Open AccessArticle

Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation

Department of Mathematics, Coimbatore Institute of Technology, Coimbatore 641014, Tamil Nadu, India

Department of Mathematics, Dr. N.G.P. Arts and Science College, Coimbatore 641048, Tamil Nadu, India

Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur 303007, Rajasthan, India

⁴

Research and Development Wing, Live4Research, Tiruppur 638106, Tamil Nadu, India

⁵

Department of Mechanical Engineering, University of West Attica, 12244 Athens, Greece

Authors to whom correspondence should be addressed.

Micromachines 2022, 13(9), 1424; https://doi.org/10.3390/mi13091424

Submission received: 30 July 2022 / Revised: 22 August 2022 / Accepted: 24 August 2022 / Published: 29 August 2022

Download

Browse Figures

Versions Notes

Abstract

The nature of this prevailing inquisition is to scrutinize the repercussion of MHD mixed convective flow of CNTs/

A l_{2} O_{3}

nanofluid in water past a heated stretchy plate with injection/suction, heat consumption and radiation. The Joule heating and viscous dissipation are included in our investigation. The Navier–Stokes equations are implemented to frame the governing flow expressions. These flow expressions are non-dimensioned by employing suitable transformations. The converted flow expressions are computed numerically by applying the MATLAB bvp4c procedure and analytically by the HAM scheme. The impacts of relevant flow factors on fluid velocity, fluid temperature, skin friction coefficient, and local Nusselt number are illustrated via graphs, tables and charts. It is unequivocally shown that the fluid speed declines when escalating the size of the magnetic field parameter; however, it is enhanced by strengthening the Richardson number. The fluid warmness shows a rising pattern when enriching the Biot number and heat consumption/generation parameter. The findings conclusively demonstrate that the surface drag force improves for a larger scale of Richardson number and is suppressed when heightening the unsteady parameter. In addition, it is evident from the outcomes that the heat transfer gradient decreases to increase the quantity of the Eckert number in the convective heating case; however, the opposite nature is obtained in the convective cooling case. Our numerical results are novel, unique and applied in microfluid devices such as micro-instruments, sleeve electrodes, nerve growth electrodes, etc.

Keywords:

SWCNTs and MWCNTs; HAM; radiation; joule heating; viscous dissipation; suction/injection

1. Introduction

The fluid thermal conductivity has been used in many scientific and technical sectors, such as microelectronics, transportation, atomic reactors, heat exchangers, cancer therapy, etc. The ordinary base fluids such as water, oils, ethylene glycol and kerosene have a smaller heat transfer phenomenon because of their weaker thermal conductivity. One of the facile ways to escalate the fluid thermal conductivity is to admix the nanoscale (1–100 nm) particles named nanoparticles into the ordinary base fluids to improve their conductivity. Imtiaz et al. [1] proved that the flow speed is enriched for larger values of the shape parameter and NPVF for the 3D flow of CNTs with the CCHF model. The problem of stagnation point flow of CNTs past a cylinder was analytically solved by Hayat et al. [2]. It has been proved that the NPVF leads to the development of the skin friction coefficient. Yacob et al. [3] noticed that the larger size of NPVF generates more heat inside the boundary for the problem of rotating flow of CNTs on a shrinking/stretching surface. The flow of water/kerosene-based CNTs over a moving plate with suction was examined by Anuar et al. [4]. Their findings unambiguously demonstrate that the SWCNTs have a more significant skin friction coefficient, and MWCNTs have a more considerable heat transfer gradient. Haq et al. [5] found that the SWCNTs have a bigger heat transfer gradient than the MWCNTs for the problem of the MHD pulsatile flow of CNTs. They consider engine oil as a base fluid.

Electrically conducting fluids play a vital role in nuclear power plants, MHD generators, plasma propulsion in astronautics, geophysics, power plants, astronomy, etc. The steady, MHD and stagnation flow of CNTs past a shrinking/stretching sheet was examined by Anuar et al. [6]. Their findings clearly demonstrate that the magnetic field parameter leads to an enhancement of the surface shear stress. Manjunatha et al. [7] examined the MHD flow of water-based CNTs on a rotating disk. It is noted that the fluid thermal profile develops when enhancing the magnetic field parameter. Acharya et al. [8] noticed a larger magnetic field parameter drop-off of the fluid heat transfer gradient in their study of the MHD flow of CNTs past a deformable sheet. The time-dependent free convective flow of Casson nanofluid past a moving plate with the impact of a magnetic field was discussed by Noranuar et al. [9]. It is evident from the outcomes that the skin friction coefficient improves when strengthening the magnetic field. The natural convective flow of MHD water-based CNTs was examined by Benos et al. [10]. Mabood et al. [11] noticed that the entropy production reduces via a magnetic field in their study of the MHD flow of Jeffrey nanofluid past a SS.

However, in the majority of the aforementioned investigations, the impact of thermal radiation on flow and heat transmission has not been considered. However, when technical procedures are carried out at high temperatures, they become significant and cannot be disregarded. The fluid flow with radiation is noteworthy in many engineering processes that occur at high temperatures in industrial processes. The heat transfer via radiation is essential in producing reliable equipment, nuclear plants, gas turbines, satellites, aircraft, missiles, spacecraft, etc. Mahabaleshwar et al. [12], in their study of the radiative flow of water-based CNTs past a stretching surface, reported that a higher radiation parameter develops a temperature gradient. The radiative flow of CNTs on a stretching sheet with a magnetic field was investigated by Shah et al. [13]. Their results unequivocally show that the radiation parameter improves the fluid warmness. Aman et al. [14] noted from the obtained results that the temperature of SWCNTs is larger than the MWCNTs for the problem of MHD water/kerosene-oil/engine-oil-based CNTs past a vertical channel. MHD mixed convective flow of CNTs on a cone with the convective heating condition was scrutinized by Sreedevi et al. [15]. Their results undoubtedly show that the skin friction coefficient develops when improving the radiation parameter. Reddy and Sreedevi [16] proved that the radiation parameter enhances the heat transfer rate in their analysis of the thermally radiative flow of CNTs inside a square chamber. The impact of radiative and unsteady MHD flow of CNTs with thermal stratification was addressed by Ramzan et al. [17]. It is noticed from their outcomes that the local Nusselt number escalates with a larger radiation parameter. Mahabaleshwar et al. ([18,19]) addressed the radiative flow of Walters’ liquid-B and couple stress fluid past an SS.

Moreover, the significance of heat generation/absorption has received considerable attention from many researchers due to its practical usage of debris, heat nuclear reactors, underground disposal of radioactive waste material and semiconductor wafers. Kataria et al. [20] detected that the heat generation/absorption parameter develops the fluid warmness in their analysis of the MHD flow of Casson fluid over an exponentially accelerated plate with heat generation/absorption. The 2D radiative flow of water-based CNTs past curved surfaces with internal heat generation was deliberated by Saba et al. [21]. It is acknowledged that the local heat flux rate declines when improving the heat generation parameter. Ojemeri and Hamza [22] unambiguously demonstrated that a higher heat source parameter improves fluid flow inside the boundary for the problem of chemically reacting MHD flow due to the effect of heat generation/absorption. Entropy optimization of MHD mixed convective flow of Cu nanofluid with a heat sink/source was portrayed by Chamkha et al. [23]. Khan and Alzahrani [24] investigated the consequences of Darcy–Forchheimer flow of nanofluid with the impact of radiation and heat generation/absorption. MHD free convective flow in a concentric annulus with heat generation/absorption was inspected by Gambo and Gambo [25]. Their results do not doubt that the changes in the heat generation/absorption parameter improve the fluid temperature.

This research aims to simulate the mixed convective flow of CNTs past a stretching plate inserted in a porous medium.

The innovation of the present exploration is:

To investigate the MHD flow over a stretchy plate inserted in a porous medium.
The impacts of Joule heating, viscous dissipation and radiation are also added to the heat expression.
These types of modeled problems are used in the thermal industry for designing equipment, such as the design of electric ovens, electric heaters, microelectronics, wind generators, etc.

2. Mathematical Formulation

The following flow hypothesis is used to simulate the fluid’s flow:

The time-dependent, 2D, incompressible, electrically conducting flow of CNTs past a stretchy plate is embedded in a porous medium.
Let the $\overset{˘}{x}$ -coordinate be delineated in the plate, the $\overset{˘}{y}$ -coordinate is normal to it, and the flow occurs when $\overset{˘}{y} > 0$ .
The surface of the plate has a constant temperature ${\overset{˘}{T}}_{w}$ , which is bigger than the ambient fluid temperature ${\overset{˘}{T}}_{\infty}$ .
The fixed magnetic field of quantity B is employed in the $\overset{˘}{y}$ -coordinate; see Figure 1.
The induced magnetic field is omitted because of the small size of the Reynold’s number.
The availability of heat consumption/generation, Joule heating, and radiation impacts are included to analyze the variations of velocity, temperature, SFC and LNN.
The characteristics of fluids are regarded as constants.

Under the above considerations, the flow model can be expressed as (see Soomro et al. [26] and Haq et al. [27])

\begin{matrix} \frac{\partial \overset{˘}{u}}{\partial \overset{˘}{x}} + \frac{\partial \overset{˘}{v}}{\partial \overset{˘}{y}} = 0 \end{matrix}

(1)

\begin{matrix} \frac{\partial \overset{˘}{u}}{\partial \overset{˘}{t}} + \overset{˘}{u} \frac{\partial \overset{˘}{u}}{\partial \overset{˘}{x}} + \overset{˘}{v} \frac{\partial \overset{˘}{u}}{\partial \overset{˘}{y}} = ν_{n f} \frac{\partial^{2} \overset{˘}{u}}{\partial {\overset{˘}{y}}^{2}} - \frac{ν_{n f}}{k_{1}^{*}} \overset{˘}{u} - \frac{σ_{n f}}{ρ_{n f}} B^{2} \overset{˘}{u} + \frac{{(ρ β)}_{n f}}{ρ_{n f}} g (\overset{˘}{T} - \overset{˘}{T_{\infty}}) \\ \frac{\partial \overset{˘}{T}}{\partial \overset{˘}{t}} + \overset{˘}{u} \frac{\partial \overset{˘}{T}}{\partial \overset{˘}{x}} + \overset{˘}{v} \frac{\partial \overset{˘}{T}}{\partial \overset{˘}{y}} = α_{n f} \frac{\partial^{2} \overset{˘}{T}}{\partial {\overset{˘}{y}}^{2}} + \frac{16 σ^{*} {\overset{˘}{T_{\infty}}}^{3}}{3 k^{*} {(ρ c_{p})}_{n f}} \frac{\partial^{2} \overset{˘}{T}}{\partial {\overset{˘}{y}}^{2}} \end{matrix}

(2)

\begin{matrix} + \frac{Q}{{(ρ c_{p})}_{n f}} (\overset{˘}{T} - {\overset{˘}{T}}_{\infty}) + \frac{μ_{n f}}{{(ρ c_{p})}_{n f}} {(\frac{\partial \overset{˘}{u}}{\partial \overset{˘}{y}})}^{2} + \frac{σ_{n f}}{{(ρ c_{p})}_{n f}} B^{2} {\overset{˘}{u}}^{2} \end{matrix}

(3)

with the boundary conditions

\begin{matrix} \overset{˘}{u} = \overset{˘}{U_{w}} = \frac{a \overset{˘}{x}}{1 - ξ t}; \overset{˘}{v} = - \frac{\overset{˘}{ν_{0}}}{\sqrt{1 - ξ t}} \\ - k_{n f} \frac{\partial \overset{˘}{T}}{\partial \overset{˘}{y}} = h_{c} [{\overset{˘}{T}}_{f} - \overset{˘}{T}] a t \overset{˘}{y} = 0 \\ \overset{˘}{u} \to 0, \overset{˘}{T} \to {\overset{˘}{T}}_{\infty} a s \overset{˘}{y} \to \infty \end{matrix}

(4)

All the symbols are shown in the nomenclature section.

Define the variables (see Upadhya et al. [28]),

\begin{matrix} Υ = \sqrt{\frac{a}{ν_{f} (1 - ξ \overset{˘}{t})}} \overset{˘}{y}; \overset{˘}{u} = \frac{a}{1 - ξ t} \overset{˘}{x} F^{'} (Υ); \overset{˘}{v} = - \sqrt{\frac{a ν_{f}}{1 - ξ t}} F (Υ); Θ = \frac{\overset{˘}{T} - {\overset{˘}{T}}_{\infty}}{{\overset{˘}{T}}_{f} - {\overset{˘}{T}}_{\infty}} \end{matrix}

(5)

Substituting Equation (5) in Equations (2) and (3), we have

\begin{matrix} A_{1} A_{2} F^{‴} (Υ) + F (Υ) F^{″} (Υ) - {F^{'}}^{2} (Υ) - A [F^{'} (Υ) + \frac{η}{2} F^{″} (Υ)] - A_{1} A_{2} λ F^{'} (Υ) \end{matrix}

\begin{matrix} - A_{2} A_{3} M F^{'} (Υ) + A_{2} A_{4} R i Θ (Υ) = 0 \\ A_{5} A_{6} \frac{1}{P r} Θ^{^{″}} (Υ) + F (Υ) Θ^{^{'}} (Υ) - F^{'} (Υ) Θ (Υ) - A [Θ (Υ) + \frac{η}{2} Θ^{'} (Υ)] + \frac{A_{6}}{P r} \frac{4}{3} R d Θ^{^{″}} (Υ) \end{matrix}

(6)

\begin{matrix} + A_{1} A_{6} E c F^{″}^{2} (Υ) + A_{3} A_{6} M E c F^{'}^{2} (Υ) + A_{6} H g Θ (Υ) = 0 \end{matrix}

(7)

with the conditions,

\begin{matrix} F (0) = f w, F^{'} (0) = 1, F^{'} (\infty) = 0 \\ Θ^{'} (0) = - \frac{B i}{A_{5}} [1 - Θ (0)]; Θ (\infty) = 0 \end{matrix}

(8)

All the parameters are shown in the nomenclature part, and the notations are defined in Appendix A.

The skin friction coefficient and local Nusselt number are expressed as follows,

\begin{matrix} \frac{1}{2} C f \sqrt{R e} = A_{1} F^{″} (0); \frac{N u}{\sqrt{R e}} = - [A_{5} + \frac{4}{3} R d] Θ^{'} (0) \end{matrix}

3. Solutions

3.1. Numerical Solutions

The amended expressions (6) and (7) and their correlated constraints (8) are numerically computed by applying the MATLAB BVP4C theory (Figure 2a) (see Eswaramoorthi et al. [29]). In this regard, initially, we convert the high-order ODEs into first-order ODEs.

Let

f = D_{1}, f^{'} = D_{2}, f^{″} = D_{3}, Θ = D_{4}, Θ^{'} = D_{5}

The system of equations is

\begin{matrix} D_{1}^{'} = D_{2} \\ D_{2}^{'} = D_{3} \\ D_{3}^{'} = \frac{D_{2}^{2} - D_{1} D_{3} + A [D_{2} + \frac{η}{2} D_{3}] + A_{1} A_{2} λ D_{2} + A_{2} A_{3} M D_{3} - A_{2} A_{4} R i D_{4}}{A_{1} A_{2}} \\ D_{4}^{'} = D_{5} \\ D_{5}^{'} = \frac{D_{2} D_{4} - D_{1} D_{5} + A [D_{4} + \frac{η}{2} D_{5}] - A_{1} A_{6} E c D_{3}^{2} - A_{3} A_{6} M E c D_{2}^{2} - A_{6} H g D_{4}}{\frac{A_{6}}{P r} [A_{5} + \frac{4}{3} R d]} \end{matrix}

with the corresponding conditions

\begin{matrix} D_{1} (0) = f w; D_{2} (0) = 1; D_{2} (\infty) = 0; D_{5} (0) = - \frac{B_{i}}{A_{5}} (1 - D_{4} (0)); D_{4} (\infty) = 0 \end{matrix}

3.2. Analytical Solutions

The amended expressions (6) and (7) and their correlated constraints (8) are analytically computed by applying the HAM scheme (Figure 2b). This method was developed by Shijun Liao in 1992, and it is a powerful mathematical method for solving highly non-linear problems (see [30,31]).

Initial approximations:

\begin{matrix} F_{0} (Υ) = f w + [1 - \frac{1}{e^{Υ}}]; Θ_{0} (Υ) = \frac{B i}{[B i + A_{5}] e^{Υ}} \end{matrix}

Linear operators:

\begin{matrix} L_{F} = F^{‴} - F^{'}; L_{Θ} = Θ^{″} - Θ \end{matrix}

Linear properties:

\begin{matrix} L_{F} [Ω_{1} + Ω_{2} e^{Υ} + Ω_{3} \frac{1}{e^{Υ}}] = 0 = L_{Θ} [Ω_{4} e^{Υ} + Ω_{5} \frac{1}{e^{Υ}}] \end{matrix}

where

Ω_{j}; j

= 1–5 are constants.

Zeroth-order deformation problems:

\begin{matrix} (1 - p) L_{F} [F (Υ, p) - F_{0} (Υ)] & = & p h_{F} N_{1} [F (Υ, p), Θ (Υ, p)] \\ (1 - p) L_{Θ} [Θ (Υ, p) - Θ_{0} (Υ)] & = & p h_{θ} N_{2} [F (Υ, p), Θ (Υ, p)] \end{matrix}

Here

p \in [0, 1]

is an embedding parameter, and

N_{1}

and

N_{2}

are non-linear operators.

The nth order problems:

\begin{matrix} F_{n} (Υ) = F_{n}^{*} (Υ) + Ω_{1} + Ω_{2} e^{Υ} + Ω_{3} \frac{1}{e^{Υ}}; Θ_{n} (Υ) = Θ_{n}^{*} (Υ) + Ω_{4} e^{Υ} + Ω_{5} \frac{1}{e^{Υ}} \end{matrix}

Here

F_{n}^{*} (Υ)

and

Θ_{n}^{*} (Υ)

are the particular solutions.

The HAM parameters (

h_{F}

and

h_{Θ}

) are responsible for the solution convergency (see Loganathan et al. [32] and Eswaramoorthi et al. [33]). The limits of

h_{F}

are [−1.15, −0.3] (SWCNTs), [−1.1, −0.35] (MWCNTs), [−1.2, −0.38] (

A l_{2} O_{3}

) and

h_{Θ}

is [−1.3, −0.3] (SWCNTs), [−1.25, −0.35] (MWCNTs), [−1.3, −0.4] (

A l_{2} O_{3}

) ( see Figure 3a,b).

4. Results and Discussion

The main objective of this segment is to show how the different pertinent flow parameters affect the fluctuations in the fluid velocity, fluid temperature, skin friction coefficient, and local Nusselt number for SWCNTs, MWCNTs and

A l_{2} O_{3}

nanofluid. The physical properties of SWCNTs, MWCNTs,

A l_{2} O_{3}

and water are presented in Table 1. Table 2, Table 3 and Table 4 clearly display the HAM order of approximations and numerical value for all cases. It can be noted from these tables that the 18th order is sufficient for all computations in all cases. The SFC for various values of A, M,

λ

R i

f w

R d

E c

and

H g

is shown in Table 5. This table plainly demonstrates that the plate surface drag force decreases when increasing the size of A, M,

λ

and

f w

. On the other hand, during development, it enhances the quantity of

R i

R d

E c

and

H g

. In addition, MWCNTs have more surface drag force than the SWCNTs and

A l_{2} O_{3}

nanofluid. Table 6 shows the consequences of A, M,

λ

R i

f w

R d

E c

and

H g

on LNN. It can be observed from this table that the HTG grows at growing the quantity of A,

R i

f w

R d

. However, it diminishes when the values of M,

λ

E c

and

H g

are magnified. Additionally, the SWCNTs have less HTG than the MWCNTs and

A l_{2} O_{3}

nanofluid.

Figure 4a–d is drawn to examine the alterations of M,

f w

R i

and

λ

on the fluid velocity distribution. It is detected from these figures that the larger measure of

R i

improves the fluid motion inside the boundary. On the contrary, the larger size of M,

f w

and

λ

reduces the fluid motion. The changes in fluid temperature for distinct quantities of M,

B i

R d

and

E c

are pictured in Figure 5a–d. These figures noticeably point out that the fluid temperature increases when the values of M,

B i

R d

and

E c

increase. Figure 6a,b is taken to analyze the change of

H g

and

ϕ

on fluid temperature distribution. It is found from these figures that the fluid temperature escalates when raising the values of

H g

and

ϕ

The contrast of the skin friction coefficient for different combinations of A and

R i

(a–b) and M and

λ

(c–d) with convective heating (a,c) and convective cooling (b,d) cases are illustrated in Figure 7a–d. A large amount of unsteady magnetic field and porosity parameters is perceived to lead to a fall out of the SFC. However, it improves when developing the Richardson number for both cases. Figure 8a–d is to used to discuss the contrast of LNN for different combinations of

E c

and

R d

(a–b) and

R d

and

f w

(c–d) with convective heating (a,c) and convective cooling (b,d) cases. It is detected from these figures that the larger magnitudes of

R d

and

f w

upsurge the heat transfer gradient, and the Eckert number weakens the LNN for the convective heating case. However, the opposite trend was attained in the convective cooling case. The contrast of LNN for different combinations of

R d

and

H g

(a–b) and

R d

and

R i

(c–d) with convective heating (a,c) and convective cooling (b,d) cases was presented in Figure 9a–d. It is seen from the graphical overview that the LNN grows when mounting the values of the radiation parameter and Richardson number, and it slumps when enhancing the heat consumption/generation parameter in the convective heating case. The reverse trend is obtained in the convective cooling case.

The diminishing percentages of SFC for A (a–b) and M (c–d) with convective heating (a,c) and convective cooling (b,d) for SWCNTs, MWCNTs and

A l_{2} O_{3}

nanofluid are drawn in Figure 10a–d. For the convective heating case, the maximum diminishing percentage

(4.14 %)

occurred in

A l_{2} O_{3}

nanofluid when changing A from 0 to

0.2

, and the minimum diminishing percentage

(3.41 %)

appeared in MWCNTs when changing A from

0.6

0.8

, see Figure 10a. In the convective cooling case, the maximum diminishing percentage

(4.21 %)

occurred in

A l_{2} O_{3}

nanofluid when changing A from

0.6

0.8

, and the minimum diminishing percentage

(3.55 %)

appeared in SWCNTs when changing A from

0.6

0.8

(see Figure 10b). For the convective heating case, the maximum diminishing percentage

(9.04 %)

occurred in MWCNTs when changing M from 0 to

0.3

, and the minimum diminishing percentage

(5.58 %)

appeared in

A l_{2} O_{3}

nanofluid when changing M from

0.9

1.2

(see Figure 10c). In the convective cooling case, the maximum diminishing percentage

(10.33 %)

occurred in MWCNTs when changing M from

0.9

1.2

, and the minimum diminishing percentage

(6.39 %)

appeared in

A l_{2} O_{3}

nanofluid when changing M from

0.6

0.8

(see Figure 10d).

Figure 11a–d is plotted to discuss the diminishing/improving percentage of SFC for

λ

(a–b) and

R i

(c–d) with convective heating (a,c) and convective cooling (b,d) for SWCNTs, MWCNTs and

A l_{2} O_{3}

nanofluid. For the convective heating case, the maximum diminishing percentage

(9.58 %)

occurred in MWCNTs when changing

λ

from 0 to

0.3

, and the minimum diminishing percentage

(5.71 %)

appeared in

A l_{2} O_{3}

nanofluid when changing

λ

from

0.9

1.2

(see Figure 11a). In the convective cooling case, the maximum diminishing percentage

(9.43 %)

occurred in MWCNTs when changing

λ

from 0 to

0.3

, and the minimum diminishing percentage

(6.22 %)

appeared in

A l_{2} O_{3}

nanofluid when changing

λ

from

0.9

1.2

(see Figure 11b). For the convective heating case, the maximum improving percentage

(1.79 %)

occurred in MWCNTs when changing

R i

from 0 to

0.2

, and the minimum improving percentage

(1.7 %)

appeared in

A l_{2} O_{3}

nanofluid when changing

R i

from

0.6

0.8

(see Figure 11c). In the convective cooling case, the maximum improving percentage

(1.42 %)

occurred in SWCNTs when changing

R i

from 0 to

0.2

, and the minimum improving percentage

(1.29 %)

appeared in

A l_{2} O_{3}

nanofluid when changing

R i

from

0.6

0.8

(see Figure 11d).

The diminishing/improving percentage of LNN for

R i

(a–b) and

R d

(c–d) with convective heating (a,c) and convective cooling (b,d) for SWCNTs, MWCNTs and

A l_{2} O_{3}

nanofluid are sketched in Figure 12a–d. For the convective heating case, the maximum improving percentage

(1 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

R i

from 0 to

0.2

, and the minimum improving percentage

(0.8 %)

appeared in MWCNTs when changing

R i

from

0.6

0.8

(see Figure 12a). In the convective cooling case, the maximum diminishing percentage

(0.82 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

R i

from 0 to

0.2

, and the minimum diminishing percentage

(0.66 %)

appeared in MWCNTs when changing

R i

from

0.6

0.8

(see Figure 12b). For the convective heating case, the maximum improving percentage

(54 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

R d

from 0 to

0.5

, and the minimum improving percentage

(18.31 %)

appeared in SWCNTs when changing

R d

from

1.5

to 2 (see Figure 12c). In the convective cooling case, the maximum diminishing percentage

(61 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

R d

from 0 to

0.5

, and the minimum diminishing percentage

(21.43 %)

appeared in SWCNTs when changing

R d

from

1.5

to 2 (see Figure 12d).

Figure 13a–d is taken to examine the diminishing/improving percentage of LNN for

E c

(a–b) and

H g

(c–d) with convective heating (a,c) and convective cooling (b,d) for SWCNTs, MWCNTs and

A l_{2} O_{3}

nanofluid. For the convective heating case, the maximum diminishing percentage

(79 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

E c

from

1.2

1.6

, and the minimum diminishing percentage

(24.28 %)

appeared in MWCNTs when changing

E c

from 0 to

0.4

(see Figure 13a). In the convective cooling case, the maximum improving percentage

(79.83 %)

occurred in

A l_{2} O_{3}

nanofluid when changing

E c

from

1.2

1.6

, and the minimum improving percentage

(24.54 %)

appeared in MWCNTs when changing

E c

from 0 to

0.4

(see Figure 13b). For the convective heating case, the maximum diminishing percentage

(11.57 %)

occurred in SWCNTs when changing

H g

from

0.2

0.4

, and the minimum diminishing percentage

(4.25 %)

appeared in MWCNTs when changing

H g

from

- 0.4

0.2

(see Figure 13c). In the convective cooling case, the maximum improving percentage

(9.85 %)

occurred in SWCNTs when changing

H g

from

0.2

0.4

, and the minimum improving percentage

(3.34 %)

appeared in MWCNTs when changing

H g

from

- 0.4

0.2

(see Figure 13d).

5. Conclusions

The impact of the thermally radiative MHD flow of CNTs/

A l_{2} O_{3}

nanofluid in water past a stretchy plate embedded in a porous medium with the availability of heat consumption/generation and injection/suction was studied. The two varieties of CNTs, such as single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs), were taken into account. The amended expressions and their correlated constraints were numerically and analytically computed via MATLAB BVP4C and HAM theory, respectively. The primary outcomes of our investigation are as follows:

Larger magnetic field and porosity parameters lead to declines in the fluid velocity.
The fluid temperature is strengthened in opposition to the larger Biot number, Eckert number and radiation parameter.
Decay in surface drag force is noted against a larger magnetic field and unsteady parameters.
The radiation parameter leads to an improvement in the heat transfer gradient in the convective heating case, while it decays in the convective cooling case.
The MWCNTs have higher skin friction values compared to SWCNTs and $A l_{2} O_{3}$ nanofluid.
The lower heat transfer gradient appears in SWCNTs compared to MWCNTs and $A l_{2} O_{3}$ nanofluid.

Author Contributions

Data correlation: S.E. and I.E.S. Formal analysis: R.P. and K.L. Investigation: S.E. and I.E.S. Methodology: S.E. and K.L. Writing—original manuscript: S.E. and K.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest, and all authors have read and agreed to the published version of the manuscript.

Nomenclature

$Symbols$	$Description$
a	$positive constant$
$α$	$thermal diffusivity$
B	$applied magnetic field$
$B_{0}$	$intensity of the magnetic field$
$C_{F}$	$skin friction coefficient$
$c_{p}$	$capacity of specific heat$
$h_{c}$	$heat transfer coefficient$
$k_{1}^{*}$	$thermal conductivity$
$k^{*}$	$the mean absorption coefficient$
$n f, f$	$subscript represents nanofluid and base fluid$
$ν$	$kinematic viscosity$
$Υ$	$dimensionless variable$
Q	$heat generation or absorption coefficient$
$Q_{0}$	$heat generation or absorption constant$
$ρ$	$density$
$\overset{˘}{T}$	$non-dimensional temperature$
${\overset{˘}{T}}_{f}$	$temperature of the hot fluid$
${\overset{˘}{T}}_{w}$	$surface temperature$
${\overset{˘}{T}}_{\infty}$	$ambient temperature$
$Θ$	$dimensionless temperature$
$\overset{˘}{u}, \overset{˘}{v}$	$velocity components$
$\overset{˘}{x}, \overset{˘}{y}$	$Cartesian coordinates$
$A (= \frac{ξ}{a})$	$unsteady parameter$
$B i (= \frac{h_{c}}{k_{f}} \sqrt{\frac{ν_{f} (1 - ξ t)}{a}})$	$Biot number$
$E c (= \frac{U_{w}^{2}}{{(c p)}_{f} ({\overset{˘}{T}}_{f} - {\overset{˘}{T}}_{\infty})})$	$Eckert number$
$f w (= \frac{v_{0}}{\sqrt{a ν_{f}}})$	$suction/injection parameter$
$H g (= \frac{Q_{0} \sqrt{(1 - ξ t)}}{a {(ρ c p)}_{f}})$	$heat consumption/generation$
$λ (= \frac{ν_{f}}{k_{1}^{*} a (1 - ξ t)})$	$porosity parameter$
$M (= \frac{σ_{f} B_{0}^{2}}{a ρ_{f}})$	$magnetic field parameter$
$P r (= \frac{ν_{f}}{α_{f}})$	$Prandtl number$
$R e (= \sqrt{\frac{a {\overset{˘}{x}}^{2}}{(1 - ξ \overset{˘}{t}) ν_{f}}})$	$Reynolds number$
$R i (= \frac{g β_{f} ({\overset{˘}{T}}_{f} - {\overset{˘}{T}}_{\infty})}{U_{w}^{2}})$	$Richardson number$
$R d (= \frac{4 σ^{} {\overset{˘}{T}}_{\infty}^{3}}{k^{} k_{f}})$	$radiation parameter$

Abbreviations

$CCHF$	$Cattaneo-Christov heat flux$
$CNTs$	$carbon nanotubes$
$HAM$	$homotopy analysis method$
$LNN$	$local Nusselt number$
$MHD$	$magnetohydrodynamics$
$MWCNTs$	$multi-wall carbon nanotubes$
$NPVF$	$nanoparticle volume fraction$
$ODE$	$ordinary differential equations$
$PDE$	$partial differential equations$
$SFC$	$skin friction coefficient$
$SS$	$stretching sheet$
$SWCNTs$	$single wall carbon nanotubes$

Appendix A

The thermophysical properties are mathematically defined as

\begin{matrix} A_{1} = \frac{μ_{n f}}{μ_{f}} = \frac{1}{{(1 - ϕ)}^{2.5}}; \\ A_{2} = \frac{ρ_{f}}{ρ_{n f}} = \frac{1}{(1 - ϕ + ϕ \frac{ρ_{C N T}}{ρ_{f}})}; \\ A_{3} = \frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{C N T}}{σ_{f}} - 1) ϕ}{(\frac{σ_{C N T}}{σ_{f}} + 2) - ϕ (\frac{σ_{C N T}}{σ_{f}} - 1)}; \\ A_{4} = \frac{{(ρ β)}_{n f}}{{(ρ β)}_{f}} = 1 - ϕ + ϕ \frac{{(ρ β)}_{C N T}}{{(ρ β)}_{f}}; \\ A_{5} = \frac{k_{n f}}{k_{f}} = \frac{(1 - ϕ) + 2 ϕ \frac{k_{C N T}}{k_{C N T} - k_{f}} l n \frac{k_{C N T} + k_{f}}{2 k_{f}}}{(1 - ϕ) + 2 ϕ \frac{k_{f}}{k_{C N T} - k_{f}} l n \frac{k_{C N T} + k_{f}}{2 k_{f}}}; \\ A_{6} = \frac{{(ρ c p)}_{f}}{{(ρ c p)}_{n f}} = \frac{1}{1 - ϕ + ϕ \frac{{(ρ c_{p})}_{C N T}}{{(ρ c_{p})}_{f}}}; \end{matrix}

References

Imtiaz, M.; Mabood, F.; Hayat, T.; Alsaedi, A. Impact of non-Fourier heat flux in bidirectional flow of carbon nanotubes over a stretching sheet with variable thickness. Chin. J. Phys. 2022, 77, 1587–1597. [Google Scholar] [CrossRef]
Hayat, T.; Farooq, M.; Alsaedi, A. Stagnation point flow of carbon nanotubes over stretching cylinder with slip conditions. Open Phys. 2015, 13, 188–197. [Google Scholar] [CrossRef]
Yacob, N.A.; Dzulkifli, N.F.; Salleh, S.N.A.; Ishak, A.; Pop, I. Rotating flow in a nanofluid with CNT nanoparticles over a stretching/shrinking surface. Mathematics 2022, 10, 7. [Google Scholar] [CrossRef]
Anuar, N.S.; Bachok, N.; Arifin, N.M.; Rosali, H. Role of multiple solutions in flow of nanofluids with carbon nanotubes over a vertical permeable moving plate. Alex. Eng. J. 2020, 59, 763–773. [Google Scholar] [CrossRef]
Haq, R.U.; Shahzad, F.; Al-Mdallal, Q.M. MHD pulsatile flow of engine oil based carbon nanotubes between two concentric cylinders. Results Phys. 2017, 7, 57–68. [Google Scholar] [CrossRef]
Anuar, N.S.; Bachok, N.; Arifin, N.M.; Rosali, H. MHD flow past a nonlinear stretching/shrinking sheet in carbon nanotubes: Stability analysis. Chin. J. Phys. 2020, 65, 436–446. [Google Scholar] [CrossRef]
Manjunatha, P.T.; Gowda, R.J.P.; Kumar, R.N.; Suresha, S.; Sarwe, D.U. Numerical simulation of carbon nanotubes nanofluid flow over vertically moving disk with rotation. Partial. Differ. Equ. Appl. Math. 2021, 4, 100124. [Google Scholar] [CrossRef]
Acharya, N.; Das, K.; Kundu, P.K. Rotating flow of carbon nanotube over a stretching surface in the presence of magnetic field: A comparative study. Appl. Nanosci. 2018, 8, 369–378. [Google Scholar] [CrossRef]
Noranuar, W.N.N.; Mohamad, A.Q.; Shafie, S.; Khan, I.; Jiann, L.Y.; Ilias, M.R. Non-coaxial rotation flow of MHD Casson nanofluid carbon nanotubes past a moving disk with porosity effect. Ain Shams Eng. J. 2021, 12, 4099–4110. [Google Scholar] [CrossRef]
Benos, L.; Karvelas, E.G.; Sarris, I.E. A theoretical model for the magnetohydrodynamic natural convection of a CNT-water nanofluid incorporating a renovated Hamilton-Crosser model. Int. J. Heat Mass Transf. 2019, 135, 548–560. [Google Scholar] [CrossRef]
Mabood, F.; Fatunmbi, E.O.; Benos, L.; Sarris, I.E. Entropy generation in the magnetohydrodynamic Jeffrey nanofluid flow over a stretching sheet with wide range of engineering application parameters. Int. J. Appl. Comput. Math. 2022, 8, 1–18. [Google Scholar] [CrossRef]
Mahabaleshwar, U.S.; Sneha, K.N.; Huang, H.N. An effect of MHD and radiation on CNTS-water based nanofluids due to a stretching sheet in a Newtonian fluid. Case Stud. Therm. Eng. 2021, 28, 101462. [Google Scholar] [CrossRef]
Shah, Z.; Bonyah, E.; Islam, S.; Gul, T. Impact of thermal radiation on electrical MHD rotating flow of carbon nanotubes over a stretching sheet. AIP Adv. 2019, 9, 015115. [Google Scholar] [CrossRef]
Aman, S.; Khan, I.; Ismail, Z.; Salleh, M.Z.; Alshomrani, A.S.; Alghamdi, M.S. Magnetic field effect on Poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with Casson fluid. AIP Adv. 2017, 7, 015036. [Google Scholar] [CrossRef]
Sreedevi, P.; Reddy, P.S.; Chamkha, A.J. Magneto-hydrodynamics heat and mass transfer analysis of single and multi-wall carbon nanotubes over vertical cone with convective boundary condition. Int. J. Mech. Sci. 2018, 135, 646–655. [Google Scholar] [CrossRef]
Reddy, P.S.; Sreedevi, P. Effect of thermal radiation and volume fraction on carbon nanotubes based nanofluid flow inside a square chamber. Alex. Eng. J. 2020, 60, 1807–1817. [Google Scholar] [CrossRef]
Ramzan, M.; Shah, S.R.Z.; Kumam, P.; Thounthong, P. Unsteady MHD carbon nanotubes suspended nanofluid flow with thermal stratification and nonlinear thermal radiation. Alex. Eng. J. 2020, 59, 1557–1566. [Google Scholar] [CrossRef]
Mahabaleshwar, U.S.; Sarris, I.E.; Lorenzini, G. Effect of radiation and Navier slip boundary of Walters’ liquid B flow over a stretching sheet in a porous media. Int. J. Heat Mass Transf. 2018, 127, 1327–1337. [Google Scholar] [CrossRef]
Mahabaleshwar, U.S.; Sarris, I.E.; Hill, A.; Lorenzini, G.; Pop, I. An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer. Int. J. Heat Mass Transf. 2017, 105, 157–167. [Google Scholar] [CrossRef]
Kataria, H.R.; Patel, H.R. Effects of chemical reaction and heat generation/absorption on magnetohydrodynamic (MHD) Casson fluid flow over an exponentially accelerated vertical plate embedded in porous medium with ramped wall temperature and ramped surface concentration. Propuls. Power Res. 2019, 8, 35–46. [Google Scholar] [CrossRef]
Saba, F.; Ahmed, N.; Hussain, S.; Khan, U.; Mohyud-Din, S.T.; Darus, M. Thermal analysis of nanofluid flow over a curved stretching surface suspended by carbon nanotubes with internal heat generation. Appl. Sci. 2018, 8, 39. [Google Scholar] [CrossRef]
Ojemeri, G.; Hamza, M.M. Heat transfer analysis of Arrhenius-controlled free convective hydromagnetic flow with heat generation/absorption effect in a micro-channel. Alex. Eng. J. 2022, 61, 12797–12811. [Google Scholar] [CrossRef]
Chamkha, A.J.; Rashad, A.M.; Mansour, M.A.; Armaghani, T.; Ghalambaz, M. Effects of heat sink and source and entropy generation on MHD mixed convection of a Cu-water nanofluid in a lid-driven square porous enclosure with partial slip. Phys. Fluids 2017, 29, 052001. [Google Scholar] [CrossRef]
Khan, M.I.; Alzahrani, F. Free convection and radiation effects in nanofluid (Silicon dioxide and Molybdenum disulfide) with second order velocity slip, entropy generation, Darcy-forchheimer porous medium. Int. J. Hydrog. Energy 2017, 46, 1362–1369. [Google Scholar] [CrossRef]
Gambo, J.J.; Gambo, D. On the effect of heat generation/absorption on magnetohydrodynamic free convective flow in a vertical annulus: An Adomian decomposition method. Heat Transf. 2020, 2020, 2288–2302. [Google Scholar] [CrossRef]
Soomro, F.A.; Haq, R.U.; Khan, Z.H.; Zhang, Q. Numerical study of entropy generation in MHD water-based carbon nanotubes along an inclined permeable surface. Eur. Phys. J. Plus 2017, 132, 12. [Google Scholar] [CrossRef]
Haq, R.U.; Nadeem, S.; Khan, Z.H.; Noor, N.F.M. Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes. Phys. B Condens. Matter 2015, 457, 40–47. [Google Scholar] [CrossRef]
Upadhya, S.M.; Raju, C.S.K.; Mahesha; Saleem, S. Nonlinear unsteady convection on micro and nanofluids with Cattaneo-Christov heat flux. Results Phys. 2018, 9, 779–786. [Google Scholar] [CrossRef]
Eswaramoorthi, S.; Alessa, N.; Sangeethavaanee, M.; Kayikci, S.; Namgyel, N. Mixed convection and thermally radiative flow of MHD Williamson nanofluid with Arrhenius activation energy and Cattaneo-Christov heat-mass flux. J. Math. 2021, 2021, 2490524. [Google Scholar] [CrossRef]
Liao, S. Homotopy Analysis Method in Nonlinear Differential Equations; Higher Education Press: Berlin/Heidelberg, Germany, 2012; pp. 153–165. [Google Scholar]
Zhao, Y.; Liao, S. On the HAM-based mathematica package BVPh for coupled nonlinear ODEs. In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA, 2012; Volume 1479, pp. 1842–1844. [Google Scholar]
Loganathan, K.; Alessa, N.; Kayikci, S. Heat transfer analysis of 3-D viscoelastic nanofluid flow over a convectively heated porous Riga plate with Cattaneo-Christov double flux. Front. Phys. 2021, 9, 379. [Google Scholar] [CrossRef]
Eswaramoorthi, S.; Loganathan, K.; Jain, R.; Gyeltshen, S. Darcy-Forchheimer 3D flow of glycerin-based Carbon nanotubes on a Riga plate with nonlinear thermal radiation and Cattaneo-Christov heat flux. J. Nanomater. 2022, 2022, 5286921. [Google Scholar] [CrossRef]

Figure 1. Physical configuration of the flow model.

Figure 2. The flow chart of bvp4c (a) and HAM (b).

Figure 3.

h

-curves of

F^{″} (0)

(a) and

Θ^{'} (0)

(b).

Figure 3.

h

-curves of

F^{″} (0)

(a) and

Θ^{'} (0)

(b).

Figure 4. The contrast of

F^{'} (Υ)

against M (a),

f w

(b),

R i

λ

(d) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 4. The contrast of

F^{'} (Υ)

against M (a),

f w

(b),

R i

λ

(d) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 5. The contrast of

Θ (Υ)

against M (a),

B i

(b),

R d

E c

(d) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 5. The contrast of

Θ (Υ)

against M (a),

B i

(b),

R d

E c

(d) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 6. The contrast of

Θ (Υ)

against

H g

(a) and

ϕ

(b) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 6. The contrast of

Θ (Υ)

against

H g

(a) and

ϕ

(b) for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 7. The contrast of the skin friction coefficient for different combinations of A and

R i

(a,b) and M and

λ

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 7. The contrast of the skin friction coefficient for different combinations of A and

R i

(a,b) and M and

λ

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 8. The contrast of LNN for different combinations of

E c

and

R d

(a,b) and

R d

and

f w

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 8. The contrast of LNN for different combinations of

E c

and

R d

(a,b) and

R d

and

f w

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 9. The contrast of LNN for different combinations of

R d

and

H g

(a,b) and

R d

and

R i

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 9. The contrast of LNN for different combinations of

R d

and

H g

(a,b) and

R d

and

R i

(c,d) with convective heating (a,c) and convective cooling (b,d) cases for SWCNTs (solid line), MWCNTs (dashed line) and

A l_{2} O_{3}

nanofluid (dotted line).

Figure 10. The diminishing percentage of SFC for A (a,b) and M (c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 11. The diminishing/improving percentage of SFC for

λ

(a,b) and

R i

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 11. The diminishing/improving percentage of SFC for

λ

(a,b) and

R i

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 12. The diminishing/improving percentage of LNN for

R i

(a,b) and

R d

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 12. The diminishing/improving percentage of LNN for

R i

(a,b) and

R d

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 13. The diminishing/improving percentage of LNN for

E c

(a,b) and

H g

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Figure 13. The diminishing/improving percentage of LNN for

E c

(a,b) and

H g

(c,d) with convective heating (a,c) and convective cooling (b,d) cases.

Table 1. Physical properties.

Physical Characteristics	SWCNTs	MWCNTs	${Al}_{2} O_{3}$	Water
k	6600	3000	40	$0.613$
$ρ$	2600	1600	3970	$997.1$
$c p$	425	796	765	4179

Table 2. HAM order and numerical value of SWCNTs.

$Order$	$- F^{''} (0)$		$- Θ^{'} (0)$
$Order$	$HAM$	$NM$	$HAM$	$NM$
1	$1.33342$		$0.17483$
5	$1.38956$		$0.15055$
10	$1.39203$		$0.14900$
15	$1.39203$	$1.39203$	$0.14882$	$0.14879$
18	$1.39203$		$0.14880$
20	$1.39203$		$0.14880$
25	$1.39203$		$0.14880$

Table 3. HAM order and numerical value of MWCNTs.

$Order$	$- F^{″} (0)$		$- Θ^{'} (0)$
$Order$	$HAM$	$NM$	$HAM$	$NM$
1	$1.32570$		$0.18125$
5	$1.36868$		$0.15715$
10	$1.37110$		$0.15554$
15	$1.37110$	$1.37110$	$0.15534$	$0.15531$
18	$1.37110$		$0.15534$
20	$1.37110$		$0.15534$
25	$1.37110$		$0.15534$

Table 4. HAM order and numerical value of

A l_{2} O_{3}

nanofluid.

Table 4. HAM order and numerical value of

A l_{2} O_{3}

nanofluid.

$Order$	$- F^{″} (0)$		$- Θ^{'} (0)$
$Order$	$HAM$	$NM$	$HAM$	$NM$
1	$1.33487$		$0.22682$
5	$1.41937$		$0.19447$
10	$1.42201$		$0.19110$
15	$1.42208$	$1.42210$	$0.19056$	$0.19044$
18	$1.42210$		$0.19046$
20	$1.42210$		$0.19046$
25	$1.42210$		$0.19046$

Table 5. The SFC for various values of A, M,

λ

R i

f w

R d

E c

and

H g

Table 5. The SFC for various values of A, M,

λ

R i

f w

R d

E c

and

H g

								$SFC$
$A$	$M$	$λ$	$Ri$	$fw$	$Rd$	$Ec$	$Hg$	$SWCNTs$	$MWCNTs$	${Al}_{2} O_{3}$
0	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$- 1.48053$	$- 1.45882$	$- 1.51221$
$0.3$								$- 1.57165$	$- 1.54773$	$- 1.60573$
$0.5$								$- 1.63077$	$- 1.60542$	$- 1.66642$
$0.8$								$- 1.71685$	$- 1.68943$	$- 1.75482$
1								$- 1.77244$	$- 1.74371$	$- 1.81193$
$0.2$	0	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$- 1.41740$	$- 1.39248$	$- 1.45261$
	$0.3$							$- 1.54159$	$- 1.51841$	$- 1.57488$
	$0.6$							$- 1.65582$	$- 1.63397$	$- 1.68758$
	$0.9$							$- 1.76229$	$- 1.74156$	$- 1.79277$
	$1.2$							$- 1.86551$	$- 1.84829$	$- 1.89280$
$0.2$	$0.3$	0	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$- 1.41056$	$- 1.38556$	$- 1.44619$
		$0.3$						$- 1.54159$	$- 1.51841$	$- 1.57488$
		$0.6$						$- 1.66138$	$- 1.63957$	$- 1.69284$
		$0.9$						$- 1.77247$	$- 1.75176$	$- 1.80246$
		$1.2$						$- 1.87707$	$- 1.8578$	$- 1.90547$
$0.2$	$0.3$	$0.3$	0	$0.3$	$0.4$	$0.5$	$- 0.4$	$- 1.6122$	$- 1.58809$	$- 1.64490$
			$0.3$					$- 1.56924$	$- 1.54571$	$- 1.60236$
			$0.5$					$- 1.54159$	$- 1.51841$	$- 1.57488$
			$0.8$					$- 1.50147$	$- 1.47875$	$- 1.53487$
			1					$- 1.47556$	$- 1.45312$	$- 1.50897$
$0.2$	$0.3$	$0.3$	$0.5$	$- 0.8$	$0.4$	$0.5$	$- 0.4$	$- 1.01101$	$- 1.00433$	$- 1.01583$
				$- 0.4$				$- 1.17455$	$- 1.16352$	$- 1.18715$
				$0$				$- 1.37146$	$- 1.35429$	$- 1.39482$
				$0.4$				$- 1.60245$	$- 1.57698$	$- 1.63935$
				$0.8$				$- 1.86531$	$- 1.82942$	$- 1.91800$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	0	$0.5$	$- 0.4$	$- 1.54375$	$- 1.52064$	$- 1.57768$
					$0.5$			$- 1.54107$	$- 1.51787$	$- 1.57420$
					1			$- 1.53859$	$- 1.5153$	$- 1.57098$
					$1.5$			$- 1.53625$	$- 1.51287$	$- 1.56795$
					2			$- 1.53402$	$- 1.51056$	$- 1.56508$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	0	$- 0.4$	$- 1.60216$	$- 1.57806$	$- 1.63474$
						$0.4$		$- 1.55337$	$- 1.53001$	$- 1.58655$
						$0.8$		$- 1.50719$	$- 1.48448$	$- 1.54070$
						$1.2$		$- 1.46335$	$- 1.44119$	$- 1.49699$
						$1.6$		$- 1.42158$	$- 1.39990$	$- 1.45524$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.5$	$- 1.54475$	$- 1.5215$	$- 1.57787$
							$- 0.3$	$- 1.53808$	$- 1.51497$	$- 1.57158$
							$0$	$- 1.52464$	$- 1.50188$	$- 1.55926$
							$0.3$	$- 1.50332$	$- 1.48148$	$- 1.54099$
							$0.5$	$- 1.47374$	$- 1.45395$	$- 1.51759$

Table 6. The LNN for various values of A, M,

λ

R i

f w

R d

E c

and

H g

Table 6. The LNN for various values of A, M,

λ

R i

f w

R d

E c

and

H g

								$LNN$
$A$	$M$	$λ$	$Ri$	$fw$	$Rd$	$Ec$	$Hg$	$SWCNTs$	$MWCNTs$	${Al}_{2} O_{3}$
0	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$0.33002$	$0.33444$	$0.34248$
$0.3$								$0.33686$	$0.34135$	$0.34981$
$0.5$								$0.34076$	$0.34530$	$0.35399$
$0.8$								$0.34579$	$0.35043$	$0.35942$
1								$0.34871$	$0.35340$	$0.36257$
$0.2$	0	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$0.36994$	$0.37517$	$0.38588$
	$0.3$							$0.33473$	$0.33919$	$0.34752$
	$0.6$							$0.30204$	$0.30583$	$0.31181$
	$0.9$							$0.27131$	$0.27450$	$0.27816$
	$1.2$							$0.24136$	$0.24339$	$0.24598$
$0.2$	$0.3$	0	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.4$	$0.34947$	$0.35424$	$0.36373$
		$0.3$						$0.33473$	$0.33919$	$0.34752$
		$0.6$						$0.32082$	$0.32500$	$0.33222$
		$0.9$						$0.30758$	$0.31150$	$0.31764$
		$1.2$						$0.29488$	$0.29855$	$0.30366$
$0.2$	$0.3$	$0.3$	0	$0.3$	$0.4$	$0.5$	$- 0.4$	$0.32704$	$0.33161$	$0.33929$
			$0.3$					$0.33175$	$0.33625$	$0.34433$
			$0.5$					$0.33473$	$0.33919$	$0.34752$
			$0.8$					$0.33897$	$0.34337$	$0.35210$
			1					$0.34165$	$0.34603$	$0.35502$
$0.2$	$0.3$	$0.3$	$0.5$	$- 0.8$	$0.4$	$0.5$	$- 0.4$	$0.32262$	$0.32433$	$0.32920$
				$- 0.4$				$0.32552$	$0.32813$	$0.33391$
				$0$				$0.330157$	$0.33381$	$0.34091$
				$0.4$				$0.33638$	$0.34110$	$0.34987$
				$0.8$				$0.34309$	$0.34875$	$0.35920$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	0	$0.5$	$- 0.4$	$0.25220$	$0.25333$	$0.24256$
					$0.5$			$0.35542$	$0.36071$	$0.37386$
					1			$0.45898$	$0.46836$	$0.50551$
					$1.5$			$0.56237$	$0.57575$	$0.63658$
					2			$0.66535$	$0.68263$	$0.76666$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	0	$- 0.4$	$0.48135$	$0.48518$	$0.51258$
						$0.4$		$0.36302$	$0.36737$	$0.37943$
						$0.8$		$0.25269$	$0.25745$	$0.25491$
						$1.2$		$0.14947$	$0.15457$	$0.13820$
						$1.6$		$0.05260$	$0.05798$	$0.02854$
$0.2$	$0.3$	$0.3$	$0.5$	$0.3$	$0.4$	$0.5$	$- 0.5$	$0.34101$	$0.34545$	$0.35443$
							$- 0.3$	$0.32784$	$0.33233$	$0.33999$
							$0$	$0.30233$	$0.30698$	$0.31247$
							$0.3$	$0.26491$	$0.27013$	$0.27372$
							$0.5$	$0.22272$	$0.22947$	$0.23333$

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Prabakaran, R.; Eswaramoorthi, S.; Loganathan, K.; Sarris, I.E. Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation. Micromachines 2022, 13, 1424. https://doi.org/10.3390/mi13091424

AMA Style

Prabakaran R, Eswaramoorthi S, Loganathan K, Sarris IE. Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation. Micromachines. 2022; 13(9):1424. https://doi.org/10.3390/mi13091424

Chicago/Turabian Style

Prabakaran, R., S. Eswaramoorthi, Karuppusamy Loganathan, and Ioannis E. Sarris. 2022. "Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation" Micromachines 13, no. 9: 1424. https://doi.org/10.3390/mi13091424

APA Style

Prabakaran, R., Eswaramoorthi, S., Loganathan, K., & Sarris, I. E. (2022). Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation. Micromachines, 13(9), 1424. https://doi.org/10.3390/mi13091424

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation

Abstract

1. Introduction

2. Mathematical Formulation

3. Solutions

3.1. Numerical Solutions

3.2. Analytical Solutions

4. Results and Discussion

5. Conclusions

Author Contributions

Funding

Conflicts of Interest

Nomenclature

Abbreviations

Appendix A

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Article Menu

Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al2O3 Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation

Abstract

1. Introduction

2. Mathematical Formulation

3. Solutions

3.1. Numerical Solutions

3.2. Analytical Solutions

4. Results and Discussion

5. Conclusions

Author Contributions

Funding

Conflicts of Interest

Nomenclature

Abbreviations

Appendix A

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Investigation on Thermally Radiative Mixed Convective Flow of Carbon Nanotubes/Al₂O₃ Nanofluid in Water Past a Stretching Plate with Joule Heating and Viscous Dissipation