Mathematical Modeling of MHD Flow of CNTs/Ag Nanoparticles Past a Heated Stretchy Surface with Nonuniform Heat Sink/Source and Thermal Radiation Impacts

Gandhimathi, T.; Eswaramoorthi, S.; Loganathan, K.; Mishra, Nirmith Kumar

doi:https://doi.org/10.1155/2023/7308650

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusions Appendix Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 7308650 | https://doi.org/10.1155/2023/7308650

Mathematical Modeling of MHD Flow of CNTs/Ag Nanoparticles Past a Heated Stretchy Surface with Nonuniform Heat Sink/Source and Thermal Radiation Impacts

T. Gandhimathi,¹S. Eswaramoorthi,²K. Loganathan,³and Nirmith Kumar Mishra⁴

Academic Editor: Taseer Muhammad

Received29 Jan 2023

Revised11 Mar 2023

Accepted13 Mar 2023

Published17 Apr 2023

Abstract

This research communication intends to evaluate the impact of time-dependent MHD Darcy–Forchheimer flow of CNTs/Ag nanoparticles on a heated stretchy surface. Water is employed as a base fluid, and two types of CNTs such as single- and multiwall carbon nanotubes are considered. The significance of nonlinear radiation and heat sink/source is added to our analysis. To accommodate the suitable variables, the governing nonlinear partial differential models are transformed into a set of ordinary differential models. These resulting models are solved analytically and numerically by utilizing the homotopy analysis technique and the bvp4c procedure in MATLAB. The distinctive behavior of pertinent physical parameters on the dimensionless profiles are displayed and discussed through diagrams, tables, and charts. It is discovered that the velocity profile decrepitude whenever there is a change in the unsteady, porosity, and injection/suction parameters. The space and temperature-dependent heat sink/source parameter cause to elevate the thermal profile. The Ag nanoparticles have a lesser surface shear stress compared to both CNTs. The heat transfer gradient develops for larger quantities of radiation and temperature ratio parameters. This research has significant applications in many industrial sectors, such as thermal exchangers, chemical reactors, microelectronics, biomedical engineering, aerodynamics, and industrial production processes.

1. Introduction

A carbon nanotube is an allotrope of carbon that resembles a tube formed of carbon atoms. Although extraordinarily strong and difficult to break, carbon nanotubes are nonetheless rather light. Due to their superior electrical, thermal, and mechanical characteristics, CNTs are one of the most studied nanoscale materials. While single-wall carbon nanotubes only have one concentric cylindrical lattice whereas multiwall carbon nanotubes have several. Many medical studies are now being conducted on carbon nanotubes (CNTs), which are widely used in the realms of effective medication administration and biosensing techniques for the diagnosis and monitoring of disorders. The heat transfer investigation of water-based CNTs on a rotating frame was studied by Hussain et al. [1]. They observed that a larger HT gradient is achieved in SWCNTs than in MWCNTs when the magnitude of the NPVF is varied. Khan et al. [2] investigated the numerical consequences of CNTs flowing through a channel. The thermal profile has been found to suppress the values of the NPVF. The stagnation point flow of CNTs past a nonlinear stretching surface was examined by Hayat et al. [3]. They used two types of CNTs like SWCNTs and MWCNTs and also considered water and kerosene oil as base fluids. They discovered that SWCNTs transferred less heat than MWCNTs in water. Ahmed et al. [4] demonstrated the flow of water-based CNTs in a rectangular channel. They proved that the SWCNTs have a greater LNN for changing the NPVF quantities. Anuar et al. [5] numerically solved the mixed convective flow of water/kerosene-based CNTs past a moving plate. It has been observed that the plate shear stress is higher in SWCNTs than in MWCNTs. Hayat et al. [6] analytically examined the stagnation point flow of CNTs past a cylinder with multiple slip effects. They noticed that the MWCNTs had a larger velocity profile compared to the SWCNTs.

Magnetic fields are very useful in a multitude of technological and engineering applications, for example, MHD generators, metal casting, fusion reactors, optical grafting, and solar wind. The MHD rotating flow of CNTs past a stretching sheet was investigated by Haider et al. [7]. They remarked that the HT rate decays when escalating the values of the magnetic field parameter. Rasool et al. [8] explored the Darcy–Forchheimer flow of Casson fluid past an SS in the presence of magnetic impact. They have seen that the fluid velocity profile declines when increasing the magnetic field parameter. The MHD and thermally radiative flow of kerosene-based nanofluid in-between two parallel plates was studied by Ghoneim et al. [9]. They recorded that the Hartmann number leads to a decline in the energy profile. Prabakaran et al. [10] analytically solved the problem of mixed convective MHD flow of CNTs/ nanofluid with the presence of radiation. Their results clearly explained that the heat transfer gradient suppresses a larger quantity of magnetic field parameter. Ali et al. [11] used the finite element method to solve the problem of MHD flow of a hybrid nanofluid past a stretching plane. They noted that the magnetic factor causes to augment the nanofluid temperature. The MHD flow of water-based CNTs past the wedge was numerically addressed by Khan et al. [12]. Nadeem et al. [13] probed the 2D MHD flow of micropolar fluid past an exponential SS. The time-dependent flow of Casson micropolar NF flow past an SS with an induced magnetic field was addressed by Amjad et al. [14]. They found that the acceleration profile decreased when raising the modified Hartmann number.

Nowadays, a significant number of researchers have shown a great deal of interest in the boundary layer flow issue via convective boundary conditions because many high-temperature industrial processes depend on it, for example, thermal energy storage, heat exchangers, nuclear plants, the petroleum industry, and drying metal. The MHD flow of Casson fluid past a wedge with CBC was inquired by Hussain et al. [15]. They announced that the larger size of the Biot number affects the SFC. Raje et al. [16] outlined the MHD flow of Jeffrey fluid past a heated porous pipe. They recorded that the Bejan number escalates when escalating the Biot number. The entropy generation of a micropolar fluid past a porous pipe with CBC was inquired by Srinivasacharya and Bindu [17]. They described that the fluid velocity decays when bolstering the quantity of Biot number. Reddy et al. [18] interrogated the flow of MHD Maxwell nanofluid past a heated exponential SS. They confirmed that the TBL was thicker with a higher Biot number. The HMT analysis of an MHD NF flow past a cone with CBC was outlined by Reddy et al. [19]. They endorsed that the SFC depresses when bettering the Biot number. Akbar et al. [20] inspected the radiative flow of MHD NF flow past a heated SS. They described that the NPVF profile improves when improving the Biot number.

Several common industrial processes rely on flow facilitated by suction or injection. A few processes are turbojets, film cooling, gas turbines, mass transfer cooling, and a plethora of additional technologies that fall under this category. Divya et al. [21] inquired about the flow of based Cu/Ag nanoparticles past a heated SS with suction/injection. They reported that the SFC decays when improving the SI parameter. The MHD flow of micropolar fluid past a porous medium with SI was reviewed by Shankar Goud [22]. He endorsed that the HT gradient intensifies when bettering the SI parameter. Mohammadein et al. [23] explored the radiative flow of CuO nanofluid past an SS with SI. They witnessed that the SI parameter causes to decay the FT. The MHD flow of Ag-nanofluid in past a plate with SI was presented by Upreti et al. [24]. They described that the SFC boosts up when raising the SI parameter. Ahmad et al. [25] surveyed the NF flow with radiation and SI. They reported that the NFT reduces when enriching the SI parameter. Eswaramoorthi et al. [26] reviewed the significance of the MHD flow of Cu/Ag-water-based NF past a 3D plate with SI. They noticed that the SI parameter develops the HTG.

Based on the aforesaid studies, the time-reliant flow of water-based CNTs/Ag nanoparticles past a heated stretchy paper with the significances of nonlinear radiation and nonuniform heat sink/source is still not reported. Therefore, our goal is to fill this void in the existing literature. The present study is to address the consequences of MHD flow of CNTs/Ag nanoparticles past a heated stretchy surface with nonuniform heat sink/source, thermal radiation, and suction/injection. Furthermore, the influence of various flow factors on fluid velocity, fluid temperature, SFC, and LNN is discussed. The extrapolation of numerical data was successfully accomplished via the use of graphs and tables. We affirmed that our computational outcomes are novel and original, and it can be applied in various fields, such as solar collectors, freezing, heat exchangers, and design of new electrical equipments.

2. Problem Formulation

We consider the 2D Darcy–Forchheimer flow of time-dependent flow of CNTs/Ag nanoparticles past a stretchy paper. Let and be the velocity factor in and directions. We set the -axis to run parallel to the paper and the -axis to run perpendicular. The flow is kept moving in the direction of ≥ 0 towards the surface. The direction is the only one in which the surface of the sheet is stretched, and the stretching velocity varies with time. The uniform magnetic field of strength that was applied in the directions as well as the induced magnetic impact was not taken into account due to the low Reynolds number, see Figure 1. Under the previous assumptions, the governing mathematical models can be defined as follows, see Madhukesh et al. [27] and Haq et al. [28].

The correlative boundary restraints are as follows:

The explanation of each of the notations that are available in our problem availed in nomenclature part.

Delineate, see Nawaz and Sadiq [29].

The following are the simplified expressions that may be derived by applying the modification denoted by equation (5) to the expressions denoted by equations (2) and (3), see Nawaz and Sadiq [29] and Subbarayudu et al. [30]:

The territorial restraints that were modified after the transformation are as follows:

The nomenclature section contains an explanation of all factors used in the previous expressions.

3. Quantities of Physical Interest

3.1. Skin Friction Coefficient

The mathematical form of skin friction coefficient may be represented as follows:

The wall shear stresses is defined as follows:

The unit-less form of skin friction coefficient is drafted as follows:

3.2. Local Nusselt Number

The following is the mathematical form of the local Nusselt number:

The wall heat flow is denoted by the variable which is formally written as follows:

The expression for the nondimensionless version of the local Nusselt number is as follows:

4. Methodology

4.1. Numerical Method

The transmitted concern problem (6) and (7) with boundary constraints (8) are numerically reckoned by employing the MATLAB bvp4c algorithm. Regarding this, changing the existing higher ODEs into a new system of first-order differential equations, see, Ishtiaq et al. [31].

Let us takewith the conditions

4.2. Analytical Method

The simplified expressions (6) and (7) with corresponding constraints (8) are determined analytically with the use of the HAM technique. This method is appropriate for analytically resolving nonlinear problems, and it offers a high degree of flexibility in selecting the starting estimates and the linear operators to structure solutions. In addition, this method is suitable for solving highly nonlinear problems that involve multiple variables. In addition to that, you may use this strategy in circumstances that call for the solution of nonlinear issues see, see Liao [32] and Zhao and Liao [33]. In addition to that, the use of this strategy may be utilized to find solutions to nonlinear issues. Initial Approximations Linear Operators Linear Propertieswhere are constants. Zeroth-Order Deformation ProblemsHere, is an embedding parameter and and are nonlinear operators. The Order Problems The and represents the particular solutions.

The and (HAM parameters) are the liability for the solution convergent, see Eswaramoorthi et al. [34]. The range values of is [−0.8, −0.28] (SWCNTs), [−0.8, −0.28] (MWCNTs), and [−0.92, −0.35] (Ag) and is [−2.1, −0.3] (SWCNTs), [−2.1, −0.3] (MWCNTs), and [−2.05, −0.35] (Ag), see Figures 2(a) and 2(b). In order to get a higher degree of precision, we set to be −0.6 and to be −1.2.

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5. Correlation Analysis

The correlation expressions play a vital role in the design of the thermal system and their performances. The correlation expressions may be acquired from the numerical values that were acquired via the process of linear regression analysis. The following are the equations that describe the association between the SFC and the LNN for both cases:

, , , and with maximum error is .

6. Results and Discussion

This section aims to examine the impacts of different physical factors on the FV, FT, SFC, and LNN. The physical properties of SWCNTs, MWCNTs, silver nanoparticles, and water are shown in Table 1. Table 2 illustrates the HAM order for SWCNTs, MWCNTs, and silver nanoparticles. According to this table, the is an appropriate factor for all the estimations. The SFC for disparate values of , , , , and for SWCNTs, MWCNTs, and Ag nanoparticles were divulged in Table 3. It can be noticed from this table that the surface drag coefficient has a declining tendency when there is a stronger presence of , , , and . In addition to this, the surface drag coefficient of the MWCNTs is much higher than that of the SWCNTs and the silver nanoparticles. Tables 4 and 5 provide the variations of LNN that occur during convective heating and cooling for various values of , , and for SWCNTs, MWCNTs, and Ag nanoparticles. In the case of heating, improving the and value causes the temperature gradient to climb. The converse tendency has been seen, with more presences of and . When the cooling scenario is taken into consideration, developing the values of , , and causes a rise in the temperature gradient. On the other hand, the larger values of indicate a declining trend.

Figures 3(a)–3(d) explore the alterations of the velocity field as a direct consequence of , , , and . This graph illustrates that the velocity field downfalls when escalating the values of , , , and . Physically, a resistive category force called the Lorentz force is created by greater magnetic field levels. This force tends to impede the flow of the fluid. The speed of a nanofluid slows down as the quantity of the porosity parameter goes up. This is because of friction and interactions between nanoparticles enhance as the porosity parameter goes up. The modifications of temperate filed for different magnitudes of , , , and are sketched in Figures 4(a)–4(d). It has been proved indisputably that the mounting values of , , , and lead to upsurging the temperature field and thicker its concern boundary layer thickness. The thermal boundary layer grows thicker as the radiation parameter is greater because this provides the fluid more heat, which in turn causes the thermal boundary layer to get thicker.

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Figures 5(a)–5(d) is captured to scrutinize the changes of SFC through , , , , and . From these figures, it is rendered that the surface drag force is reduced due to more presences of , , , , and . The nature of LNN for disparate combos of , , and for convective heating and convective cooling cases are taken in Figures 6(a)–6(d). From the heating case, it is detected from these figures that the behavior of the LNN seems to be improving when a bigger quantity of the and . Furthermore, it is the fallout when strengthening the values of . In the cooling case, the reverse behavior is accounted. Figures 7(a)–7(d) present the conversions of LNN with respect to , , , and for convective heating and convective cooling cases. In heating case, it is noted that the higher extent of creates a more temperature gradient inside the boundary and reduce the temperature gradient for larger values of and in the heating case. The quite opposite trend is obtained in the cooling case. The suction/injection parameter leads to enriching the temperature gradient in both cases.

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The declining rate of SFC for the disparate quantity of (a), (b), (c), and (b) for SWCNTs, MWCNTs, and silver nanomaterials are provided in Figures 8(a)–8(d). In SWCNTs, the maximum declining percentage (4.26) is collected when is shifted from 0 to 0.3, and the minimum declining percentage (3.49) is collected when is shifted from 0.9 to 1.2. In MWCNTs, the maximum declining percentage (4.24) is collected when is shifted from 0 to 0.3 and the minimum declining percentage (3.47) is collected when is shifted from 0.9 to 1.2. In silver nanomaterials, the maximum declining percentage (4.39) is collected when is shifted from 0 to 0.3 and the minimum declining percentage (3.6) is collected when is shifted from 0.9 to 1.2, see Figure 8(a). For variations in SWCNTs, the maximum declining percentage (12.6) is collected when is shifted from 0 to 0.5 and the minimum declining percentage (3.5) is collected when is shifted from 1.5 to 2, in MWCNTs, the maximum declining percentage (12.9) is collected when is shifted from 0 to 0.5 and the minimum declining percentage (1.96) is collected when is shifted from 1.5 to 2, in silver nanomaterials, the maximum declining percentage (10.4) is collected when is shifted from 0 to 0.5 and the minimum declining percentage (5.92) is collected when is shifted from 1.5 to 2, see Figure 8(b). For variations in SWCNTs, the maximum declining percentage (8.58) is collected when is shifted from 0 to 0.4 and the minimum declining percentage (3.9) is collected when is shifted from 1.2 to 1.6, in MWCNTs, the maximum declining percentage (8.79) is collected when is shifted from 0 to 0.4 and the minimum declining percentage (3.2) is collected when is shifted from 1.2 to 1.6, in silver nanomaterials, the maximum declining percentage (7.24) is collected when is shifted from 0 to 0.4 and the minimum declining percentage (4.79) is collected when is shifted from 1.2 to 1.6, see Figure 8(c). For variations in SWCNTs, the maximum declining percentage (6.59) is collected when is shifted from 0 to 0.5 and the minimum declining percentage (4.16) is collected when is shifted from 1.5 to 2, in MWCNTs, the maximum declining percentage (6.55) is collected when is transferred from 0 to 0.5 and the minimum declining percentage (4.05) is collected when is shifted from 1.5 to 2, in silver nanomaterials, the maximum declining percentage (6.82) is collected when is shifted from 0 to 0.5 and the minimum declining percentage (4.62) is collected when is shifted from 1.5 to 2, see Figure 8(d).

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Figures 9(a)–9(d) is taken for analyzing the declining percentage of SFC for disparate values of (a) & (b) and soaring/declining percentage of LNN for disparate values of with convective heating (c) and convective cooling (d) cases. The SFC in SWCNTs, the maximum declining percentage (10.28) is collected when is shifted from 0.3 to 0.6 and the minimum declining percentage (10.01) is collected when is shifted from to . The SFC in MWCNTs, the maximum declining percentage (10.01) is collected when is shifted from 0 to to 0.6 and the minimum declining percentage (9.82) is collected when is shifted from to . The SFC in silver nanomaterials, the maximum declining percentage (11.8) is collected when is shifted from 0 to to 0.6 and the minimum declining percentage (11.4) is collected when is shifted from to , see Figure 9(a). The SFC in SWCNTs, the maximum declining percentage (4.44) is collected when is shifted from 0.06 to 0.08 and the minimum declining percentage (4.41) is collected when is shifted from 0 to to 0.04. The SFC in MWCNTs, the maximum declining percentage (11.33) is collected when is shifted from 0 to 0.02 and the minimum declining percentage (8.75) is collected when is shifted from 0.06 to 0.08. The SFC in silver nanomaterials, the maximum declining percentage (3.72) is collected when is shifted from 0.06 to 0.08 and the minimum declining percentage (3.51) is collected when is shifted from 0 to 0.02, see Figure 9(b). In the heating surface, the LNN in SWCNTs, the maximum declining percentage (9.24) is collected when is shifted from 0 to 0.02 and the minimum declining percentage (3.64) is collected when is shifted from 0.06 to 0.08. In MWCNTs, the maximum declining percentage (7.01) is collected when is shifted from 0 to 0.02 and the minimum declining percentage (3.74) is collected when is shifted from 0.06 to 0.08. In silver nanomaterials, the maximum declining percentage (2.19) is collected when is shifted from 0 to 0.02 and the minimum declining percentage (1.91) is collected when is shifted from 0.06 to 0.08, see Figures 9(c). In the cooling surface, the LNN in SWCNTs, the maximum soaring percentage (9.59) is collected when is shifted from 0 to 0.02 and the minimum soaring percentage (3.8) is collected when is shifted from 0.06 to 0.08. In MWCNTs, the maximum soaring percentage (7.22) is collected when is shifted from 0 to 0.02 and the minimum soaring percentage (3.82) is collected when is shifted from 0.06 to 0.08. In silver nanomaterials, the maximum soaring percentage (2.18) is collected when is shifted from 0 to 0.02 and the minimum soaring percentage (1.9) is collected when is shifted from 0.06 to 0.08, see Figure 9(d).

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The soaring/declining percentage of LNN for disparate values of (a-b) (c and d) with convective heating (a, c) and convective cooling (b, d) cases are demonstrated in Figures 10(a)–10(d). In SWCNTs in heating surface, the LNN attained the maximum soaring percentage (53.52) when is shifted from 0 to 0.6 and the minimum soaring percentage (20.38) is collected when is shifted from 1.8 to 2.4. In MWCNTs in heating surface, the LNN attained the maximum soaring percentage (55.22) when is shifted from 0 to 0.6 and the minimum soaring percentage (20.62) is collected when is shifted from 1.8 to 2.4. In silver nanomaterials in heating surface, the LNN attained the maximum soaring percentage (74.72) when is shifted from 0 to 0.6 and the minimum soaring percentage (22.74) is collected when is shifted from 1.8 to 2.4, see Figure 10(a). In SWCNTs in cooling surface, the LNN attained the maximum declining percentage (53.06) when is shifted from 0 to 0.6 and the minimum declining percentage (20.42) is collected when is shifted from 1.8 to 2.4. In MWCNTs in cooling surface, the LNN attained the maximum declining percentage (54.73) when is shifted from 0 to 0.6 and the minimum declining percentage (20.66) is collected when is shifted from 1.8 to 2.4. In silver nanomaterials in cooling surface, the LNN attained the maximum declining percentage (74.69) when is shifted from 0 to 0.6 and the minimum declining percentage (22.87) is collected when is shifted from 1.8 to 2.4, see Figure 10(b). In SWCNTs in heating surface, the LNN attained the maximum declining percentage (2.82) when is transferred from 0.9 to 1.2 and the minimum declining percentage (2.51) is collected when is varied from 0 to 0.3. In MWCNTs in heating surface, the LNN attained the maximum declining percentage (2.63) when is shifted from 0.9 to 1.2 and the minimum declining percentage (2.35) is collected when is shifted from 0 to 0.3. In silver nanomaterials in heating surface, the LNN attained the maximum declining percentage (2.18) when is shifted from 0.9 to 1.2 and the minimum declining percentage (1.97) is collected when is shifted from 0 to 0.3, see Figure 10(c). In SWCNTs in cooling surface, the LNN attained the maximum soaring percentage (2.73) when is shifted from 0.9 to 1.2 and the minimum soaring percentage (2.42) is collected when is shifted from 0 to 0.3. In MWCNTs in cooling surface, the LNN attained the maximum soaring percentage (2.54) when is shifted from 0.9 to 1.2 and the minimum soaring declining percentage (2.27) is collected when is shifted from 0 to 0.3. In silver nanomaterials in cooling surface, the LNN attained in maximum soaring percentage (2.07) when is shifted from 0.9 to 1.2 and the minimum soaring percentage (1.86) is collected when is shifted from 0 to 0.3, see Figure 10(d).

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The soaring/declining percentage of LNN for disparate values of (a and b) (c-d) with convective heating (a, c) and convective cooling (b, d) cases are presented in Figures 11(a)–11(d). In SWCNTs in heating surface, the LNN attained the maximum declining percentage (0.23) when is shifted from 0.9 to 1.2 and the minimum declining percentage (0.16) is collected when is shifted from 0 to 0.3. In MWCNTs in heating surface, the LNN attained the maximum declining percentage (0.2) when is shifted from 0.9 to 1.2 and the minimum declining percentage (0.14) is collected when is shifted from 0 to 0.3. In silver nanomaterials in heating surface, the LNN attained the maximum declining percentage (0.16) when is shifted from 0.9 to 1.2 and the minimum declining percentage (0.12) is collected when is shifted from 0 to 0.3, see Figure 11(a). In SWCNTs in cooling surface, the LNN attained the maximum soaring percentage (0.04) when is shifted from 0.9 to 1.2 and the minimum soaring percentage (0.01) is collected when is shifted from 0 to 0.3. In MWCNTs in cooling surface, the LNN attained the maximum soaring percentage (0.03) when is shifted from 0.9 to 1.2 and the soaring declining percentage (0.01) is collected when is shifted from 0 to to 0.6. In silver nanomaterials in cooling surface, the LNN attained the maximum declining percentage (0.13) when is shifted from 0.9 to 1.2 and the minimum declining percentage (0.03) is collected when is shifted from 0 to 0.3, see Figure 11(b). In SWCNTs in heating surface, the LNN attained the maximum soaring percentage (1.06) when is shifted from 1.6 to 1.8 and the minimum soaring percentage (0.98) is collected when is shifted from 1 to 1.2. In MWCNTs in heating surface, the LNN attained the maximum soaring percentage (1.08) when is shifted from 1.6 to 1.8 and the soaring declining percentage (1) is collected when is shifted from 1 to 1.2. In silver nanomaterials in heating surface, the LNN attained the maximum soaring percentage (1.44) when is shifted from 1.6 to 1.8 and the minimum soaring percentage (1.32) is collected when is shifted from 1 to 1.2, see Figure 11(c). In SWCNTs in cooling surface, the LNN attained the maximum soaring percentage (0.42) when is transferred from 1 to 1.2 and the minimum soaring percentage (0.4) is collected when is shifted from 1.4 to to 1.8. In MWCNTs in cooling surface, the LNN attained the maximum soaring percentage (0.46) when is shifted from 1 to 1.2 and the soaring percentage (0.43) is collected when is shifted from 1.6 to 1.8. In silver nanomaterials in cooling surface, the LNN attained the maximum soaring percentage (0.78) when is shifted from 1 to 1.2 and the minimum soaring percentage (0.72) is collected when is shifted from 1.6 to 1.8, see Figure 11(d).

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7. Conclusions

This work evaluates the impact of time-dependent MHD Darcy–Forchheimer flow of CNTs/Ag nanoparticles past a heated stretchable surface. The water is the base fluid, and there are two varieties of carbon nanotubes (CNTs) such as single-wall carbon nanotubes and multiwall carbon nanotubes are considered. The research includes nonuniform heat sources and sinks along with their applicability. To account for the provided suitable variables, the governing model of nonlinear partial differential expressions is transformed into a set of ordinary differential expressions. These resulting equations are solved analytically and numerically using the homotopy analysis method and the MATLAB bvp4c algorithm. The following is a rundown of the most significant discoveries made:(i)The greater quantity of unsteady, suction/injection, magnetic field, and porosity parameters leads to decline the fluid velocity.(ii)The fluid temperature rises when escalating the magnitudes of radiation, nanoparticle volume fraction, and space and time-dependent heat source/sink parameters.(iii)The stronger values of Forchheimer number, unsteady, porosity, and magnetic field parameters leads to reduce skin friction coefficient.(iv)The radiation parameter plays an opposite-opposite role in heating and cooling cases of local Nusselt number.

Appendix

The mathematical form of thermophysical properties are as follows:

Nomenclature

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Data Availability

The data used to support the findings of this study are made available by the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to this work. All the authors have read and approved the final version of the manuscript.

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Copyright © 2023 T. Gandhimathi et al. 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.

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