$ . ²⁴ A selection of initial gasses is

$$\begin{array}{l}{F}_0\left(\eta \right)=\frac{S{\left(\varphi -1\right)}^2\left(1-e^{-x}\right)}{1+\chi {\left(1-\varphi \right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-2\varphi +{\varphi }^2},\hfill \\ G_0\left(\eta \right)=\frac{e^{-x}{\left(\varphi -1\right)}^2}{1+\chi {\left(1-\varphi \right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-2\varphi +{\varphi }^2}\hfill \\ {\beta }_0\left(\eta \right)=\frac{\gamma k_f}{\gamma k_f+e^x{k}_{nf}}\hfill \end{array}$$

(14)

$L_F,L_G$ , and $L_{\beta }$ are linear operators such that

$$L_F=\frac{d^{3}F\left(\eta \right)}{d{\eta }^3},L_G=\frac{d^{2}G\left(\eta \right)}{d{\eta }^2},\mathrm{and}{L}_{\beta }=\frac{d^2\beta \left(\eta \right)}{d{\eta }^2}$$

(15)

The general result of $L_F,L_G$ , and $L_{\beta }$ is

$$L_F(P_1+P_2\eta +P_3{\eta }^2)=0,L_G(P_5+P_6\eta )=0,\mathrm{and}{L}_{\beta }(P_7+P_8\eta )=0$$

(16)

Zero-order problem

Here, the embedding variable $\Lambda \in [01]$ is with auxiliary variables $h_F,h_G$ , and $h_{\beta }$ . So, equations (9)–(12) deform for zero order as

$$(1-\Lambda )L_F(F(\eta ,\Lambda )-F_0(\eta ))=\Lambda h_F{M}_F(F(\eta ,\Lambda ),G(\eta ,\Lambda ))$$

(17)

$$(1-\Lambda )L_G(G(\eta ,\Lambda )-G_0(\eta ))=\Lambda h_G{M}_G[F(\eta ,\Lambda ),G(\eta ,\Lambda )]$$

(18)

$$(1-\Lambda )L_{\beta }(\beta (\eta ,\Lambda )-{\beta }_0(\eta ))=\Lambda h_{\beta }{M}_{\beta }[F(\eta ,\Lambda ),G(\eta ,\Lambda ),\beta (\eta ,\Lambda )]$$

(19)

$$\begin{array}{c}\multicolumn{1}{c}{F(0,\Lambda )=0,F\prime (0,\Lambda )=S+\chi {(1-\varphi )}^{-2.5}F″(0,\Lambda ),}\\ \multicolumn{1}{c}{G(0,\Lambda )=1+\chi {(1-\varphi )}^{-2.5}G\prime (0,\Lambda )}\\ \multicolumn{1}{c}{\beta \prime (0,\Lambda )=-\frac{k_f}{k_{\mathrm{nf}}}\gamma (1-\beta (0,\Lambda ))}\\ \multicolumn{1}{c}{F\prime (\infty ,\Lambda )=0,G(\infty ,\Lambda )=0,\beta (\infty ,\Lambda )=0}\end{array}$$

(20)

Therefore, the resulting nonlinear operators are

$$\begin{array}{c}\multicolumn{1}{c}{M_F(F(\eta ;\Lambda ),G(\eta ;\Lambda ))=\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\{2F_{\eta \eta \eta }(\eta ;\Lambda )\}}\\ \multicolumn{1}{c}{+G^2(\eta ;\Lambda )-F_{\eta }(\eta ;\Lambda )}\\ \multicolumn{1}{c}{-2F_{\eta \eta }(\eta ;\Lambda )·F(\eta ;\Lambda )-M{F}_{\eta }(\eta ;\Lambda )}\end{array}$$

(21)

$$\begin{array}{c}\multicolumn{1}{c}{M_G(F(\eta ;\Lambda ),G(\eta ;\Lambda ))=\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\{2G_{\eta \eta }(\eta ;\Lambda )\}}\\ \multicolumn{1}{c}{-2F_{\eta }(\eta ;\Lambda )·G(\eta ;\Lambda )}\\ \multicolumn{1}{c}{+2F(\eta ;\Lambda )G_{\eta }(\eta ;\Lambda )-\mathrm{MG}(\eta ;\Lambda )}\end{array}$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{M_{\beta }(\beta (\eta ;\Lambda ),F(\eta ;\Lambda ))=(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4R_d}{3}){\beta }_{\eta \eta }(\eta ;\Lambda )}\\ \multicolumn{1}{c}{+\mathrm{Pr}(1-\varphi +\frac{{\left(\rho c_p\right)}_{\mathrm{CNT}}\varphi }{{\rho }_f})}\\ \multicolumn{1}{c}{F(\eta ;\Lambda ){\beta }_{\eta }(\eta ;\Lambda )}\end{array}$$

(23)

By using Taylor’s series, ²⁴ $F(\eta ;\Lambda ),G(\eta ;\Lambda ),\mathrm{and}\beta (\eta ;\Lambda )$ are expressed in terms of $\Lambda$

$$\begin{array}{c}\multicolumn{1}{c}{F(\eta ;\Lambda )=F_0(\eta )+\sum _{i=1}^{\infty }{F}_i(\eta )}\\ \multicolumn{1}{c}{G(\eta ;\Lambda )=G_0(\eta )+\sum _{i=1}^{\infty }{G}_i(\eta )}\\ \multicolumn{1}{c}{\beta (\eta ;\Lambda )={\beta }_0(\eta )+\sum _{i=1}^{\infty }{\beta }_i(\eta )}\end{array}$$

(24)

where

$$F_i(\eta )={\frac{1}{i!}(F_{\eta }(\eta ,\Lambda ))|}_{\Lambda =0},G_i(\eta )={\frac{1}{i!}(G_{\eta }(\eta ,\Lambda ))|}_{\Lambda =0},{\beta }_i(\eta )={\frac{1}{i!}({\beta }_{\eta }(\eta ,\Lambda ))|}_{\Lambda =0}$$

(25)

ith-order deformation problem

Differentiating the zeroth order equations $i\mathrm{th}$ time to achieve the $i\mathrm{th}$ -order deformation equations with respect to $\Lambda$

$$\begin{array}{c}\multicolumn{1}{c}{L_F(F_i(\eta )-{\zeta }_i{F}_{i-1}(\eta ))=h_f(R_i^f(\eta ))}\\ \multicolumn{1}{c}{L_G(G_i(\eta )-{\zeta }_i{G}_{i-1}(\eta ))=h_G(R_i^G(\eta ))}\\ \multicolumn{1}{c}{L_{\beta }({\beta }_i(\eta )-{\zeta }_i{\beta }_{i-1}(\eta ))=h_{\beta }(R_i^{\beta }(\eta ))}\end{array}$$

(26)

The corresponding resultant boundary conditions are

$$\begin{array}{c}\multicolumn{1}{c}{F_i=0,{F\prime }_i=\chi {(1-\varphi )}^{-2.5}{F″}_i,G_i=\chi {(1-\varphi )}^{-2.5}{G\prime }_i,}\\ \multicolumn{1}{c}{{\beta \prime }_i=-\frac{k_f}{k_{\mathrm{nf}}}\gamma (1-{\beta }_i)\mathrm{at}\eta =0}\\ \multicolumn{1}{c}{{F\prime }_i=0,G_i=0,{\beta }_i=0\mathrm{at}\eta =\infty }\end{array}$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{R_i^f(\eta )=\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\left\{2\frac{d^3{F}_{i-1}}{d{\eta }^3}\right\}+{\left(G_{i-1}\right)}^2}\\ \multicolumn{1}{c}{-\sum _{j=0}^{i-1}\frac{d{F}_{i-1}}{d\eta }-2\left(\sum _{j=0}^{i-1}\left(F_{i-1-j}\right)\frac{d^2{F}_j}{d{\eta }^2}\right)}\\ \multicolumn{1}{c}{-M\left(\sum _{j=0}^{i-1}\frac{d{F}_{i-1}}{d\eta }\right)-{\lambda }_r{\left(\sum _{j=0}^{i-1}\frac{d{F}_{i-1}}{d\eta }\right)}^2}\end{array}$$

(28)

$$R_i^f(\eta )=\frac{{(1-\varphi )}^{-2.5}}{(1-\varphi +\frac{{\rho }_{\mathrm{CNT}}\varphi }{{\rho }_f})}\left\{2\frac{d^2{g}_{i-1}}{d{\eta }^2}\right\}-2G_{i-1}.\left(\sum _{j=0}^{i-1}\frac{d{f}_{i-1}}{d\eta }\right)+2F_{i-1}.\left(\sum _{j=0}^{i-1}\frac{d{G}_{i-1}}{d\eta }\right)-M{G}_{i-1}$$

(29)

$$R_i^{\beta }(\eta )=(\frac{k_{\mathrm{nf}}}{k_f}+\frac{4R_d}{3})\frac{d^2{\beta }_{i-1}}{d{\eta }^2}+\mathrm{Pr}(1-\varphi +\frac{{\left(\rho c_p\right)}_{\mathrm{CNT}}\varphi }{{\rho }_f})F_{i-1}.\left[\sum _{j=0}^{i-1}\frac{d{\beta }_{i-1}}{d\eta }\right]$$

(30)

where

$${\zeta }_i=\{\begin{array}{c}1,\mathrm{if}\Lambda >1\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}\Lambda ⩽1}\end{array}$$

(31)

Convergence of HAM

HAM scheme has the auxiliary constants $h_F,h_G$ , and $h_{\beta }$ that control and modify the convergence of the solution. For appropriate value of $h_F,h_G$ , and $h_{\beta }$ , we perform 20th order approximation. The appropriate region of $h_F,h_G$ , and $h_{\beta }$ lies between $-2⩽h_F⩽1,-3.7⩽h_G⩽2.5,\mathrm{and}-3.8⩽h_{\beta }⩽2.6$ , and it is plotted in Figures 2–4, respectively.

Figure 2.

Sketch of h-curve for $F″(\eta )$ .

Figure 3.

Sketch of h-curve for $G\prime (\eta )$ .

Figure 4.

Sketch of h-curve for $\beta \prime (\eta )$ .

Graphical analysis

This section demonstrates the behavior of various physical parameters like $\varphi$ , $M$ , $R_d$ , and $\gamma$ on $F\prime (\eta )$ , $G(\eta )$ , $\beta (\eta )$ , $C_F\sqrt{R{e}_{\alpha }},C_G\sqrt{R{e}_{\alpha }}$ , and $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ versus SWCNT nanoparticles. The physical interpretations of model flow and coordinate axis are elaborated through Figure 1. Figures 2–4 represent h-curves for $F″(\eta )$ , $G\prime (\eta )$ , and $\beta \prime (\eta )$ . The comparison of HAM and numerical scheme has been shown graphically in Figures 5–7 for velocities $F\prime (\eta )$ and $G(\eta )$ and temperature $\beta (\eta )$ , respectively. The characteristics of various pertinent physical quantities like $M,\varphi$ versus $F\prime (\eta )$ for SWCNTs are addressed in Figures 8 and 9. Figure 8 is sketched to display the variation in $F\prime (\eta )$ for various values of $M$ . An increase in $M$ (magnetic field) leads to decrease in $F\prime (\eta )$ . This is due to the fact that the presence of magnetic field in the flow creates a force known as the Lorentz force, which acts as a retarding force, and consequently, the momentum boundary layer thickness decelerates throughout the flow on the rotating disk. $M\ne 0$ corresponds to hydro-magnetic flow and $M=0$ displays hydro-magnetic flow situation. Impact of $\varphi$ on $F\prime (\eta )$ profile in the presence of SWCNTS is shown in Figure 9. From Figure 9, we can perceive that $F\prime (\eta )$ displays the increasing trend corresponding to higher magnitude of $\varphi$ . Actually, $\varphi$ directly relates with convective flow. The features of different appropriate physical factors $M,\varphi$ against $G(\eta )$ filed for SWCNTs nanoparticles are elaborated through Figures 10 and 11. Figure 10 demonstrates the variation in $G(\eta )$ for varying $M$ (magnetic parameter). It has been observed that $G(\eta )$ field declines when $M$ increases. The impact of $\varphi$ on $G(\eta )$ in the presence of SWCNTS is revealed in Figure 11. From Figure 11, we can witness that $G(\eta )$ displays snowballing behavior corresponding to large value of $\varphi$ . Figures 12 and 13 are sketched to disclose the performance of various suitable model factors like $\gamma ,R_d$ against thermal field $\beta (\eta )$ with SWCNT nanoparticles. The outcome of $R_d$ on $\beta (\eta )$ is established in Figure 12 in the presence of SWCNTs nanoparticles. Here, the larger magnitude $R_d$ corresponds to the increase in $\beta (\eta )$ field and enhances the related viscosity of thermal film. Physically, strengthening $R_d$ transmits extra heat to operational nanoliquids. Figure 13 addresses the significance of $\gamma$ on $\beta (\eta )$ profile and thermal boundary film with SWCNTs as nanoparticles. The $\beta (\eta )$ field boots when magnitude of $\gamma$ is enhanced. The stronger $\gamma$ or disk convection parameter leads to the increase in the surface temperature that permits the thermal effect to penetrate deeper into the quiescent nanofluid.

Figure 5.

The comparison between HAM and numerical solution for $F\prime (\eta )$ profile.

Figure 6.

The comparison between HAM and numerical solution for $G(\eta )$ profile.

Figure 7.

The comparison between HAM and numerical solution for $\beta (\eta )$ profile.

Figure 8.

Plots of $F\prime (\eta )$ via $M$ (magnetic parameter).

Figure 9.

Plots of $F\prime (\eta )$ via $\varphi$ (nanoparticle volume fraction).

Figure 10.

Plots of $G(\eta )$ via $M$ (magnetic parameter).

Figure 11.

Plots of $G(\eta )$ via $\varphi$ (volume fraction).

Figure 12.

Plots of $\beta (\eta )$ via $R_d$ (radiation parameter).

Figure 13.

Plots of $\beta (\eta )$ via $\gamma$ (Biot number).

Table 1 portrays certain thermophysical properties of water and nanoparticles specifically SWCNT. Different numerical characteristics of $\varphi ,\chi ,M$ , and $S$ against $C_F\sqrt{R{e}_{\alpha }}\mathrm{and}{C}_G\sqrt{R{e}_{\alpha }}$ for SWCNTs water are defined in Table 2. Clearly, Table 2 shows that $C_F\sqrt{R{e}_{\alpha }}$ of SWCNTs is reduced for various increasing values of $\chi ,M$ , and $S$ but $C_F\sqrt{R{e}_{\alpha }}$ shows increments for higher approximation of $\varphi$ . Similarly, from Table 2, we can also observe the behavior of $\varphi ,\chi ,M$ , and $S$ against $C_G\sqrt{R{e}_{\alpha }}$ . Table 3 explains the influence of $\varphi ,R_d,\mathrm{and}\mathrm{Pr}$ against $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ for both cases of nanoparticles (SWCNTs). From Table 3, we noticeably observed that the model parameters $\varphi ,R_d,\mathrm{and}\mathrm{Pr}$ relate to $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ enrichment. The comparison of HAM and numerical solutions for the velocity fields is shown in Tables 3 as well as in Table 4, and from both results, a close agreement between these two methods has been perceived (Table 5).

Table 1.

Particular thermophysical properties of water and nanoparticles (SWCNT).^8,9

Physical properties	Base liquid	Nanoparticle
	Water	SWCNT
$\rho (\mathrm{kg}{\mathrm{m}}^{-3})$	997	2600
$C_p(\mathrm{J}(\mathrm{kg}\mathrm{K}{)}^{-1})$	4197	425
$k(\mathrm{W}({\mathrm{mK})}^{-1})$	0.613	6600
$\sigma (\mathrm{U}{\mathrm{m}}^{-1})$	0.05	106

SWCNT: single-walled carbon nanotubes.

Table 2.

Mathematical values for $C_F\sqrt{R{e}_{\alpha }}\mathrm{and}{C}_G\sqrt{R{e}_{\alpha }}$ through various value of $\varphi ,\chi ,M$ , and $S$ .

$\varphi$	$\chi$	$M$	$S$	$C_F\sqrt{R{e}_{\alpha }}$	$C_G\sqrt{R{e}_{\alpha }}$
				SWCNTs	SWCNTs
0.1	0.2	0.5	0.2	−0.335389	−2.03564
0.2				−0.369341	−2.83737
0.3				−0.437817	−3.45985
0.1	0.2			−0.335389	−2.03564
	0.3			−0.406780	−1.96204
	0.4			−0.477508	−1.89315
	0.2	0.1		−0.335389	−2.03564
		0.3		−0.368797	−2.00117
		0.5		−0.402067	−1.96772
		0.5	0.2	−0.335389	−2.03564
			0.3	−0.350384	−2.02359
			0.4	−0.365330	−2.01182

SWCNT: single-walled carbon nanotubes.

Table 3.

Mathematical values for $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ through various values of $\varphi ,R_d,\mathrm{Pr}$ , and $\gamma$ .

$\varphi$	$R_d$	$\mathrm{Pr}$	$\gamma$	$\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$
				SWCNTs
0.2	0.2	6.2	8	−0.104190
0.3				−0.104115
0.4				−0.104039
0.2	0.2			−0.104190
	0.4			−0.103618
	0.6			−0.103068
	0.2	6.2		−0.104190
		6.3		−0.106642
		6.4		−0.109085
		6.2	8	−0.104190
			8.1	−0.106602
			8.2	−0.109014

SWCNT: single-walled carbon nanotubes.

Table 4.

The relationship between HAM and ND-Solve for velocity field $F\prime (\eta )$ .

$\eta$	HAM solution $F\prime (\eta )$	Numerical solution $F\prime (\eta )$	Absolute error $F\prime (\eta )$
0	0.860613	0.771675	$8.8\times {10}^{-2}$
0.5	0.537335	0.504868	$3.2\times {10}^{-2}$
1.0	0.332745	0.333967	$1.2\times {10}^{-3}$
1.5	0.204600	0.222710	$1.8\times {10}^{-2}$
2.0	0.125178	0.149294	$2.4\times {10}^{-2}$
2.5	0.076335	0.100258	$2.3\times {10}^{-2}$
3.0	0.046454	0.067045	$2.1\times {10}^{-2}$
3.5	0.028233	0.044015	$1.6\times {10}^{-2}$
4.0	0.017145	0.027217	$1.0\times {10}^{-2}$
4.5	0.010407	0.013543	$6.3\times {10}^{-3}$
5.0	0.006315	0.006378	$1.3\times {10}^{-3}$

HAM: homotopy analysis method.

Table 5.

The relationship between HAM and ND-Solve is shown for $G(\eta )$ .

$\eta$	HAM solution of $G(\eta )$	Numerical solution $G(\eta )$	Absolute error $G(\eta )$
0	0.776819	0.729503	$4.7\times {10}^{-2}$
0.5	0.455582	0.424247	$3.1\times {10}^{-2}$
1.0	0.270592	0.245929	$2.4\times {10}^{-2}$
1.5	0.162014	0.142277	$1.9\times {10}^{-2}$
2.0	0.097490	0.082196	$1.5\times {10}^{-2}$
2.5	0.058845	0.047421	$1.1\times {10}^{-2}$
3.0	0.035586	0.027297	$8.3\times {10}^{-3}$
3.5	0.021545	0.015641	$5.9\times {10}^{-3}$
4.0	0.013054	0.008865	$4.1\times {10}^{-3}$
4.5	0.007912	0.004891	$3.0\times {10}^{-3}$
5.0	0.004797	0.002512	$2.2\times {10}^{-3}$
5.5	0.002909	0.001021	$1.8\times {10}^{-3}$
6.0	0.001764	0.00000	$1.7\times {10}^{-3}$

HAM: homotopy analysis method.

Conclusion

Here, in present communication, the significance of MHD-radiated three-dimensional flows of nanofluid containing CNT (SWCNTs) nanoparticles due to stretchable rotating disk with velocity slip impact is described. Heat transport mechanism is explained by convective boundary condition. The critical points of this work are pinpointed in the following

• The acts of $\varphi$ against $F\prime (\eta )\mathrm{and}G(\eta )$ field are qualitatively comparable.

• An increase in $M$ leads to decline in $F\prime (\eta )\mathrm{and}G(\eta )$ fields.

• Both $F\prime (\eta )\mathrm{and}G(\eta )$ fields drop for large magnitude of $\chi$ (velocity slip factor).

• High value of $S$ (stretching strength factor) displays an increase in $F\prime (\eta )$ , but opposite trend is observed in $G(\eta )$ .

• Higher values of $R_d$ yield an enhancement in $\beta (\eta )$ .

• Thermal field increases with increase in the magnitude of $\varphi$ .

• The thermal layer is enhanced via higher value of $\gamma$ .

• $C_F\sqrt{R{e}_{\alpha }}\mathrm{and}{C}_G\sqrt{R{e}_{\alpha }}$ is enhanced for larger value of $\varphi$ .

• $\mathrm{Nu}(R{e}_{\alpha }{)}^{-1/2}$ is upraised for longer magnitude of $\varphi$ for both cases of nanoparticles.

• The accuracy of the HAM results has been verified via numerical scheme.