**1. Introduction**

The idea of nanofluid was first introduced by Choi [1] in 1995. In his pioneering study, Choi named nanofluids as one of the most essential type of fluids for an enhanced heat transfer rate. Nanofluids are formed by suspending nanoparticles of interested metals in the base fluid. To date, di fferent types of nanoparticles as well as the base fluids are used in the literature. Some of them are magnetic nanoparticles, polymeric nanoparticles, carbon nanotubes, liposomes, quantum dots, metallic nanoparticles, dendrimers, polymeric nanoparticles, and many others. The base fluids are always water, oil, and/or ethylene glycol. The metallic nanoparticles are alumina, carbides, copper, metal oxides, and nitrides, whereas non-metallic nanoparticles are graphite and the well-known carbon nanotubes. Researchers have used di fferent combinations of nanoparticles and base fluids, however, nobody gave a final decision about which combination of nanoparticles and base fluid can give a better enhancement in the heat transfer rate (see, for example, the works of [2–18]).

In addition to the above discussion, each type of nanoparticle and base fluid has its unique importance. In this work, carbon nanotubes (CNTs) were used as nanoparticles suspended in Kerosene oil, chosen as base fluid. CNTs are elongated, tubular structure, and 1–2 nm in diameter (see, for example, the works of [19–23]). However, the best CNTs are those arranged in the form of hexagonal network of carbon atoms rolled up to form a steam-less hollow cylinder (Choi and Zhang [24]). CNTs were first discovered by a Japanese physicist Sumio Iijma in 1991 for multiple wall nanotubes (Sumio [25]). However, it took less than two years before single wall nanotubes were discovered. Several researchers these days are taking keen interest in studying CNTs owing to their unique nanostructures, high thermal conductivity, and exceptional mechanical strength and corrosion resistance. These novel characteristics of CNTs make them useful in industry such as solar cell, nanotube transistors, lithium ion batteries, chemical sensors, and so on. The theoretical and experimental researchers usually use two types of CNTs, namely single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs). Choi [26] found anomalous thermal conductivity enhancement in nanotubes' suspensions. Hone [27] studied, in details, the thermal properties of CNTs. Kamali and Binesh [28] numerically investigated heat transfer enhancement in non-Newtonian nanofluids using CNTs. Prajapati et al. [29], Kumaresan and Velraj [30], and Wang et al. [31] also provided some interesting studies on CNTs. Khan et al. [32] examined heat transfer using CNTs and fluid flow with Navier slip boundary conditions. Noreen et al. [33] used CNTs and analyzed thermal and velocity slips on Magnetohydrodynamics (MHD) peristaltic flow in an asymmetric channel. Noreen and Khan [34] studied heat transfer using individual MWCNTs owing to the metachronal beating of cilia. Ebaid and Sharif [35] suspended CNTs in a base fluid and studied the e ffect of a magnetic field on fluid motion and enhanced the rate of heat transfer of nanofluids using CNTs (also see the works of [36–39]). Zhang et al. [40] examined the e ffects of surface modification on the thermal conductivity and stability of the suspension formulated using the CNTs. Aman et al. [41] suspended CNTs in four di fferent types of molecular liquid and studied heat transfer enhancement in the free convection flow of Maxwell nanofluids. Zhang experimentally investigated the heat transfer of CNTs membranes. Soleimani et al. [42] studiedthe impact of carbon nanotubes-based nanofluid on oil recovery e fficiency using core flooding. Details about the MWCNTs were provided by Taheriang et al. [43] for enhanced thermophysical properties of nanofluids in a critical review. Wang et al. [44], in their review paper, studied mechanisms and applications of CNTs in terahertz devices. Pop et a. [45] studied the well-known stagnation point fluid flow over a stretching sheet involving the heat transfer factor due to radiation. Sharma and Singh [46] investigated stagnation point flow past a linearly stretching sheet with additional e ffects of variable thermal conductivity, heat source/sink, and MHD. Some other important studies with some interesting experimental findings are given in the works of [47–60] and cross references cited therein.

The basic objective of this examination is to report an MHD flow of nanofluid with Kerosene oil as base fluid and CNTs as nanoparticles over a variable thicker surface. Such formulation is not found in the literature so far. Two types of CNTs (SWCNTs and MWCNTs) are chosen. The problem is first arranged in suitable nonlinear di fferential equations using reasonable transformations. Analytic solution is obtained by utilizing a notable procedure called the optimal homotopy analysis technique (O-HAM). Several plots are generated to discuss the physical behavior of embedded parameters on the dimensionless velocity field and thermal distribution. The results for skin-friction (wall-drag) coefficient and rate of heat transfer (Nusselt factor) are computed in tabular data form. The present results are successfully reduced to the published results in the literature when compared.

#### **2. Mathematical Formulation**

The present communication reports a theoretical study on the magnetohydrodynamic flow and heat exchange of carbon nanotubes (CNTs)-based nanoliquid over a variable thicker surface. Kerosene oil is taken as the base fluid for the suspension of nanoparticles. Two types of carbon nanotubes (CNTs) are accounted for saturation in base fluid. Flow phenomenon is investigated in the presence of applied magnetic field. SWCNTs and MWCNTs are utilized as nanomaterials and kerosine oil is as base liquid. The impact of radiation and viscous dissipation are considered in the heat analysis. The thickness of the surface mentioned by *y* = *<sup>B</sup>*(*x* + *b*) 1−*<sup>m</sup>* 2 is variable. The ambient temperature is taken to be constant. The physical model can be seen in Figure 1.

**Figure 1.** Physical flow model. MWCNT, multi-walled carbon nanotube; SWCNT, single-walled carbon nanotube.

The boundary layer equations for the aforementioned problem (see, for example, the works of [49–52]) are as follows:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial v}{\partial y} = 0,
\tag{1}
$$

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \mathcal{U}\_{\text{cf}}\frac{d\mathcal{U}\_{\text{cf}}}{dx} + \nu\_{\text{nf}}\frac{\partial^2 u}{\partial^2 y} - \frac{\sigma B\_0^2}{\rho\_{\text{nf}}}(\mathcal{U} - \mathcal{U}\_{\text{cf}}),\tag{2}$$

$$u\frac{\partial T}{\partial \mathbf{x}} + v\frac{\partial T}{\partial y} = a\_{\text{nf}}\frac{\partial^2 T}{\partial^2 y} - \frac{1}{\rho \mathbb{C}\_p} \frac{\partial q\_r}{\partial y} + \frac{\nu\_{\text{nf}}}{\rho \mathbb{C}\_p} (\frac{\partial u}{\partial y})^2 \tag{3}$$

where velocity components are (*<sup>u</sup>*, *v*) along the *x*- and *y*-axes, respectively; *T* denotes the temperature; ν*n f* , α*n f* denotes the effective kinematic viscosity and thermal diffusivity of nanoliquid, respectively; (*Uw*, *Ue*) are defined as the stretching surface and free stream velocity, respectively; *qr* defines the radiative heat flux; and *Cp* defines the specific heat. The fruitful characteristics of nanoliquids may be defined using the properties of base liquid and carbon nanotubes and the solid volume fraction of CNTs in the base liquids (see, for example, the works of [49–52]) as follows:

$$\begin{aligned} \mu\_{nf} &= \frac{\mu\_f}{(1-q)^{2.5}}, \; \nu\_{nf} = \frac{\mu\_{nf}}{\rho\_{nf}}, \; \rho\_{nf} = (1-q) \; \rho\_f + q\rho\_s \text{(C}\_p)\_{\text{CNT}},\\ \alpha\_{nf} &= \frac{k\_{af}}{\rho\_{nf} \left(\text{C}\_p\right)\_{nf}}, \; \frac{k\_{af}}{k\_f} = \frac{(1-q) + 2q \frac{k\_{\text{CNT}}}{\text{kCN}-k\_f} \ln \frac{k\_{\text{CNT}}+k\_f}{2k\_f}}{(1-q) + 2q \frac{k\_f}{\text{kCN}-k\_f} \ln \frac{\ln \text{CNT}^{-k\_f}}{2k\_f}} \end{aligned} \tag{4}$$

where viscosity of nanoliquid is defined by μ*n f* ; nanoparticle volume fraction is defined by ϕ; and density of liquid and CNTs are defined by ρ*f* and ρCNT, respectively. Base liquid's thermal conductivity is defined by *k f* ; thermal conductivity of nanoliquids is defined by *kn f* ; the specific heat of nanoliquids, base liquid, and carbon nanotubes are defined by -*Cpn f* , -*Cpf* , and -*Cp*CNTs, respectively; while thermal conductivity of CNTs is defined by *k*CNT, with boundary conditions as follows:

$$\begin{aligned} \text{Let } y &= B(\mathbf{x} + \mathbf{b})^{\frac{1-\mathfrak{m}}{2}}, \; \mathcal{U}\_{\mathbf{w}}(\mathbf{x}) = \mathcal{U} = \mathcal{U}\_{0}(\mathbf{x} + \mathbf{b})^{\mathfrak{m}}, \; T = T\_{\mathfrak{w}\prime} & \; \mathfrak{v} = \mathbf{0}, \\\ \text{as } y &\to \infty, \; \mathfrak{u} \to \mathcal{U}\_{\mathbf{c}\prime}(\mathbf{x} + \mathbf{b})^{\mathfrak{m}} = \mathcal{U}\_{\mathbf{c}(\mathbf{x})\prime} & \; T \to T\_{\mathrm{co}}. \end{aligned} \tag{5}$$

Using Rosseland approximation, we ge<sup>t</sup> the accompanying articulation

$$q\_{I} = \frac{-4}{3} \frac{\sigma^\*}{K^\*} \frac{\partial T^4}{\partial y} = -\frac{16\sigma^\*}{3K^\*} T^3 \frac{\partial T}{\partial y'} \tag{6}$$

where the mean absorption coefficient is defined by *k*<sup>∗</sup>, the Stefan–Boltzmann constant is defined by <sup>σ</sup><sup>∗</sup>, and *T*<sup>4</sup> defined using Taylor series expansion about *T*∞ is

$$T^4 = 4T\_{\alpha\nu}^3 T - 3T\_{\alpha\nu}^4$$

The state of surface firmly relies on *m*. It should be noted that, for *m* = 1, the surface is flat; the thickness of the wall rises for *m* < 1 and the surface shape becomes of the outer convex type. The wall thickness decreases for *m* > 1 and, consequently, the surface shape becomes of the inner concave type; *m* is accountable for the motion type, that is, for *m* = 0, the motion is linear with constant velocity. Motion deceleration and accelerated are defined by *m* < 1 and *m* > 1. Employing the transformations

$$\begin{aligned} \xi &= y \sqrt{\frac{\mathbb{U}\_{\boldsymbol{\theta}}(\mathbf{x}+\mathbf{b})^{m-1}}{2\boldsymbol{v}\_{f}}} (\boldsymbol{m}+\mathbf{1}), \quad \boldsymbol{\psi} = \sqrt{\frac{1}{m+1} 2\boldsymbol{v}\_{f}! \mathcal{U}\_{\boldsymbol{\theta}}(\mathbf{x}+\mathbf{b})^{m+1}} \mathcal{F}(\boldsymbol{\xi}), \ \boldsymbol{\Theta}(\boldsymbol{\xi}) &= \frac{\boldsymbol{T} - \boldsymbol{T}\_{\boldsymbol{w}}}{\boldsymbol{T}\_{\boldsymbol{w}} - \boldsymbol{T}\_{\boldsymbol{w}}}, \\ \boldsymbol{u} &= \mathcal{U}\_{\boldsymbol{\theta}}(\mathbf{x}+\mathbf{b})^{m} \mathcal{F}'(\boldsymbol{\xi}), \ \boldsymbol{v} = -\sqrt{\frac{m+1}{2}\boldsymbol{v}\_{f}! \mathcal{U}\_{\boldsymbol{\theta}}(\mathbf{x}+\mathbf{b})^{m-1}} \Big[\mathcal{F} + \boldsymbol{\xi} \mathcal{F}' \frac{m-1}{m+1}\Big], \end{aligned} \tag{7}$$

the incompressibility condition is consequently fulfilled and Equations (2), (3), and (5) are lessened to the following:

$$\left(\frac{1}{(1-q)^{2.5}\Big(1-q+q\frac{\rho\_{\text{CNOT}}}{\rho\_f}\Big)}\right)F'' + FF' - \frac{2m}{m+1}(F')^2 + \frac{2m}{m+1}A^2 - M^2(F'-A) = 0,\tag{8}$$

$$\left(\frac{k\_{nf}/k\_f}{\left(1-\wp+\wp\frac{\left(\rho c\_p\right)\_{\text{CNI}}}{\left(\rho c\_p\right)\_f}\right)}\right)\left(1+\frac{4}{3k\_{nf}/k\_f}R\_d\right)\Theta''+\text{PrF}\Theta'+\text{PrEc}\left(F''\right)^2=0,\tag{9}$$

$$\begin{aligned} F'(a) &= 1, \; F(a) = a \frac{1 - m}{1 + m}, \; \Theta(a) = 0, \; \text{at } a = B \sqrt{\frac{m + 1}{2} \frac{lL\_0}{\nu\_f}}, \\\ F'(\infty) &\to A, \; \Theta(\infty) \to 1 \quad a \to \infty, \end{aligned} \tag{10}$$

where α = *B m*+<sup>1</sup> 2 *U*0 <sup>ν</sup>*f* is the wall thickness parameter. Putting

$$F(\xi) = f(\eta) = f(\xi - a).$$

The final equations in one variable form are given here:

$$\left(\frac{1}{\left(1-\wp\right)^{2S}\left(1-\wp+\wp \frac{\text{CNOT}}{\rho\_f}\right)}\right)f''' + ff' - \frac{2m}{m+1}(f')^2 + \frac{2m}{m+1}A^2 - M^2(f'(\eta)-A) = 0,\tag{11}$$

$$\left(\frac{k\_{nf}/k\_f}{\left(1-\rho+\varrho\frac{\left(\rho\_r\right)\_{\text{CAT}}}{\left(\rho\_r\right)\_f}\right)}\right)\left(1+\frac{4}{3k\_{nf}/k\_f}R\_d\right)\theta''+\text{Pr}f\theta'+\text{PrEc}(f'')^2=0,\tag{12}$$

$$f'(0)=1,\ f(0)=a\frac{1-m}{1+m'},\ \theta(0)=0,\ \text{at }\eta=0,\tag{13}$$

$$\begin{array}{ccccc} \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ\\ f'(\infty) \to A, \; \theta(\infty) \to 1 & \eta \to \infty, & & & \end{array} \tag{13}$$

where *A* is defined as ratio parameter, *M* is defined as magnetic parameter, *Rd* is defined as radiation estimator, Pr is defined as Prandtl number, and *Ec* is defined as the well-known Eckert number. Mathematically,

$$A = \frac{\mathcal{U}\_{\rm co}}{\mathcal{U}\_0}, \; M = \sqrt{\frac{\sigma B\_0^2}{\rho c\_p}}, \; R\_d = \frac{4 \sigma^\* T^3}{k\_f K^\*}, \; \text{Pr} = \frac{\mu c\_p}{k}, \; Ec = \frac{\mathcal{U}\_w^2}{c\_p (T\_w - T\_\infty)}.$$

The expression of friction coefficient (wall-drag) and local Nusselt (the heat transfer) number are

 $C\_f = \frac{\pi\_w}{\rho f L\_w^2}$   $\mathcal{N}u\_x = \frac{(x+b)q\_w}{k\_f(T\_{\rm av} - T\_w)}$   $\tau\_w = u\_{\rm nf} \left(\frac{\partial \mathcal{U}}{\partial y}\right)\_{y = B(x+b)}$   $\frac{1-\mathfrak{u}}{2}$   $\mathcal{V}\_{\rm av} = -\kappa\_{\rm nf} \left(\frac{\partial T}{\partial y}\right)\_{y = B(x+b)}$   $\frac{1-\mathfrak{u}}{2}$ 

The dimensionless forms of the above parameters are

$$\mathcal{C}\_f \text{Re}\_x^{\frac{1}{2}} = \frac{1}{\left(1 - \varphi\right)^{2.5}} \sqrt{\frac{1}{2}(m+1)} f'''(0), \text{ } Nu\_x \text{Re}^{\frac{1}{2}} = -\frac{k\_{nf}}{k\_f} \sqrt{\frac{1}{2}(m+1)} \theta'(0).$$

where Re*x* = *Uw*(*x*+*b*) <sup>ν</sup>*f* denotes Reynolds number.

#### **3. Mathematical Analysis**

#### *3.1. OHAM (BVPh 2.0)*

The governing problems are explained using BVPh 2.0, via the homotopy analysis method (HAM)-based Mathematica package. The BVPh 2.0 is simple to utilize. It simply needs to compose the required governing problems. For each governing equation, we select the proper auxiliary linear operators and accurate initial guess for every undetermined function. Expression of linear operators and initial guesses are

$$f\_0(\eta) = A\eta + (1 - A)(1 - \exp(-\eta)) - a \frac{m - 1}{m + 1},\tag{14}$$

$$\theta\alpha(\eta) = 1 - \exp(-\eta),\tag{15}$$

$$L\_f(f) = \frac{d^3f}{d\eta^3} - \frac{df}{d\eta'}, \\ L\_0(\theta) = \frac{d^2\theta}{d\eta^2} - \theta,\tag{16}$$

with

$$L\_f[E\_1 + E\_2 \exp(\eta) + E\_3 \exp(-\eta)] = 0,\tag{17}$$

$$L\_0[E\_4 \exp(\eta) + E\_5 \exp(-\eta)] = 0,\tag{18}$$

above *Ei*(*i* = 1, 2, ... , 5) shows arbitrary constants.

#### *3.2. Optimal Convergence Analysis*

The estimations of convergence control parameters *hf* , *h*θin kerosene oil nanoliquids for both type of nanotubes, that is, SWCNTs and MWCNTs, are calculated through the boundary value problem solver package BVPh 2.0. We now continue to attain the solution of governing equations via Boundary value problem solver package BVPh 2.0. These governing equations hold two unknown convergence estimates *hf* , *<sup>h</sup>*θ. Optimal estimates of these parameters are calculated by the total minimum error. It ought to be seen that the convergence estimates assume an essential part in the frame of the homotopy analysis method (HAM), and HAM differs from other analytical techniques. To enormously diminish the Central Processing Unit (CPU) time, the average residual error at the *k*th-order of estimate is characterized by

$$\varepsilon\_{\mathcal{O}}^f(h\_{f\prime}, h\_{\mathcal{O}}) = \frac{1}{N+1} \sum\_{l=0}^N \left[ \sum\_{i=0}^N \left( f\_i \right)\_{\eta = j\pi} \right]^2 \tag{19}$$

and

$$\varepsilon\_{\varnothing}^{f}(h\_{f'}, h\_{\varnothing}) = \frac{1}{N+1} \sum\_{f=0}^{N} \left| \sum\_{i=0}^{k} \left( f\_i \right)\_{\eta = j\pi'} \sum\_{i=0}^{k} \left( \theta\_i \right)\_{\eta = j\pi} \right|^2. \tag{20}$$

The optimal estimates of *hf* and *h*θ for single-walled (SWCNT) kerosene oil are *hf* = −0.338076, *h*θ = −0.112645 and those for multi-walled (MWCNT) kerosene oil are *hf* = −0.373325, *h*θ = −0.122976. Figures 2 and 3 are drawn to see the relative total residual errors for SWCNT and MWCNT kerosene oil, respectively.

**Figure 3.** Total error vs. order of approximations.

#### **4. Results and Discussion**

The flow and heat transfer of CNTs (SWCNTs and MWCNTs) with kerosene oil as a base liquid are investigated. The governing set of nonlinear di fferential equations is numerically solved. The impact of *A* on *f* (η) is plotted in Figure 4 for kerosene oil for SWCNTs and MWCNTs. It is noticed that the velocity field rises for the increment in *A* for both *A* > 1 and *A* < 1. On the other side, for *A* > 1 and *A* < 1, the related thickness of boundary layer has a reverse trend, but for *A* = 1, no boundary layer is found. This means that the surface and ambient velocities are the same. The velocity field is dominant for single-walled tubes as compared with multi-walled tubes. Figure 5 illustrates the impact of φ on *f* (η). It is observed that velocity distribution is the mounting function for the increment in φ for SWCNTs and MWCNTs. The increment in nanomaterial volume fraction leads to rise in the convective flow. It is likewise noticed that *f* (η) enhances for kerosene oil nanoliquid for MWCNT as compared with SWCNT. The significance of magnetic parameter on the velocity field is outlined in Figure 6. The velocity *f* (η) and related boundary layer decrease for larger estimates of the magnetic estimator. The increment in *M* demonstrates the rise in resistive power (Lorentz force) and, therefore, the velocity of the liquid reduces. It is additionally noticed that velocity distribution is dominated for MWCNT as compared with SWCNT kerosene oil. Figure 7 is drawn for the behavior of *m* on *f* (η). It is analyzed that, for the higher power index, the velocity profile shows reduction for SWCNTs and MWCNTs using base liquid kerosene oil.

An analysis of *Ec* on <sup>θ</sup>(η) is portrayed in Figure 8. It is worth mentioning that <sup>θ</sup>(η) becomes higher for the increment in *Ec* for both SWCNT and MWCNT. The increment in Eckert number leads to larger drag forces between the fluid materials. Consequently, more heat is induced and the temperature distribution increases. The influence of φ on <sup>θ</sup>(η) is drawn in Figure 9. Here, temperature distribution reduces with the increase in φ. Additionally, the increment in the nanomaterial volume fraction causes the improvement of the convective heat phenomenon from heated liquid along the cold surface, and consequently, temperature reduces. It is additionally noted that the temperature distribution is dominant for MWCNT as compared with SWCNT. Figure 10 outlines the significance of the temperature in light of an adjustment in the estimations of the radiation parameter *Rd* for MWCNT and SWCNT. Obviously, the temperature distribution and related boundary layer thickness improve for higher values of the radiation estimator *Rd*. It is obvious that the surface heat flux increments under the e ffect of thermal radiation. Consequently, temperature enhances inside the boundary layer region.

Table 1 displays the thermophysical characteristics of the base liquid kerosene oil with carbon nanotubes. Table 2 is set up for the square average residual errors of governing problems at various orders of approximations. It is noticed that the square average residual error diminishes as the request of estimation rises for SWCNTs and MWCNTs using kerosene oil. Table 3 is drawn for the numerical values of the friction estimator for various estimates of di fferent related parameters. It is examined that the friction estimator is larger for higher estimates of *m*, *M*, φ, and α, while it decreases for larger *A* for both SWCNTs and MWCNTs. Table 4 is set up for the numerical estimates of local Nusselt number for di fferent values of di fferent appropriate parameters. It is examined that the local Nusselt parameter rises for larger *A*, *Rd*, Ec, and φ, while it lessens for increments in α and *M* for both single-walled and multi-walled carbon nanotubes. Table 5 exhibits the relative investigation of the skin fraction coe fficient with the past work of Pop et al. [45] and Sharma and Singh [46] in limiting cases. It is established that all the outcomes have a decent understanding.

**Figure 4.** Impact of *A* on velocity *f*(η).

**Figure 5.** Impact of φ on velocity *f*(η).

**Figure 6.** Impact of *M* on velocity *f*(η).

**Figure 7.** Impact of *m* on velocity *f*(η).

**Figure 8.** Impact of *Ec* on velocity <sup>θ</sup>(η).

**Figure 9.** Impact of φ on velocity <sup>θ</sup>(η).

**Figure 10.** Impact of *Ec* on velocity <sup>θ</sup>(η).

**Table 1.** Data of thermophysical properties for the given particles and fluid. MWCNT, multi-walled carbon nanotube; SWCNT, single-walled carbon nanotube.



**Table 2.** Mean square residual errors.




**Table 4.** Heat transfer (Nusselt number) data.

**Table 5.** Comparison with previous literature.

