Thermal Behavior of the Time-Dependent Radiative Flow of Water-Based CNTs/Au Nanoparticles Past a Riga Plate with Entropy Optimization and Multiple Slip Conditions

This communication deliberates the time-reliant and Darcy–Forchheimer flow of water-based CNTs/gold nanoparticles past a Riga plate. In addition, nonlinear radiation, heat consumption and multiple slip conditions are considered. Entropy generation is computed through various flow parameters. A suitable transformation with symmetry variables is invoked to remodel the governing mathematical flow models into the ODE equations. The homotopy analysis scheme and MATLAB bvp4c method are imposed to solve the reduced ODE equations analytically and numerically. The impact of sundry flow variables on nanofluid velocity, nanofluid temperature, skin friction coefficient, local Nusselt number, entropy profile and Bejan number are computed and analyzed through graphs and tables. It is found that the nanofluid velocity is reduced by greater porosity and slip factors. The thickness of the thermal boundary layer increases with increasing radiation, temperature ratio, and heat consumption/generation parameters. The surface drag force is reduced when there is a higher Forchheimer number, unsteadiness parameter and porosity parameter. The amount of entropy created is proportional to the radiation parameter, porosity parameter and Reynolds number. The Bejan number profile increases with radiation parameter, heat consumption/generation parameter and the Forchheimer number.


Introduction
Due to their intriguing thermal transport features, nanofluids have fascinated a large number of researchers in recent decades. The customary heat transmittal fluids, such as water, ethylene glycol and oil, have lesser fundamental heat transport properties. In order to boost the thermal conductivity of customary heat transmittal fluids, nanofluids (the homogeneous mingling of solid nanoparticles of dimension 1-100 nm in customary heat transmittal fluids) are often used. When it comes to heat conductivity, carbon nanotubes (CNTs) are far and away the best nano-element, with a wonderful capability of conducting heat. CNTs are rolled-up graphene sheets with the simplest atomic bonding arrangement. Because of their nanoscale size, they exhibit exceptional structural and mechanical characteristics and excellent electrical and thermal conductivity. The flow of CNTs upon a stretchy sheet with suction was probed by Saleh et al. [1]. They detected that MWCNTs have a higher temperature than the SWCNTs. Anuar et al. [2] found that kerosene-based CNTs have a larger HT gradient than water-based CNTs for the flow of CNTs past a moving plate with stability analysis. The 2D flow of water-based CNTs past a curved surface was numerically computed by Hayat et al. [3]. Kamali and Binesh [4] detected that the addition of MWCNTs leads to improving the HT coefficient for the problem of water-based CNTs of non-Newtonian nanofluids. The heat transmission of water-based SWCNTs inside a heated circular pipe was numerically addressed by Saeed et al. [5]. They concluded that the maximum improvement of the Nusselt number was attained at a 0.05% volume fraction of SWCNTs. Hussain et al. [6] proved that water-based SWCNTs generate a larger HT rate than water-based MWCNTs on their study of the flow of CNTs between two spinning plates. The 3D DF flow of CNTs in a heated rotating frame was explored by Hayat et al. [7]. They found that NPVF leads to improve the nanofluid temperature. The flow of graphene-based nanofluid past a cylinder was investigated by Ghani et al. [8]. They found that increased heat flux was obtained in graphene-water than the SWCNT-water.
Magnetic fields are essential in several scientific, technical and industrial operations. However, commonly used electrically conducting fluids, such as plasma, electrolytes and liquid metals, are poor electrical conductors. Enhancing conductivity is a challenging endeavor for researchers. Incorporating an external agent is one of the most straightforward approaches to enhancing electrical conductivity. A Riga plate is an example of an external material that may be used to boost the electrical conductivity of fluids. This plate is made by combining an implanted magnetic bar or magnets with electrodes that are arranged in alternating positions. The flow of chemically reacting second-grade fluid over a heated Riga plate was presented by Rasool et al. [9]. They demonstrated that the changed Hartmann number leads to decrease the fluid temperature. Vishnu Ganesh et al. [10] proved that the temperature gradient decreases when improving the modified Hartmann number for their problem of the flow of a nanofluid past a Riga plate. The slip impact of nanofluid flow through a Riga plate was addressed by Nadeem Abbas et al. [11]. They noticed that the SFC decreases when enriching the modified Hartmann number. Hanumesh Vaidya et al. [12] identified that a nanofluid's temperature drops when increasing the modified Hartmann number in a study of the impact of mixed convection of nanofluid flow over a Riga plate. The effect of a heat source/sink on a flow of Maxwell nanofluid past a heated Riga plate was analyzed by Ramesh et al. [13]. Madhukesh et al. [14] deliberated the salient features of slip effects of water-based SWCNTs past a Riga plate with microorganisms. They noticed that the density of motile microorganism decreased when the modified Hartmann number was increased.
In recent years, many scientists have been willing to scrutinize the radiation impact because of its widespread usage in various fields such as power plants, atomic plants, glass making, projectiles, propulsion devices, space exploration and gas turbines. The linearized Rosseland approximation is widely used; however, it only works when the fluid and surrounding temperatures are close. However, in many industrial situations, the difference between the fluid and surrounding temperature is high, and this method is insufficient. In this situation, a non-linear Rosseland approximation is used. The non-linear radiative 2D flow of nanofluid past a Riga plate was inspected by Waqas et al. [15]. They noticed that the nanofluid temperature increases with higher temperature ratio parameter. Ghasemi et al. [16] noted that a thicker thermal boundary layer occurs for larger values of thermal radiation for the problem of radiative flow of nanofluid flow over an SS with magnetic impact. The non-linear radiative 3D flow of nanofluid past a Riga plate was presented by Abdul Hakeem et al. [17], and they uncovered that increased temperature ratio improved the HT rate. Gautam et al. [18] detected that the thicker thermal boundary layer occurs in a Maxwell fluid than in a Casson fluid when changing the temperature ratio parameter for the problem of MHD flow of Maxwell/Casson fluid flow past an SS with radiation. The MHD flow of a tangent hyperbolic hybrid nanofluid past a heated SS was examined by Rashid et al. [19]. They noticed that a higher temperature ratio leads to improve the nanofluid temperature. Eswaramoorthi et al. [20] achieved that the heat transfer gradient is enhanced with increasing temperature ratio for 3D DF flow of CNTs past a Riga plate with glycerin as a base fluid.
Entropy production is a physical phenomenon inherent in all heat-transmission configurations linked to thermodynamic irreversibility. Any thermal system with a high entropy production rate has its usable work destroyed and its efficiency drastically reduced. The term "entropy generation minimization" (EGM) was first introduced by Bejan [21,22] to quantify and optimize the creation of order from chaos in a wide range of processes, including cryogenics, heat transfer, heat exchangers, storage, turbomachinery and electronic cooling. The entropy optimization of a water-based nanofluid past an SS with heat consumption was addressed by Eswaramoorthi et al. [23]. They used Cu and Ag nanoparticles in their study and found that the Bejan number increases when the radiation parameter raises. Dadheech et al. [24] found that the Brinkman number improves the entropy profile for their problem of radiative Williamson fluid flow in a porous stretching surface. The entropy generation of a steady, MHD Carreau fluid flow past a microchannel with radiation was deliberated by Srinivas Reedy et al. [25]. They proved that the Biot number creates more entropy generation and a high Bejan number. The flow of MHD viscous fluid past a porous microchannel with entropy generation was addressed by Abbas et al. [26]. They ascertained that a higher entropy generation rate is obtained for stronger magnetic field and injection parameters. Ibrahim et al. [27] proved that the higher values of Prandtl number decrease the entropy profile for the analysis of the 3D mixed convective flow of a couple-stress nanofluid. Entropy minimization of MHD flow of a Carreau-Yasuda fluid past a SS was illustrated by Khan et al. [28].
According to the aforementioned literature survey, the variations of entropy and HT of water-based CNTs with non-linear radiation past a Riga plate have not been fully investigated. Therefore, our study's primary goal is to fill this information gap. The primary objective of this body of study is to investigate the impact of time-dependent DF flow of water-based CNTs/gold nanoparticles past a Riga plate with non-linear thermal radiation and heat consumption. The effects of a variety of different factors on the skin friction coefficient, local Nusselt number, entropy generation and the Bejan number are discussed using tabular and graphical representations. Our results are unique and can be used in many industrial areas, including the design of electrical equipment, solar collectors, storage of harmful heat exchangers, freezing, and growing of crystal and glass.

Mathematical Formulation
The time-dependent DF flow of water-based CNTs/gold nanoparticles past a Riga plate is investigated. These components are assembled on a flat surface to make a Riga plate. Let u and v are the X and Y component velocities, respectively. The velocity and thermal slip impacts are included in our analysis. The SWCNTs (single-wall carbon nanotubes) and MWCNTs (multi-wall carbon nanotubes) are the two most-prevalent types of CNTs. In the energy expressions, the consequences of heat consumption are also taken into account. In energy expression, nonlinear radiation can be derived with the help of the nonlinear Rosseland approximation theory. Moreover, the free stream temperature and plate temperature are denoted as T ∞ and T w , respectively. The Riga plate and physical configuration of the flow model are displayed in Figure 1a,b. The mathematical modeling of continuity, momentum and energy expressions are defined in the following format (see Shafiq et al. [29], Ijaz Khan et al. [30] and Soomro et al. [31]): The corresponding constraints are The description of each of the notations is offered in the nomenclature segment. Define (see Upadhya et al. [32]) Through the implementation of change (5) in expressions (2) and (3), the simplified expressions that are obtained are as follows: The converted boundary constraints are An explanation of each non-dimensional number used in the preceding statements is provided in the nomenclature section.

Skin Friction Coefficient
The definition of the skin friction coefficient is expressed as Here, τ X Y represents the wall shear stresses, which may be mathematically written as The skin friction coefficient, in its dimensionless form, is written as

Local Nusselt Number
The local Nusselt number is expressed as where q w represents the amount of wall heat flux and is mathematically represented by The dimensionless representation of the local Nusselt number is formulated as The reduced mathematical expressions (6) and (7) with boundary constraints (8) are analytically computed by employing the HAM procedure. This approach is suitable for solving nonlinear problems analytically and provides a great degree of flexibility in picking starting estimates and linear operators for structuring solutions (see Zeeshan et al. [33]). As a result, we have: Initial approximations: Linear operators: Linear properties: where ψ i ; i = 1, 2, . . . , 5 are constants. Zeroth-order deformation problems: Here, Q ∈ [0, 1] is an embedding parameter and R 1 and R 2 are non-linear operators (see Khan et al. [34]). The nth-order problems are: Here, F * n ( ) and θ * n ( ) are the particular solutions. The parameters h F and h θ are the ones that take responsibility for the solution's convergent implementation (see Prabakaran et al. [35] and Sajjad Haider et al. [36]).

Numerical Method
The transferred nonlinear ODEs (6) and (7) with the boundary constraints under consideration (8) are numerically computed by implementing the MATLAB bvp4c solution procedure. In this context, we begin by transforming the system of higher-order ODEs into a system of first-order differential equations (see Ghani et al. [37] and Safak Kayikci et al. [38]).
The system of equations is with the corresponding conditions We solve the aforementioned problem by implementing the MATLAB bvp4c function with an error of 10 −5 and a step size of 0.05.

Entropy Analysis
Entropy creation may be expressed mathematically as follows: The non-dimensional form of the entropy generation equation is The Bejan number may be determined by taking the entire amount of entropy created and dividing it by the amount of entropy produced due to heat transfer.

Results and Discussion
The purpose of this segment is to scrutinize the consequences of the various emerging parameters on nanofluid velocity, nanofluid temperature, skin friction coefficient, local Nusselt number, entropy profile and Bejan number. All graphical results are made by implementing the HAM method, and the bvp4c scheme is used only for comparative purposes. Table 1 represents the physical characteristics of CNTs, gold nanoparticles and water. The mathematical definitions of the thermophysical characteristics are expressed in Table 2. Table 3 presents the comparison of − f (0) with A = Fr = φ = Ha = K 1 = 0 for different values of λ to the results of Akbara et al. [39] for different values of λ and are in good agreement. The HAM order of approximations of SWCNTs, MWCNTs and gold nanoparticles are demonstrated in Table 4. It is perceived that the 13th order is to be sufficient for all computations. Table 5 provides the analytical and numerical comparison of SFC for assorted estimates of A, K 1 , Ha, λ, Fr and φ for SWCNTs, MWCNTs and gold nanoparticles. It is found that greater values A, λ, Fr and φ lead to decrease the surface shear stress for all cases. However, higher values of K 1 and Ha causes to increase the surface shear stress. In addition to this, the surface shear stress of MWCNTs is noticeably greater than that of SWCNTs and gold nanoparticles. The analytical and numerical computation of LNN for assorted estimates of A, R, Λ, K 2 , Hg and φ for SWCNTs, MWCNTs and gold nanoparticles are described in Table 6. This table shows that the temperature gradient increases with higher values of A, R and Λ, and the opposite is attained for larger K 2 , Hg and φ values. Figure 3a-d show the impact of A (a), Ha (b), K 1 (c) and λ (d) on the NF velocity profile for all cases. It is seen that the NF velocity enhances when higher modified Hartmann number and it decreases for higher A, K 1 and λ values. Physically, a larger value for the porosity parameter creates more resistance to fluid flow, and this slow the fluid motion. A higher modified Hartmann number strengthens the external electric field, and this improves the fluid motion. Further, it is noted from these figures that MWCNTs have larger momentum boundary thickness compared to SWCNTs and gold nanoparticles because MWCNTs have low density values. The repercussions of Ha (a), K 2 (b), φ (c) and Hg (d) on the NF temperature profile are sketched in Figure 4a-d. It is noticed that upgrading the values of Hg increases the temperature of the nanofluid, but increasing the concentrations of Ha and K 2 has the opposite effect. NPVF causes the temperature of the nanofluid to decrease in the vicinity of the surface, while the temperature increases away from the surface. The responses of R and Λ on the temperature profile are captured in Figure 5a,b. Both R and Λ contribute to improving the nanofluid temperature. Physically, heat energy transport is improved because of higher values of the radiation parameter. Thus, the fluid becomes warmer and thicker the thermal boundary layer thickness.

Property Symbol Definition
Viscosity Thermal conductivity k n f Carbon nanotube: Au nanoparticles: Kinematic viscosity ν n f ν n f = µ n f ρ n f Table 3.      The changes to SFC versus A, λ, Fr and Ha for SWCNTs (solid line), MWCNTs (dashed line) and gold nanoparticles (dotted lines) are captured in Figure 6a   The increase/decrease percent of SFC for A (a), K 1 (b), Ha (c) and λ (d) for SWCNTs, MWCNTs and gold nanoparticles is displayed in Figure 10a-d. For unsteady parameter: In SWCNTs, the maximal decline percent (7.14) is attained when A is altered from 0 to 0.2, and the minimal decline percent (4.88) is attained when A is varied from 0.6 to 0.8. In MWCNTs, the maximal decline percent (7.16) is attained when A is altered from 0 to 0.2, and the minimal decline percent (4.89) is attained when A is varied from 0.6 to 0.8. In gold nanoparticles, the maximal decline percent (6.88) is attained when A is altered from 0 to 0.2, and the minimal decline percent (4.63) is attained when A is varied from 0.6 to 0.8; see Figure 10a. For slip parameter: In SWCNTs, the maximal ascent percent (45.4) is attained when K 1 is altered from 0 to 0.5, and the minimal ascent percent (18.32) is attained when K 1 is varied from 1.5 to 2. In MWCNTs, the maximal ascent percent (45.18) is attained when K 1 is altered from 0 to 0.5, and the minimal ascent percent (18.29) is attained when K 1 is varied from 1.5 to 2. In gold nanoparticles, the maximal ascent percent (48.48) is attained when K 1 is altered from 0 to 0.5, and the minimal ascent percent (18.79) is attained when K 1 is varied from 1.5 to 2; see Figure 10b. For the modified Hartmann number: In SWCNTs, the maximal ascent percent (23.05) is attained when Ha is altered from 0.6 to 0.8, and the minimal ascent percent (15.5) is attained when Ha is varied from 0 to 0.2. In MWCNTs, the maximal ascent percent (23.5) is attained when Ha is altered from 0.6 to 0.8, and the minimal ascent percent (15.69) is attained when Ha is varied from 0 to 0.2. In gold nanoparticles, the maximal ascent percent (17.45) is attained when Ha is altered from 0.6 to 0.8, and the minimal ascent percent (12.97) is attained when Ha is varied from 0 to 0.2; see Figure 10c. For porosity parameter: In SWCNTs, the maximal decline percent (12.2) is attained when λ is altered from 0 to 0.3, and the minimal decline percent (5.95) is attained when λ is varied from 0.9 to 1.2. In MWCNTs, the maximal decline percent (12.5) is attained when λ is altered from 0 to 0.3, and the minimal decline percent (6.03) is attained when λ is varied from 0.9 to 1.2. In gold nanoparticles, the maximal decline percent (8.76) is attained when λ is altered from 0 to 0.3, and the minimal decline percent (4.88) is attained when λ is varied from 0.9 to 1.2; see Figure 10d. The decline percent of SFC for Fr (a) and φ (b) and LNN for Fr (c) and φ (d) for SWC-NTs, MWCNTs and gold nanoparticles are plotted in Figure 11a-d. For the Forchheimer number: In SWCNTs, the maximal decline percent (10.1) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (4.78) is attained when Fr is varied from 1.5 to 2. In MWCNTs, the maximal decline percent (10.1) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (4.81) is attained when Fr is varied from 1.5 to 2. In gold nanoparticles, the maximal decline percent (9.08) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (4.34) is attained when Fr is varied from 1.5 to 2; see Figure 11a. For φ: In SWCNTs, the maximal decline percent (6.66) is attained when φ is altered from 0 to 0.04, and the minimal decline percent (6.01) is attained when φ is varied from 0.12 to 0.16. In MWCNTs, the maximal decline percent (5.15) is attained when φ is altered from 0.12 to 0.16, and the minimal decline percent (5.02) is attained when φ is varied from 0 to 0.04. In gold nanoparticles, the maximal decline percent (27.94) is attained when φ is altered from 0 to 0.04, and the minimal decline percent (8.4) is attained when φ is varied from 0.12 to 0.16; see Figure 11b. For the Forchheimer number: In SWCNTs, the maximal decline percent (0.72) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (0.47) is attained when Fr is varied from 1.5 to 2. In MWCNTs, the maximal decline percent (0.72) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (0.47) is attained when Fr is varied from 1.5 to 2. In gold nanoparticles, the maximal decline percent (0.79) is attained when Fr is altered from 0 to 0.5, and the minimal decline percent (0.51) is attained when Fr is varied from 1.5 to 2; see Figure 11c. For φ: In SWCNTs, the maximal decline percent (10.4) is attained when φ is altered from 0 to 0.04, and the minimal decline percent (1.95) is attained when φ is varied from 0.12 to 0.16. In MWCNTs, the maximal decline percent (9.67) is attained when φ is altered from 0 to 0.04, and the minimal decline percent (2.03) is attained when φ is varied from 0.12 to 0.16. In gold nanoparticles, the maximal decline percent (4.8) is attained when φ is altered from 0 to 0.04, and the minimal decline percent (2.8) is attained when φ is varied from 0.12 to 0.16; see Figure 11d. In SWCNTs, the maximal ascent percent (71.37) is attained when R is altered from 0 to 0.7, and the minimal ascent percent (18.16) is attained when R is varied from 2.1 to 2.8. In MWCNTs, the maximal ascent percent (73.01) is attained when R is altered from 0 to 0.7, and the minimal ascent percent (18.19) is attained when R is varied from 2.1 to 2.8. In gold nanoparticles, the maximal ascent percent (88.7) is attained when R is altered from 0 to 0.7, and the minimal ascent percent (17.3) is attained when R is varied from 2.1 to 2.8; see Figure 12a. For slip parameter: In SWCNTs, the maximal decline percent (58) is attained when K 2 is altered from 0 to 0.5, and the minimal decline percent (22.1) is attained when K 2 is varied from 1.5 to 2. In MWCNTs, the maximal decline percent (57.6) is attained when K 2 is altered from 0 to 0.5, and the minimal decline percent (22) is attained when K 2 is varied from 1.5 to 2. In gold nanoparticles, the maximal decline percent (55.7) is attained when K 2 is altered from 0 to 0.5, and the minimal decline percent (21.9) is attained when K 2 is varied from 1.5 to 2; see Figure 12b. For heat consumption/generation parameter: In SWCNTs, the maximal decline percent (7.38) is attained when Hg is altered from 0.2 to 0.4, and the minimal decline percent (2.79) is attained when Hg is varied from −0.4 to −0.2. In MWCNTs, the maximal decline percent (7.24) is attained when Hg is altered from 0.2 to 0.4, and the minimal decline percent (2.8) is attained when Hg is varied from −0.4 to −0.2. In gold nanoparticles, the maximal decline percent (9.57) is attained when Hg is altered from 0.2 to 0.4, and the minimal decline percent (3.09) is attained when Hg is varied from −0.4 to −0.2; see Figure 12c. For the modified Hartann number: In SWCNTs, the maximal ascent percent (1.67) is attained when Ha is altered from 0 to 0.4, and the minimal ascent percent (0.62) is attained when Ha is varied from 1.2 to 1.6. In MWCNTs, the maximal ascent percent (1.69) is attained when Ha is altered from 0 to 0.4, and the minimal ascent percent (0.52) is attained when Ha is varied from 1.2 to 1.6. In gold nanoparticles, the maximal ascent percent (1.65) is attained when Ha is altered from 0 to 0.4, and the minimal ascent percent (1.03) is attained when Ha is varied from 1.2 to 1.6; see Figure 12d.

Conclusions
This current study explores the influence on time-dependent DF flow of water-based CNTs and gold nanoparticles across a Riga plate under various slip situations. In addition, nonlinear radiation as well as heat consumption are taken into consideration. The creation of entropy may be estimated via the use of a variety of flow parameters. The controlling mathematical flow models are remodeled into ODE equations using an appropriate transformation with symmetry variables. The simplified ODE models are solved analytically and numerically using the homotopy analysis method and the MATLAB bvp4c approach. The following is a summary of the significant findings of our study: • Higher unsteady parameter, porosity and slip parameters reduce the velocity of the nanofluid.