Verification of Dual Solutions for Water and Kerosene-Based Carbon Nanotubes over a Moving Slender Needle

Abstract

This article focuses on the boundary layer for an axisymmetric flow and heat transfer of a nanofluid past a moving slender needle with single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). In this study, the streamlines of the flow are symmetrically located along the needle’s surface. Water and kerosene are two types of base fluids that are considered in this study. This analysis is presented with needle thickness, the ratio of velocity, nanoparticle volume fraction, and Prandtl number. The partial differential equations (PDEs) are transformed into dimensionless ordinary differential equations (ODEs) by adopting relevant similarity transformations. The bvp4c package is implemented in MATLAB R2018a to solve the governing dimensionless problems numerically. The behaviors of various sundry variables on the flow and heat transfer are observed and elaborated further. The magnitude of the skin friction, heat transfer rate, as well as velocity and temperature distributions are demonstrated in graphical form and discussed. It is worth mentioning that kerosene-based CNTs have the largest skin friction coefficient and heat transfer rate compared to water-based CNTs. The thin wall of the needle and the single-walled carbon nanotubes also contributes to high drag force and heat transfer rate on the surface. It is revealed from the stability analysis that the first solution exhibits a stable flow. Obtained results are also matched with the present data in the restricting situation, and excellent agreement is noticed.

1. Introduction

Nanofluids are defined as emulsions of nanomaterials in regular fluid, namely, oil, water, and ethylene glycol. Meanwhile, carbon nanotubes, oxides, metals, or carbides are examples of nanomaterials used in nanofluids. The novelty of nanofluids has been widely recognized as a catalyst for thermal energy transfer escalation in many industrial purposes, including automotive, biomedicine, electronic devices, and food industries. For instance, such applications are in radiators, cooling of microchips tools in cell phones and laptops, lubrication, and refrigeration (see Najib et al. [1]). Additionally, the nanofluid applications in biomedical applications, such as fluid drug transport and antibacterial and biomedical treatment, have been reviewed by Sheikhpour et al. [2]. Several studies on nanofluids that have recently been conducted are in line with findings that found that nanofluids are still relevant to the improvement of other fluids as well as effects, see [3,4,5,6,7,8,9].

Carbon nanotubes (CNTs) possess unique electronic behaviors due to great strength, lightweight, high stability, and unusual electronic structures that allow them to be metals or semiconductors relying on their chirality (i.e., conformational variation) (see Che et al. [10]). The most commonly used carbon nanotubes are single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The difference between single and multi-walled carbon nanotubes is the number of layers of graphene cylinder. For SWCNTs, the graphene cylinder is coated by a single layer, whereas MWCNTs are coated by multi-layers. The consideration of a low concentration of CNTs nanofluids has great applications in heat transfer systems [11,12]. A different approach was conducted by Aman et al. [13], who analyzed the heat transfer escalation of CNTs in Maxwell nanofluids using several kinds of molecular liquids. The impacts of magnetohydrodynamics and convective variables on flow, mass, and heat transfer containing SWCNTs and MWCNTs past a vertical cone were studied by Sreedevi et al. [14]. According to their findings, they concluded that the coefficient of the skin friction, Sherwood number, and Nusselt number augmented the increment of the convective variable. Apart from that, a numerical result on the behavior of the skin friction for both SWCNTs and MWCNTs with the slip effect at the boundary has been reported by Anuar et al. [15]. They discovered that higher density in SWCNTs caused higher skin friction compared to MWCNTs. A few studies of the slip effect on the boundary can be found in Reddy et al. [16] and Xia et al. [17]. Other important studies considering some interesting fluids and different types of surface geometry, as well as the unsteadiness state, were given in the works of [18,19,20,21,22,23,24,25].

The study on boundary layer and thermal energy transfer behavior towards a slender needle has received much attention from researchers due to its substantial practical importance. Lee [26] was the pioneer who discussed and computed the numerical outcomes on boundary layer flow over a slender needle. He discovered that the displacement thickness and drag per unit length diminished leisurely and ultimately became zero as the needle vanished. After that, many investigations of the flow and heat transfer past a slender needle in several liquids have been discussed. One of the investigations was performed by Grosan and Pop [27], where they studied the fluid flow and heat transfer characteristics when a slender needle is sunk in nanofluids. From the study, they concluded that the size or volume of nanoparticles obviously influenced the heat transfer and flow characteristics. Meanwhile, the impact of magnetohydrodynamics nanofluid flow past a moving slender needle with a frictional heating impact has been investigated using Tiwari and Das, and Buongiorno models (see Sulochana et al. [28,29]). Apart from considering the Ag-water and Ag-Kerosene nanofluids [29], Krishna et al. [30] extended the idea by analyzing the boundary layer and heat transfer behavior with the existence of Cu nanoparticles embedded in methanol. In the same year, Soid et al. [31] investigated the flow and heat transfer on a moving slender needle by considering Cu nanoparticles immersed in the base fluid. They also determined the stability of the solutions acquired using the stability valuation. The studies on moving thin needles in nanofluids in various effects, such as when the needle is heated and in the presence of a slippery wall, were conducted by Salleh et al. [32,33]. They found that the local wall heat flux expedites with the increment of the convective parameter and slip parameter. Recently, such research on boundary layer flow towards a slender or thin needle in hybrid nanofluid has been extended to investigating the flow and heat behavior when the slender needle is embedded in hybrid nanofluid (see Aladdin et al. [34] and Mousavi et al. [35]).

As mentioned above, there are three elements that we are trying to combine in this present study which is the slender needle coated by carbon nanotubes immersed in a nanofluid filled with water and kerosene. Generally, we can say that the investigation is about the behavior of the flow and heat transmission of fluid in a slender needle. The slender needle is known as a surface or geometry that is used in this study where the radius or thickness of the needle will have an important impact on the behavior of the fluid flow and heat transfer. Further, the needle itself is coated with carbon nanotubes which exhibit extraordinary mechanical properties such as high thermal capacity and are also good conductors of heat and electrical conductivity. The good performance of carbon nanotubes will become excellent when there is a better fluid flow through them. Hence, nanofluid (water and kerosene) is used because of its excellent heat transfer performance.

In summary, the present study investigates the behavior of skin friction and heat transmission rate of moving slender needles coated by SWCNTs and MWCNTs filled with water and kerosene. Although many studies regarding slender needles coated by CNTs have been carried out, we found there is a significant gap in terms of research findings. We noticed that the study of CNTs in water and kerosene had not been done yet; thus, we conducted this present study to create measuring data for the other extension research. Due to the consideration of the moving needle (opposing and assisting flow), we are expecting the solutions to be obtained in dual. In possession of dual, the stability analysis is performed to verify and prove that the first solution is stable and can be physically realized. Some past studies on stability analysis using several surfaces can be seen in [36,37,38,39,40,41].

2. Problem Description

2.1. Governing Equations

The boundary layer for an axisymmetric flow of a nanofluid past a moving slender needle using water and kerosene-based fluids is considered. The axisymmetric flow is a two-dimensional flow with a line of symmetry along the needle surface or a flow in which the streamlines are symmetrically located around an axis. Figure 1 shows a schematic diagram of the current study.

It is assumed that the needle wall is fixed at a constant temperature $T_n$ where $T_n$ is greater than the free stream temperature $T_{\infty }.$ $x$ and $r$ are the axial and radial coordinates, respectively, in which $r={({\nu }_{f}ex/U)}^{1/2}=R(x)$ is denoted as the needle radius. The needle surface is considered to move with a uniform velocity $U_n$ in a similar and opposite direction to the fluid motion of a constant velocity $U_{\infty }.$ Following the assumptions, the system of equations for this problem is [31]

$${(ru)}_x+{(rv)}_r=0,$$

(1)

$$u{u}_x+v{u}_r={\left(\frac{\mu }{\rho }\right)}_{nf}{r}^{-1}{(r{u}_r)}_r,$$

(2)

$$u{T}_x+v{T}_r={\alpha }_{nf}{r}^{-1}{(r{T}_r)}_r,$$

(3)

and the apparent boundary conditions are

$$\begin{array}{c}u=U_n, v=0, T=T_n \mathrm{at} r=R(x),\\ u\to U_{\infty }, T\to T_{\infty } \mathrm{as} r\to \infty .\end{array}$$

(4)

Here, $u$ and $v$ are the velocity components along the coordinates $x$ and $r,$ respectively, $\mu$ is the dynamic viscosity, $\rho$ is the density, $\alpha$ is the thermal diffusivity, and $T$ is the fluid temperature. Terms with subscript ‘$nf$’ are referred to as the nanofluid term. Mathematically, the applied relation of nanofluid [42] can be specified as:

$$\begin{array}{cc}Dynamic\; Viscosity:\hfill & {\mu }_{nf}=\frac{{\mu }_f}{{(1-\phi )}^{2.5}},\end{array}$$

(5)

$$\begin{array}{cc}Density:\hfill & {\rho }_{nf}=(1-\phi ){\rho }_f+\phi {\rho }_{CNT},\end{array}$$

(6)

$$\begin{array}{cc}Thermal\; Diffusivity:\hfill & {\alpha }_{nf}=\frac{k_{nf}}{{(\rho C_p)}_{nf}},\end{array}$$

(7)

$$\begin{array}{cc}Thermal\; Conductivity:\hfill & \frac{k_{nf}}{k_f}=\frac{1-\phi +2\phi \frac{k_{CNT}}{k_{CNT}-k_f}\ln \frac{k_{CNT}+k_f}{2k_f}}{1-\phi +2\phi \frac{k_f}{k_{CNT}-k_f}\ln \frac{k_{CNT}+k_f}{2k_f}},\end{array}$$

(8)

$$\begin{array}{cc}Specific\; Heat\; Capacitance:\hfill & {(\rho C_p)}_{nf}=(1-\phi ){(\rho C_p)}_f+\phi {(\rho C_p)}_{CNT},\end{array}$$

(9)

where $\phi$ is the nanoparticle volume fraction parameter, and the subscript ‘$f$’ and ‘$CNT$’ are referred to as the fluid and carbon nanotubes, respectively. The thermophysical properties of CNTs used are based on the work of Hayat et al. [43].

In order to simplify the system of Equations (1)–(4), the following dimensionless quantities are introduced [31]:

$$\psi \left(x,r\right)={\nu }_{f}xf\left(\xi \right),\hspace{1em}\xi =\frac{U{r}^2}{{\nu }_{f}x},\hspace{1em}\theta \left(\xi \right)=\frac{T-T_{\infty }}{T_n-T_{\infty }}.$$

(10)

Here, $\psi$ is an axisymmetric stream function that defined $u$ and $v$ as follows:

$$u=r^{-1}\frac{\partial \psi }{\partial r}\hspace{1em}\mathrm{and}\hspace{1em}v=-r^{-1}\frac{\partial \psi }{\partial x}$$

(11)

and ${\nu }_f$ is the fluid kinematic viscosity. Further, $U=U_{\infty }+U_n$ is the composite velocity between the free stream and needle, $\xi$ is the similarity variable, and $\theta (\xi )$ is the dimensionless temperature. Then, substituting Equation (10) and making use of Equation (11) into Equations (1)–(4) leads to the nondimensional form of momentum and energy equations below:

$$\frac{2}{A}\left(\xi f^{‴}+f^{″}\right)+f{f}^{″}=0,$$

(12)

$$\frac{2}{\mathrm{Pr}}\frac{B}{C}\left(\xi {\theta }^{″}+{\theta }^{\prime }\right)+f{\theta }^{\prime }=0,$$

(13)

and the boundary restrictions are

$$\begin{array}{c}f\left(e\right)=\frac{\epsilon }{2}e, f^{\prime }\left(e\right)=\frac{\epsilon }{2}, \theta \left(e\right)=1,\\ f^{\prime }\left(\xi \right)\to \frac{1-\epsilon }{2}, \theta \left(\xi \right)\to 0 \mathrm{as} \xi \to \infty .\end{array}$$

(14)

The prime is referred to the derivatives with respect to $\xi .$ From Equations (12) and (13), the arbitrary constants are given by

$$A={\left(1-\phi \right)}^{2.5}\left[1-\phi +\phi \frac{{\rho }_{CNT}}{{\rho }_f}\right],\hspace{1em}B=\frac{k_{nf}}{k_f},\hspace{1em}C=1-\phi +\phi \frac{{\left(\rho C_p\right)}_{CNT}}{{\left(\rho C_p\right)}_f}.$$

(15)

Other significant parameters involved are Prandtl number $\mathrm{Pr}$, needle thickness $e,$ and the velocity ratio parameter between the flow and the needle $\epsilon ,$ which can be written, respectively, as:

$$\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f},\hspace{1em}\xi =e,\hspace{1em}\epsilon =\frac{U_n}{U},$$

(16)

where $\epsilon >0$ denotes the needle moves parallel to the free stream direction, and $\epsilon <0$ denotes the needle moves against the free stream direction. The physical quantities involved, namely, skin friction coefficient and heat transfer rate (local Nusselt number) are given as:

$$C_f=\frac{{\tau }_n}{{\rho }_f{U}^2},\hspace{1em}{\mathrm{Nu}}_x=\frac{x{q}_n}{k_f\left(T_n-T_{\infty }\right)},$$

(17)

where ${\tau }_n$ is the wall shear stress, and $q_n$ is the wall heat flux that can be defined as follows:

$${\tau }_n={\mu }_{nf}{\left(\frac{\partial u}{\partial r}\right)}_{r=e},\hspace{1em}{q}_n=-k_{nf}{\left(\frac{\partial T}{\partial r}\right)}_{r=e}.$$

(18)

Plugging Equations (5) and (11) into Equation (18) and using Equation (17), one can obtain the following:

$${\mathrm{Re}}_x^{1/2}{C}_f=\frac{4}{{\left(1-\phi \right)}^{2.5}}{e}^{1/2}{f}^{″}\left(e\right),\hspace{1em}{\mathrm{Re}}_x^{-1/2}{\mathrm{Nu}}_x=-2\frac{k_{nf}}{k_f}{e}^{1/2}{\theta }^{\prime }\left(e\right),$$

(19)

where ${\mathrm{Re}}_x=U_x/{\nu }_f$ is the local Reynolds number.

2.2. Stability Analysis

There exist multiple branches of the solution, namely, dual solutions against a single variable in this study. It is important to know the physical reliability of these solutions where they can be applied for industrial and engineering purposes. Thus, the stability analysis of solutions is performed in this research. To carry out this analysis, Equations (2) and (3) must be considered in an unsteady-state flow by proposing the dimensionless time parameter $\tau =2Ut/x.$ Therefore, the unsteady-state equations are given by [44,45]

$$u_t+u{u}_x+v{u}_r={\left(\frac{\mu }{\rho }\right)}_{nf}{r}^{-1}{(r{u}_r)}_r,$$

(20)

$$T_t+u{T}_x+v{T}_r={\left(\frac{k}{\rho C_p}\right)}_{nf}.$$

(21)

The new similarity variables are introduced as follows:

$$\psi \left(x,r\right)={\nu }_{f}xf\left(\xi ,\tau \right), \theta \left(\xi ,\tau \right)=\frac{T-T_{\infty }}{T_n-T_{\infty }},\hspace{1em}\xi =\frac{U{r}^2}{{\nu }_{f}x},\hspace{1em}\tau =\frac{2Ut}{x}.$$

(22)

Using Equation (22) into Equations (20) and (21), one can obtain

$$\frac{2\eta }{A}\frac{{\partial }^{3}f}{\partial {\xi }^3}+\frac{2}{A}\frac{{\partial }^{2}f}{\partial {\xi }^2}+f\frac{{\partial }^{2}f}{\partial {\xi }^2}-\tau \frac{{\partial }^{2}f}{\partial {\xi }^2}\frac{\partial f}{\partial \tau }+\tau \frac{\partial f}{\partial \xi }\frac{{\partial }^{2}f}{\partial \xi \partial \tau }-\frac{{\partial }^{2}f}{\partial \xi \partial \tau }=0,$$

(23)

$$\frac{2\eta B}{\mathrm{Pr}C}\frac{{\partial }^2\theta }{\partial {\xi }^2}+\frac{2B}{\mathrm{Pr}C}\frac{\partial \theta }{\partial \xi }+f\frac{\partial \theta }{\partial \xi }-\tau \frac{\partial f}{\partial \tau }\frac{\partial \theta }{\partial \xi }+\tau \frac{\partial f}{\partial \xi }\frac{\partial \theta }{\partial \tau }-\frac{\partial \theta }{\partial \tau }=0,$$

(24)

together with the conditions below:

$$\begin{array}{c}f\left(e,\tau \right)=\tau \frac{\partial f}{\partial \tau }\left(e,\tau \right)+\frac{\epsilon e}{2},\hspace{1em}\frac{\partial f}{\partial \xi }\left(e,\tau \right)=\frac{\epsilon }{2},\hspace{1em}\theta \left(e,\tau \right)=1,\\ \frac{\partial f}{\partial \eta }(\xi ,\tau )\to \frac{1-\epsilon }{2},\hspace{1em}\theta (\xi ,\tau )\to 0,\hspace{1em}\mathrm{as}\hspace{1em}\xi \to \infty .\end{array}$$

(25)

As suggested by Merkin [44] and Weidman et al. [45], to test the stability of the steady flow solutions $f=f_0(\xi )$ and $\theta ={\theta }_0(\xi )$ that satisfying the boundary value problems (20) and (21), the following equations must be introduced:

$$f(\xi ,\tau )=f_0(\xi )+e^{-\gamma \tau }F(\xi ,\tau ),\hspace{1em}\theta (\xi ,\tau )={\theta }_0(\xi )+e^{-\gamma \tau }H(\xi ,\tau ).$$

(26)

Here functions $F(\xi ,\tau )$ and $H(\xi ,\tau )$ are small relative to $f_0(\xi )$ and ${\theta }_0(\xi ),$ respectively, and $\gamma$ is the eigenvalue parameter.

Using Equations (22) and (26) into Equations (23)–(25), the final equations are yielded as follows:

$$\frac{2\eta }{A}{F}_0{}^{‴}+\frac{2}{A}{F}_0{}^{″}+f_0{F}_0{}^{″}+f_0{}^{″}{F}_0+\gamma F_0{}^{\prime }=0,$$

(27)

$$\frac{2\eta B}{\mathrm{Pr}C}{H}_0{}^{″}+\frac{2B}{\mathrm{Pr}C}{H}_0{}^{\prime }+f_0{H}_0{}^{\prime }+{\theta }_0{}^{\prime }{F}_0+\gamma H_0=0,$$

(28)

and the boundary conditions became

$$\begin{array}{c}{F}_0(e)=0, F_0{}^{\prime }(e)=0, H_0(e)=0,\\ F_0{}^{\prime }(\xi )\to 0, H_0(\xi )\to 0 \mathrm{as} \xi \to \infty .\end{array}$$

(29)

To further continue, we assume $\tau =0$ indicated an initial growth or decay of the solution (26). Hence, $F(\xi ,\tau )$ and $H(\xi ,\tau )$ can be rewritten as $F_0(\xi )$ and $H_0(\xi ),$ respectively. As described in the published article by Harris et al. [46], the boundary conditions can be relaxed on $F_0{}^{\prime }(\xi )$ or $H_0(\xi )$ to find the smallest eigenvalue $\gamma .$ In this analysis, the condition $F_0{}^{\prime }(\xi )\to 0$ as $\xi \to \infty$ is selected for relaxation and Equations (27)–(29) are computed along the new condition $F_0{}^{″}(e)=1.$

2.3. Method of Solutions

There are two methods for solving the modeling of the system in the form of ODEs and PDEs, which are an analytical method and a numerical method. In this research, the best way to solve our model, which represents a problem in real life, is to use the numerical approach. Many systems possess complex functionality, and it is difficult to track system behavior by formulas. Thus, numerical methods are applied to solve the equations. In the numerical method, a computer must be used to perform thousands of repetitive calculations to give the solution. Although the analytical method gives us more exact solutions than the numerical method, it works only for simple models and is very limited in finding solutions compared to numerical ones. Due to these reasons, many researchers nowadays have considered numerical methods more than analytical methods.

The significance of using the numerical method for this kind of flow problem is that it is very helpful in finding solutions for the proposed model. Although it gives approximate solutions, the behavior of the physical quantities of interest, such as skin friction coefficient and heat transfer rate obtained, is similar to the analytical and experimental studies.

In this research, the use of the bvp4c solver via MATLAB R2018a is very helpful in finding the numerical outcomes for Equations (12)–(14). The first procedure to apply this package is to determine the initial guess at the starting mesh point. Next, the step size must be altered to give suitable accuracy. In spite of that, it is essential to reduce the differential equations to the first-order system of equations. Hence, the following variables are proposed to reduce the equations:

$$y(1)=f(\xi ), y(2)=f^{\prime }(\xi ), y(3)=f^{″}(\xi ), y(4)=\theta (\xi ), y(5)={\theta }^{\prime }(\xi ),$$

(30)

where

$$y^{\prime }(1)=y(2), y^{\prime }(2)=y(3), y^{\prime }(4)=y(5).$$

(31)

Using Equations (30) and (31), Equations (12) and (13) can be expressed as:

$$f^{‴}=-\frac{1}{\xi }\left[y(3)+\frac{A}{2}y(1)y(3)\right],$$

(32)

$${\theta }^{″}=-\frac{1}{\xi }\left[y(5)+\frac{\mathrm{Pr}C}{2B}y(1)y(5)\right],$$

(33)

and the boundary conditions take the following form:

$$ya(1)-\frac{\epsilon e}{2}=0, ya(2)-\frac{\epsilon }{2}=0, ya(4)-1=0,$$

$$yb(2)-\frac{1-\epsilon }{2}=0, yb(4)=0,$$

(34)

in which $a$ and $b$ refer to the conditions on the wall $\xi =0,$ and far field $\xi ={\xi }_{\infty },$ respectively. These boundary conditions have to be considered in the general form to implement the BVP in MATLAB software. In this analysis, an appropriate finite value of ${\xi }_{\infty }$ is taken as ${\xi }_{\infty }=60$ which depends on the values of the considered variables. The value of the Prandtl number, needle thickness, velocity ratio, and nanoparticle volume fraction parameters are varied throughout the entire study. The numerical outcomes can be obtained by assuming the inputs of unfixed variables that satisfied the condition $f^{\prime }(\xi )\to (1-\epsilon )/2,\hspace{1em}\theta (\xi )\to 0$ as $\xi \to \infty .$ The process must be reiterated until the converged outcome meets a tolerance limit of ${10}^{-6}.$ To perform the stability valuation, Equations (27) and (28) with the boundary condition (29) must be applied in the bvp4c function. If the result shows a positive value for the eigenvalue, this will indicate a stable solution (or flow). In contrast, if the result shows a negative value for the eigenvalue, it will imply that the solution yielded is unstable.

In the bvp4c solver, there are four major codes involved in computing the results. The first code (code a), also known as the initial guess code, is used to find the initial guess for the first and second solutions. In this regard, two sets of initial guesses are needed to estimate the initial solutions for both the first and second solutions. Second is the continuation code (code b), where the same ODEs and boundary conditions as in the first code are used to compute the numerical solution of the problem. This code is used to find the other solutions which are near to the initial guess in the first code. Meanwhile, the third code (first solution code or code c) and the fourth code (second solution code or code d) are used to identify the minimum eigenvalues for the first and second solutions, respectively.

The summary of the steps for solving this problem is provided in Figure 2 and Figure 3 for both numerical solutions and eigenvalues.

2.4. Code of Validation

Before computing the numerical results, the code in the bvp4c function was tested by comparing the values of the shear stress $f^{″}(e)$ for certain values of needle thickness with the previous studies.

Table 1. Comparison values of the shear stress for several values of $e$ when $\mathrm{Pr}=1,\epsilon =-1,$ and $\phi =0$ (regular fluid).

$\mathbf{Needle} \mathbf{Size}, \mathit{e}$	Soid et al. [31]	Current Result
0.01	26.599394	26.599394
0.1	3.703713	3.703713
0.2	2.005424	2.005424

displays the comparison result between the current study with those discussed by Soid et al. [31]. As can be noticed, the values of $f^{″}(e)$ are in outstanding agreement with the existing study. Therefore, we are confident that our numerical procedure is correct.

According to previously published work by Narain and Uberoi [47], the maximum size of the needle is determined by the ratio of ${\delta }^{*}/e\ge 1$ where ${\delta }^{*}$ is the displacement thickness and $e$ is the needle size. In their work, they noticed that for a needle size $e={10}^{-1}$, this ratio is around 6. Thus, they restricted the needle thickness below ${10}^{-1}.$ Meanwhile, the value of $e=0.2$ is taken from Ishak et al. [48]. In their study, they want to show what will happens to the numerical outcomes if the needle size is larger than that considered by Narain and Uberoi [47].

3. Discussion of Results

This section presents the surface shear stress $f^{″}(e),$ local heat flux $-{\theta }^{\prime }(e),$ skin friction coefficient ${({\mathrm{Re}}_x)}^{1/2}{C}_f$, heat transfer rate ${({\mathrm{Re}}_x)}^{-1/2}{\mathrm{Nu}}_x$, velocity $f^{\prime }(\xi )$, and temperature $\theta (\xi )$ profiles, as well as the stability of the solutions gained. For this purpose, we have prepared Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 and

Table 2. Numerical values of the skin friction coefficient ${({\mathrm{Re}}_x)}^{1/2}{C}_f$ for several values of $\phi ,e$ and Pr.

Types of CNTs	$\mathit{\phi }$	$\mathbf{Water} (\mathbf{Pr}=6.2)$		$\mathbf{Kerosene} (\mathbf{Pr}=21)$
		$\mathit{e}=0.1$	$\mathit{e}=0.2$	$\mathit{e}=0.1$	$\mathit{e}=0.2$
SWCNT	0	0.981533	0.819587	0.981533	0.819587
	0.004	0.990611	0.827080	0.991242	0.827676
	0.008	0.999795	0.834656	1.001061	0.835851
	0.012	1.009089	0.842320	1.010989	0.844116
MWCNT	0	0.981533	0.819587	0.981533	0.819587
	0.004	0.989719	0.826238	0.990109	0.826606
	0.008	0.998008	0.832968	0.998790	0.833707
	0.012	1.006397	0.839780	1.007576	0.840893

,

Table 3. Numerical values of the local Nusselt number ${({\mathrm{Re}}_x)}^{-1/2}{\mathrm{Nu}}_x$ for several values of $\phi ,e$ and Pr.

Types of CNTs	$\mathit{\phi }$	$\mathbf{Water} (\mathbf{Pr}=6.2)$		$\mathbf{Kerosene} (\mathbf{Pr}=21)$
		$\mathit{e}=0.1$	$\mathit{e}=0.2$	$\mathit{e}=0.1$	$\mathit{e}=0.2$
SWCNTs	0	2.526693	2.241342	3.731503	3.530355
	0.004	2.749760	2.445966	4.147080	3.942872
	0.008	2.979034	2.656954	4.579506	4.374513
	0.012	3.214483	2.874291	5.028795	4.825371
MWCNTs	0	2.526693	2.241342	3.731503	3.530355
	0.004	2.728731	2.426610	4.113931	3.909935
	0.008	2.935992	2.617217	4.510960	4.306061
	0.012	3.148460	2.813160	4.922633	4.718850

and

Table 4. Minimum eigenvalues for several values of needle thickness $e$ and velocity ratio parameter $\epsilon$.

$\mathit{e}$	$\mathit{\epsilon }$	First Solution	Second Solution
	−4.2598	0.0168	−0.0165
0.10	−4.259	0.0215	−0.0211
	−4.25	0.0507	−0.0482
	−3.2969	0.0064	−0.0064
0.15	−3.296	0.0151	−0.0148
	−3.29	0.0387	−0.0369
	−2.7738	0.0077	−0.0077
0.20	−2.773	0.0147	−0.0144
	−2.77	0.0286	−0.0274

. All calculations in this paper have been computed for an extensive range of values of controlling variables;$e (0.1\le e\le 0.2),$ $\phi (0\le \phi \le 0.02),$$\epsilon (-4.4<\epsilon \le 1.0),$$\mathrm{Pr}=6.2$ (water), and $\mathrm{Pr}=21$ (kerosene).

The influence of the needle thickness on the variation of the shear stress for both Water-SWCNTs and MWCNTs versus the velocity ratio parameter is examined in Figure 4a,b. Meanwhile, Figure 5a,b demonstrates the velocity fields for both Water-SWCNTs and MWCNTs with several values of $e$ when $\epsilon =-2.8.$ It is observed from Figure 4 that an increase in the value of $e$ diminishes the surface shear stress. This behavior occurs due to the enhancement of the momentum boundary layer thickness on the wall as the value of $e$ increases (see Figure 5). Other than that, the range of the solution decreases for thicker needles. It is obviously seen that the dual solutions exist when the direction of the fluid flow opposes the direction of the needle $(\epsilon <0).$ Nevertheless, the dual solutions appear only in the range of ${\epsilon }_c<\epsilon \le -0.8.$ The range of the solution for Water-SWCNTs is quite smaller compared to Water-MWCNTs. This implies that the single-walled carbon nanotubes accelerate the boundary layer separation in the flow. This criterion is proven by critical points ${\epsilon }_c$ that unify both first and second solutions. It is noted in Figure 5 that the velocity distributions decrease for the thicker needles. This decrement is affected by an increase in the momentum boundary layer thickness at the needle surface.

Figure 6a,b shows the effect of the needle thickness on the local heat flux for both Water-SWCNTs and MWCNTs versus the velocity ratio parameter, while Figure 7a,b presents the temperature fields for both Water-SWCNTs and MWCNTs with several values of $e$ when $\epsilon =-2.8.$ The local heat flux reduces with larger values of $e.$ In fact, the enhancement in the thermal boundary layer thickness as the needle thickness increase prevents heat from diffusing from the needle wall to the surrounding fluid (see Figure 7). Furthermore, an increase in the thermal boundary layer thickness, as in Figure 7, also enhances the temperature distributions in the flow. This situation leads to the reduction of the local heat flux at the needle wall.

Figure 8a,b shows the variation of the surface shear stress and the local heat flux versus $\epsilon$ for some values of nanoparticle volume fraction variable $\phi$ when $\mathrm{Pr}=21$ (kerosene) and $e=0.15$ for SWCNTs. Meanwhile, Figure 9 presents the velocity and temperature fields for some values of $\phi$ when $\epsilon =-2.0.$ The higher value of $\phi$ enhances the shear stress at the needle wall. The presence of more suspended nanoparticles in the flow causes a stronger collision between the base fluid particles and nanoparticles, which leads to the occurrence of drag forces on the surface. Thus, it increases the shear stress at the needle surface. In the meantime, the higher values of $\phi$ increase the local heat flux for $\epsilon >0$, which denotes that the needle moves parallel to the free stream direction. The contrary trend is observed for the local heat flux in the region of $\epsilon <0$ (needle moves facing the free stream flow) as $\phi$ increases. Interestingly, dual solutions are noticed to exist in the region of $\epsilon <0$. In Figure 8a, the range of the solutions exist enlarges with larger values of $\phi .$ Meanwhile, the range of the solutions in Figure 8b gets smaller with larger $\phi .$ In Figure 9a, the velocity distributions reduce with greater nanoparticle volume fraction as influenced by thickening in the momentum boundary layer thickness. Additionally, Figure 7b depicts that the temperature distribution increases for low boundary layer thickness, while it decreases for high boundary layer thickness as the parameter $\phi$ enhances.

The numerical values of the skin friction coefficient and the heat transfer rate are given in Table 2 and Table 3, respectively, for various values of $\phi ,e,$ and $\mathrm{Pr}$ for both SWCNTs and MWCNTs. It is seen in Table 2 that the coefficient of skin friction increases with larger values of the nanoparticle volume fraction parameter. As we discussed before, the inclusion of nanoparticles in the base fluid will create a larger drag force due to the collision between the suspended particles in the flow. Hence, it will enhance the skin friction coefficient on the wall. Additionally, the thinner wall of the needle gives the highest value of the skin friction coefficient compared to a thicker wall. Other than that, a higher skin friction coefficient occurs when the SWCNTs are considered. In fact, the thinner surface of the needle and the thinner wall of carbon nanotubes provide large space for suspended particles to collide, which leads to a higher skin friction coefficient on the wall. Additionally, kerosene-based CNTs $(\mathrm{Pr}=21)$ show better performance on the skin friction coefficient than water-based CNTs. The fact that the density of kerosene is less than that of water makes the nanoparticles float and bump into each other and, consequently, increases skin friction on the needle wall. It is revealed in Table 3 that the heat transfer rate ascends when more nanoparticles are imposed in the flow. More nanoparticles in the flow accelerate the continuous collision between the suspended nanoparticles, which assists the heat transfer swiftly from the wall to the fluid flow. Additionally, the heat transfer rate is higher for the thinner wall of the needle and the thinner wall of carbon nanotubes (SWCNTs). Physically, heat is more effortlessly transmitted via a thin wall than a thick one. It is noticed that kerosene-based CNTs offer a greater heat transfer rate than water-based CNTs. The lower density of kerosene helps reduce the thermal boundary layer thickness, and at the same time facilitates the heat transfer process from the wall to the fluid.

Figure 10 demonstrates the minimum eigenvalues $\gamma$ with different values of $\epsilon .$ It is found from the figure that a positive value of $\gamma$ is obtained for the first solution, which indicates the solution is stable. In contrast, a negative value of $\gamma$ is obtained for the second solution, which implies the solution is unstable. In the stable flow, there is an initial decay of disturbances (see Weidman et al. [45]). Meanwhile, in the unstable flow, there is an initial growth of disturbances. In Figure 10, $\gamma =-2.7741$ is a transition point from the unstable flow to the stable flow. Table 4 presents the values of $\gamma$ for different values of $e$ and $\epsilon .$ Same as Figure 10, in this table, the first solution also gives the positive eigenvalues, while the second solution gives the negative eigenvalues.

4. Concluding Remarks

The boundary layer for an axisymmetric flow of CNTs along a moving horizontal slender needle has been explored for both kerosene-based CNTs and water-based CNTs. Kerosene-based CNTs show better performance on the skin friction coefficient and heat transfer rate compared to water-based CNTs. The single-walled carbon nanotubes and the thin wall of the needle give the highest value of skin friction coefficient and heat transfer rate. The presence of more suspended nanoparticles in the flow increases the skin friction coefficient as well as the heat transfer rate at the needle’s surface. The outcomes show that two solutions were obtained in the opposing flow region; meanwhile, the solutions were discontinued when $\epsilon <{\epsilon }_c.$ It is confirmed from the stability analysis that the first solution is stable; meanwhile, the second solution is unstable. Based on the results obtained, the significance of using both the thin wall of the needle and single-walled carbon nanotubes in kerosene-based CNTs is capable of enhancing the drag forces that occur between the needle surface and the fluid particles. This phenomenon is suitable for some applications that require high frictional force because it helps to reduce the speed of the moving surface. In addition, the advantage of using the current model is that it provides a high heat transfer rate where the surface will cool down quickly. This behavior is appropriate for the cooling processes.