Comparative analysis on single- and multi-walled carbon nanotubes suspended for enhanced thermal energy transfer past stretching surfaces

Bhargavi, N.; Sreenivasulu, P.; Poornima, T.

doi:10.1063/5.0232829

Flat and parabolic trough solar collectors concentrate sunlight onto a receiver tube containing the heat transfer liquid. Particularly, CNT nanofluids enhance the efficacy of energy harvesting in these systems. Flat plate collectors are simple in design and cheaper than parabolic shape collectors. Based on this idea, the present investigation focuses on how energy transmission helps harvest solar energy. Thus, three-dimensional, electrically conducting carbon nanotubes suspended in engine oil formed nanofluid flowing past a stretching porous surface with thermal slip are investigated numerically. Through the utilization of similarity transformations, the governing nonlinear partial differential equations are converted into a set of coupled ordinary differential equations. After that, the shooting approach is applied to these equations together with the fourth-order Runge–Kutta method to solve them. The study investigates the influence of various flow parameters on the velocity, temperature, skin friction coefficients, and Nusselt number near the wall. A comparative study of single- and multi-walled carbon nanotubes is made. From the theoretical calculations, the momentum of flow is higher, and temperature is lower for multi-walled carbon nanotubes than single-walled ones. The heat transfer rate enhances with an increase in Pr and S, and it is opposite when R, St, and δ increases. This study shows that the energy transmission rate is better in multi-walled carbon nanotubes, which act as better cooling liquid.

NOMENCLATURE

Dimensionless parameters

C_fx: Coefficient of skin friction in the x-direction
C_fy: Coefficient of skin friction in the y-direction
G: Porous permeability
M: Magnetic parameter
Nu_x: Local Nusselt number
p′: Velocity in the x-direction
Pr: Prandtl’s number
q′: Velocity in the y-direction
R: Thermal radiation parameter
Re: Local Reynold’s number
S: Stretching parameter
S_f: Velocity slip parameter
S_t: Thermal slip parameter
η: Dimensionless variable
θ: Temperature
ω: Rotation parameter

Subscripts

f: Base fluid

Abbreviations

SWCNTs: Single-walled carbon nanotubes

Dimensional parameters

A: Thermal diffusivity (m² s⁻¹)
Cp: Specific heat (J kg⁻¹ K)
K: Thermal conductivity (W m⁻¹ K⁻¹)
MWCNTs: Multi-walled carbon nanotubes
nf: Nanofluid
N: Kinematic viscosity (m² s⁻¹)
P: Density (kg m⁻³)
q_w: Heat flux (W m⁻²)
T: Temperature (K)
T_w: Surface temperature (K)
T_∞: Ambient temperature (K)
u,v,w: Velocity components in x,y,z directions (m s⁻¹)
U_w: Stretching velocity in the x-direction (m s⁻¹)
V_w: Stretching velocity in the y-direction (m s⁻¹)
x,y,z: Cartesian coordinates (m)
_Ω: Angular velocity (m.s⁻¹)
μ: Dynamic viscosity (Pa s)
τ_w: Shear stress at the wall (N m⁻²)

I. INTRODUCTION

Nanofluids are now widely used in several sectors, including automotive, solar cells, atomic reactor coolants, electronics, engines, propellant combustion, fuel purification, welding, and biomedicine. Their study has attracted attention for their potential applications in several areas, including geophysics, chemistry, biology, mechanical industries, and petroleum. Moreover, due to their superior stability and thermal conductivity, they are widely used in industrial research and development. Choi introduced the phrase “nanofluid” it indicates nano-sized particles dropping within a base liquid. Base fluids are often made of di-hydrogen oxide or complex hydrocarbon compounds (nano-lubricants). Since that time, multiple investigations have been undertaken on them. Numerous research studies^1–7 performed investigations on the thermal conductivity characteristics of nanofluids. Rashid et al.⁸ studied the aligned MHD impact on radiated engine oil-based Casson nanofluid with carbon nanotubes along a shrinking sheet. Mahmood and Khan⁹ studied the impact of nanoparticle aggregation on the unstable flow of stagnation of nanofluids based on hydrogen oxide.

Examining fluid flow and its thermal distribution properties on rotating and overstretching surfaces is a fascinating subject in fluid mechanics. It involves the applications of torsional vibration analysis, including Flettner ships, bioreactors, and many more. Numerous scholars^10–12 have investigated fluid flow and its thermal distribution properties on rotating and overstretched surfaces. Raju et al.¹³ investigated the dual solutions to the MHD flow of a nanofluid in three dimensions across a stretched sheet that is nonlinearly permeable. Archana et al.¹⁴ studied the impacts of a magnetic field and nonlinear heat radiation on three-dimensional flow across a stretched surface of a Maxwell nanofluid. Mohd Sohut et al.¹⁵ examined the unsteady three-dimensional flow across a stretched sheet in a spinning hybrid nanofluid. Tarakaramu et al.¹⁶ studied the effects of viscous dissipation and joule heating on 3D magnetohydrodynamics transportation of Williamson nanofluid in a porous medium across a stretched surface under melting conditions.

Heat exchange study in a system investigates the formation, conversion, exchange, and usage of energy via diverse mechanisms such as conduction/diffusion, advection, radiation, and convection. The ability of this process is determined by the dimensions and form of nanoparticles. Carbon nanotubes (CNTs) are small cylindrical pieces made from rolling layers of single carbon atoms, also known as graphene. CNTs are classified into two types: single-walled carbon nanotubes (SWCNTs), with widths less than 1 nm, and multi-walled carbon nanotubes (MWCNTs), which have sizes greater than 100 nm, as shown in Fig. 1. These nanotubes have a wide range of applications, including fire alarms, anthropogenic heat generation, Doppler, biosensors, sympathetic cooling, smart meters, thermostats, improving thermal transmittance, and retaining magnetic characteristics. Numerous studies^17–20 investigated the nanofluid flow suspended in MWCNTS and SWCNTs. Abideen and Saif²¹ studied the impacts of internal heat production and heat radiation on a Casson nano-fluid passing via a flexible, curved surface that was suspended in carbon nanotubes (CNTs). Ganie et al.²² examined the impacts of non-linear thermal radiation on the three-dimensional unsteady flow of a nanofluid suspended in carbon nanotubes with varying radii and lengths across a stretching material.

FIG. 1.

Structure of (a) SWCNTs and (b) MWCNTs.23 Reproduced from Norizan et al. RSC Adv. 10(71), 43704–43732, 2020, with the permission of Royal society of Chemistry This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence.

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Structure of (a) SWCNTs and (b) MWCNTs.²³ Reproduced from Norizan et al. RSC Adv. 10(71), 43704–43732, 2020, with the permission of Royal society of Chemistry This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence.

The term “thermal radiation” describes the electromagnetic waves that a surface emits as a result of its temperature. Recognizing thermal radiation is crucial for formulating effective heating and cooling systems, maximizing energy efficiency, and managing temperatures in diverse industrial procedures and uses. Many researchers^24–28 have studied the effect of heat radiation on the flow of fluids. Sudarmozhi et al.²⁹ investigated the double diffusion of MHD Maxwell fluid in a porous medium using thermal radiation, heat production, and chemical reactions. Yashkun et al.³⁰ investigated the flow and thermal exchange of a nanofluid via a porous material as a result of stretching/shrinking a sheet by suction, a magnetic field, and heat radiation. Aminuddin et al.³¹ investigated the effect of heat radiation on MHD GO-Fe₂O₄/EG movement and the exchange of heat across a moving surface. Jalili et al.³² investigated the physical properties of cilia propulsion that electro-osmotically interacts on symmetric and asymmetric conduit flow of coupling stress fluid with heat radiation and heat exchange.

Based on a survey of the literature, the purpose of this study is to demonstrate the impacts of thermal radiation on MHD nanofluid (SWCNTs/engine oil or MWCNTs/engine oil) slip flow past a stretching porous surface, which has not been previously investigated. The analysis is given in dimensionless form, and the shooting approach is applied in conjunction with the fourth-order Runge–Kutta method to solve the resulting problem. The temperature and tangential velocity calculations were graphically presented to understand the influence of various physical factors in the scenario. In addition, a full evaluation of the physical interpretation has been carried out, which includes a presentation and discussion of the skin friction coefficients and the rate at which heat changes the area around the boundary layer. This work will yield new insights that are beneficial for solar thermal applications. The present investigation focuses on

how energy transmission helps harvest solar energy and
finding out which of SWCNTs or MWCNTs have a better energy transmission rate and which of these two act as a better cooling liquid.

II. MATHEMATICAL REPRESENTATION

A 3D incompressible steady MHD nanofluid flow serves as the physical model for the current problem across a stretched sheet. The liquid occupies z > 0, and the movement is generated by the sheet being stretched z = 0 along the y-axis and x-axis having velocities V_w = by and U_w = ax, where b and a > 0 are the sheet stretch rates. A stable magnetic field of magnitude B₀ is imposed normally to the stretching surface. An extremely low magnetic Reynolds number is chosen to neglect the impact of the induced magnetic field. A constant temperature T_w is maintained for the sheet, where T_w > T_∞ and T_∞ is the temperature of the ambient fluid. The following are the flow assumptions used in the current analysis:

The movement is rotational having a fixed angular velocity Ω and through the z-axis, as illustrated in Fig. 2.
While the surface is placed in a porous medium containing MHD nanofluid, the thermal radiation impact is taken into account.
CNTs are used as nanoparticles, and engine oil is the base fluid.
Approximations of boundary layers.
Both velocity and thermal slips are taken into consideration.

FIG. 2.

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Physical representation of the problem.

The flow problems contain the following equations:¹⁰

Continuity equation:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 .

(1)

x-direction momentum equation:

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - 2 Ω v = \frac{1}{ρ_{n f}} (μ_{n f} \frac{\partial^{2} u}{\partial z^{2}} - σ B_{0}^{2} u - \frac{μ_{n f}}{k} u) .

(2)

y-direction momentum equation:

u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} + 2 Ω u = \frac{1}{ρ_{n f}} (μ_{n f} \frac{\partial^{2} v}{\partial z^{2}} - σ B_{0}^{2} v - \frac{μ_{n f}}{k} v) .

(3)

Temperature equation:

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = α_{n f} \frac{\partial^{2} T}{\partial z^{2}} - \frac{1}{{(ρ C_{p})}_{n f}} \frac{\partial q_{r}}{\partial z} .

(4)

Boundary limitations are¹⁰

\begin{gathered} u = k ν_{f} \frac{\partial u}{\partial z} + U_{w}, v = k ν_{f} \frac{\partial v}{\partial z} + V_{w}, w = 0, T = l \frac{\partial T}{\partial z} + T_{w}, \\ a t z = 0, u \to 0, v \to 0, T \to T_{\infty}, a t z \to \infty \cdot \end{gathered}

(5)

Radiative heat flux is considered and can be estimated utilizing the Rosseland approximation evaluated from the idea of diffusion. This can be easily expressed as follows:³³

q_{r} = - \frac{4 σ^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial z},

(6)

where k* is the mean absorption coefficient and σ* is the Stephan–Boltzmann constant. Using Taylor series expansion and ignoring higher-order terms other than the first degree in (T − T_∞), give

T^{4} ≅ 4 T T_{\infty}^{3} - 3 T_{\infty}^{4} .

(7)

Using Eqs. and in Eq. gives

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z} = (α_{n f} + \frac{1}{{(ρ C_{p})}_{n f}} \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*}}) \frac{\partial^{2} T}{\partial z^{2}} .

(8)

Quantities without dimensions are as follows:¹⁰

\begin{gathered} η = z {(\frac{a}{ν_{f}})}^{1 / 2}, u = a x p^{'} (η), v = a y q^{'} (η), \\ w = - {(a ν_{f})}^{1 / 2} (p (η) + q (η)), θ (η) = \frac{T - T_{\infty}}{T_{w} - T_{\infty}} . \end{gathered}

(9)

Using Eq. in Eqs. –, , and gives

\frac{A_{1}}{A_{2}} (p^{'''} - G p^{'}) - \frac{M}{A_{2}} p^{'} - {(p^{'})}^{2} + (p + q) p^{″} + 2 ω δ q^{'} = 0,

(10)

\frac{A_{1}}{A_{2}} (q^{'''} - G q^{'}) - \frac{M}{A_{2}} q^{'} - {(q^{'})}^{2} + (p + q) q^{″} - 2 \frac{ω}{δ} p^{'} = 0,

(11)

\frac{(A_{4} + R)}{A_{3} \Pr} θ^{″} + (p + q) θ^{'} = 0,

(12)

\begin{gathered} p (0) = 0, p^{'} (0) = 1 + S_{f} p^{″} (0), q (0) = 0, q^{'} (0) = S + S_{f} q^{″} (0), \\ θ (0) = 1 + S_{t} θ^{'} (0), a t η = 0, p^{'} \to 0, q^{'} \to 0, θ \to 0, a s η \to \infty \cdot \end{gathered}

(13)

Here, prime denotes the derivative with η, and dimensionless parameters are

\begin{gathered} M = \frac{σ B_{0}^{2}}{a ρ_{f}}, G = \frac{ν_{f}}{a k}, ω = \frac{Ω}{a}, δ = \frac{y}{x}, \Pr = \frac{{(μ C_{p})}_{f}}{k_{f}}, \\ R = \frac{16 σ T_{\infty}^{3}}{3 k^{*} k_{f}}, S = \frac{b}{a}, S_{f} = k {(a ν_{f})}^{1 / 2}, S_{t} = l {(\frac{a}{ν_{f}})}^{1 / 2}, \\ A_{1} = \frac{μ_{n f}}{μ_{f}}, A_{2} = \frac{ρ_{n f}}{ρ_{f}}, A_{3} = \frac{{(ρ C_{p})}_{n f}}{{(ρ C_{p})}_{f}}, A_{4} = \frac{k_{n f}}{k_{f}} . \end{gathered}

From an engineering and mathematical perspective, the skin friction coefficients and Nusselt number in dimensional form are as follows:¹⁰

C_{f x} = {\frac{τ_{w x}}{ρ_{f} U_{w}^{2}}|}_{z = 0}, C_{f y} = {\frac{τ_{w y}}{ρ_{f} U_{w}^{2}}|}_{z = 0}, N u = {\frac{x q_{w}}{k_{f} (T_{w} - T_{\infty})}|}_{z = 0} .

(14)

Here, $τ_{w x} = μ_{n f} (\frac{\partial u}{\partial z}), τ_{w y} = μ_{n f} (\frac{\partial v}{\partial z}), a n d q_{w} = - k_{n f} (1 + R) (\frac{\partial T}{\partial z})$ ⁠.

Using Eq. in Eq. , the skin friction coefficients and Nusselt number without dimensions are acquired as ¹⁰

\begin{matrix} C_{f x} {(R e)}^{1 / 2} = A_{1} p^{″} (0), δ^{- 1} C_{f y} {(R e)}^{1 / 2} = A_{1} q^{″} (0), \\ N u {(R e)}^{- 1 / 2} = - A_{4} (1 + R) θ^{'} (0) . \end{matrix}

(15)

Here,

{Re}^{1 / 2} = \sqrt{\frac{a}{υ_{f}}} x

⁠.

III. NUMERICAL PROCEDURE

A non-linear boundary value problem is constructed using dimensionless Eqs. –. This set of ordinary differential equations that are non-linear is resolved utilizing a combination of the shooting approach and the fourth-order Runge–Kutta method, and a boundary value solver (BVP4C) incorporated in the MATLAB program. Figure 3 depicts the flow chart describing the numerical processes performed by the MATLAB-based boundary value problem. Before coding, several assumptions must be considered,

\begin{gathered} p = f (1), p^{'} = f (2), p^{″} = f (3), q = f (4), \\ q^{'} = f (5), q^{″} = f (6), θ = f (7), θ^{'} = f (8) . \end{gathered}

(16)

FIG. 3.

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Flow chart of the numerical procedure.

Utilizing the assumptions made in , the following set of first-order ordinary differential equations is derived using Eqs. – and the boundary conditions as

\begin{cases} f^{'} (1) = f (2), \\ f^{'} (2) = f (3), \\ f^{'} (3) = \frac{A_{2}}{A_{1}} ({(f (2))}^{2} - (f (4) + f (1)) f (3) - 2 ω δ f (5)) \\ + \frac{M}{A_{1}} f (2) + G f (2), \\ f^{'} (4) = f (5), \\ f^{'} (5) = f (6), \\ f^{'} (6) = \frac{A_{2}}{A_{1}} ({(f (5))}^{2} - (f (1) + f (4)) f (6) + 2 \frac{ω}{δ} f (2)) \\ + \frac{M}{A_{1}} f (5) + G f (5), \\ f^{'} (7) = f (8), \\ f^{'} (8) = - \frac{\Pr A_{3}}{(A_{4} + R)} (f (1) + f (4)) f (8), \end{cases}\}

(17)

\begin{gathered} f a (1) = 0, f a (2) = 1 + S_{f} f a (3), f a (4) = 0, f a (5) = S + S_{f} f a (6), \\ f a (7) = 1 + S_{t} f a (8), f b (2) \to 0, f b (5) \to 0, f b (7) \to 0 \cdot \end{gathered}

(18)

Once the MATLAB code has been converted, the aforementioned system is run to obtain the necessary results displayed in tables and graphs. For each result, a tolerance error of 10⁻⁶ is imposed throughout the calculation.

IV. DISCUSSION ON OUTCOMES

Utilizing the fourth-order Runge-Kutta method in conjunction with the shooting approach, the system of coupled non-linear ordinary differential Eqs. (10)–(12) is employed along with a new set of conditions subject to (13). Using Tables I and II, the influence of the impacts of many dimensionless constants on velocities and temperature is shown in Figs. 4–23 and discussed in detail. In addition, the influence of the impacts of many dimensionless constants on the coefficients of skin friction and Nusselt number is presented in Tables III–V and discussed in detail. The default values for the many physical constants that are used in this investigation are M = 3, G = 0.4, S = 1, S_f = 0.1, S_t = 0.5, ω = 0.2, δ = 0.2, Pr = 7.56, and R = 3.

TABLE I.

Thermophysical correlations of nanofluid used in this study. Reproduced with permission Bhaskar Reddy et al. Int. J. Eng. Math. 2014, 1–10. Copyright^© 2014. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Properties	Nanofluid
Dynamic viscosity	$μ_{n f} = \frac{μ_{f}}{{(1 - ϕ)}^{2.5}}$
Density	$ρ_{n f} = ρ_{f} (1 - ϕ) + ρ_{s} ϕ$
Thermal conductivity	$K_{n f} = K_{f} (\frac{2 K_{f} + K_{s} - 2 ϕ (K_{f} - K_{s})}{2 K_{f} + K_{s} + ϕ (K_{f} - K_{s})})$
Heat capacity	$\begin{gathered} {(ρ C_{p})}_{n f} = {(ρ C_{p})}_{f} (1 - ϕ) + {(ρ C_{p})}_{s} ϕ \end{gathered}$

Properties	Nanofluid
Dynamic viscosity	$μ_{n f} = \frac{μ_{f}}{{(1 - ϕ)}^{2.5}}$
Density	$ρ_{n f} = ρ_{f} (1 - ϕ) + ρ_{s} ϕ$
Thermal conductivity	$K_{n f} = K_{f} (\frac{2 K_{f} + K_{s} - 2 ϕ (K_{f} - K_{s})}{2 K_{f} + K_{s} + ϕ (K_{f} - K_{s})})$
Heat capacity	$\begin{gathered} {(ρ C_{p})}_{n f} = {(ρ C_{p})}_{f} (1 - ϕ) + {(ρ C_{p})}_{s} ϕ \end{gathered}$

TABLE II.

Thermophysical values of the engine oil, SWCNTs, and MWCNTs used in this study. Reproduced with permission from [Rashid et al., Micromachines 13(9), 1501, (2022)]. Copyright licensed 2022, Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY 4.0) license.⁸

Physical parameters	Engine oil	SWCNTs	MWCNTs
$ρ (kg / m^{3})$	884	2600	1600
$k (W / mK)$	0.145	6600	3000
$C p (J / kg K)$	1910	425	796

FIG. 4.

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$p^{'} (η)$ via magnetic parameter (M).

FIG. 5.

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$θ (η)$ via magnetic parameter (M).

FIG. 6.

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$p^{'} (η)$ via porous permeability (G).

FIG. 7.

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$q^{'} (η)$ via porous permeability (G).

FIG. 8.

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$θ (η)$ via porous permeability (G).

FIG. 9.

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$p^{'} (η)$ via rotation parameter (ω).

FIG. 10.

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$q^{'} (η)$ via rotation parameter (ω).

FIG. 11.

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$θ (η)$ via rotation parameter (ω).

FIG. 12.

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$p^{'} (η)$ via axis ratio parameter (δ).

FIG. 13.

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$q^{'} (η)$ via axis ratio parameter (δ).

FIG. 14.

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$θ (η)$ via axis ratio parameter (δ).

FIG. 15.

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$p^{'} (η)$ via velocity slip parameter (S_f).

FIG. 16.

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$q^{'} (η)$ via velocity slip parameter (S_f).

FIG. 17.

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$θ (η)$ via velocity slip parameter (S_f).

FIG. 18.

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$p^{'} (η)$ via stretching parameter (S).

FIG. 19.

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$q^{'} (η)$ via stretching parameter (S).

FIG. 20.

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$θ (η)$ via stretching parameter (S).

FIG. 21.

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$θ (η)$ via thermal slip parameter (S_t).

FIG. 22.

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$θ (η)$ via Prandtl’s number (Pr).

FIG. 23.

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$θ (η)$ via thermal radiation parameter (R).

TABLE III.

Variations in −Re^1/2C_fx for SWCNTs and MWCNTs.

M	S	S_f	ω	δ	−Re^1/2C_fx (SWCNTs)	−Re^1/2C_fx (MWCNTs)
3	1	0.1	0.2	0.2	2.086 354	2.065 476
2					1.891 281	1.866 727
1					1.659 810	1.629 822
	1.0				2.086 354	2.065 476
	1.5				2.108 518	2.085 905
	2.0				2.130 021	2.105 773
		0.1			2.086 354	2.065 476
		0.3			1.54 5416	1.536 205
		0.5			1.234 924	1.229 966
			0.2		2.086 354	2.065 476
			0.4		2.069 946	2.050 301
			0.6		2.060 001	2.040 561
				0.2	2.086 354	2.065 476
				0.4	2.073 226	2.053 071
				0.6	2.057 394	2.038 375

M	S	S_f	ω	δ	−Re^1/2C_fx (SWCNTs)	−Re^1/2C_fx (MWCNTs)
3	1	0.1	0.2	0.2	2.086 354	2.065 476
2					1.891 281	1.866 727
1					1.659 810	1.629 822
	1.0				2.086 354	2.065 476
	1.5				2.108 518	2.085 905
	2.0				2.130 021	2.105 773
		0.1			2.086 354	2.065 476
		0.3			1.54 5416	1.536 205
		0.5			1.234 924	1.229 966
			0.2		2.086 354	2.065 476
			0.4		2.069 946	2.050 301
			0.6		2.060 001	2.040 561
				0.2	2.086 354	2.065 476
				0.4	2.073 226	2.053 071
				0.6	2.057 394	2.038 375

TABLE IV.

Variations in −δ ⁻¹Re^1/2C_fy for SWCNTs and MWCNTs.

M	S	S_f	ω	δ	−δ ⁻¹Re^1/2C_fy (SWCNTs)	−δ ⁻¹Re^1/2C_fy (MWCNTs)
3	1	0.1	0.2	0.2	2.548 105	2.489 723
2					2.432 076	2.365 191
1					2.326 074	2.246 971
	1.0				2.548 105	2.489 723
	1.5				3.709 171	3.633 073
	2.0				4.947340	4.848 190
		0.1			2.548 105	2.489 723
		0.3			1.822 007	1.790 582
		0.5			1.421 903	1.401 830
			0.2		2.548 105	2.489 723
			0.4		3.001 079	2.904 698
			0.6		3.462 816	3.326 851
				0.2	2.548 105	2.489 723
				0.4	2.328 774	2.288 136
				0.6	2.256 563	2.221 657

M	S	S_f	ω	δ	−δ ⁻¹Re^1/2C_fy (SWCNTs)	−δ ⁻¹Re^1/2C_fy (MWCNTs)
3	1	0.1	0.2	0.2	2.548 105	2.489 723
2					2.432 076	2.365 191
1					2.326 074	2.246 971
	1.0				2.548 105	2.489 723
	1.5				3.709 171	3.633 073
	2.0				4.947340	4.848 190
		0.1			2.548 105	2.489 723
		0.3			1.822 007	1.790 582
		0.5			1.421 903	1.401 830
			0.2		2.548 105	2.489 723
			0.4		3.001 079	2.904 698
			0.6		3.462 816	3.326 851
				0.2	2.548 105	2.489 723
				0.4	2.328 774	2.288 136
				0.6	2.256 563	2.221 657

TABLE V.

Variations in Re^−1/2N_ux for SWCNTs and MWCNTs.

Pr	R	S	S_t	ω	Re⁻¹^/2N_ux (SWCNTs)	Re⁻¹^/2N_ux (MWCNTs)
7.56	3	1	0.5	0.2	3.194 361	3.243 626
6					2.815 878	2.865 132
5					2.531 406	2.579 638
	3				3.194 361	3.243 626
	2				2.729 792	2.765 792
	1				2.128 480	2.150 909
		1.0			3.194 361	3.243 626
		1.5			3.565 354	3.613 157
		2.0			3.857 601	3.905 353
			0.5		3.194 361	3.243 626
			1.0		2.458 205	2.487 260
			1.5		1.997 781	2.016 923
				0.2	3.194 361	3.243 626
				0.4	2.990 942	3.063 586
				0.6	2.718 617	2.830 147

Pr	R	S	S_t	ω	Re⁻¹^/2N_ux (SWCNTs)	Re⁻¹^/2N_ux (MWCNTs)
7.56	3	1	0.5	0.2	3.194 361	3.243 626
6					2.815 878	2.865 132
5					2.531 406	2.579 638
	3				3.194 361	3.243 626
	2				2.729 792	2.765 792
	1				2.128 480	2.150 909
		1.0			3.194 361	3.243 626
		1.5			3.565 354	3.613 157
		2.0			3.857 601	3.905 353
			0.5		3.194 361	3.243 626
			1.0		2.458 205	2.487 260
			1.5		1.997 781	2.016 923
				0.2	3.194 361	3.243 626
				0.4	2.990 942	3.063 586
				0.6	2.718 617	2.830 147

Figure 4 shows $p^{'} (η)$ for various M values. It can be observed that by increasing the M values, the velocity decreases. The fact is that applying a magnetic field that is transverse produces a force of drag equivalent to Lorentz’s that restricts the flow of liquid and thus lower its velocity. In addition, the thickness of the momentum boundary layer diminishes. Figure 5 illustrates $θ (η)$ for various values of M. The temperature is seen to increase in increments of M. Due to its thick thermal boundary layer, strong thermal conductivity, and consistent heat variations in the nanofluid, the velocity profile is more, and the temperature profile is less for M in MWCNTs/engine oil than SWCNTs/engine oil. Figures 6, 7 and 8 show the impact of G on $p^{'} (η)$ ⁠, $q^{'} (η)$ ⁠, and $θ (η)$ ⁠. It is evident that a porous media makes the flow of the nanofluid more constrictive, and this modulates its motion. In this manner, when G increases, the resistance following fluid motion increases and subsequently reduces both velocities $p^{'} (η)$ and $q^{'} (η)$ ⁠. The opposite nature occurs in $θ (η)$ ⁠. Velocities are high and temperature is less for G in MWCNTs/engine oil than SWCNTs/engine oil. Figures 9, 10 and 11 illustrate the impact of ω on $p^{'} (η) q^{'} (η)$ and $θ (η)$ ⁠. $p^{'} (η)$ and $θ (η)$ profiles rise in conjunction with an increase in ω, but the $q^{'} (η)$ profile decreased. Both velocity profiles are high, and the temperature profile is less for ω in MWCNTs/engine oil than SWCNTs/engine oil.

Figures 12, 13 and 14 show the effect of δ on $p^{'} (η)$ ⁠, $q^{'} (η)$ ⁠, and $θ (η)$ ⁠. Both velocity profiles increase with an increase in δ whereas the temperature profile has the opposite tendency. Both velocities profiles are more, and temperature profile is less for δ in MWCNTs/engine oil than SWCNTs/engine oil. Figures 15, 16 and 17 illustrate the influence of S_f on $p^{'} (η)$ ⁠, $q^{'} (η)$ ⁠, and $θ (η)$ ⁠. The velocity distributions $p^{'} (η)$ and $q^{'} (η)$ are greatly reduced, and temperature $θ (η)$ is increased by the velocity slip parameter S_f. The reason behind this is that the velocity of the fluid close to the surface does not continue in the same way as the stretched surface velocity inside the sight of a slip. As a result, the slip velocity increases as the slip velocity parameters are raised. When there is slip, the fluid can only learn about the deformation of the stretched surface, which lowers the fluid velocity. Both velocity profiles are high, and the temperature profile is less for S_f in MWCNTs/engine oil than SWCNTs/engine oil.

Figures 18, 19 and 20 illustrate the influence of S on $p^{'} (η)$ ⁠, $q^{'} (η)$ ⁠, and $θ (η)$ ⁠. It is clear that when the stretching ratio parameter increases, both the temperature $θ (η)$ and the velocity $p^{'} (η)$ are less, while the velocity $q^{'} (η)$ behaves in another way. Increasing the stretching parameter generally results in increased fluid flow pressure and a drop in temperature. Raising the radiation parameter causes the fluid’s temperature in the boundary layer to rise at a distance from the surface and to drop in the closest area of the surface. Both velocity profiles are high, and the temperature profile is less for S in MWCNTs/engine oil than SWCNTs/engine oil. Figure 21 reveals the impact of S_t on $θ (η)$ ⁠. The temperature slip parameter is increased to lower the temperature profile. The temperature drops as the temperature slip parameter increases because low energy is physically transmitted from the surface to the fluid. The temperature profile is less for S_t in MWCNTs/engine oil than SWCNTs/engine oil. Figure 22 shows the influence of Pr on $θ (η)$ ⁠. It is seen that the temperature profile $θ (η)$ and the corresponding boundary layer width decrease with increasing Pr values. The drop in thermal diffusivity is correlated with the Prandtl number growth. It is commonly known that fluids at lower temperatures have reduced thermal diffusivity. A thicker thermal boundary layer and lower temperature are indicated by such reduced thermal diffusivity. The temperature profile is less for Pr in MWCNTs/engine oil than SWCNTs/engine oil. Figure 23 illustrates the effect of R on $θ (η)$ ⁠. The temperature distribution at the boundary layer clearly increases as the heat radiation constant values increase. Heat radiation has the effect of accelerating heat transfer because it causes the energy boundary layer to expand. As a result, it has been said that the reduction of heat radiation needs to happen more quickly. The temperature profile is less for R in MWCNTs/engine oil than SWCNTs/engine oil.

Table III displays variations in the surface skin friction along the x-axis for several physical parameters. The friction coefficients in the x-direction are found to decrease as S increases and grows with M, S_f, ω, and δ increase in both SWCNTs and MWCNTs. In addition, variations in the surface skin friction along the x-axis is more in MWCNTs/engine oil than SWCNTs/engine oil. Table IV displays variations in the surface skin friction along the y-axis for several physical parameters. The friction coefficients in the y-direction are found to decrease as S and ω increase and grows with M, S_f, and δ increase in both SWCNTs and MWCNTs. In addition, variations in the surface skin friction along the x-axis is less in MWCNTs/engine oil than SWCNTs/engine oil. Table V displays variations in the heat transfer rate for several physical parameters. The heat transfer rate is found to decrease as R, S_t, and ω increase and grows as S and Pr increase in both SWCNTs and MWCNTs. In addition, variations in the heat transfer rate is more in MWCNTs/engine oil than SWCNTs/engine oil.

V. CONCLUSIONS

The current work examines when a stretched porous sheet is in the presence of thermal radiation with slips in front of an MHD 3D nanofluid (SWCNTs/engine oil or MWCNTs/engine oil). Tables and graphs are used to display numerical findings. The following conclusions are obtained:

The velocities $p^{'} (η)$ and $q^{'} (η)$ decrease with an increase in M, G, and S_f and are quite opposite when δ increases.
The velocity $p^{'} (η)$ grows with an increase in ω and is opposite in $q^{'} (η)$ ⁠.
The velocity $p^{'} (η)$ drops with an increase in S and is opposite in $q^{'} (η)$ ⁠.
The temperature $θ (η)$ grows with an increase in M, G, R, S_f, and ω and is opposite when δ and Pr increases.
Velocities are more and temperature is less in MWCNTs/engine oil than in SWCNTs/engine oil.
The friction factor in both x and y axes is reduced with an increase in M, S_f, and δ and is opposite for S.
The friction factor in the x-axis reduces with an increase in ω and is opposite in the y-axis.
The heat transfer rate enhances with an increase in Pr and S, and it is opposite when R, S_t, and δ increase.
This work will yield new insights that are beneficial for solar thermal applications. This study shows that the energy transmission rate is better in multi-walled carbon nanotubes, which act as better cooling liquid.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

N. Bhargavi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). P. Sreenivasulu: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). T. Poornima: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Comparative analysis on single- and multi-walled carbon nanotubes suspended for enhanced thermal energy transfer past stretching surfaces

NOMENCLATURE

Dimensionless parameters

Subscripts

Abbreviations

Dimensional parameters

I. INTRODUCTION

II. MATHEMATICAL REPRESENTATION

III. NUMERICAL PROCEDURE

IV. DISCUSSION ON OUTCOMES

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Comparative analysis on single- and multi-walled carbon nanotubes suspended for enhanced thermal energy transfer past stretching surfaces

NOMENCLATURE

Dimensionless parameters

Subscripts

Abbreviations

Dimensional parameters

I. INTRODUCTION

II. MATHEMATICAL REPRESENTATION

III. NUMERICAL PROCEDURE

IV. DISCUSSION ON OUTCOMES

V. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

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