Computational study of cross-flow in entropy-optimized nanofluids

Nomenclature

A b ⁎

exponentially decaying space coefficients

B b ⁎

temperature-dependent heat source/sink

Br b

Brinkman number

c p

specific heat at constant pressure (J kg⁻¹ K⁻¹)

C f

skin friction

Nu x

local Nusselt number

f wa

suction parameter

G '

dimensionless velocity

H

dimensionless temperature

A b ⁎ , B b ⁎ > 0

heat source parameter

A b ⁎ , B b ⁎ < 0

heat sink parameter

Pr

Prandtl number

Re x

local Reynolds number

T a

temperature of the nanofluid (K)

T wa

constant temperature (K)

T ∞

free stream temperature (K)

u wa ( x a )

variable velocity at the surface of the sheet (m s⁻¹)

x a , y a

Cartesian coordinates (m)

v wa ( x a )

mass suction/injection or transpiration velocity at the surface of the sheet

u a , v a

velocities in the x ̅ a ̅ , and y ̅ a ̅ directions, respectively (m s⁻¹)

Greek symbols

ε b

positive constant

γ b

expanding/contracting factor

μ

absolute viscosity (Pa s)

υ

kinematic viscosity (m² s⁻¹)

k

thermal conductivity (W mK⁻¹)

ρ

density (kg m⁻³)

φ

nanoparticle volume fraction

Ω b

dimensionless temperature difference

η

pseudo-similarity variable

Acronyms

BCs

boundary conditions

EG

entropy generation

2D

two-dimensional flow

Bvp4c

boundary value problem of fourth-order

TiO₂

titania nanoparticles

NFD

nanofluid

PDEs

partial differential equations

ODEs

ordinary differential equations

SBES

stable branch solutions

USBES

unstable branch solutions

Subscripts

nf

nanofluid

f

regular fluid

w

wall boundary condition

∞

far-field condition

Superscript

‘

derivative w.r.t. η

1 Introduction

There are many nanoparticles and nanometer-sized molecules present in a nanofluid (NFD). Although the structure and content of nanomolecules depend on carbides, metals, and carbon nanotubes, tiny particles are dispersed (added) in fluids to improve their heat transfer properties. Most often, the components that make up an NFD are mixed with average-sized nanoscale particles. NFDs are frequently utilized to create nanotubes, nanofibers, nanowires, nanoparticles, nanosheets, nanorods, etc. The most effective and practical methods have been developed and put into practice for the modeling of NFD flow models, while the addition of specific materials of a particular kind has increased the thermal conductivity of various fluids. Several engineering procedures use nanoparticles in fluids to improve heat transfer. Equipment like heat exchangers in heavy machinery, automobiles, and industries is highly dependent on effective energy transfer. With these applications in mind, Choi and Eastman [1] developed NFDs to enhance the heat transfer capabilities of ordinary fluids. Eastman et al. [2] discovered a 40% increase in thermal conductivity when copper nanoparticles are added to ethylene glycol at a 0.3% volume concentration. Khan and Pop [3] have numerically addressed the issue of laminar fluid flow that develops when a flat surface is stretched in an NFD. It was discovered that the reduced heat transfer is a diminishing function of every single dimensionless number. The numerical solution of the boundary layer flow caused by a linearly extending sheet in an NFD was inspected by Makinde and Aziz [4]. They observed that the convective heating, thermophoresis, and Brownian motion all get more intense as the local temperature rises, which causes the thermal boundary layer to thicken. The thermal conductivity performance of carbon nanotubes in fluid flow over a stretching sheet was discovered by Haq et al. [5]. Sheikholeslami et al. [6] discovered the impacts of Lorentz forces on free convective flow in the presence of NFD with thermal radiation. A two-dimensional time-independent flow conveying an NFD toward a thin needle was deliberated by Soid et al. [7] where the existence of multiple solutions was reported. Bakar et al. [8] looked into the stability analysis of mass suction impacts through a shrinking/stretching cylinder considering the nanoparticles. Kamal et al. [9] investigated the flow of a nanomaterial through a stretchable/shrinkable sheet with a chemical reaction effect. A two-dimensional magneto mixed convection flow induced by a shrinking/stretching plate in an NFD was investigated by Jumana et al. [10]. Johan and Mansur [11] examined the features of dusty nanomaterial flow and thermal transport analysis past a stretchable sheet with a slip boundary. They used three types of nanoparticles, namely, copper, alumina, and titania. Shahzad et al. [12] inspected the impact of the slip effect on the flow and heat transfer induced by a copper NFD with different shape factors through a heated stretched sheet and found that the platelet shape factor particles have a greater rate of heat transfer compared to other shape factors. The significant impacts of viscosity dependent on the temperature through the slip flow with the assessment of bioconvection by involving a Maxwell NFD across a stretchy sheet were examined by Khan et al. [13]. Yasmin et al. [14] investigated the 3D unsteady flow of an NFD across a Riga plate with convective and slip conditions. They observed that the slip effect reduces the velocity, whereas the Biot number enhances the temperature of the NFD. The effect of activation energy on the 3D buoyancy flow of Sutterby fluid from an extended bidirectional surface with dissipation and thermophoresis effects was inspected by Yasmin et al. [15]. Yasmin et al. [16] scrutinized the blood flow by utilizing a Casson hybrid NFD across a heated surface in porous media. The temperature profile is accelerated by increasing values of the space-dependent heat source factor, constant heat source, and Eckert number. Hussain et al. [17] inspected the impact of chemically reactive and heat-absorbing effects on the dissipative flow of an NFD driven by a stretchable surface with a radiation effect. Their results indicate that the strengths of radiation, magnetic effect, and dissipation effects lead to enhancement of the temperature.

The problem of sheet stretching or shrinking in a viscous fluid has received a lot of attention since it has numerous applications in physics, engineering, and other scientific disciplines. It frequently occurs in practical issues that have attracted a lot of research attention because of their wide range of significance in fields like the production of glass fiber, glass blowing, metal extrusion, transportation, microfluidics, paper production, hot rolling, space, and acoustics (see Fisher [18]). The boundary layer flow over an ongoing solid kind surface flowing at uniform motion was initially studied by Sakiadis [19] in light of these applications. Numerous authors [20,21,22] have thought about different elements of this problem and found similarity solutions since the groundbreaking research conducted by Crane [23], who provided an exact solution for the 2D steady flow caused by a stretchable surface in a quiescent fluid. Instead of focusing on the scenario of a stretched sheet, researchers instead looked at the scenario of a shrinking sheet. According to Goldstein [24], this new kind of flow of shrinking sheet is fundamentally a backward flow. The steady flow across a shrinkable sheet was studied by Miklavčič and Wang [25]. They discovered that mass suction is necessary to continue the flow across a shrinkable sheet. Waini et al. [26] examined the dependency of time-varying flow along with thermal transport across a shrinking/stretching sheet incorporated with hybrid NFDs and provided multiple solutions.

The examination of cross-flow began following the early studies by Prandtl [27] and Blasius [28] that included the laminar flow over a flat surface caused by thin viscosity. It appears that Prandtl [29] is the first to present the results for uniform gradients in pressure flow across a finite yawed cylinder. Cooke and Hall [30] and Eichelbrenner [31] examined various theoretical and numerical techniques that have been developed for simulating a generalized boundary layer 3D flow. Weidman [32] examined the boundary layer via cross-flow generated by plate motions that are transverse. The work of Weidman was recently expanded by Roşca et al. [33] by taking into account rotational stagnation point flow that transports the hybrid NFD along a permeable shrinking or stretching surface. It was discovered that both stretching and shrinking surfaces can have multiple solutions to the fundamental similarity equations.

The phenomena of an irregular heat sink or source have applications in both engineering and medicine, including the cooling of metallic sheets, the design of thrust bearings, the recovery of unrefined oil, etc. In the presence of an irregular heat sink/source, Tawade et al. [34] addressed the motion of the MHD unsteady thin film and heat transfer past a stretchable sheet. It was determined that irregular heat parameters are crucial to the effectiveness of heat transfer. Thumma et al. [35] revealed that the stretching of a sheet caused the MHD convective motion of an NFD to have a changeable heat sink or source. To obtain the solution, a well-known Keller-Box numerical approach was utilized. Kumar et al. [36] looked at the movement of a hybrid ferrofluid film and heat transfer in the inclusion of radiation and an erratic heat source/sink (EHS/EHS). The rate of heat transfer is believed to be greater in hybrid ferrofluids than in ferrofluids. In addition, the velocity of the fluid and temperature tend to decline as the film thickness increases. Areekara et al. [37] investigated the impact of an irregular heat source/sink on the fluid flow of an NFD past a nonlinear stretching sheet. They observed that positive correlations exist between the radiative heat flux and the rate of heat transfer. Negative sensitivity to the rate of heat transfer is shown by the exponential heat source. Akram et al. [38] discussed the concepts of non-linear stretching and EHS/SHS to describe the heat transfer through the stretchable cylinder. It is discovered that the temperature distribution in the fluid region is being disrupted by the non-linear stretching rate and the source of heat.

The scrutiny of second law analysis (SLA) or entropy generation (EG) in fluid flow and heat transport is a prominent area of study. Energy losses resulting from diffusion, chemical reactions, friction between solid surfaces, and fluid viscosity all contribute to the production of entropy in thermodynamic systems. As a result, the formation of EG has drawn a lot of attention to applications, including heat exchangers, turbo machinery, and electronics cooling. Aiboud and Sauoli [39] scrutinized the entropy optimized in viscoelastic flow through a flexible surface subjected to the magnetic field. The effects of slip across a heated vertical surface in an entropy-optimized flow were calculated by Butt et al. [40]. Slips have been found to allow for the control and adjustment of entropy formation in thermal systems. Abolbashari et al. [41] employed HAM to examine EG in a magneto NFD flow near an unsteady stretched surface with a water-based fluid and several types of nanoparticles. Tlau and Ontela [42] examined the role of magnetohydrodynamics in a nanomaterial entropy-optimized flow through an inclined channel with a heat source/sink embedded in a porous media. Entropy optimization of nanomaterial flow across two stretchable rotating disks with effects of bioconvection was examined by Khan et al. [43]. Mondal et al. [44] used trapezoidal lid-driven enclosures with EG to study the Lorentz forces on the constant buoyant flow of Al₂O₃ nanoparticles. According to the calculations, the average Nusselt and Sherwood numbers and aspect ratios all decrease with increasing percentage of nanoparticle volume. Hussain [45] analyzed the EG on the unsteady magneto flow driven by non-Newtonian hybrid NFDs from a stretchable surface with a slip effect.

The literature that is currently accessible indicates that no studies have been conducted on the EG of the cross-flow induced by a nanomaterial past a stretching/shrinking sheet with significant impact on environmental and non-uniform heat generation/absorption. The present investigation fills a research gap by demonstrating cross-flow and heat transfer toward a stagnation point of NFD via a stretching/shrinking sheet with an irregular heat source/sink. Dual solutions are provided for certain values of parameters such as the mass suction and shrinking sheet. The proposed model is originally described via a highly nonlinear system of PDEs. The PDEs are converted into a set of ODEs by using appropriate similarity variables and then solved by employing a bvp4c solver.

The format of the article is as follows: Section 2 describes the description of the model. An SLA is developed in Section 3. Section 4 describes the numerical methodology, and Section 5 discusses the results of the problem. Finally, concluding remarks are provided in Section 6.

2 Description of the flow problem

The NFD stagnation point cross flow and thermal system characteristics past a stretching/shrinking surface with the mutual influence of mass suction and EHSE/EHSK are taken in this study. As portrayed in Figure 1, ( x a , y a ) are Cartesian coordinates measured along the horizontal and vertical surfaces of the sheet, respectively, with the flow occurring in the domain y a ≥ 0 . An NFD is a mixture of regular fluid (water) and titania (TiO₂) nanoparticles. The investigational features of the (water/TiO₂) nanomaterials are assumed to be uniform. In addition, the horizontal surface of the sheet is assumed to have a variable velocity of u a = ε b 2 / 3 ν f 1 / 3 x a 1 / 3 γ b = u wa ( x a ) γ b , where γ b refers to the expanding/contracting factor with γ b < 0 , γ b > 0 , and γ b = 0 signifying the particular cases of shrinking, stretching, and stationary/static sheets, respectively. ε b indicates the positive constant and υ f is the kinematic viscosity of the regular (water) fluid. The ambient or far-field (NFD) is also supposed to have a linear velocity of u ex ( y a ) = ε b y a , see Weidman [32]. Moreover, the mass suction/injection or transpiration velocity at the surface of the sheet is v wa ( x a ) , with v wa ( x a ) < 0 referring to the case of injection and v wa ( x a ) > 0 referring to the case of suction, while v wa ( x a ) = 0 indicating the impermeable surface of the sheet. It is also supposed that the constant temperature of the sheet is T wa , while T ∞ represents the free stream temperature (inviscid fluid). With the help of these aforesaid assumptions, the governing equations in the Cartesian form are written as [32,33]:

with the following boundary conditions (BCs):

Figure 1

Physical model of the problem.

In equations (1)–(4), u a and v a are the NFD velocities in the corresponding x a and y a directions, respectively; T a refers to the temperature of the NFD; A b ⁎ refers to the exponentially decaying space coefficients; and B b ⁎ refers to the temperature-dependent heat source/sink. Therefore, the heat source or absorption phenomenon is produced due to the positive value of A b ⁎ and B b ⁎ , while the phenomenon of heat generation or sink is found by the negative value of both A b ⁎ and B b ⁎ .

Furthermore, k nf indicates the electrical conductivity (EC) of the essential posited NFD, ( ρ c p ) nf indicates the heat capacitance of the NFD, ρ nf indicates the density of the NFD, and μ nf indicates the absolute viscosity of the NFD. The correlation of these NFDs is written as follows:

Here, k f , ρ f , and μ f refer to the thermal conductivity, the density, and the absolute viscosity of the base (water) fluid, respectively, while the heat capacity at constant pressure is represented as c p . Therefore, φ TiO 2 symbolizes the volume fraction of TiO₂ nanoparticles, and the special case φ TiO 2 = 0 reduces equation (5) to a normal or a regular fluid (water). In addition, Table 1 displays the physical data of TiO₂ nanoparticles and the regular fluid (water).

For the considered model, the similarity transformations that can be expressed to further simplify the procedure for mathematical analysis are as follows:

which provides the opportunity to describe flows extremely broadly, regardless of the system size. Also, the prime corresponds to the derivative with respect to η , H is the non-dimensional temperature distributions profile, G describes the non-dimensional quantities, and G ' is the dimensionless velocity profile. However, the mass suction/injection velocity at the surface of the sheet is written as:

In equation (7), f wa is the constant mass suction/blowing constraint with f wa = 0 , f wa < 0 , and f wa > 0 which describe the phenomena of impermeable, blowing, and suction, respectively.

With the help of similarity transformations (6), the continuity equation (1) of the governing model is satisfied, while the rest of equations (2) and (3) change to the resulting known ordinary (similarity) differential equations (ODEs) as:

along with BCs:

In addition, equation (8) for the case of φ TiO 2 = 0 is the same as equation (6.1) in the study of Weidman [32] when α = 1 , but equation (9) with some special effects has been not taken in the same reference paper. Moreover, the dimensionless model comprised the following distinct parameters such as the suction/injection f wa , the expanding/contracting γ b , and the Prandtl number Pr = υ f / α f .

3 SLA

EG, also known as SLA, is a necessary instrument for measuring the energy loss and depreciation in the effectiveness of engineering and industrial systems, such as rate and transport operations. As a result, the systems expend less energy, making EG analysis and comprehension crucial. The scenario of EG is taken into account for viscous Newtonian liquids with the inclusion of nanoparticles.

Two fundamental elements are principally responsible for the EG in the contemplated cross-flow of viscous Newtonian NFDs. The first term in the statement, which is on the right-hand side, denotes the ensuing local heat transfer, and the final term, the consequent fluid friction or viscous dissipation. The SLA is defined as follows in the dimensionless form:

The following formulas are obtained by incorporating the similarity transformations from equation (6) into equation (13). Hence,

where Ω b is the dimensionless temperature difference and B r b is the Brinkman number. Consequently, they are represented mathematically as:

4 Methodology

This section demonstrates the analysis of the assumed cross-flow and the suspension of NFD for heat transfer. The requisite model equations are expressed as highly nonlinear ODEs (8) and (9) along with BCs (10) using similarity variables (6). A built-in function named bvp4c included in the MATLAB software is used to work out these equations numerically. It ought to be noted that the scheme of the finite difference is the foundation for the bvp4c package, which is further highlighted by the three-stage Lobatto IIIA procedure. To instigate the bvp4c method, the transmuted ODEs are modified into a first-order system by launching new-fangled variables. By establishing this process, let

Substituting equation (17) into (8) and (9) along with BCs (10), we obtain first-order ODEs as follows:

with BCs

The code desired initial estimations at the posited mesh point to solve equation (18) and the corresponding conditions (19). The polynomial used in the collective type yields a continuous result. A fourth-order precision set that is equally distributed over the spatial intervals where the function is integrated provides the result. The limitation at a distance ( η → ∞ ) is replaced by the value ( η = η ∞ = 8 ) in many successful boundary layer theory applications, and the relative error tolerance is pre-defined as 10⁻⁶. Additionally, the residual of the smooth output serves as the basis for both the mesh selection and error control. The initial mesh comprises four identical discredited points in the range ( 0 , η ∞ = 8 ) , and the mesh selection is then automatically changed using the bvp4c package. The problem now has two possible solutions, which means that the bvp4c package needed two alternative guesses for the unstable and stable solutions. The early initial guess for the first solution is quite straightforward, while choosing a guess for the second solution is fairly challenging. Merkin [46] and Weidman et al. [47] claim that the first solution is physically stable and reliable, but the second solution is unstable and not physically dependable since the outcome only exists for a specific range of shrinking sheet.

5 Analysis of results

This section presents the binary (stable and unstable) solutions for a certain region or area of the dimensionless mass suction parameter as well as the contracting parameter γ b due to the variations in one physical parameter at the time of computation while the other factors are assumed to be fixed. To ease analysis, the values of the basic physical influential parameters are set as φ TiO 2 = 0.025 , γ b = − 2.0 , f wa = 1.5 , A b ⁎ = 0.1 , and B b ⁎ = 0.1 . Table 1 reveals the experimentation data of TiO₂ nanoparticles and the water-based liquid whose Prandtl number is 6.2. The comparison of the upshots for the unusual case is demonstrated in Table 2. The outcomes of the friction factor, heat transfer, and EG are captured in various graphs (Figures 2–8) of the NFD for the unstable and stable branches owing to the influence of several factors, and their quantitative outputs are shown in Tables 3 and 4. The branches of stable solutions (SBES) as well as those of unstable solutions (USBES) are indicated by the black solid and red dash lines, respectively, in the figures. The position in the graph or picture where both (SBES and USBES) curves meet at a single point is called the bifurcation or critical point. In this study, the SBES and USBES are invented only for the case of shrinking parameter.

Figure 2

Impact of f wa on C f Re x 1 / 2 versus γ b .

Figure 3

Impact of f wa on Nu x Re x − 1 / 2 versus γ b .

Figure 4

Impact of γ b on C f Re x 1 / 2 versus f wa .

Figure 5

Impact of γ b on Nu x Re x − 1 / 2 versus f wa .

Figure 6

Impact of Br b on NG ⁎ versus η .

Figure 7

Impact of φ TiO 2 on NG ⁎ versus η .

Figure 8

Impact of Ω b on N G ⁎ versus η .

The numerical data of gradients (friction factor and heat transfer) with the impression of several distinguished factors corresponding to water-based TiO₂ NFD are illustrated in Tables 3 and 4 for the SBSE and USBES, respectively. Upshots divulge that the friction factor upsurges for the SBSE owing to the superior values of φ TiO 2 and f wa , while for the branch of USBES it behaves distinctly with variations in the mass suction parameter f wa but similarly with higher impacts of φ TiO 2 . Notably, the shear stress of the NFD is the highest and lowest for the SBES and USBES with mass suction parameter f wa . In contrast, the heat transfer escalates for both (SBES and USBES) results with superior consequences of φ TiO 2 . Therefore, owing to the rise in the heat source parameter A b ⁎ , B b ⁎ > 0 , the heat transfer shrinkages in the SBES as well as the USBES, and it is endlessly enriching due to the higher role of the heat sink parameter A b ⁎ , B b ⁎ < 0 . In addition, the lowest and highest heat transfer approximations are perceived for the SBES and USBES with the following selected values A b ⁎ , B b ⁎ = 0.9 and φ TiO 2 = 0.035 .

Figures 2 and 3 show the impact of f wa on the shear stress and heat transfer corresponding to (TiO₂/water) NFDs for the SBES as well as the USBES, respectively. In this study, the dual (SBES and USBES) outcomes are possible to occur for a certain domain of specific set of physical parameters. Therefore, the non-unique outcomes in either pictures or graphs exist for a posited shrinkable sheet. Moreover, it is clear from the above graphs that the position where both solution curves meet is at a point called the critical point. Mathematically, this point is expressed by the symbol γ b C , where the solutions are unique γ b = γ b C . The non-unique solutions (dual) and no solutions are possible to exist for the range γ b C < γ b < ∞ and − ∞ < γ b < γ b C , respectively. Besides, the outcomes refer to the shear stress and heat transfer escalating for the SBES due to the larger impact of f wa , while they are declined for the USBES. Physically, the motion of the NFD stops due to the inspiration of f wa shifting the particles of the liquid moving toward the surface of the sheet and sticking with it. Hence, the friction and motion/velocity of the NFD hold the inverse relations; as a result, the shear stress is enhanced. Furthermore, the next eight distinct critical values −2.1903, −2.2941, −2.4016, −2.5125, −2.6274, −2.7463, −2.8686, and −2.9951 are obtained for the respective change value of f wa . The rise of f wa causes an increment in the absolute value of ∣ γ b C ∣ . This behavior corresponds to the superior inclusion of f wa decelerating the boundary layer separation.

The impact of γ b on the shear stress and heat transfer versus f wa of the (TiO₂/water) NFD for both (SBES and USBES) results are presented in Figures 4 and 5, respectively. The dual (SBES and USBES) results are shown in both graphs for the case of f wa due to the variations in the shrinking parameter. In both graphs, it is seen that the SBES and USBES curves meet at a point called the critical point, which is denoted as f wa C . Meanwhile, the outcomes are unique for the case when f wa = f wa C , but the phenomena f wa C < f wa < ∞ and − ∞ < f wa < f wa C indicate the non-unique outcome and no solution, respectively. Besides, the shear stress decays and rises for the SBES due to the higher values of γ b , while it shrinks for the USBES. Alternatively, with the increase of γ b , the thermal transport phenomenon uplifts for the SBES and declines for the USBES. More significantly, it is understood from the diagrams that bifurcation values like 2.1615, 2.0760, 1.9888, 1.8986, 1.8057, 1.7096, 1.6098, and 1.5066 are found due to the several values of γ b . Also, it is noted here that the magnitude of the critical values ∣ f w a C ∣ is weakened once the shrinking parameter is boosted. This further specifies that the growth in the impacts of γ b hastens the boundary layer (BL) separations.

With the assistance or support of EG or SLA, the thermodynamic system performance of water-based TiO₂ nanoparticles can be improved. Figures 6–8 describe the effect of parameters Br b , φ TiO 2 , and Ω b on the SLA corresponding to the posited NFD for the SBES and USBES, respectively. As premeditated in Figures 6 and 7, an improvement in both parameters of Br b and φ TiO 2 results in an improvement of SLA for the branch of stable as well as unstable solutions. The SLA is also more susceptible to changes in this parameter Br b value at the place close to the wall surface of the sheet. As increases η , susceptibility rapidly decreases. Therefore, the physical data and prior research findings are compatible with this occurrence. Conversely, Figure 8 is designed to inspect the parameter Ω b on SLA of the (TiO₂/water) NFD for both (SBES and USBES). In general, the lower temperature difference between the wall and its surroundings is generally caused by larger values of Ω b , which reduces SLA or EG. The aforementioned discussion leads to the conclusion that by altering the related parameters, the system’s SLA value can be decreased to increase the solar radiation utilization system’s effectiveness.

6 Conclusions

The theoretical inspection on a cross flow- and heat transfer-incorporated water-based TiO₂ NFD through a permeable stretching/shrinking sheet with an irregular heat source/sink has been explored. EG was used to analyze the heat transfer process after the development of a requisite computational model. Combined impacts of pertaining governing parameters like suction, expanding/contracting parameter, irregular heat source/sink parameter, and volume fraction of the nanoparticles on shear stress and heat transfer have been analyzed. The important points are gathered as follows:

• The shear stress is enhanced due to f wa and φ TiO 2 , while the heat transfer is accelerated due to φ TiO 2 .

• The domain of dual solutions is increased and boundary layer separation is delayed with higher impacts of f wa ; however, it is decreased with larger γ b .

• The heat transfer accelerates due to the heat source ( A b ⁎ , B b ⁎ > 0 ) and decelerates due to the heat sink ( A b ⁎ , B b ⁎ < 0 ) .

• EG increases in the presence of Brinkman number and nanoparticle volume fraction in both solutions, while it decreases due to the higher impacts of the difference in temperature parameter in both solutions.

7 Practical applications

NFDs are becoming better understood as a result of the substantial boost in thermal efficiency advances and the rate of energy exchange employed in requisite fuel dynamics and automotive coolants. In addition, the study of EG or irreversibility analysis 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 night-time when solar radiation is insufficient. In addition, the significant applications of NFDs by incorporating the impact of EG have been studied theoretically in this article, which is within the scope of nanotechnology reviews.

8 Future direction

The present work can be further inspected by considering the time-dependent flow with different aspects like thermal radiation, magnetic field, viscous dissipation, etc. Also, the mass transfer with chemical reaction is incorporated. Innovative combinations of hybrid NFDs and ferrofluids can be explored by solving current industrial difficulties and integrating a deep understanding of the physical atmosphere.