The Rotating Flow of Magneto Hydrodynamic Carbon Nanotubes over a Stretching Sheet with the Impact of Non-Linear Thermal Radiation and Heat Generation/Absorption

Abstract

The aim of this research work is to investigate the innovative concept of magnetohydrodynamic (MHD) three-dimensional rotational flow of nanoparticles (single-walled carbon nanotubes and multi-walled carbon nanotubes). This flow occurs in the presence of non-linear thermal radiation along with heat generation or absorption based on the Casson fluid model over a stretching sheet. Three common types of liquids (water, engine oil, and kerosene oil) are proposed as a base liquid for these carbon nanotubes (CNTs). The formulation of the problem is based upon the basic equation of the Casson fluid model to describe the non-Newtonian behavior. By implementing the suitable non-dimensional conditions, the model system of equations is altered to provide an appropriate non-dimensional nature. The extremely productive Homotopy Asymptotic Method (HAM) is developed to solve the model equations for velocity and temperature distributions, and a graphical presentation is provided. The influences of conspicuous physical variables on the velocity and temperature distributions are described and discussed using graphs. Moreover, skin fraction coefficient and heat transfer rate (Nusselt number) are tabulated for several values of relevant variables. For ease of comprehension, physical representations of embedded parameters such as radiation parameter $\left(Rd\right)$, magnetic parameter $\left(M\right)$, rotation parameter ($K$), Prandtl number ($Pr$), Biot number $\left(\lambda \right)$, and heat generation or absorption parameter $\left(Q_h\right)$ are plotted and deliberated graphically.

1. Introduction

Single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) are similar in certain aspects, but they also have some striking differences. SWCNTs are an allotrope of sp² hybridized carbon, and are similar to fullerenes. Single-walled carbon nanotubes (SWCNTs) have unique character due to their unusual structure. The structure of SWCNTs demonstrates significant optical and electronic features, tremendous strength and flexibility, and high thermal and chemical stability. As a result, SWCNTs are expected to dramatically impact several industries, particularly with respect to electronic displays, health care, and composites. MCWNTs are also suitable for many applications due to their high conductivity. Applications include electrostatic discharge protection in wafer processing and fabrication, plastic components for automobiles, plastics rendered conductive to enable the electrostatic spray painting of automobile body parts. The carbon nanotube (CNT) is composed of allotropes of carbon, and has a cylinder-shaped nanostructure. The molecules of carbon in cylindrical form have remarkable physiognomies that are useful for applied sciences, optics, physical sciences, fabric sciences, energy, health sciences, and manufacturing [1]. Terminology with respect to carbon nanotubes (CNTs) was first presented by Iijima [2] in 1991, who explored multi-walled carbon nanotubes (MWCNTs) using the Krastschmer and Huffman technique. Single-walled carbon nanotubes (SWCNTs) were reported by Donald Bethune in 1993 [3]. It is acknowledged that nanofluids will reduce the difficulties connected with materials with low thermal conductivities, like coal oil, water, engine oil, gasoline etc. Nanofluids contain a homogeneous combination of nano-size materials, with lengths ranging from 1 nm to 100 nm in a base liquid. Nanofluids have applications in automobile engine freezing, heat interchangers, solar heating water, bio-medicine, cooling of micro-electronic chips, and enhanced productivity of diesel engines. Expressions with respect to nanofluids were initially presented by Choi [4] using the inconsistent thermal conductivity of liquid (water) by depositing copper nanomaterial. Said et al. [5] reported that the SWCNT nano-liquids improve the thermal performance of solar collectors in place of water, in a context where one of the most manageable and contamination-free source of renewable energy is solar energy, and solar aerials. It can transform solar radiation into heat energy. In the textile industry, much attention has been paid to coating CNTs because of their uses in the sound industry, for example in megaphones and headsets. Xiao et al. [6] reported that very thin CNT films radiate lurid sound, and therefore prepared thin film CNT megaphones. Halelfadel et al. [7] examined the productivity of water-based nanofluid CNTs and deliberated the consequences of stumpy volume fraction nanoparticles in a range from $0.0055\%\mathrm{to}0.278\%$ on the physical properties (density, viscosity, thermal conductivity, and specific heat) of nanofluids. Aman et al. [8] investigated the influence of magnetohydrodynamics on Poiseuille flow and heat analysis of CNTs with a Casson fluid vertical channel. Further studies on carbon nanotubes can be appraised in [9,10,11,12,13,14].

Hayat et al. [15] investigated the three-dimensional (3D) rotating flow of carbon nanotubes with a Darcy–Frochheimer porous medium. Manevitch et al. [16] studied nonlinear optical vibrations of single-walled carbon nanotubes. Imtiaz et al. [17] discussed the convective flow of carbon nanotubes between rotating stretchable disks with thermal radiation effects in his research article. In an article by Tran et al. [18] the purification and dissolution of carbon nanotube fiber spun using the floating catalyst method were reported.

In applied sciences and engineering, flows with respect to nonlinear materials are essential. To explore the properties of stream and heat analysis, a number of rheological models are available for second-, third-, and fourth-grade fluid, micro polar fluid, Jaffery fluid, and Maxwell fluid models. One such type of general model is the Casson model [19] presented to calculate the flow behaviors and display yield stress. When the produced stress is greater than the shear stress, the Casson fluid acts similarly to a solid. Examples of these (Casson) fluids are chocolate, honey, blood, jelly, tomato sauce, and soup. Dash et al. [20] studied the performances of Casson liquids over a permeable medium restricted by a circular cylinder. Pramanik [21] inspected the impact of radiation on the Casson liquid stream and heat through a porous extending medium. Hassanan et al. [22] investigated the unsteady Casson fluid layer and Newtonian heat through an oscillating surface. Asma et al. [23] examined the magnetohydrodynamic (MHD) unsteady flow of Casson fluid over an oscillating plate fixed in porous media. Walicka and Falicki [24] studied the influence of Reynold number on the stream of Casson liquid. Nadeem et al. [25] deliberated three-dimensional flows with convective boundary conditions of a Casson fluid. Specific studies related to Casson liquid flow can be found in [26,27].

The flow behavior is profoundly affected by the strength and placement of applied magnetic field. The applied magnetic field affects the deferred nanoparticles and reforms their absorption inside the fluid, which substantially modifies the flow features of heat analysis. The analysis of magnetohydrodynamic (MHD) flow of an electrically directing liquid flow has several major uses in engineering fields, such as in atomic reactor freezing, magnetohydrodynamic power producers, plasma studies, and the petroleum industry. Arial [28] studied the impact of MHDs on the two-dimensional (2D) flow of a gummy liquid imposing normal to the plane, and analyzed the influence of the Hartmann number. Ganapathirao and Ravindran [29] investigated the suction/injunction effect on 2D MHD steady flow with the chemical reaction along with mixed convection. Rahman et al. [30] and Srinivasacharya et al. [31] deliberated the impression of heat generation/absorption on 2D nanofluid flow and reported that the Nusselt number was enhanced with large values of the Biot number and the slip parameter. Hameed et al. [32] investigated the combined magnetohydrodynamic and electric field effect on an unsteady Maxwell nanofluid flow over a stretching surface under the influence of variable heat and thermal radiation. Recently, Shah et al. [33,34] studied the effects of hall current on three dimensional non-Newtonian nanofluids and micropolar nanofluids in a rotating frame.

In various engineering processes, the boundary film stream on a stretching/extending sheet is of great importance. In metallurgical and polymer productions, stretching is a significant phenomenon in the construction of polymer sheets, food processing, film coating, and drawing of copper wire. Crane [35] is the pioneering author who introduced the fluid flow by means of the stretching sheet. The work of Crane was expanded upon by many scholars, like Vajraelu and Roper [36] who investigated the stream of second-grade fluids and stretching sheets. Rosca and Pop [37] and Sajid et al. [38] deliberated the vicious unsteady motion due to a shrinking/stretching curved medium. Further research on the liquid stream due to the stretching sheet can be found in several studies [39,40]. Similarly the thermal radiation is significant in certain applications where radiant discharge depends on temperature and nanoparticle volume fraction. Pooya et al. [41] performed a thermal exploration of single phase nanofluid flow. Afify et al. [42] considered the impact of thermal radiation on 2D boundary layer flow. Brinkman [43] has investigated the viscosity of concentrated suspensions and solution. Xue [44] has discussed the model for thermal conductivity of CNTs based composites.

Here, the main aim is to deal the heat generation/absorption along with thermal radiation in a steady MHD 3D rotational stream of a Casson nanofluid with an entirely different base liquid. The movement of liquid is created by means of a stretching sheet. Two classes of carbon nanotubes (CNTs), specifically single-walled carbon nanotubes (SWCNTs) and multiple-walled carbon nanotubes (MWCNTs), are employed to explore the behavior of nanofluids. The acceptable transformation implies alterations the system of partial differential equations (PDEs) to a mandatory system of ordinary differential equations (ODEs).

The Homotopy Asymptotic Method (HAM) [45,46,47,48] involves calculation expansion for velocities and the energy profile. Here, the expressions of velocities and temperature are mathematically and graphically examined. Furthermore, the coefficient of skin friction and Nesselt number are explored numerically, and the results are discussed graphically.

2. Problem Formulation

We assume a three-dimensional, viscous, steady, MHD rotating Casson nanofluid flow comprising single- and multi-walled CNTs over a stretching sheet. The flows of SWCNTs and MWCNTs are taken under the influence of non-linear thermal radiation and heat absorption/generation. We adopt the coordinate structure in such a manner that the surface associated with the $xy$-plane and fluid is placed in $z>0$. The stretching rate of the surface and the angular velocity of fluid are assumed as $c>0$ and $\mathsf{\Omega }$, respectively. The surface temperature is a result of the convective heating process, which involves the hot fluid temperature $T_f$ and coefficient of heat transfer $h_f$. The geometry of the problem shown in Figure 1.

The necessary constitutive model of a Casson fluid [19,22,24] is as follows:

$$\begin{array}{l}\frac{{\tau }_{mn}^{c.f}}{2e_{mn}}={\mu }_{\gamma }^{c.f}+\frac{p_y}{\sqrt{2{\pi }_d}};{\pi }_d>{\pi }_{c.f},\\ \frac{{\tau }_{mn}^{c.f}}{2e_{mn}}={\mu }_{\gamma }^{c.f}+\frac{p_y}{\sqrt{2{\pi }_{c.f}}};{\pi }_d<{\pi }_{c.f}.\end{array}$$

(1)

where ${\tau }_{mn}^{c.f}$ is the shear stress tensor of the Casson fluid, ${\mu }_{\gamma }^{c.f}$ is the Casson fluid plastic viscosity, ${\pi }_d$ is the rate of deformation, and ${\pi }_{c.f}$ is the critical amount of current product that depends on the Casson fluid. We assume a situation in which a certain liquid requires the gradual growth of the the shear stress to sustain the constant rate of strain. In the case ${\pi }_d>{\pi }_{c.f}$ the Casson fluid possesses the kinematic viscosity ${\upsilon }^{c.f}$ as a function of ${\mu }_{\gamma }^{c.f},{\rho }^{c.f}$, and Casson parameter $\beta$, such that

$$\begin{array}{l}{\mu }^{c.f}={\mu }_{\gamma }^{c.f}+\frac{p_y}{\sqrt{2{\pi }_d}},p_y=\frac{{\mu }_{\gamma }^{c.f}\sqrt{2{\pi }_d}}{\beta },{\upsilon }^{c.f}=\frac{{\mu }^{c.f}}{{\rho }^{c.f}},\\ {\upsilon }^{c.f}=\frac{{\mu }^{c.f}\left(1+\frac{1}{\beta }\right)}{{\rho }^{c.f}},\frac{\partial {\tau }^{c.f}}{\partial z}=\left(1+\frac{1}{\beta }\right)u_{zz}.\end{array}$$

(2)

Here ${\mu }^{c.f}$ represents the dynamic viscosity of the Casson fluid, $\beta =\frac{{\mu }_{\gamma }^{c.f}\sqrt{2{\pi }_d}}{p_y}$, and $p_y$ shows the yield stress of the liquid. From the denotation of $\beta$ describing the ratio of the product of ${\mu }_{\gamma }^{c.f}$ and $\sqrt{2{\pi }_d}$ to the yield stress $p_y$, it can be seen that the resistivity in liquid is greater if $\beta >o$, and the liquid presents an inviscid behavior if $\beta <o$, which eliminates the viscosity of the Casson fluid.

Taking into account the above declared rules, the primary equation takes the following form [8,25]:

$$u_x+v_y+w_z=0,$$

(3)

$${\rho }_{nf}\left\{u{u}_x+v{u}_y+w{u}_z-2\mathsf{\Omega }v\right\}={\mu }_{nf}\left(1+\frac{1}{\beta }\right)\left(u_{zz}\right)-{\sigma }_{nf}{B}_0^2\left(u\right),$$

(4)

$${\rho }_{nf}\left\{u{v}_x+v{v}_y+w{v}_z-2\mathsf{\Omega }u\right\}={\mu }_{nf}\left(1+\frac{1}{\beta }\right)\left(v_{zz}\right)-{\sigma }_{nf}{B}_0^2\left(v\right),$$

(5)

$$u{T}_x+v{T}_y+w{T}_z={\alpha }_{nf}\left(T_{zz}\right)-\frac{\left(q_z^{red}\right)}{{\left(\rho c_p\right)}_f}+\frac{Q_0\left(T-T_0\right)}{{\left(\rho c_p\right)}_f}.$$

(6)

The allied boundary conditions [25] are defined as

$$\begin{array}{l}{u}_w=cx,v=0,w=0,-k_{nf}{T}_z=h_f\left(T_f-T\right)atz=0,\\ u=v=w=0,T=T_{\infty },atz=\infty .\end{array}$$

(7)

where $u,v$ and $w$ specify velocities in the $x,y,z$ directions, respectively, ${\rho }_{nf}$ represents the nanofluid density, ${\mu }_{nf}$ the nanofluid dynamic viscosity, and ${\sigma }_{nf}$ the electrical conductivity of the nanofluid. $T$ shows the local temperature, $k_{nf}$ is the nanofluid thermal conductivity, and ${\left(\rho c_p\right)}_{nf}$ is the nanofluid specific heat capacity.

Brinkman [43] defined the dynamic viscosity of nano-liquid in term of base liquid as:

$${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\phi \right)}^{2.5}}.$$

(8)

The density, specific heat capacity, and rate of thermal conductivities of the nanofluid [7,8] are expressed as:

$$\begin{array}{l}{\rho }_{nf}={\rho }_f\left(1-\phi \right)+\phi {\rho }_{CNT},{\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\phi \right)}^{2.5}},\\ {\alpha }_{nf}=\frac{{\mu }_f}{{\left(\rho c_p\right)}_{nf}},{\left(\rho C_p\right)}_{nf}=\left(1-\phi \right){\left(\rho C_p\right)}_f+\phi \left(\rho C_p\right){}_{CNT},\\ {\sigma }_{nf}={\sigma }_f\left(\frac{1+3\left(\sigma -1\right)\phi }{\left(\sigma +2\right)-\left(\sigma -1\right)\phi }\right).\end{array}$$

(9)

Here, the model by Xue [44] is implemented to calculate the thermal conductivities

$$\frac{k_{nf}}{k_f}=\frac{1-\phi +2\phi \left(\frac{k_{CNT}}{k_{CNT}-k_f}ln\left(\frac{k_{CNT}+k_f}{2k_f}\right)\right)}{1-\phi +2\phi \left(\frac{k_f}{k_{CNT}-k_f}ln\left(\frac{k_{CNT}+k_f}{2k_f}\right)\right)}.$$

(10)

where ${\rho }_{CNT}$ shows the density of carbon nanotube, ${\left(\rho c_p\right)}_{CNT}$ denotes the specific heat of the carbon nanotube, and ${\rho }_f,{\left(\rho c_f\right)}_f$ represent the density and specific heat of base fluid, respectively. $q^{rad}=-\frac{4{\sigma }^{\ast }}{3k^{\ast }}\frac{\partial {\mathrm{T}}^4}{\partial z}$ has been achieved using the Roseland approximation [43], where ${\sigma }^{\ast }$ is the Stefan Boltzmann constant, and $k^{\ast }$ is the absorption coefficient. We can observe that temperature difference in liquid motion is suitably negligible, so $T^4$ can be expressed in linear form using the Taylor’s series, neglecting the higher terms: $T^4=4T_{\infty }^3-3T_{\infty }^4$. Therefore, we have $q^{rad}=-\frac{16{\sigma }^{\ast }}{3k^{\ast }}{\mathrm{T}}^3\frac{\partial T}{\partial z}.$

Using the following suitable similarities variable [13],

$$\begin{array}{l}u=cx{f}^{\prime }(\eta ),v=cxg(\eta ),w=-\sqrt{c{\upsilon }_f}f(\eta ),\\ \mathsf{\Theta }(\eta )=T_{\infty }+\left(T_f-T_{\infty }\right)\mathsf{\Theta }(\eta ),\eta ={\left[\frac{c}{{\upsilon }_f}\right]}^{\frac{1}{2}}.\end{array}$$

(11)

Equation (3) is verified identically and Equations (4)–(10) take the following system when we use the similarities variables:

$$\left(1+\frac{1}{\beta }\right)f^{‴}\left(\eta \right)-{\zeta }_1{\zeta }_2[f\left(\eta \right)f^{″}\left(\eta \right)+{\left(f^{\prime }\left(\eta \right)\right)}^2+2Kg\left(\eta \right)-{\zeta }_{4}M{f}^{\prime }\left(\eta \right)]=0,$$

(12)

$$\left(1+\frac{1}{\beta }\right)g^{″}(\eta )-{\zeta }_1{\zeta }_2[f\left(\eta \right)g^{\prime }(\eta )+f^{\prime }(\eta )g\left(\eta \right)-2K{f}^{\prime }(\eta )-{\zeta }_{4}Mg\left(\eta \right)]=0,$$

(13)

$$\frac{d}{d\eta }\left({\zeta }_5+\frac{4}{3}Rd{\left[1+\left(\mathsf{\Theta }\left(\eta \right)-1\right)\mathsf{\Theta }\left(\eta \right)\right]}^3\right){\mathsf{\Theta }}^{\prime }\left(\eta \right)+{\zeta }_{3}Pr\left[f(\eta ){\mathsf{\Theta }}^{\prime }\left(\eta \right)+Q_h\mathsf{\Theta }\left(\eta \right)\right],$$

(14)

$$\begin{array}{l}f\left(\eta \right)=g\left(\eta \right)=0,f^{\prime }\left(\eta \right)=1,{\mathsf{\Theta }}^{\prime }\left(\eta \right)=\frac{k_f}{k_{nf}}\left(1-\mathsf{\Theta }\left(\eta \right)\right)\lambda ,at\eta =0,\\ f^{\prime }\left(\eta \right)=g\left(\eta \right)=\mathsf{\Theta }\left(\eta \right)=0,at\eta =\infty .\end{array}$$

(15)

Here $K$ represent the rotation parameter, $M$ represents the magnetic parameter, $Pr$ represents the Prandtl number, $Q_h$ denotes the heat generation or absorption parameter, $Rd$ represents radiation parameter, and $\lambda$ is the Biot number, and are expressed as follows:

$$K=\frac{\mathsf{\Omega }}{c},M=\frac{{\sigma }_f{B}_0{}^{2}g}{c},Pr=\frac{{\left(\rho c_p\right)}_f}{k_f},Q_h=\frac{Q}{c{\left(\rho c_p\right)}_f},Rd=\frac{4{\sigma }^{\ast }{\mathrm{T}}_{\infty }^3}{3k^{\ast }{k}_f},\lambda =\frac{h_f}{k_f}\sqrt{\frac{{\upsilon }_f}{c}}.$$

(16)

$$\begin{array}{l}{\zeta }_1={\left(1-\phi \right)}^{2.5},{\zeta }_2=\left\{(1-\phi )+\frac{{\rho }_{CNT}\phi }{{\rho }_f}\right\},{\zeta }_3=\left[(1-\phi )+\frac{{(\rho c_p)}_{CNT}}{{(\rho c_p)}_f}\right],\\ {\zeta }_4={\sigma }_f\left(\frac{1+3\left(\sigma -1\right)\phi }{\left(\sigma +2\right)-\left(\sigma -1\right)\phi }\right),{\zeta }_5=\frac{k_{nf}}{k_f}=\frac{1-\phi +\left(2\frac{k_{CNT}}{k_{CNT}-k_f}ln\frac{k_{CNT}+k_f}{2k_f}\right)\phi }{1-\phi +\left(2\frac{k_f}{k_{CNT}-k_f}ln\frac{k_{CNT}+k_f}{2k_f}\right)\phi }.\end{array}$$

(17)

3. Physical Quantities of Interest

The physical quantities of interest such as Skin friction $C_f$ is defined as $C_f=\left(1+\frac{1}{\beta }\right)\frac{{\mu }_{nf}}{{\rho }_f{u}_w^2}{\left(u_y\right)}_{y=0}$ and local Nusselt number $N{u}_x,$ is defined as $N{u}_x=\frac{x{k}_{nf}}{k_f\left(T_w-T_{\infty }\right)}{\left(T_y\right)}_{y=0}$, the dimensionless forms of these main design quantities are as follows:

$${\tilde{C}}_{fx}=\frac{1}{{\left(1-\phi \right)}^{2.5}}\left(1+\frac{1}{\beta }\right)f^{″}(0),{\tilde{C}}_{fy}=\frac{1}{{\left(1-\phi \right)}^{2.5}}\left(1+\frac{1}{\beta }\right)g^{\prime }(0),Nu=-\frac{k_{nf}}{k_f}{\mathsf{\Theta }}^{\prime }\left(0\right).$$

(18)

4. Solution Methodology

In this section, the system of coupled nonlinear differential Equations (12)–(14), along with specific boundary conditions (Equation (15)) are solved analytically by using the Homotopy Analysis Method (HAM). The idea of the HAM was first presented by Liao [44,45] using the concept of homotopy. For solution development, the HAM scheme has certain advantages, such as the fact that it is free from small or large emerging parameters. The HAM offers an easy way to confirm the convergence of solution, and it provides liberty in terms of the right selection of the base function and the auxiliary parameter.

In the HAM scheme, the assisting parameter $h$ is used to regulate and control the convergence of the solutions. The initial estimates are:

$$f_0(\eta )=1-e^{-x},g_0(\eta )=e^{-x},{\mathsf{\Theta }}_0(\eta )=\frac{k_f\gamma }{k_f\gamma k_{nf}}{e}^{-x}.$$

(19)

The linear operators $L$ are as follows:

$$L_f(f)=f_{\eta \eta \eta },L_g(g)=g_{\eta \eta },L_{\mathsf{\Theta }}(\mathsf{\Theta })={\mathsf{\Theta }}_{\eta \eta }.$$

(20)

For the above stated differential operators, constants are shown as:

$$\begin{array}{l}{L}_f(C_1+C_2\eta +C_3{\eta }^2)=0,\\ L_g(C_4+C_5\eta )=0,\\ L_{\mathsf{\Theta }}(C_6+C_7\eta )=0.\end{array}$$

(21)

4.1. Zeroth Order Deformation of the Problem

$Ϋ\in [0,1]$ is an embedded parameter with auxiliary parameters $h_f,h_g$ and $h_{\mathsf{\Theta }}$. The problem deforms for the zero order as follows:

$$(1-Ϋ)L_g\left(f(\eta ,Ϋ)-f_0(\eta )\right)=Ϋh_f{\mathrm{N}}_f\left\{f(\eta ,Ϋ),g(\eta ,Ϋ)\right\},$$

(22)

$$(1-Ϋ)L_g\left(g(\eta ,Ϋ)-g_0(\eta )\right)=Ϋh_g{\mathrm{N}}_g\left\{f(\eta ,Ϋ),g(\eta ,Ϋ)\right\},$$

(23)

$$(1-Ϋ)L_{\mathsf{\Theta }}\left(\mathsf{\Theta }(\eta ,Ϋ)-{\mathsf{\Theta }}_0(\eta )\right)=Ϋh_{\mathsf{\Theta }}{\mathrm{N}}_{\mathsf{\Theta }}\left\{f(\eta ,Ϋ),g(\eta ,Ϋ),\mathsf{\Theta }(\eta ,Ϋ)\right\}.$$

(24)

The specific boundary conditions are obtained as:

$$\begin{array}{l}f(0,Ϋ)=f^{\prime }(0,Ϋ)=g(0,Ϋ)=\mathsf{\Theta }(0,Ϋ)=0,\\ f(1,Ϋ)=f^{\prime }(1,Ϋ)=g(1,Ϋ)=\mathsf{\Theta }(1,Ϋ)=0.\end{array}$$

(25)

The resultant nonlinear operators are:

$$\begin{array}{l}{\mathrm{N}}_f\left(f(\eta ;Ϋ),g(\eta ;Ϋ)\right)=\left(1+\frac{1}{\beta }\right)f_{\eta \eta \eta }(\eta ;Ϋ)+{\zeta }_1{\zeta }_2{[\left\{f_{\eta \eta }(\eta ;Ϋ)\right\}}^2+f(\eta ;Ϋ)f_{\eta \eta }(\eta ;Ϋ)\\ +2Kg(\eta ;Ϋ)-{\zeta }_{4}M{f}_{\eta }(\eta ;Ϋ)],\end{array}$$

(26)

$$\begin{array}{l}{\mathrm{N}}_g\left(g(\eta ;Ϋ),f(\eta ;Ϋ)\right)=\left(1+\frac{1}{\beta }\right)g_{\eta \eta }(\eta ;Ϋ)-{\zeta }_1{\zeta }_2[f_{\eta }\left\{(\eta ;Ϋ)g(\eta ;Ϋ)\right\}+\\ \left\{f(\eta ;Ϋ)g_{\eta }(\eta ;Ϋ)\right\}-2K{f}_{\eta }(\eta ;Ϋ)-{\zeta }_{4}Mg(\eta ;Ϋ)],\end{array}$$

(27)

$$\begin{array}{l}{\mathrm{N}}_{\mathsf{\Theta }}\left(\mathsf{\Theta }(\eta ;Ϋ),f(\eta ;Ϋ)\right)=\frac{d}{d\eta }\left({\zeta }_5+\frac{4}{3}Rd{\left[1+\left(\mathsf{\Theta }\left(\eta \right)-1\right)\mathsf{\Theta }\left(\eta \right)\right]}^3\right){\mathsf{\Theta }}_{\eta }(\eta ;Ϋ)+\\ Pr{\zeta }_3\left[\left\{\left(f(\eta ;Ϋ)\right){\mathsf{\Theta }}_{\eta }(\eta ;Ϋ)\right\}+Q_h\mathsf{\Theta }(\eta ;Ϋ)\right].\end{array}$$

(28)

Taylor’s series expresses $f(\eta ;Ϋ),g(\eta ;Ϋ)\mathrm{and}\mathsf{\Theta }(\eta ;Ϋ)$. In terms of $Ϋ$ we have:

$$\begin{array}{l}f(\eta ,Ϋ)=f_0(\eta )+{\displaystyle \sum _{i=1}^{\infty }{f}_i(\eta ),}\\ g(\eta ,Ϋ)=g_0(\eta )+{\displaystyle \sum _{i=1}^{\infty }{g}_i(\eta ),}\\ \mathsf{\Theta }(\eta ,Ϋ)={\mathsf{\Theta }}_0(\eta )+{\displaystyle \sum _{i=1}^{\infty }{\mathsf{\Theta }}_i(\eta ).}\end{array}$$

(29)

where

$$f_i(\eta )={\frac{1}{i!}\left\{f_{\eta }(\eta ,Ϋ)\right\}|}_{Ϋ=0},g_i(\eta )={\frac{1}{i!}\left\{g_{\eta }(\eta ,Ϋ)\right\}|}_{Ϋ=0},{\mathsf{\Theta }}_i(\eta )={\frac{1}{i!}\left\{{\mathsf{\Theta }}_{\eta }(\eta ,Ϋ)\right\}|}_{Ϋ=0}$$

(30)

4.2. ith-Order Deformation Problem

Now we differentiate the zeroth component equations ith time to achieve the ith order deformation equations with respect to $Ϋ$, so we get:

$$\begin{array}{l}{L}_f\left(f_i(\eta )-{\xi }_i{f}_{i-1}(\eta )\right)=h_f\left(R_i^f(\eta )\right),\\ L_g\left(g_i(\eta )-{\xi }_i{g}_{i-1}(\eta )\right)=h_g\left(R_i^g(\eta )\right),\\ L_{\mathsf{\Theta }}\left({\mathsf{\Theta }}_i(\eta )-{\xi }_i{\mathsf{\Theta }}_{i-1}(\eta )\right)=h_{\mathsf{\Theta }}\left(R_i^{\mathsf{\Theta }}(\eta )\right).\end{array}$$

(31)

The resultant boundary conditions are:

$$\begin{array}{l}{f}_i={f^{\prime }}_i=g_i={\mathsf{\Theta }}_i=0,at\eta =0,\\ f_i=f^{\prime }=g_i={\mathsf{\Theta }}_i=0at\eta =1.\end{array}$$

(32)

$$\begin{array}{l}{R}_i^f(\eta )=\frac{d^3{f}_{i-1}}{d{\eta }^3}+{\zeta }_1{\zeta }_2[{\left({\displaystyle \sum _{j=0}^{i-1}\frac{d^2{f}_{i-1}}{d{\eta }^2}}\right)}^2+\left({\displaystyle \sum _{j=0}^{i-1}\left(f_{i-1-j}\right)}\frac{d^2{f}_j}{d{\eta }^2}\right)\\ +2K{g}_{i-1}-{\zeta }_{4}M\left({\displaystyle \sum _{j=0}^{i-1}\frac{d{f}_{i-1}}{d\eta }}\right)],\end{array}$$

(33)

$$R_i^f(\eta )=\frac{d^2{g}_{i-1}}{d{\eta }^2}+{\zeta }_1{\zeta }_2[\left({\displaystyle \sum _{j=0}^{i-1}\frac{d{f}_{i-1}}{d\eta }{g}_j}\right)+\left({\displaystyle \sum _{j=0}^{i-1}{f}_{i-1-j}}\frac{d{g}_j}{d\eta }\right)-{\zeta }_{4}M{g}_{i-1}-2K\left({\displaystyle \sum _{j=0}^{i-1}\frac{d{f}_{i-1}}{d\eta }}\right)],$$

(34)

$$\begin{array}{l}{R}_i^f(\eta )=\frac{d}{d\eta }\left(\zeta +\frac{4}{3}Rd{\left[1+\left(\mathsf{\Theta }\left(\eta \right)-1\right)\mathsf{\Theta }\left(\eta \right)\right]}^3\right)\frac{d{\mathsf{\Theta }}_{i-1}}{d\eta }+\\ Pr{\zeta }_3\left[{\displaystyle \sum _{j=0}^{i-1}{f}_{i-1-j}\frac{d{\mathsf{\Theta }}_j}{d\eta }}+Q_h{\displaystyle \sum _{j=0}^{i-1}{\mathsf{\Theta }}_{i-1-j}}\right].\end{array}$$

(35)

where

$${\xi }_i=\{\begin{array}{l}1,\mathrm{if}Ϋ>1\\ 0,\mathrm{if}Ϋ\le 1.\end{array}$$

(36)

5. HAM Scheme Convergence

When we compute the series solutions of the velocity and temperature functions to use the HAM, the assisting parameters $h_f,h_g$, and $h_{\mathsf{\Theta }}$ appear. These assisting parameters are responsible for adjusting the convergence of these solutions. In the possible region of velocities and temperature profiles,$f^{″}(0),g^{\prime }(0)$, and ${\mathsf{\Theta }}^{\prime }(0)$ for the 15th-order approximation are plotted, for different values of the embedding parameter. The $\hslash$-curves consecutively display the valid region.

6. Results and Discussion

The present research has been carried out in order to study magnetohydrodynamics in the three-dimensional rotational flow of nanoparticles (single- and multi-walled carbon nanotubes) in the existence of non-linear thermal radiation along with heat generation or absorption based on the Casson fluid model over a stretching sheet. In this subsection we examine the physical outcomes of dissimilar embedded parameters with respect to the velocity distributions $f^{\prime }(\eta ),g(\eta )$ and temperature distribution $\mathsf{\Theta }(\eta ).$ The influence of specific model quantities, such as $K,M,Pr,Q_h,Rd$, and $\lambda$ (the rotation parameter, magnetic parameter, Pandtl number, heat generation or absorption parameter, radiation parameter, and Biot number, respectively) on $f^{\prime }(\eta ),g(\eta )$, and $\mathsf{\Theta }\left(\eta \right)$ for SWCNTs and MWCNTs as nanoparticles is highlighted in the present discussion. Figure 1 shows the schematic flow geometry of the problem. Figure 2 and Figure 3 present the convergence and probable choice of h-curve for $f^{″}\left(0\right),{\mathsf{\Theta }}^{\prime }\left(0\right)$ and $g^{\prime }\left(0\right).$ The effects of model values such as $K,M,Pr,Q_h,Rd,\lambda$ on $f^{\prime }(\eta ),g(\eta )$ and $\mathsf{\Theta }\left(\eta \right)$ are illustrated in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 and consequence are considered. Figure 4 demonstrates the consequence of rotation parameter $K$ on the velocity profile $f^{\prime }(\eta )$ for SWCNTs and MWCNTs. The increasing values of $K$ depreciate the value of $f^{\prime }(\eta )$ and result in a thinner boundary coating for SWCNTs and MWCNTs. Physically, when the value of rotation parameter $K$ enlarges, the rate of rotation becomes dominant as compared to the stretching rate. Hence, there is a larger effect of rotation parameter $K$ delivering resistance to the fluid stream. For the maximum value of the rotation parameter $K$, the values of velocity-filled $f^{\prime }(\eta )$ become small near the boundary, and the layer becomes thinner, so the x-coordinate of velocity component is a decreasing function of the rotation parameter $K$. The similar impact for the high value of magnetic parameter $M$ is shown in Figure 5 for both CNTs. It is obvious from the sketch that an increment in the magnetic parameter $M$ depreciates the value of $f^{\prime }(\eta )$. Usually, high values of magnetic parameter $M$ improve the contradictory force (friction force), known as the Lorentz force. The corresponding force has the propensity to condense the value of $f^{\prime }(\eta )$. The influence of volume fraction of nanoparticle $\phi$ on the velocity profile $f^{\prime }(\eta )$ is exhibited in Figure 6 for SWCNTs and MWCNTs. We observe that increment in $f^{\prime }(\eta )$ increases the quantity of $\phi$ for SWCNTs and MWCNTs. In fact, adding the nanoparticle $\phi$ enhances the energy and cohesive force between the atoms of the liquid, causing it to become frail and halt the faster liquid. Figure 7 illustrates the behavior of the Casson fluid parameter $\beta$ on $f^{\prime }\left(\eta \right)$ for SWCNTs and MWCNTs. With increasing values of the Casson fluid parameter $\beta$ decreases the velocity field $f^{\prime }\left(\eta \right)$ in the boundary layer. It is noted that to increases the value of the Casson fluid parameter $\beta$ implies a reduction in the yield stress of the Casson liquid, and hence there is a clear acceleration in the motion of the boundary layer stream neighboring the stretching surface of the sheet. Moreover, it has been found that Casson fluids are similar to Newtonian fluids for large values of $\beta \to \infty$. The impact of nanoparticle volume fraction $\phi$ on $g(\eta )$ has been illustrated in Figure 8 for SWCNTs and MWCNTs. It is observed that the large value of $\phi$ decreases the transverse velocity $g(\eta )$ for SWCNTs and MWCNTs. In Figure 9 the velocity distribution $g(\eta )$ is intended for various values of rotation $K$. It is demonstrated that $K$ has an energetic character in accelerating the flow in the y-plane. The suitably bulky value of $K$ corresponds to an oscillatory pattern close to the wall of the sheet for SWCNTs and MWCNTs. The core reason is that the sheet is extended only along the x-plane, and due to rotation the liquid flow is solely possible in the y-direction. It can be observed that a high value of $K$ reduces the magnitude of $g(\eta )$. Figure 10 shows that the features of the magnetic parameter $M$ on $g(\eta )$ are qualitatively equivalent to those of $K$. Generally, the large value of $M$ improves the resistance to the motion of liquid, which depreciates the value of $g(\eta )$. In Figure 11 it can be seen that the increment in $Pr$ generates a fall in temperature $\mathsf{\Theta }\left(\eta \right)$ due to the weaker thermal diffusivity. Physically, thermal layer thickness and $\mathsf{\Theta }\left(\eta \right)$ form the reducing function of $Pr$. For the high $Pr$, the fluid retains low thermal conductivities, which causes the thinner thermal boundary layer. Consequently, the heat rate is increased. The fluids which have a small $Pr$ possess excellent thermal conductivity. Figure 12 presents the impact of radiation parameter $Rd$ on $\mathsf{\Theta }\left(\eta \right)$ for SWCNTs and MWCNTs as nanoparticles. It seems that the increase in $Rd$ boosts $\mathsf{\Theta }\left(\eta \right)$, which is associated to the thickness of the boundary film. Basically, high $Rd$ delivers more heat to functional nanofluids, which indicates an increment in $\mathsf{\Theta }\left(\eta \right)$ in the presence of SWCNTs and MWCNTs. Figure 13 characterizes the temperature field $\mathsf{\Theta }\left(\eta \right)$ for heat generation parameter $Q_h$. The value of $\mathsf{\Theta }\left(\eta \right)$ and the thermal thickness of boundary layer are improved for a large value of $Q_h$. In this manner, the rate of heat is enhanced as the $Q_h$ value is increased. Consequently, we note that the thermal thickness of boundary film is a function of $Q_h$.

Variation in $\mathsf{\Theta }\left(\eta \right)$ due to the Biot number $\lambda$ is represented in the Figure 14 for both CNTs. The stronger value of $\lambda$ leads to an enhanced temperature field $\mathsf{\Theta }\left(\eta \right)$ and a thickness of the thermal film that allows the thermal impact to infiltrate deeper into the dormant liquid. In fact, $\lambda$ depends on the heat transfer coefficient $h_f$, which is maximum for a massive value of $\lambda$, illustrating an increase in temperature.

Table Discussion

Table 1. The thermo physical properties of carbon nanotubes (CNTs) of some base fluids. SWCNT: single-walled carbon nanotube; MWCNT: multi-walled carbon nanotube

Physical Properties		Density $\mathit{\rho }{(\mathbf{kg}/\mathbf{m}}^3)$	Thermal Conduct $\mathit{k}(\mathbf{W}/\mathbf{mk})$	Specific Heat ${\mathit{c}}_{\mathit{p}}{(\mathbf{kg}}^{-1}{/\mathbf{k}}^{-1})$
Base fluid	Water	997	0.613	4197
	Kerosene (lamp) oil	783	0.145	2090
	Engine oil	884	0.144	1910
Nanoparticles	SWCNT	2600	6600	425
	MWCNT	1600	3000	796

illustrates various physical properties of certain base liquids (water, kerosene oil and engine oil) and nanoparticles such as SWCNTs and MWCNTs. Xue [44] calculated dissimilar values of effective thermal conductivities of CNT nanofluids for several values of the volume fraction φ of the CNTs shown in

Table 2. Thermal conductivity values of CNTs with different volume fraction values.

Volume Fraction $\mathit{\phi }$	Thermal Conductivity ${\mathit{k}}_{\mathit{n}\mathit{f}}$ for SWCNT	Thermal Conductivity ${\mathit{k}}_{\mathit{n}\mathit{f}}$ for MWCNT
0	0.145	0.145
0.01	0.174	0.172
0.02	0.204	0.2
0.03	0.235	0.228
0.04	0.266	0.257

. It can be seen in Table 2 that the thermal conductivity is improved with inspiring volume fraction φ of CNTs. For the same values of φ, can be noted that the nanofluids with SWCNTs have higher effective thermal conductivities as compared to those with MWCNTs. The logic for this is the elevated value of thermal conductivity of SWCNTs (6600 Wm⁻¹/k⁻¹) as opposed to MWCNTs (3000 Wm⁻¹/k⁻¹), as given in Table 1.

Physical values such as the skin friction co-efficient $\left(C_f\right)$ and the Nusselt number $\left({\rm N}u\right)$ for the engineering purposes are calculated through

Table 3. Numerical data of the skin-friction coefficient for dissimilar values of $M,K,\phi$, and $\beta$.

$\mathit{M}$	$\mathit{K}$	$\mathit{\phi }$	$\mathit{\beta }$	$-{\mathit{C}}_{\mathit{f}\mathit{x}}$		$-{\mathit{C}}_{\mathit{f}\mathit{y}}$
				SWCNT	MWCNT	SWCNT	MWCNT
0.0	0.1	0.1	0.1	$1.08611$	1.03456	$1.24004$	1.14526
0.1				$1.15356$	1.12677	$1.27582$	1.19870
0.3				$1.19726$	1.14340	$1.34626$	1.24358
0.1	0.0			$1.19564$	1.04566	$1.19564$	1.13657
	0.3			$0.98477$	0.23561	$1.44063$	1.17452
	0.5			$0.85341$	0.02537	$1.61105$	1.32435
	0.1	0.0		$0.96929$	0.23409	$1.10111$	1.03452
		0.3		$1.54113$	1.45473	$1.23251$	1.10034
		0.5		$1.68492$	1.52435	$1.69792$	1.37342
		0.1	0.1	1.69211	1.57235	1.72340	1.22349
			0.3	$1.73421$	1.62345	$1.88191$	1.31902
			0.5	1.92345	1.81189	1.90823	1.59321

and

Table 4. Numerical data of Nusselt numbers for dissimilar values of $M,K,\lambda ,Rd$, and $Q_h$ when $Pr=6.4$.

$\mathit{M}$	$\mathit{K}$	$\mathit{\lambda }$	$\mathit{R}\mathit{d}$	${\mathit{Q}}_{\mathit{h}}$	$\mathit{N}{\mathit{u}}_{\mathit{x}}$
					SWCNT	MWCNT
0.0	0.1	0.1	0.1	0.1	$0.116964$	0.231567
0.1					$0.116974$	0.232390
0.3					$0.116988$	0.233321
0.1	0.0				$0.116965$	0.134136
	0.3				$0.116966$	0.134342
	0.5				$0.116967$	0.134351
	0.1	0.3			$0.116953$	0.261532
		0.5			$0.116943$	0.261531
		0.8			$0.116926$	0.261530
		0.1	0.5		$0.118451$	0.156382
			1.0		$0.120623$	0.234521
			1.5		$0.122502$	0.267373
			0.1	0.5	$0.116945$	0.234536
				1.0	$0.116928$	0.198342
				1.5	$0.116895$	0.162435

, respectively. Table 3 presents the numerical values of skin friction $\left(C_{fx}\right)$ and $\left(C_{fy}\right)$. It can be observed that the large values of $M, \phi$ and $\beta$ increase the skin friction $\left(C_{fx}\right)$ and $\left(C_{fy}\right)$ of both SWCNTs and MWCNTs, where the large value of $K$ reduces the skin friction $\left(C_{fx}\right)$ of both SWCNTs and MWCNTs and increases skin friction along the y-axis $\left(C_{fy}\right)$. Table 4 presents the numerical results of the Nusselt number $\left({\rm N}u\right)$. It can be noted that the large values of $M$ and $K$ increase the $\left({\rm N}u\right)$ of both SWCNTs and MWCNTs, where the large values of $\lambda ,Rd$ and $Q_h$ reduce the Nusselt number of both SWCNTs and MWCNTs.

7. Conclusions

We address the three-dimensional rotational flow of Casson fluids containing carbon nanotubes (SWCNTs and MWCNTs) as nanoparticles in the presence of thermal radiation along with heat generation over a stretching sheet. The effects of embedded parameters are observed and studied graphically. Moreover, the variation of the skin friction, Nusselt number, and their influences on the velocity and heat distributions are examined. The main points of the present study are shown below:

• For CNTs, large values of $K$ (the rotation parameter) produce smaller velocities $f^{\prime }\left(\eta \right)$ and $g\left(\eta \right)$, but the opposite tendency is detected for temperature profile $\mathsf{\Theta }\left(\eta \right)$.
• Increasing the nanoparticle volume fraction $\phi$ into the working liquid boosted $f^{\prime }\left(\eta \right)$, $g\left(\eta \right)$ and $\mathsf{\Theta }\left(\eta \right)$ values for SWCNTs and MWCNTs.
• A strong magnetic parameter $M$ reduces $f^{\prime }\left(\eta \right)$ and $g\left(\eta \right)$, while increases the temperature $\mathsf{\Theta }\left(\eta \right)$.
• The thermal layer rises through heat generation for SWCNTs and MWCNTs.
• The large values of Prandlt number $Pr$, reduces nanoparticle temperature $\mathsf{\Theta }\left(\eta \right)$.
• Large values of Biot number $\lambda$ boost temperature and thicken the boundary film concentration.
• The degree of coefficient of skin friction increases for larger $\phi$ values. The results show that skin friction in case of SWCNTs is higher than in MWCNTs.
• Heat transfer rate is controlled by large values of $\lambda$ and $Pr.$
• Nusselt number is greater with SWCNTs as opposed to MWCNTs.