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Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings

  • Adebowale Martins Obalalu , Umair Khan EMAIL logo , Olalekan Adebayo Olayemi , Aurang Zaib , Anuar Ishak and El-Sayed M. Sherif
Published/Copyright: March 14, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

The primary heat source from the sunlight is solar energy (SE), which is used in photovoltaic (PV) panels, solar power plates, PV, streetlights, and solar-based hybrid nanocomposites. Currently, research is focused on analyzing and improving the efficiency of SE, particularly for powering aircraft, by combining solar power with nanotechnology advancements. As such, this study focuses on examining concentrated solar power and proposes a method to improve the performance of solar airplanes by employing nanotechnology. Furthermore, the work is based on the investigation of the flow rate, thermal distribution, and entropy generation of the magnetized tangent hyperbolic hybrid nanofluid (HNF) along the interior parabolic solar trough collector of an aircraft wing. This work utilizes similarity variables to simplify the partial derivative model into ordinary differential equations. These equations are then solved using the Galerkin weighted residual approach with the help of MATHEMATICA 11.3 software. From the obtained outcomes, it is reflected that the HNFs have high thermal conductivity than the NF. Intensification of Weissenberg number improves the performance of airplane wings subjected to heat transmission. Therefore, this research contributes to improved thermal management in advanced nanotechnology and solar aircraft.

Keywords: tangent hyperbolic hybrid nanofluids; solar aircraft and concentrated solar power; Cattaneo–Christov heat flux

Nomenclature

B i

Biot number

Br

Brinkmann number

b

preliminary extending ratio

b *

rate of thermal differences

C f

skin resistance

Ec

Eckert number

h g

thermal conductivity of solid

h g

heat transfer coefficient

K

Porosity parameter

N G

entropy generation

Nr

solar radition

Nu x

heat gradient

n

power-law index

Pr

Prandtl number

Re

Reynolds number

S

suction/injection parameter

TH

tangent hyperbolic

V w

surface permeability

We

Weissenberg number

χ

temperature variance

Θ

fluid temperature

Θ ∞

ambient temperature

ϕ a , ϕ b , ϕ c , ϕ d

thermo-physical properties of TH hybrid nanofluid

ϕ

volume fraction

ε *

thermal relaxation time

γ

velocity slip parameter

Ω

difference in temperatures

1 Introduction

The transportation sector is a major contributor to pollution, responsible for emitting 27% of greenhouse gases. Emissions from numerous kinds of global transportation, like buses, cars, trucks, aircraft, ships, and boats, discharge carbon dioxide (CO2) and other harmful substances into the air [1]. These emissions play an important role in causing climate warming and ecological challenges. Moreover, it stands as the most ecologically harmful sector in China, Russia, and the United States [2]. Scientific research has proved that the production and utilization of energy is the main contributor to CO2 emissions, which are a significant cause of pollution [3]. Nowadays, the main sources of energy are carbon-based fuels, nuclear power, and sustainable energy [4]. These sources play a crucial role in meeting our energy needs. Sustainable energy resources presently account for only 9% of the world’s energy demand, while the remaining 91% is covered by carbon-based fuels and nuclear energy [5]. The combustion of carbon-based fuels for electricity production, supplying energy for households and industry is the major contributor to global carbon dioxide emissions [6]. The raising levels of carbon releases in the atmosphere are contributing to the increasing risk of climate change, which poses a dangerous threat to life in our planet. Solar energy (SE) is the only sustainable and nature-friendly energy source qualified to fulfill our future energy requirements [7]. Based on the predictions provided by the Intergovernmental Panel on Climate Change, SE can fulfill the entire worldwide electricity needs by 2050 [8]. This forecast can be accomplished due to the reduced cost and enhanced efficiency of SE. Over the past 10 years, the cost of solar panels has fallen by 80%, making SE the most economical alternative for producing new electricity in numerous regions globally [9]. New research conducted by the United States Department of Energy revealed that the SE reaching the earth’s surface in a mere 90 min is more than sufficient to fulfill the universal energy requirements for an entire year [10]. Additionally, the sun is located approximately 148 million kilometers away from the Earth and emits a tremendous amount of energy. Every hour, about 430 quadrillion kilojoules of this energy reach the earth [11]. Therefore, the main focus at present is to determine the most efficient approach to harness SE.

Solar power is produced by the sun in the form of light and heat, contributing to processes such as heating, initiating chemical reactions, or generating electrical energy [12]. The conversion of SE into electricity primarily involves two key methods: the photovoltaic (PV) system and concentrated SE. The important uses of the PV system include: (a) powering plants [13], (b) serving as the main power source for satellites in space [14], and (c) functioning as independent power devices [15], such as for parking meters, roadway lighting, and radio transmitters. The main component of the PV surface (solar panel) is a semiconductor, housing multiple electrons. When sunlight is absorbed by the surface of the solar cell, it induces electron vibration, and their flow through the semiconductor generates electricity.

The Sunrise, which was the world’s first solar aircraft, was created, built, and operated via Astro Flight, Inc. On November 4, 1974, the Sunrise I, a pioneering solar aircraft prototype, effectively completed its first journey. This incredible outcome exhibited an important innovation in aeronautics history, as it was the first time an aircraft had ever flown with SE. The Sunrise I flew for 20 min, reaching an altitude of 330 feet, successfully showcasing the feasibility and potential of solar aircraft. This achievement aided as a critical breakthrough in the development of solar aircraft technology. The Sunrise I aircraft possessed a wingspan of 9.8 m, a length of 4.4 m, a wing area of 8.4 m2, and a total weight of 12.3 kg [16]. During a test flight in 1975, the Sunrise I experienced destruction caused by a windstorm [17]. A new version, known as the Sunrise II, was then established. The Sunrise II has the same design as its previous version, with the extra improvement of being 13% lighter, weighing just 10.3 kg [18]. Furthermore, it showcased a remarkable 33% improvement in power compared to the previous Sunrise I solar aircraft. The command-and-control system failure resulted in the Sunrise II falling significantly short of its anticipated distance. Instead of reaching 50,000 km, as initially expected, it only managed to cover a distance of 5.2 km. This incident occurred on September 27, 1975. The Sunrise II was estimated to travel a distance of 50,000 km on September 27, 1975. However, due to a malfunction in the flight control system, it was only able to cover a distance of 5.2 km [19]. The massive damage of the solar aircraft led to the dissolution of the testing aircraft program. The solar panels that were initially installed on the Sunrise II were detached after 5 years and then securely attached to the Gossamer Penguin, a different solar aircraft that is operated by human pilots. On July 26, 2015, Bertrand Piccard and André Borschberg, Swiss aeronauts, commenced a remarkable operation to accomplish the feat of being the first persons to travel around the globe using an aircraft powered by SE. The Solar Impulse 2, which they piloted, was an airplane particularly engineered to drive using only SE as its energy source [20]. On July 26, 2015, Bertrand Piccard and André Borschberg, Swiss aeronauts, began a mission to make history by being the first individuals to circumnavigate the globe using an aircraft powered solely by SE [21]. The duration of the flight was 505 days, which covered a total distance of 26,000 miles. The aircraft, piloted by Piccard and Borschberg, showcased the incredible abilities of airplanes with a weight of 2.3 tons and a wingspan of 236 ft, this solar aircraft showcased the technological improvement and emphasized the abilities of SE in transfiguring the transportation sector [22]. Figure 1 exhibits a demonstration of the solar-powered airplanes. The study conducted via Shukla and Nema [23] provides a thorough explanation of the research carried out on solar aircraft systems. Khan et al. [24] analyzed solar-powered aircraft, covering their historical development and the potential challenges they may face in the future. A review study on design and development procedures that can be utilized to heat transport on solar-powered airplanes was conducted by Sajid et al. [25]. Development in the heat transfer properties of nanofluid (NF) due to the interaction of inclined magnetic field and non-uniform heat source was conducted by Jena et al. [26]. Influence of velocity slip on the MHD flow of a micropolar fluid over a stretching surface was conducted by Pattnaik et al. [27].

Figure 1 Solar aircraft.
Figure 1

Solar aircraft.

NFs have shown a great capability to improve the heat transport efficiency of SE systems. This innovative kind of fluid holds fantastic potential in increasing the performance of SE systems [28]. NFs possess unique thermal properties that aid in more capable thermal transport compared to standard fluids. One of these remarkable characteristics is their high thermal conductivity, which means that they can transport heat more efficiently. This is vital for SE systems since it allows them to boost the speed and performance of transforming SE into electricity power. NFs can increase the heat transport performance of SE systems by up to 30%, resulting in substantial energy and cost savings [29]. This enhancement indicates a better conversion of SE into electricity power. NFs are a combination of liquid that contains nanoparticles (NPs) made of metallic materials such as iron, aluminum, and copper, or non-metallic materials like iron(iii) oxide, polyvinyl chloride, aluminum oxide, and silica [30]. These NPs are suspended in a base fluid (BF) [31]. NFs have a wide range of relevance, such as chemical processing, food production, and powering SE systems. The concept of “nanotechnology” was first coined in 1974 by Norio Taniguchi, a Japanese scientist from the Tokyo University of Science. In his discussion on nanotechnology, he studied the prospective for producing materials at the nanometric scale [32].

In recent times, researchers have shown interest in a novel category of NFs known as hybrid (HNFs) [33]. These are created by dispersing two types of NPs within a fluid medium. HNFs exhibit the potential for enhanced thermal and rheological properties. The thermal properties of water-based HNF (Cu-Al2O3) beyond an inclined plane was conducted by Mohd et al. [34]. Soret–Dufour effects on the water-based HNF flow with NPs of alumina and copper was conducted by Isa et al. [35]. Stagnation bioconvection flow of titanium and aluminum alloy NF containing gyrotactic microorganisms over an exponentially vertical sheet was conducted by Soid et al. [36].

In the present era, many researchers are focused on scrutinizing the measurable and physical properties of manufacturing liquids because of their critical roles in mechanical and industrial sectors [22]. These properties fall under the classification of non-Newtonian fluids (NNFs), which include petroleum oils and lubricants, cement, pharmaceutical goods, and production adhesives [37]. Finding a correct solution is often difficult due to the highly nonlinear nature of the differential equations governing NNFs. Using NNFs in the industrial sector provides numerous benefits. These liquids possess a range of viscosities, which allows the customization of flow characteristics that can be useful in numerous production processes [31]. NNFs can enhance the behavior of products by offering distinct properties such as dilatant armor and pseudoplastic lubricants [38]. NNFs are utilized in many food and healthcare products including ketchup, toothpaste, blood polymers, and drug delivery systems [39]. Additionally, NNFs play an important role in the creation of paints and coatings as they assist in controlling the thickness and smoothness of the materials. Various categories of NNF model, such as Maxwell fluid, Williamson fluid, second grade and grade three liquids, Carreau fluid, and so on are commonly utilized to depict the performance of NNFs. The NNFs tangent hyperbolic fluid (THF) model holds importance among these NNF models [40]. This is because it has the significant capability to correctly calculate the phenomenon of high shear conditions. The THF model is the NNFs approach used to describe the performance of liquids that experience shear thinning. Shear thinning describes the property of liquids where their viscosity reduces as the shear rate increases [41]. This performance is commonly observed in several practical applications including wire drawing, blood, paper production, polymer solutions, and paints [42]. According to Jamshed et al. [48], the extra stress tensor for the THF model [43,44] can be expressed in the following manner:

(1) τ ⃡ = δ ̇ [ α ∞ + ( α 0 + α ∞ ) tanh ( Γ δ ̇ ) n ] ,

where δ ̇ is expressed as

(2) δ ̇ = 1 2 ∑ a ∑ b δ ̇ ab δ ̇ ab = π 2

with

π = 1 2 trace ( grad V → + ( grad V → ) T ) 2 .

To investigate the shear thinning properties based on theoretical principles, we examine the conditions of α ∞ = 0 and Γ α ̇ < 1. The necessary form can be acquired in the following manner:

(3) τ ⃡ = α 0 [ ( Γ δ ̇ ) n ] δ ̇ = α 0 [ ( 1 + Γ δ ̇ − 1 ) n ] δ ̇ = α 0 [ 1 + n ( Γ δ ̇ − 1 ) ] δ ̇ .

1.1 Aim and originality of the current study

The use of SE has become gradually popular as a major method of harnessing heat from the sun. This is achieved through the use of advanced technologies, including electricity generation, PV lighting, receiver designs, space heating and cooling, and solar water pumping. The aim of these current findings is to investigate the SE and increase the performance of solar-powered airplanes by including nanotechnology-based energy. The study focuses on inspecting how heat is transported on the solar aircraft wing, specifically within the parabolic trough solar collector (PTSC) configuration via employing an HNF. The main objective is to enhance the performance of solar aircraft wings. This solar aircraft wing is constructed in a special design known as a PTSC, which is a technology employed for capturing, heat transmission fluid, and concentrating SE. The objective of this research is to enhance the design of solar-powered airplanes by presenting novel technologies like HNF to boost the performance of thermal transport. Previous research work on SE utilization has not fully included novel technologies such as HNF, which can increase thermal transport in airplane wings. The purpose of this work is to link this gap by studying the applicability and advantages of using a new HNF, comprising of copper ( Cu ) and zirconium dioxide ( Zr O 2 ) NPs suspended in ethylene glycol ( EG ) BF, to increase the thermal transport of their performance. Entropy creation analysis is performed to calculate the effect of flow. However, the study includes the use of hybrid NPs. The group of equations describing momentum, energy, and entropy generation were solved using the Galerkin weighted residual method.

1.2 Applications of the proposed study

The proposed method of using ZrO2–Cu/EG HNF to increase the thermal expansion in solar airplanes has various potential applications in the fields of solar thermal technology and the aircraft industry. The following are some practical usages of this proposed method: Temperature regulation and control, manufacturing engines, sundry industrial operations, biomedical applications, solar technology industries, and radiator cooling.

1.3 Present theoretical experiment for model one

The utilization of HNF flow in a parabolic trough solar collector is proposed in this theoretical experiment as a means to boost the performance of solar airplane wings. Numerous PTSCs are installed in several locations on the wing of the airplane. The motive behind the current work is mentioned below:

  • In the proposed model, a PTSC is employed as a replacement for PV cell sheets.

  • PTSC has a cylindrical shape, unlike absolute PV cell sheets. Therefore, PTSC has a better capability to absorb SE because of its greater surface area.

  • Investigators in the aviation sector are increasingly exploring SE as a sustainable alternative to decrease dependency on fossil fuels, given its gradual price rise.

  • Solar airplane aircrafts have the potential to be developed and maintained at a low cost, thus placing them as a good option within the aviation sector.

  • Based on the experiment’s findings, the introduction of an HNF into the fluid flowing through PTSC leads to an improvement in thermal transport and generates more power. The higher thermal transport is attributed to numerous factors, including, thermal radiation and viscous dissipation phenomena.

  • solar airplanes are nature-friendly as they do not emit any harmful substances into the atmosphere. The present theoretical experiment is visually depicted in Figure 2.

Figure 2 Present theoretical experiment.
Figure 2

Present theoretical experiment.

2 Model development

  • The ethylene glycol (EG) is taken as a base fluid (BF).

  • Non-Newtonian tangent hyperbolic fluid, Cattaneo–Christov heat flux

  • Thermal radiative flow, Porous medium, NF scheme of Tiwari-Das

  • Convective conditions and slippery velocity

  • Zirconium dioxide ( Zr O 2 ), boundary-layer approximations

  • Two-dimensional laminar steady flowing

  • Viscous dissipation, HNFs.

2.1 Physical description

Using flow analysis, this study describes the movement of a horizontal surface via utilizing non-regularly extending velocity.

(4) U w ( x , t ) = bx ,

where Θ w ( x , 0 ) = Θ ∞ + b * x is the insulated surface temperature, Θ ∞ is the thermal variant rate, b is the preliminary extending ratio, and b * is the rate of thermal differences.

2.2 Geometric flow structure

Figure 3 displays the flow dimension configuration.

Figure 3 Flow model description.
Figure 3

Flow model description.

Given the assumptions stated earlier, the equations of two-dimensional steady HNF flow together with the factors of THF model, magnetic field, velocity slip, solar radiation, and suction/injection are as follows [45]:

Continuity equation:

(5) ∂ ς 1 ∂ x + ∂ ς 2 ∂ y = 0 ,

Momentum equations:

(6) ς 1 ∂ ς 1 ∂ x + ς 2 ∂ ς 2 ∂ y = ( 1 − n ) + n √ 2 ϱ ∂ ς 2 ∂ y μ hnf ρ hnf ∂ 2 ς 1 ∂ y 2 − μ hnf ρ hnf k ς 1 ,

Energy equation:

(7) ς 1 ∂ Θ ∂ x + ς 2 ∂ Θ ∂ y = k hnf ( ρ C p ) hnf ∂ 2 Θ ∂ y 2 + μ hnf ( ρ C p ) hnf ∂ ς 1 ∂ x 2 − ω * ς 1 ∂ ς 1 ∂ x ∂ Θ ∂ x + ς 2 ∂ ς 2 ∂ y ∂ Θ ∂ y + ς 1 ∂ ς 2 ∂ x ∂ Θ ∂ x + ς 2 ∂ ς 1 ∂ y ∂ Θ ∂ y + ς 1 2 ∂ 2 Θ ∂ x 2 + ς 2 2 ∂ 2 Θ ∂ x 2 + 2 ς 1 ς 2 ∂ 2 Θ ∂ x ∂ y − 1 ( ρ C p ) hnf ∂ q r ∂ y

subjected to:

(8) ς 1 ( x , 0 ) = U w + μ hnf ∂ ς 1 ∂ x , ς 1 ( x , 0 ) = V w , − k g ∂ Θ ∂ y , = h g ( Θ w − Θ ) , ς 1 ⟶ 0 , Θ ⟶ Θ ∞ , as y ⟶ ∞ .

Table 1 presents information on the thermodynamic properties of the HNF as well as the notation employed in the present study. When considering the heating of a surface through convection, it is essential to account for the heat lost due to conduction, which is described as Newtonian heating. This feature holds important relevance in assessing the movement of thermal energy. Insight into the performance of the fluid near the surface is important, as it can be influenced via the slip condition. Therefore, this describes how the liquids interact with solid boundaries, influencing the velocity of the fluid close to the surface. Two NPs, Cu and ZrO2, were added to EG BF, each of them as a concentration factor size φ Cu and φ Zr O 2 .

Table 1

Thermo-physical features of the nanofluid and hybrid nanofluid

Features NF HNF
Viscosity μ nf = μ f ( 1 ‒ ϕ ) ‒ 2.5 μ hnf = μ f ( 1 ‒ ϕ Cu ) ‒ 2.5 ( 1 ‒ ϕ Zr O 2 ) ‒ 2.5
Density ρ nf = ( 1 ‒ ϕ ) ρ f ‒ ϕ ρ p s ρ hnf = ( 1 ‒ ϕ Zr O 2 ) [ ( 1 ‒ ϕ Cu ) ρ f + ϕ Cu ρ p 1 ] + ϕ Zr O 2 ρ p 2
Heat capacity ( ρ C p ) nf = ( 1 ‒ ϕ ) ( ρ C p ) f ‒ ϕ ( ρ C p ) s ( ρ C p ) hnf = ( 1 ‒ ϕ Zr O 2 ) [ ( 1 ‒ ϕ Cu ) ( ρ C p ) f + ϕ Cu ( ρ C p ) p 1 ] ϕ Zr O 2 ( ρ C p ) p 1
Thermal conductivity k gf k f = k s + 2 k f ‒ 2 ϕ ( k f ‒ k s ) k s + 2 k f ‒ ϕ ( k f ‒ k s ) k hnf k gf = ( k p 2 + 2 k gf ) ‒ 2 ϕ Zr O 2 ( k gf ‒ k p 2 ) ( k p 2 + 2 k gf ) + ϕ Zr O 2 ( k gf ‒ k p 2 )

2.3 Thermodynamics and physical properties of the non-Newtonian tangent hyperbolic HNF

The HNF is composed of a BF ( EG and NPs [ Cu and Zr O 2 ] made up of two elements). By introducing this mixture of NPs into a regular fluid, the rate of heat transfer is enhanced as a result of its capability to provide a higher amount of thermal energy. The thermodynamic properties of the tangent hyperbolic HNF are presented in Table 1 [46,47].

2.4 NP features and BF

Heat exchangers (HEs) are widely used in numerous engineering applications, such as power generation, renewable energy systems, heat recovery ventilation, and thermal management in electronics. These devices are responsible for efficiently transferring heat between two or more fluids. The operation of HE is important in systems engineering. However, HE plays a crucial role, and their efficiency is of greatest significance. Many approaches have been utilized to increase their efficiency; however, the thermal conductivity of the transport fluid is usually restricted in achieving important progress. EG has attracted the interest of scientists due to its multi-functional chemical with utilizations in automotive cooling systems, fuel additives, HVAC systems, and potential carriers for hydrogen gas. The improvement of catalysts for synthesizing EG using copper as a base metal is greatly anticipated, regardless of the large industrialization in this field. EG is relevant and useful in various applications: it has a low freezing temperature, which is of assistance for preventing the fluid from freezing in regions with lower temperatures or during nighttime when solar radiation is insufficient. An earlier study has considerably examined the improvement in thermal conductivities of EG systems, resulting in the development of several significant findings. Table 2 shows the thermal characteristics of Cu , Zr O 2 NPs, and EG BF.

Table 2

Thermal characteristics of Cu and Zr O 2 NPs, and EG BF

Physical property EG ZrO2 Cu
ρ ( kg / m 3 ) 1,110 5,680 8,933
k ( W / mK ) 0.253 1.7 401.00
C p ( J / kg K ) 22,000 502 385

2.5 Approximate Roseland form

The Rosseland approximation (RA) is suitable for handling slight temperature differences between the surface and the surrounding fluid. The energy equation shows a high nonlinearity in the temperature ( Θ ) of the fluid, which is hard to describe through computational means. Nevertheless, by minimizing the thermal gradients within the stream, an important simplification can be attained. Given the appropriate conditions, the suggested expression via Rosseland can be simplified and represented in terms of Θ ∞ via substituting Θ 3 with ( Θ ∞ ) 3 . The RA is employed in Eq. (4) to incorporate the influences of radiation. This inclusion is attained through the use of the following expression:

(9) qr = − 4 σ * 3 k * ∂ Θ ∂ y ,

where k * is the absorption coefficient and σ * denotes the Stefan–Boltzmann constant.

3 The considered problem is solved for tangent hyperbolic HNF

The governing equations mentioned above are solved by applying appropriate transformations. The stream functions and similarity quantities are defined as follows [48]:

(10) ς 1 = ∂ ψ ∂ y , ς 2 = − ∂ ψ ∂ x , Φ ( χ ) = Θ − Θ ∞ Θ w − Θ ∞ , ψ ( x , y ) = υ f b xp ( χ ) , and χ ( x , y ) = b υ f y ,

where Φ and p are functions of χ By replacing Eq. (10) in Eqs (5)–(7), the following outcome is obtained:

(11) p ‴ ( 1 − n ) + n We p ‴ p ″ + ϕ a ϕ b P ″ p − ϕ a ϕ b p ′ 2 − K p ′ = 0 ,

(12) Φ ″ 1 + 1 ϕ d PrNr + Pr ϕ e ϕ d p Φ ′ + ϕ c PrEc ϕ d p ′ 2 − ϕ c Pr ε * ϕ d p p ′ Φ ′ − ϕ c Pr ε * ϕ d p 2 Φ ″ = 0 ,

with boundary conditions (BCs)

(13) p ( 0 ) = S , p ′ ( 0 ) = 1 + γ ϕ a p ″ ( 0 ) , p ′ ( χ ) ⟶ 0 , Φ ′ ( 0 ) = − B i ( 1 − θ ( 0 ) ) , Φ ( χ ) ⟶ 0 , as χ ⟶ ∞ ,

where

ϕ a = ( 1 − ϕ Cu ) 2.5 ( 1 − ϕ Zr O 2 ) 2.5 , ϕ b = ( 1 − ϕ Zr O 2 ) ( 1 − ϕ Cu ) + ϕ 1 ρ p 1 p f + ϕ Zr O 2 ρ p 1 p f ,

ϕ c = ( 1 − ϕ Zr O 2 ) ( 1 − ϕ Cu ) + ϕ Cu ( ρ C p ) p 1 ( ρ C p ) f + ϕ Zr O 2 ( ρ C p ) p 2 ( ρ C p ) f ,

ϕ d = k hnf k f = ( k p 2 + 2 k nf ) − 2 ϕ Zr O 2 ( k nf − k p 2 ) ( k p 2 + 2 k nf ) + ϕ Zr O 2 ( k nf − k p 2 ) x ( k p 1 + 2 k f ) + ϕ Cu ( k f − k p 1 ) ( k p 1 + 2 k f ) + 2 ϕ Cu ( k f − k p 1 ) .

Moreover, Table 3 displays the formula and parameters in the governing equations.

Table 3

Formula and parameters in the governing equations

Formula Parameters Formula Parameters
Pr = υ f α f , Prandtl number ϕ a , ϕ b , ϕ c , ϕ d NP concentration
Ec = U w 2 ( C p ) f ω w ‒ ω ∞ Eckert number h g Thermal conductivity of solid
W e = ς 2 b 3 υ f Weissenberg number k hnf Thermal conductance
V w Surface permeability
B i = h g k g υ f b Biot number ρ C p Specific heat
K = υ f bk Porous medium ε * = b λ o Thermal relaxation time
S = ‒ V w 1 υ f b Suction/injection k g Thermal conductivity of solid
Nr = 16 3 σ * ω ∞ 3 k * υ f ( ρ C p ) f Solar-radiation parameter σ hnf Electrical conductivity
γ = b μ f Velocity slip parameter ς 1 , ς 2 Velocity component

3.1 Physical quantities of engineering interest

The parameters that engineers analyze and consider when designing systems are known as engineering quantities of interest. These physical quantities play an important role in the field of engineering. Several fields of engineering depend on two important factors, known as skin resistance ( C f ) and local Nusselt number ( Nu x ). The C f is important in fields like fluid mechanics and aerospace as it determines the amount of resistance that a solid object experiences when it moves through a fluid. The Nu x is an essential aspect of HE design as it determines the rate at which heat is transmitted between a liquid and a solid surface. The two parameters serve the purpose of enhancing the effectiveness of engineering designs. The skin resistance ( C f ) together with the local Nusselt number ( Nu x ) are mathematically expressed as follows:

(14) C f = τ w ρ hnf U w 2 and Nu x = x q w k f ( Θ w − Θ ∞ ) ,

After utilizing equation (10) in equation (14), the following dimensionless form of the skin friction and heat transfer rate can take place as:

(15) Re x 1 / 2 C f = 1 ϕ a ϕ b ( 1 − n ) f ″ ( 0 ) + n W e 2 ( f ″ ( 0 ) ) 2 ,

where τ w = μ hnf ( 1 − n ) ∂ υ 1 ∂ y y = 0 + n ς x ∂ υ 1 ∂ y 2 y = 0

and

(16) Re x 1 / 2 Nu x = − k hnf k f [ ( 1 − Nr ) θ ′ ( 0 ) ] ,

where q w = − k hnf 16 3 σ * Θ ∞ 3 k * υ f ( ρ C p ) f ∂ Θ ∂ y .

3.2 Entropy generation analysis

Entropy generation is a term used to describe the amount of irreversibility that happens during physical or chemical processes. This concept is crucial in thermodynamics as it helps to measure how effective energy conversion processes are. In SE systems, understanding entropy generation is essential since it allows researchers to enhance system performance and efficiency. Various factors contribute to entropy generation in SE applications such as heat transfer losses, fluid friction losses, and radiative losses. By scrutinizing these factors using entropy generation analysis, experts can pinpoint areas where enhancements can be made to improve system efficiency and minimize waste. For example, analyzing entropy generation can help optimize the design of solar collectors by minimizing heat transfer losses and enhancing overall performance. The expression of entropy generation ( N G ) is as follows:

(17) N G = Re ϕ d ( 1 − Nr ) Φ ′ 2 + Br Ω 1 ϕ a ( p ″ 2 + Kp ′ 2 ) ,

where N G = b 2 u ∞ 2 E G k f ( Θ w − Θ ∞ ) 2 , Br = μ f u w 2 k f ( Θ w − Θ ∞ ) , Re = u w b 2 x υ f , Ω = Θ w − Θ ∞ Θ ∞ are the entropy generation, Brinkmann number, Reynolds number, and the difference in temperatures.

4 Galerkin weighted residual scheme (GWRS): A highly effective numerical scheme

The GWRS is a computational technique employed in the field of computational fluid dynamics (CFD) for solving partial differential equations (PDEs) that govern the performance of fluid flow. In this computational technique, we utilize an approximate solution to the PDEs by introducing a trial function that effectively satisfies the BCs. In addition, the weighting functions are chosen to fulfill the mathematical properties. In conclusion, the GWRS provides an effective solution for solving issues related to CFD by offering applicability to non-linear issues, convergence solutions, and reliable outcomes.

4.1 Application of GWRS

The GWRS is employed to calculate the numerical solutions for formulas (10–12). The necessary steps of the GWRS are presented below:

The first step, consider the expressions (18) and (19)

(18) p ‴ ( 1 − n ) + n We p ‴ p ″ + ϕ a ϕ b P ″ p − ϕ a ϕ b p ′ 2 − K p ′ = 0 ,

(19) Φ ″ 1 + 1 ϕ d PrNr + Pr ϕ e ϕ d p Φ ′ + ϕ c PrEc ϕ d p ′ 2 − ϕ c Pr ε * ϕ d p p ′ Φ ′ − ϕ c Pr ε * ϕ d p 2 Φ '' = 0 .

In the second step, the trial function is employed to simulate the numerical outcome of Eqs (18) and (19):

(20) p ′ ( χ ) = w 0 + ⅇ − χ / 3 w 1 + ⅇ − 2 χ / 3 w 2 + ⅇ − χ w 3 + ⅇ − 4 χ / 3 w 4 + ⅇ − 5 χ / 3 w 5 + ⅇ − 2 χ w 6 + ⅇ − 7 χ / 3 w 7 + ⅇ − 8 χ / 3 w 8 + ⅇ − 3 χ w 9 + ⅇ − 10 χ / 3 w 10 + ⅇ − 11 χ / 3 w 11 + ⅇ − 4 χ w 12 + ⅇ − 13 χ / 3 w 13 + ⅇ − 14 χ / 3 w 14 + ⅇ − 5 χ w 15 + ⅇ − 16 χ / 3 w 16 + … … + w n ⅇ − χ / 3 = ∑ k = 0 n w k * ⅇ − x k 3 ,

(21) Φ ′ ( χ ) = ⅇ − χ / 3 s 1 + ⅇ − 2 χ / 3 s 2 + ⅇ − χ s 3 + ⅇ − 4 χ / 3 s 4 + ⅇ − 5 χ / 3 s 5 + ⅇ − 2 χ s 6 + ⅇ − 7 χ / 3 s 7 + ⅇ − 8 χ / 3 s 8 + ⅇ − 3 χ s 9 + ⅇ − 10 χ / 3 s 10 + ⅇ − 11 χ / 3 s 11 + ⅇ − 4 χ s 12 + ⅇ − 13 χ / 3 s 13 + ⅇ − 14 χ / 3 s 14 + ⅇ − 5 χ s 15 + ⅇ − 16 χ / 3 s 16 + … … … + w n ⅇ − χ / 3 = ∑ k = 0 n s k * ⅇ − x k 3 .

In the third step, it is important to make sure that the trial solutions proposed in the GWRS satisfy the boundary conditions for expression (13) as required:

(22) ∑ k = 0 n w k ⅇ − k χ / 3 − S χ = 0 = 0 , d d χ ∑ k = 0 m w k ⅇ − k χ / 3 − γ d 2 d χ 2 ∑ k = 0 m w k ⅇ − k χ / 3 − 1 χ = 0 = 0 , d d χ ∑ k = 0 m s k ⅇ − k χ / 3 + B i ( 1 − d d χ ∑ k = 0 n s k ⅇ − k χ / 3 χ = 0 = 0 .

In the fourth step, we construct the residual vectors for velocity flow p ′ ( χ ) , and thermal distribution Φ ′ ( χ ) . This is done by including the trial solutions (20) into the expression that was earlier introduced in the first step.

(23) R p = p ‴ ( 1 − n ) + n We p ∼ ‴ p ∼ ″ + ϕ a ϕ b p ∼ ‴ p ∼ − ϕ a ϕ b p ∼ ′ 2 − K p ∼ ′ ≅ 0 ,

(24) R Φ = Φ ∼ ″ 1 + 1 ϕ d PrNr + Pr ϕ e ϕ d p Φ ∼ ″ + ϕ c PrEc ϕ d p ∼ ′ 2 − ϕ c Pr ε * ϕ d p ∼ p ∼ ′ Φ ∼ ′ − ϕ c Pr ε * ϕ d p 2 Φ ∼ ″ ≅ 0 .

In the fifth step, to improve the accuracy of our findings, we begin by reducing the residual errors through a procedure of putting the integral of the residual functions and weight functions ⅇ − k χ / 3 , k = 0, 1, …, M−2 to zero, i.e.,

∫ 0 ∞ R p ⅇ − k χ / 3 d χ ≈ ∑ k = 1 m [ B k ( R p ⅇ − k χ / 3 ) χ = y k ] = 0 ,

and

(25) ∫ 0 ∞ R Φ ⅇ − k χ / 3 d χ ≈ ∑ k = 1 m [ B k ( R Φ ⅇ − k χ / 3 ) χ = y k ] = 0 ,

where B k is expressed as follows:

(26) B k = 1 U ′ i ( y k ) ∫ 0 ∞ ‍ U i ( y ) e − y y − y k d y = ( n ! ) 2 y k ( U ′ j ( y k ) ) 2 , U i = e y d i d y i ( e − y y i ) .

The algebraic equations were computed using a mathematical tool, and the flow chart of the GWRM is shown in Figure 4.

Figure 4 The GWRS flow chart.
Figure 4

The GWRS flow chart.

5 Results and discussion

In this section, the effect of controlling parameters on velocity f ′ ( η ) , temperature θ ( η ) , entropy generation N G , drag coefficient ( C f ) , and heat transfer rate ( Nu x ) for Cu -engine oil and Ag − e ngine oil NF are presented graphically. The ranges of influencing parameters are: Weissenberg number parameter ( We = 1 , 2 , 3 ) , Reynolds number ( Re = 0.2 , 0.4 , 0.6 ), Solar radiation parameter ( Nr = 1 , 2 , 3 ), Brinkman number ( Br = 0.2 , 0.4 , 0.6 ), NPs parameter ( ϕ , ϕ h = 0.02 , 0.06 , 0.1 ), Injection parameter (S < 0 = −(0.1, 0.2, 0.3)), Eckert number ( Ec = 1, 3, 5), velocity slip ( γ = 0.1 , 0.2 , 0.3 ), and suction parameter (S > 0 = 0.1, 0.2, 0.3). Comparison of consistencies available in studies is summarized in Table 4. However, highly accurate outcomes about the present analysis are obtained.

Table 4

Comparison of Nusselt number ( Nu x ) for varying values of the Prandtl number (Pr)

Pr 0.72 1 3 10
Rauf et al. [49] 0.8086 1.0000 1.9237 3.7207
Shahzad et al. [50] 0.8086 1.0100 1.9211 3.7204
Obalalu et al. [6] 0.8086 1.0000 1.9237 3.7207
Current result 0.8085 1.0000 1.9236 3.7203

5.1 Cu-EG NF and Zr O 2 - Cu / EG HNF for relative thermal transport rate

The skin resistance ( C f ) and local Nusselt number ( Nu x ) for two different fluid compositions, Zr O 2 - Cu //EG and Cu-EG, were examined through parametric investigation. The outcome of this investigation displayed in Table 4 provides a valuable understanding of the connection between C f and Nu x . The rate of fluid motion is substantially impacted via C f , mostly in conditions where viscous-based THFs are in combination with NFs and HNF. In addition to the thermal features revealed via the components within the system, these variations in C f have the power to influence the rate at which thermal transport is carried out as a result of the manipulations made to the fluid motion. The parameter We plays a role in reducing C f resulting in smoother fluid motion by improving the lubrication of the surface. Other physical characteristics such as K and S parameters also add to the reduction in C f for both Zr O 2 - Cu / EG HNF and Cu - EG NF. The HNF of Zr O 2 - Cu / EG had higher skin resistance compared to a single suspended Cu - EG NF due to the existence of more particle mixtures in the fluid system. In addition, the effectiveness of thermal transport from a solid surface to a fluid motion is quantified via the Nu x . The Nu x values obtained from Table 4 present evidence of the anticipated findings, confirming the expectation that utilizing Zr O 2 - Cu / EG HNF combinations would produce a better heat transport rate compared to the Cu- EG NF. The parameter B i increases the Nu x . Meanwhile, the Nu x is reduced through parameters such as We , K , and Ec . The combination of Zr O - Cu / EG HNF demonstrates superior thermal transport efficiency with the key features. This improvement makes it a beneficial choice for increasing heat transport processes (Table 5).

Table 5

Values of C f and Nu x for Cu -EG and Zr O 2 - Cu /EG

Ec We k B i S C f Cu -EG C f Zr O 2 - Cu /EG Nu x Cu -EG Nu x Zr O 2 - Cu /EG
0.2 0.1 0.1 0.2 0.5 5.2711 5.5371 4.6124 4.7229
0.3 0.1 0.1 0.2 0.5 5.2711 5.5371 4.4639 4.5789
0.4 0.1 0.1 0.2 0.5 5.2711 5.5371 4.4361 4.5340
0.2 0.1 0.1 0.2 0.5 5.5212 5.5593 4.4758 4.8107
0.2 0.2 0.1 0.2 0.5 5.5194 5.5343 4.5741 4.7328
0.2 0.3 0.1 0.2 0.5 5.5126 5.5241 4.5619 4.8589
0.2 0.1 0.1 0.2 0.5 5.2371 5.4899 4.4158 4.5431
0.2 0.1 0.2 0.2 0.5 5.2543 5.4902 4.4281 4.5486
0.2 0.1 0.3 0.2 0.5 5.2744 5.5014 4.4327 4.5507
0.2 0.1 0.1 0.2 0.5 5.4369 5.5891 4.0723 4.1559
0.2 0.1 0.1 0.4 0.5 5.4369 5.5891 4.0739 4.1637
0.2 0.1 0.1 0.6 0.5 5.4369 5.5891 4.0735 4.1689
0.2 0.1 0.1 0.2 0.1 5.3187 5.4943 4.2101 4.5240
0.2 0.1 0.1 0.2 0.3 5.4862 5.5270 4.3217 4.5392
0.2 0.1 0.1 0.2 0.5 5.4716 5.5481 4.4958 4.5487

5.2 Cu-EG NF and Zr O 2 -Cu/ EG HNF thermal performance against solar radiation (Nr)

Thermal radiation refers to the electromagnetic radiation emitted by a body due to its temperature. As shown in Figure 5, the thermal boundary layer thickness of Zr O 2 - Cu / EG HNF is smaller compared to that of Cu-EG NF mixture. The higher value of Nr shows that the intensity of radiation exceeds that of the convection. However, the release of heat by the NPs into the surrounding environment contributes to an improvement in fluid temperature. Additionally, the inclusion of NPs can lead to increased solar radiation, further increasing the temperature distribution within the system. In physical application, the combination NFs performance is higher in cooling and heating processes. Higher solar radiation levels indicate a greater absorption of radiative thermal energy via the system. Engineers can improve the productivity and usefulness of systems by understanding the effect of the solar radiation parameter on the thermal distribution of NFs, which is important for designing solar collectors. From the outcome presented in Figure 6, it shows that the internal energy within the system goes through an irreversible process. It is needed to correctly model and calculate entropy production in these fluid systems by understanding the importance of the Nr . However, the ratio at which entropy is created serves as a measure to calculate the level of irreversibility process. When the value of Nr increases, the rate at which entropy is created will also improve. In order to create energy innovations that are both environmentally friendly and energy-efficient, it is imperative to have an understanding of the necessary physical processes involved and how to minimize the creation of entropy. Figure 6 demonstrates that the Cu-EG NFs have a greater capacity for heat transfer when compared to the Zr O 2 - Cu / EG HNF.

Figure 5 Thermal propagation variation against Nr . {\rm{Nr}}.
Figure 5

Thermal propagation variation against Nr .

Figure 6 Influence of Nr {\rm{Nr}} on the entropy generation.
Figure 6

Influence of Nr on the entropy generation.

5.3 Cu-EG NF and Zr O 2 - Cu / EG HNF for velocity and thermal performance against Weissenberg number (We)

The Weissenberg number (We) is a measure that helps us understand how much a material deforms or stretches in a specific direction. It is especially useful when describing processes like natural shear. This number highlights the non-Newtonian behavior of the fluid, indicating that the fluid becomes tougher and less flowy due to frictional manipulations during stretching. This is evident in Figure 7, where the flow becomes restricted at higher values of the Weissenberg number (We). The results presented in Figure 7 show that as the values of We increase, the fluid velocity of the Zr O 2 - Cu / EG HNF is higher compared to the Cu - EG NF mixture. The Weissenberg number is relevant in understanding the flow behavior of drilling fluids, muds, and other NNFs used in oil and gas exploration. It helps in designing drilling processes and equipment to account for the non-Newtonian nature of these fluids. The improved Weissenberg number (We) makes the fluid move more slowly, giving it enough time to absorb heat from the surface. This causes the heat to spread out more, as shown in Figure 8. Therefore, the results shown in Figure 8 display that as the values of We increase, the temperature distribution of the Cu - EG NF is higher compared to the Zr O 2 - Cu / EG HNF.

Figure 7 Effect of increase in We {\rm{We}} on the velocity distribution.
Figure 7

Effect of increase in We on the velocity distribution.

Figure 8 Thermal field response to We {\rm{We}} .
Figure 8

Thermal field response to We .

5.4 Cu-EG NF and Zr O 2 - Cu / EG HNF velocity against Eckert number (Ec)

The Ec characterizes the balance between kinetic energy and enthalpy variations in a fluid movement, providing an understanding of the implication of kinetic energy compared to enthalpy variations in a thermodynamic process. The significance of Ec can be observed in Figures 9 and 10, where the Cu- EG NF exhibits a higher potential for heat transfer compared to the Zr O 2 - Cu / EG HNF. When ( Ec > 1 ), it implies that the kinetic energy of the fluid exceeds the enthalpy difference during flow. The kinetic energy happens to be the main factor in the thermal transport process. In these situations, the kinetic energy becomes important in the heat transport process. Also, the conversion of kinetic energy into thermal energy becomes more efficient, increasing thermal propagation. This indicates that the liquid encounters a large temperature rise as its kinetic energy is converted into heat. Physically, the Ec finds a realistic advantage in the design of heat transport systems. However, the present outcome can be employed to advance the thermal transport system, ensuring that the fluid continues at a safe temperature with no overheating.

Figure 9 Temperature field of various Ec {\rm{Ec}} .
Figure 9

Temperature field of various Ec .

Figure 10 Irreversibility field of various Ec {\rm{Ec}} .
Figure 10

Irreversibility field of various Ec .

5.5 Cu-EG NF and Zr O 2 - Cu / EG HNF velocity against slip velocity ( γ )

The slip velocity signifies the speed difference between a liquid and a solid surface that are in contact with one another. It represents the motion of the liquid relative to the solid boundary. The velocity slip has a noticeable effect in numerous fluid dynamics and has influential outcomes, remarkably when working with small scales and solid surfaces. Figure 11 shows the impact of γ on the velocity distribution. The findings exhibit that γ leads to a decrease in the velocity distribution. However, when slip occurs, the changes in fluid motion are examined, particularly focusing on the phenomenon where the liquid close to the solid surface moves at a higher speed. When the slip in velocity increases, the velocity flow in the far-field shows a decrease. As a result, the forces exerted on the stretched surface are reduced, leading to a diminished ability to transfer energy to the system. Figure 11 illustrates that the Cu − EG NF mixture exhibits greater heat transfer capability in comparison to the Zr O 2 - Cu / EG HNF.

Figure 11 Effect of the increase in γ \gamma on the velocity distribution.
Figure 11

Effect of the increase in γ on the velocity distribution.

5.6 Cu-EG NF and Zr O 2 - Cu / EG HNF thermal performance against Biot number ( B i )

The value of B i is a non-dimensional value that quantifies the correlation between the thermal transport resistance within a solid matter and the thermal transport resistance at its surface. However, as the temperature of the internal body responds to temperature differences at the surface, the degree of temperature regulator within the body goes through substantial changes over time. In circumstances where a uniform temperature distribution within the surface is needed, B i substantially greater than 1 ( B i > 1 ) is considered. On the other hand, when B i is much less than 1 ( B i < 1) shows that the temperature distribution within the surface is non-uniform. Figure 12 provides three scenarios illustrating the impact of the B i on temperature distributions. The first situation corresponds to a B i < 1. However, low value of B i signifies that the internal resistance to thermal flow is more important than the surface convective resistance. This indicates that thermal transport from the surface to the surrounding fluid is effective, resulting in an enhanced fluid temperature (as observed in the case of a nano polymer surface). In the second scenario, when B i > 1, the temperature inside the body reveals non-uniform performance. The third situation, denoted via an infinitely large ( B i = ∞), signifies a situation where the wall temperature remains constant all through (as depicted in Figure 12). The findings shown in Figure 13 explain that the irreversibility field of the Zr O 2 - Cu / EG HNF is greater than that of the Cu-EG NF mixture as the values of B i increase.

Figure 12 Influence of the increase in B i {B}_{{\rm{i}}} on the temperature distribution.
Figure 12

Influence of the increase in B i on the temperature distribution.

Figure 13 Effect of the increase in B i {B}_{{\rm{i}}} on the irreversibility field.
Figure 13

Effect of the increase in B i on the irreversibility field.

5.7 Cu-EG NF and Zr O 2 - Cu / EG HNF entropy generation against Brinkman number (Br) and Reynolds number (Re)

In the study of fluid mechanics, the Br acts as a dimensionless parameter to evaluate the relative implication of viscous forces in relationship to inertia forces within a fluid. Its value helps in evaluating the importance of these forces in fluid dynamics. The rate of entropy creation reveals an increment as the Br raises, as observed in Figure 14. Physically, the implementation of solar thermal energy in aircraft has the prospective to decrease reliance on non-renewable energy sources and subsequently lower the destructive ecological impression of the aircraft industry. The efficiency of PV cells and the availability of sunshine are important aspects that determine the effectiveness of SE. As the value of Re grows, the rate of entropy creation in the boundary layer rises. This observation is shown in Figure 15. Regulating the Re within the boundary layer of the system allows the physical optimization of solar panel productivity and the creation of entropy in a sun-powered aircraft. This method can be utilized to increase the performance of the fluid system.

Figure 14 Effect of the increase in Re on the irreversibility field.
Figure 14

Effect of the increase in Re on the irreversibility field.

Figure 15 Impact of the increase in Br on the irreversibility field.
Figure 15

Impact of the increase in Br on the irreversibility field.

5.8 Cu-EG NF and Zr O 2 - Cu / EG HNF velocity and thermal performance against volume fraction of NPs ( ϕ , ϕ h )

The performance of liquids, including their fluid motion and thermal distribution, is extremely influenced by the existence of ( ϕ , ϕ h ) . However, the ( ϕ , ϕ h ) in the BF appears to be the basic contributing factor impacting the effectiveness of both the HNF and NFs. The addition of NPs to the NFs leads to an increase in their viscosity, which subsequently decreases the velocity distribution. NPs exhibit better thermal conductivity compared to the BF, as a result, they possess the power to effectively dissipate heat from the surface, thereby lowering the temperature gradient within the NFs. Subsequently, this leads to a reduced velocity distribution. Generally, a lower fluid flow increases thermal transport by letting the liquid have more time to interact with the surface. However, it is vital to note that a lesser fluid flow can also cause a rise in pressure drop. The fractional enhancement boosts the Cu- EG NF mixture over the Zr O 2 - Cu / EG HNF combination, as demonstrated via the lower flow rate observed in Figure 16. When a liquid has extreme flowability, its interaction with the surface is constrained, leading to minimal thermal transport. Physically, there are two key reasons for this phenomenon of heat dispersion. First, via raising the ( ϕ , ϕ h ) , the load power is intensified, resulting in a reduction in fluid movement. Therefore, the thermal absorption efficiency of the system is improved when transferring heat from the surface. Second, the concentration of NPs improves the liquid’s thermophysical characteristics, thereby boosting its capacity to drive heat energy. The flow of the NF substances demonstrates significant coefficients for both convection and conduction, enabling efficient thermal transport. Furthermore, as illustrated in Figure 17, the addition of NPs leads to an enhancement in the temperature distribution. Consequently, Zr O 2 - Cu / EG HNF is favored via the concentration of NPs compared to the Cu- EG NF. Physically, NPs have the power to capture solar thermal energy and subsequently improve the temperature of the NFs. The substantial implication of this feature is noticeable in applications involving the exposure of NPs to sunlight, such as in solar collectors. When NPs absorb sunlight, they become hot and transmit this heat to the fluid. This process leads to a better thermal distribution in the HNF. Figure 18 exhibits that the higher impacts of the nanoparticles volume fraction increases the profiles of the entropy generation. The entropy creation of the Zr O 2 - Cu / EG HNF is found to be minimal compared to the Cu – EG NF, as exhibited in Figure 18. NFs have been found to be less useful in regulating the thermodynamic system when compared to HNF, according to this finding.

Figure 16 Velocity field response to various ( ϕ , ϕ h ) (\phi \left,{\phi }_{{\rm{h}}}) .
Figure 16

Velocity field response to various ( ϕ , ϕ h ) .

Figure 17 Impact of the increase in ( ϕ , ϕ h ) (\phi \left,{\phi }_{{\rm{h}}}) on the temperature field.
Figure 17

Impact of the increase in ( ϕ , ϕ h ) on the temperature field.

Figure 18 Impact of the increase in ( ϕ , ϕ h ) (\phi \left,{\phi }_{{\rm{h}}}) on the irreversibility field.
Figure 18

Impact of the increase in ( ϕ , ϕ h ) on the irreversibility field.

5.9 Cu-EG NF and Zr O 2 - Cu / EG HNF velocity and thermal performance against suction (S > 0) and injection (S < 0)

The process of liquid removal from a system, which can be achieved by the use of a pump or vacuum, is identified as suction ( S > 0 ). On the other hand, injection ( S < 0 ) involves adding liquid into the system. Figures 19 and 22 depict the influence of suction ( S > 0 ) on the velocity, and thermal distribution. On the other hand, Figures 20 and 21 demonstrate the impact of injection ( S < 0 ) on the velocity and thermal distribution. The surface is experiencing the effect of pulling for both Zr O 2 - Cu / EG HNF and Cu-EG due to the increasing ( S > 0 ) values. When compared to the ZrO2-Cu/EG NF, Figures 19 and 22 reveal that the single NF Cu − EG , with suspended NPs, is more easily pushed down by suction. Injection ( S < 0 ) increases in flow velocity when large amounts of liquid are injected through the surface, as shown in Figure 20. However, the fluid temperature exhibits opposing influences. The fluid temperature is resisted via suction ( S > 0 ), resulting in a reduction in flow depicted in Figure 20. It is important to note that greater fluidity produces better results for injection, and despite increased thermal dispersion, it performs well. In conclusion, controlling fluid flow through suction and injection parameters directly influences the velocity distribution and thermal performance.

Figure 19 Impact of the increase in ( S > 0 ) (S\left\gt 0) on the velocity distribution.
Figure 19

Impact of the increase in ( S > 0 ) on the velocity distribution.

Figure 20 Impact of the increase in ( S < 0 S\lt 0 ) on the velocity distribution.
Figure 20

Impact of the increase in ( S < 0 ) on the velocity distribution.

Figure 21 Impact of the increase in ( S < 0 S\lt 0 ) on the temperature field.
Figure 21

Impact of the increase in ( S < 0 ) on the temperature field.

Figure 22 Impact of the increase in ( S > 0 ) (S\left\gt 0) on the temperature field.
Figure 22

Impact of the increase in ( S > 0 ) on the temperature field.

6 Concluding remarks

In summary, the study examines the effect of different parameters on the tangent hyperbolic HNFs past a PTSC installed on the solar aircraft wings. The enhancement of thermal transport performance in solar aircraft has been exhibited via a computational study using two kinds of NFs: Cu and Zr O 2 in EG . The study also utilized the entropy generation and non-Newtonian tangent hyperbolic model. The effect of several variables on the performance of a PTSC is shown through the utilization of tables and graphs to exhibit the outcomes. The design of solar-powered tractors, solar streetlights, and commercial power projects are some of the useful applications of the framework. The following are the main findings obtained from the present study:

  • The Zr O 2 - Cu /EG HNFs have a smoother flow compared to the Cu -EG NFs.

  • Based on the physical parameter utilized, the Galerkin weighted residual method produces a series of converging solutions for the stated problem.

  • Velocity distribution is influenced via several parameters, including the (S < 0). On the other hand, ( ϕ , ϕ h ) , γ , and We parameters have a negative influence on velocity distribution.

  • The fluid temperature was improved due to the increase in the value of solar radiation.

  • The thermal distribution of the Zr O 2 - Cu /EG HNF is positively influenced by the ( S < 0 ), Nr , ( ϕ , ϕ h ) , B i , and We .

  • The velocity distribution reduces as the values of RM , K , δ , and M are increased.

  • The Zr O 2 - Cu /EG HNF significantly elevates the entropy production compared to the Cu -EG NFs.

  • The Cu -EG NFs produce a greater level of solar radiation characteristics.

  • The heat transfer rate of the Zr O 2 - Cu /EG HNF is strongly elevated with a relative percentage of 2.6% compared to the Cu -EG NFs.

7 Future direction

Future research should include experimental validation of the theoretical findings. Conducting experiments to verify the performance of solar collectors using magnetized tangent hyperbolic HNFs under realistic conditions would strengthen the credibility and applicability of the study’s outcomes. Investigate the use of alternative nanomaterials and NFs for improved thermal properties. The study focuses on magnetized tangent hyperbolic HNFs, but exploring other nanomaterials could lead to innovations in thermal management.

  1. Funding information: This research has been funded by the Universiti Kebangsaan Malaysia project number “DIP-2023-005.” Also, the authors extend their appreciation to the Researchers Supporting Project Number (RSP2024R33), King Saud University, Riyadh, Saudi Arabia.

  2. Author contributions: A.M.O., U.K., and O.A.O.: conceptualization, methodology, software, formal analysis, validation, and writing – original draft. A.Z.: writing – original draft, data curation, investigation, visualization, and validation. A.I.: conceptualization, writing – original draft, writing – review and editing, supervision, and resources. El-S.M.S.: validation, investigation, writing – review and editing, formal analysis, project administration, funding acquisition, and software. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-12-17
Revised: 2024-02-04
Accepted: 2024-02-13
Published Online: 2024-03-14

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2023-0207
Keywords for this article
tangent hyperbolic hybrid nanofluids; solar aircraft and concentrated solar power; Cattaneo–Christov heat flux
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
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  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
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  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
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  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
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  24. Study of fractional variable-order lymphatic filariasis infection model
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  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
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  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
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  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
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  105. Review Article
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  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
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  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
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  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
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  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
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  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
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  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
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  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
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