Entropy minimization of GO–Ag/KO cross-hybrid nanofluid over a convectively heated surface

Nomenclature

B 0

transfer magnetic field

Bi

Biot number

C f x

skin friction coefficient

C p

specific heat ( J ( kg ) − 1 K − 1 )

F'

velocity

g

acceleration due to gravity ( m s − 2 )

h

coefficient of heat transfer ( m − 2 W K − 1 )

k

thermal conductivity ( kg m K − 1 s − 3 )

K*

mean absorpbtion

Kr

chemical reaction parameter

M

magnetic field

Nu x

Nusselt number

n

power index number

Pr

Prandtl number

q r

radiative heat flux ( W m − 2 )

Rd

radiation parameter

S

suction/injection parameter

Sc

Schmidt number

Sh _x

Sherwood number

T

fluid temperature ( K )

T w

wall temperature ( K )

T ∞

ambient temperature ( K )

T f

surface heat K

u , v

component of velocity ( m s − 1 )

v f

kinematic viscosity of KO ( m 2 s − 1 )

ϑ

dimensionless temperature field

We

Weissenberg number

σ

electrical conductivity

ϕ

dimensionless concentration field

φ 1

nanoparticle volume fraction of silver

φ 2

nanoparticle volume fraction of graphene oxide

τ w

wall shear stress ( kg m − 1 s − 2 )

μ f

dynamic viscosity ( kg m − 1 s − 1 )

α f

thermal diffusivity ( m 2 s − 1 )

ρ f

density of nanofluid ( kg m − 3 )

σ *

Stefan–Boltzmann constant ( W m − 1 K − 1 )

1 Introduction

Cross-hybrid nanofluid flow refers to the behavior and characteristics of nanofluids containing nanoparticles from different materials or compositions as they flow through conduits or channels. The study of such flows is important for understanding the transport properties and optimizing their performance in various applications. When cross-hybrid nanofluids flow, the presence of nanoparticles, significantly affects the fluid dynamics and heat transfer capabilities. The nanoparticles alter the rheological properties of the nanofluid, such as viscosity and shear stress, which can impact the flow behavior. Cross-hybrid nanofluid flow has potential applications in various fields. In microfluidics and biomedical applications, the unique flow behavior of these nanofluids can be utilized to optimize drug delivery systems or enhance mixing processes. The concept of mixing nanoparticles in pure fluid has been familiarized primarily by Choi and Eastman [1] to optimize the thermal flow features of pure fluid. Khan et al. [2] observed the Agrawal flow of nanofluid at a stagnant point on a movable disk, incorporating Smoluchowski temperature and slip conditions at the boundary and provided a comprehensive framework to evaluate the flow characteristics and thermal performance of nanofluids in complex scenarios. Wakif [3] studied numerically the 2D magnetohydrodynamic flow of radiative Maxwell nanofluids in the vicinity of a convective heated vertical surface and proved that thermal characteristics are influenced by nanoparticles concentration and magnetic factor. Nazir et al. [4] examined the hydrothermal characteristics of magnetite nanoparticles for naturally convective flow in a square channel and highlighted that cobalt oxide showed a 30% growth in heat transfer rate, while magnetite exhibited an 18% escalation. Similarly, manganese–zinc–ferrite demonstrated 15% intensification, and cobalt ferrite showed a 12% rise in heat transfer rate, respectively. Nazeer et al. [5] performed an entropy production inquiry on blood-gold nanofluid flowing in a wavy horizontal channel, considering velocity and thermal slips, and explored the potential applications of their analysis in the context of skin diseases. Nayak [6] inspected 3D MHD nanofluid flow with thermal transportation on a shrinking sheet using thermal radiative effects. Shaw et al. [7] deliberated on hydro-magnetic flow for cross-hybrid nanofluid with different thermally radiative effects and arbitrary Prandtl numbers.

Because they have less thermal conductivity and moderate heat transfer capability, base and pedestal liquids are unable to satisfy the requirements of new technical and technological developments. This is a serious obstacle. The combination of a broad variety of nano-sized materials can boost the efficiency of the base liquid. Because nanomaterials boost the chemical properties as well as augment the thermal transport behavior to a larger extent, various scholars conducting research on the thermal transport of nanomaterial flows have increased dramatically over the past few years. This is due to the abundance of features in the existing industry technology such as electronic equipment and cooling of engines, nuclear system, medical science, the textile industry, and many other fields. In the beginning, Choi [8] presented the concept of a nanoliquid that boosts the thermal conductivity of the liquid via dispersing nano-scaled particles throughout the liquid. After that, several researchers from a variety of institutions carried out extensive examinations of nanomaterials, taking into account their structure, dimensions, and physicochemical characteristics. An evaluation study for thin needle of the comparative analysis on Ag–H₂O and silver–kerosene oil (Ag–KO) was carried out by Sulochana et al. [9], and the researchers concluded that the role of the heat of Ag–H₂O nanoliquids is superior to that of silver-kerosene nanoliquid. Mosayebidorcheh and Hatami [10] studied the peristaltic nanoliquid and the characteristics of heat transmission in a wavy channel. Using the KVL model, Gireesha et al. [11] noted the dusty nanofluid as it passed an extended surface. They discovered that the use of dusty particles has a vital role in significantly accelerating the rate of heat transmission. Borah and Pati [12] noted the effect that uneven heat had on the flow of nanoliquids by employing an Eulerian-Lagrangian model. Elboughdiri et al. [13] developed a new model for EMHD dissipative flow at the stagnant point for nanofluid and discussed 2D time-dependent second-grade homogeneous flow to explore the behavior of the nanofluids in their configuration. Nazir et al. [14] presented the hybrid nanofluid with aluminum and titanium alloys due to thermal analysis and square cavity. Nazir et al. [15] described the MHD flow of nanoparticles due to forced convective flow and square cavity. Nazeer et al. [16] examined the second law of thermodynamics on Eyring Powell nano liquid in the presence of a magnetic field due to a stretching sheet.

To make it more cost-effective, a novel kind of hybrid nanoliquid is utilized. An unexpected improvement in thermal characteristics can be achieved by incorporating minute quantities of metal nanoparticles or nanotubes into an oxide or metal nanostructure that is already submerged in a base liquid. The characteristics of a “hybrid nanofluid” include a large competent thermal conductivity, solidness, improved heat exchange, and recognition of an excessive perspective proportion. These characteristics, in addition to the significant drawbacks of separate suspensions and the synergistic notion of the nanoframe, are referred to as inordinate perspective proportions. The excellent thermal conductivity of nanofluid results in improvements in energy productivity, reduced operating costs, and significantly enhanced performance. The majority of the fields that deal with heat transfer, including cooling of the generator, biofluid, nuclear reactor, electronic and transformer cooling, solar, thermal frame, lubrication, building cooling and airing, alcohol management, safety, heating pump, spaceship, and aircraft are all potential applications for hybrid nanofluids. Because hybrid nanomaterials offer performance that is far higher than that of nanofluids, their application in industrial settings is becoming increasingly important. These characteristics piqued the interest of numerous academics that were working towards a hybrid nanofluid. The use of these kinds of hybrid nanofluid systems has been the subject of several experiments, and those experiments have produced astonishing results. The LSM approach was utilized by Usman et al. [17] to visualize the importance of different levels of thermal conductivity for the flow of hybrid nanomaterial. Arani and Aberoumand [18] further studied the stagnation point for the stability analysis due to a hybrid nanoliquid induced via a stretch sheet. This flow was caused by stretching the surface. They observed that the suction and injection parameters on the hybrid nanofluid considerably increased the heat transfer rate. Elsaid and Abdel-wahed [19] researched the thermal cooling process of hybrid nanoliquid. They found that using hybrid nanoparticles increased the liquid’s cooling effectiveness by 18–28%. This was found via the investigation of the implication of hybrid nanomaterials in the thermal cooling system. Within a zigzag channel, Alnaqi et al. [20] used the rheology of non-Newtonian fluid characteristics with the PL-nN model to characterize the hybrid liquid.

In magnetohydrodynamics (MHD), investigators examine the behavior of electrically conducting fluids in association with the magnetic fields. The fundamental equations of MHD include the Navier–Stokes equation [21], which describe the motion of the fluid, coupled with the magnetic induction equation [22–24]. These equations, known as the MHD equations, are derived by combining the equations of fluid dynamics and electromagnetism. Dey et al. [25] evaluated multiple solutions for MHD fluid flow and thermal as well as mass transportations on an exponentially stretching sheet with porous effects. Abbas et al. [26] studied the MHD nanofluid flow and thermal transference on a nonlinear stretched sheet with porous effects and other flow conditions. Yaseen and co-workers [27] described the MHD fluid flow of hybrid nanofluid due to movable spongy vertical surface and noted that velocity distribution was opposed/assisted while thermal characteristics have been supported by an upsurge in a magnetic factor. Abbas et al. [28] explained the impact of the third-grade fluid on Darcy–Forchheimer MHD dissipative with the thermal transmission on porous surfaces using Joule heating effects and a computational technique for the evaluation of modeled equations. Mahabaleshwar et al. [29] examined the MHD fluid flow with the effects of CNTs on a shrinking/elongating sheet with thermally radiative effects. Rehman et al. [30] discussed a comparative investigation regarding the thermal transmission for MHD fluid flow on plane and cylindrical surfaces. Abbas et al. [31] probed the thermal conductivity of Casson fluid with an application in solar radiation effect over the exponential sheet. Shankar et al. [32] inspected the combined effect of heat generation with chemical species of significance of MHD nanofluid due to a rotating disk. Mahabaleshwar [33] carried out the significance of CNT for the flow of MHD Casson nanomaterial in a porous surface. Wakif et al. [34] produced a mixed convective flow of MHD Maxwell nanoliquid due to convective heated surface. Nazeer et al. [35] analyzed the second law of thermodynamics due to the Eyring-Powell nanofluid of the magnetic field through the Riga plate. Farooq et al. [36] scrutinized the effect of the nonlinear thermal radiative flow of MHD on Joule heating using a numerical study. Chu et al. [37] examined the numerical assessment of MHD Eyring-Powell fluid subject to the magnetic field.

Thermal radiation refers to the emission of electromagnetic waves, specifically in the form of infrared radiation, by a material or object due to its temperature. This type of radiation is produced by the thermal energy of the atoms and molecules within the material, which causes them to vibrate and emit photons. It is responsible for phenomena such as heat transfer through radiation, blackbody radiation, and the emission and absorption of light by objects based on their temperature. Yu and Wang [38] used thermal radiative effects for convective fluid flow with CNTs combination and found that MWCNTs intensified the scheme rapidly and improved the Nusselt number. Ibrahim et al. [39] simulated computationally 2D thermally radiative fluid flow and observed that the Reynolds, Prandtl, and Eckert numbers have increased the temperature distribution. Tarakaramu et al. [40] inspected 3D couple stress fluid flow past a starched permeable sheet using a heat source and nonlinear thermally radiative effects. Lim et al. [41] debated on thermally radiative effects on fluid flow on a stretched sheet and noted that all the flow profiles have augmented with growth in cylinder curvature. Raza et al. [42] established dual solutions for convective and radiative fluid flow using the effects of injection and suction. Agarwal et al. [43] discussed computationally the thermal transmission for fluid flow on widening and permeable sheets with thermally radiative effects. Waqas et al. [44] studied heat transmission for radiative nanofluid flow on a stretching surface and noticed that thermal distribution has amplified for the upsurge in radiative factor and Biot number. Zeeshan [45] discussed the production of entropy and thermal transmission for MHD fluid flow on an expandable porous sheet with Brownian motion effects and thermal radiation.

Fluid flow on a stretching sheet has been an area of significant research interest. When a flat sheet is stretched or pulled, it induces a flow of fluid over its surface. This phenomenon is confronted in various applications, like the manufacturing of polymer films and fibers, coating processes, and heat transfer systems. The behavior of the fluid flow on the widening surface is influenced by several factors, including the stretching rate, fluid properties, and surface conditions. Understanding the characteristics of the flow, such as velocity and boundary layer development, is crucial for optimizing these processes. Researchers have employed analytical, numerical, and experimental techniques to investigate the flow patterns, heat transfer, and mass transfer associated with fluid flow on a stretching sheet. The insights gained from these studies contribute to the development of efficient and controlled processes in various industrial applications. Alzharani et al. [46] used a stretching surface, with prime emphasis on thermal properties for fluid flow upon the sheet. Cui et al. [47] inspected different models for nanofluid flow past an extending surface using thermal flow generation effects. Sharma and Shaw [48] discussed MHD fluid flow on an elongating sheet using various flow conditions and observed that the velocity slip constraint decreased the skin friction. Nadeem et al. [49] inspected computationally the model for nanofluid on a stretching sheet and noted that fluid motion is opposed while thermal transportation was supported by increase in the Brownian motion factor. Vishalakshi et al. [50] studied the impacts of MHD on the transportation of heat for fluid flow on an elongating surface with porous media effects. Gireesha et al. [51] discussed melting heat transportation for MHD dusty fluid flow on a stretched surface with the effects of the heat flux model proposed by Cattaneo–Christov. Neethu et al. [52] examined the MHD bio-convective flow of nanofluid on an exponentially stretched sheet with dissipative and radiative effects. Patel and Patel [53] discussed mass and heat transmissions for the magnetic field of mixed convective flow of fluid on a nonlinear starched surface immersed by a porous medium with impacts of non-uniform heat absorption and generation. Hussain et al. [54] elaborated the heat generation for the MHD convective flow subject to the nonlinear thermal radiative flow.

According to the aforementioned studies, most scientists are working on elucidating the magnetic field of suction/injection flow at a specified wall temperature with a hybrid nanofluid, but they have not analyzed cross-hybrid nanofluid due to a stretching sheet. As a result, filling this gap is the main part of our argument. Our research, therefore, sheds light on the consequences of the second law analysis on a cross-hybrid nanofluid when subjected to convective flow, heat radiation, and chemical reactivity. Entropy generation optimization enhances heat transmission of cross-hybrid nanofluid. Using boundary layer analysis and similarity alteration, the system of governing expression is transmitted into dimensionless ordinary differential expression. MATLAB software is used to communicate with the bvp4c and the shooting technique to numerically solve governing equations that exhibit higher order nonlinearity. This is done to solve the equations numerically. The results of the model that was envisioned with comparative explorations are revealed by graphics and tables. The goal of this research is to answer the following interrelated questions:

1. At different values of MHD, how does the velocity of them effect in the presence of Ag and graphene oxide (GO) nanoparticles.

2. To what levels does the addition of Ag and GO nanoparticles to the existing hybrid nanofluid, volume proportion of the total hybrid nanofluid, and heat source temperature gradients in the domain are influenced by chemically reactive KO carrying Ag and GO nanoparticles.

2 Formulation of the model

The generalized Newtonian liquid model that explains the characteristics of shear-thinning and shear-thickening liquids has governing formulas for a cross viscosity model that are expressed as follows:

where B 1 , p 1 , μ 1 , and I , represent the first Rivlin–Ericksen tensor, the pressure, the viscosity, and the identity matrix; the viscosity rheological equation for cross fluid is as follows:

By assuming viscosity infinite shear rate to zero μ ∞ , the stress tensor now becomes

In the presence of simulation, the shear rate of the cross-viscosity model is given as follows:

Cross-hybrid nanofluids are a special instance.

For a 2D flow, the Cauchy stress tensor field is denoted by τ 1 . The fundamental Newtonian model is achieved if we set Γ = 0. For values of n between 0 and 1, n is the power law index for the cross fluid that determines whether or not the fluid acts as a shear-thinning fluid, and it is clear that the cross fluid makes this decision. The current analysis is performed under the following presumptions:

• 2D, laminar, and incompressible flows.

• The gravitational effect is neglected.

• The induced magnetic field and, thus, the electric field are not taken into account.

2.1 Mathematical expressions

Let us consider 2D, steady flow of incompressible MHD flow of radiative flow of cross-hybrid nanofluid due to a stretching sheet. It is necessary to take into account the thermal radiation, heat source/sink, convective heat, and suction/injection effects to evaluate the capacity of hybrid nanofluid flow to transfer heat. The current model’s geometry and physical domain are shown in Figure 1, where x and y are the Cartesian coordinates. The y-axis is vertical to the surface, whereas the x-axis is measured parallel to it. Uw = bx ( b > 0 , constant) describes how fast the sheet wall expands in both directions from the origin. The energy equation also makes use of thermal radiation and heat absorption. The chemical reaction is added to the concentration equation. The wall temperature and concentration are represented by T w and C w . T ∞ and C ∞ are symbolized by free stream temperature and concentration.

Figure 1

The Schematic flow model.

The standard boundary layer equations for the flow of cross-hybrid nanofluids, which adhere to the aforementioned norms, are as follows:

(6) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(7) u ∂ u ∂ x + v ∂ u ∂ y = ν hnf ∂ ∂ y ∂ u ∂ y 1 + Γ ∂ u ∂ y n + σ hnf ρ hnf B 0 2 u 2 ,

(8) u ∂ T ∂ x + v ∂ T ∂ y = k hnf ( ρ C P ) hnf ∂ 2 T ∂ y 2 − 1 ( ρ C P ) hnf ∂ q r ∂ y + Q 0 ( T − T ∞ ) ( ρ C P ) hnf ,

(9) u ∂ C ∂ x + v ∂ C ∂ y = D b ∂ 2 C ∂ y 2 − kc ( C − C ∞ ) .

Boundary condition is

(10) u = U w ( x ) = bx , v = − V w ( x ) , − k hnf ∂ T ∂ y = h f ( T f − T ) , C = C w at y = 0 , u = 0 , T → T ∞ , C → C ∞ as y → ∞ ,

where x- and y-axes have the velocity components u and v , n is the power index, g is the gravitational acceleration, Γ is the cross time constant, k hnf is the thermal conductivity of hybrid nanofluid, ν hnf is the kinematics viscosity of hybrid nanofluid, ρ hnf is the density of hybrid nanofluid, q r is the radiative heat fluid, and σ hnf is the electrical conductivity of hybrid nanofluid. Finally, the thermophysical characteristics of the hybrid nanofluid are deliberated as follows [41]:

(11) μ hnf μ f = 1 ( 1 − φ 1 ) 2 5 ( 1 − φ 2 ) 2 / 5 , ρ hnf ρ F = ( 1 − φ 2 ) ( 1 − φ 1 ) + φ 1 ρ 1 S ρ f + φ 2 ρ 2 S ρ f , k hnf = k nf * k S 2 + 2 k f − 2 φ 2 ( k f − k S 2 ) k S 2 + 2 k f + 2 φ 2 ( k f − k S 2 ) , k nf = k f * k S 1 + 2 k f − 2 φ 1 ( k f − k S 1 ) k S 1 + 2 k f + 2 φ 1 ( k f − k S 1 ) , ( ρ c P ) hnf ( ρ c P ) f = ( 1 − φ 2 ) ( 1 − φ 2 ) + φ 1 ( ρ c P ) 1 S ( ρ c P ) f + φ 2 ( ρ c P ) 2 S ( ρ c P ) f , σ hnf = σ nf σ 2 ( 1 + 2 φ 2 ) + 2 σ nf ( 1 − φ 2 ) σ 2 ( 1 − φ 2 ) + σ nf ( 1 + φ 2 ) , σ nf = σ f σ 1 ( 1 + 2 φ 1 ) + 2 σ f ( 1 − φ 1 ) σ 1 ( 1 − φ 1 ) + σ f ( 1 + φ 1 ) .

The ensuing alterations in similarity are displayed [8,9] as follows:

(12) ζ = bx v f y , u = bx F ′ ( ζ ) , v = − a v f F ( ζ ) , ϑ ( ζ ) = T − T ∞ T w − T ∞ , Φ ( ζ ) = C − C ∞ C w − C ∞ .

To turn into an ordinary differential equation, we utilize the following [9]:

(13) Y 1 [ ( 1 + ( 1 − n ) ( W F ″ ) n ) F ‴ ] + Y 2 [ F F ″ − ( F ) 2 ] + { 1 + ( W F ″ ) n } 2 − MF' Y 3 { 1 + ( W F ″ ) n } 2 = 0 ,

(14) ϑ ″ Y 4 + 4 3 Rd + Y 5 Pr ( F ϑ ′ ) + Pr Q ϑ = 0 ,

(15) φ ″ + Sc F φ ′ − ScKr φ .

Boundary constraints associated with this conversion are as follows [9]:

(16) F ( 0 ) = S , F ′ ( 0 ) = 1 , Λ 4 ϑ ′ ( 0 ) = − Bi ( 1 − ϑ ( 0 ) ) , φ ( 0 ) = 1 F ′ ( ∞ ) = 0 , ϑ ( ∞ ) → 0 , φ ( ∞ ) → 0 .

The associated non-dimensionless parameters are defined as follows:

Weissenberg number We = Γ Re , Thermal Radiation parameter Rd = 4 σ * T ∞ 3 k * k F , Prandtl number Pr = μ f ( C p ) f k f , Biot number parameter Bi = h f υ f a × 1 k f . Magnetic parameter A = σ f B 0 2 ρ f u , heat source/sink parameter Q = Q 0 u ( ρ C p ) f , Prandtl number Pr = u ( ρ C p ) f k f , S denotes the fluid suction and injection parameter, chemical reaction Kr = k c b .

Table 1 deliberates the thermophysical properties of the hybrid nanoliquid.

Dynamic viscosity = Y 1 = μ hnf μ f = 1 ( 1 − ϕ 1 − ϕ 2 ) 5 / 2 , Density = Y 2 = ( 1 − φ 2 ) ( 1 − φ 1 ) + φ 1 ρ 1 S ρ F + φ 2 ρ 2 S ρ f , Electrical conductivity = Y 3 ( σ ) hnf ( σ ) f = σ hnf = σ nf σ 2 ( 1 + 2 φ 2 ) + 2 σ nf ( 1 − φ 2 ) σ 2 ( 1 − φ 2 ) + σ nf ( 1 + φ 2 ) , σ nf = σ f σ 1 ( 1 + 2 φ 1 ) + 2 σ f ( 1 − φ 1 ) σ 1 ( 1 − φ 1 ) + σ f ( 1 + φ 1 ) , Thermal Conductivity = Y 4 = k hnf = k nf * k S 2 + 2 k f − 2 φ 2 ( k F − k S 2 ) k S 2 + 2 k f + 2 φ 2 ( k F − k S 2 ) , k nf = k f * k S 1 + 2 k f − 2 φ 1 ( k f − k S 1 ) k S 1 + 2 k f + 2 φ 1 ( k f − k S 1 ) , Y 5 = ( 1 − φ 2 ) ( 1 − φ 1 ) + φ 1 ( ρ c P ) 1 S ( ρ c P ) f + φ 2 ( ρ c P ) 2 S ( ρ c P ) f .

Table 1

Thermophysical features of the hybrid nanoliquid [11]

Physical properties	C p ( J kg − 1 K )	ρ ( kg m − 3 )	k ( W m − 1 K )	σ
KO	2,090	783	2,090	21 × 10−6
GO	717	1,800	5,000	6.3 × 107
Ag	235	10,500	429	63 × 10−6

Drag force C fx , Nusselt number Nu x , and Sherwood number Sh are defined as follows:

(17) C f = τ w ρ f u w 2 , Nu = x q w k f ( T w − T ∞ ) , Sh = x q m k f ( C w − C ∞ ) .

Wall shear stress and heat flux are defined as follows:

(18) τ w = μ hnf ∂ u ∂ y / 1 + Γ ∂ u ∂ y n y = 0 , q w = − k hnf 1 + 16 σ * T ∞ 3 3 k * k f ∂ T ∂ y y = 0 , q m = ∂ C ∂ y y = 0 .

Using equation (12) in equation (18), we obtain

(19) Re x 1 2 C fx = Y 1 F ″ ( 0 ) 1 + ( W F ″ ( 0 ) ) n y = 0 , Re x − 1 2 Nu x = Y 4 + 4 3 Rd ϑ ′ ( 0 ) y = 0 , Re x − 1 / 2 Sh x = − Φ ( 0 ) .

3 Result and discussion

Numerical outcomes have been accessible to examine and visualize the physical impact of governing variables such as the Weissenberg number ( We ), Power index number (n), the magnetic parameter ( M ), suction parameter (S), thermal radiation parameter (Rd), Biot number (Bi), Prandtl number (Pr), Schmidth number, and the chemical species in Figures 2–9. We verify our result and the methodology employed by this study by comparing it with previous findings for ϑ ′ ( 0 ) for varying magnitude of the Prandtl number Pr in Table 2.

Figure 2

(a–d) Plot for F ′ ( ζ ) for the various values of n , M , We , and S .

Figure 3

(a–c) Plot of ϑ ( ζ ) for the various values of We , Bi , and Rd .

Figure 4

(a–c) Plot of Φ ( ζ ) for the various values of Kr and Sc .

Figure 5

(a–c) Plot of Ns ( ζ ) for the various values of We , M , and Rd .

Figure 6

(a–c) Plot of Be ( ζ ) for the various values of We , M , and Rd .

Figure 7

(a–c) Fluctuation of skin friction, Nusselt number, and Sherwood number for S , Rd , and κ .

Figure 8

(a and b) Hybrid nanofluid and nanofluid viewed by contour plot.

Figure 9

Bar graph of: (a) skin friction and (b) Nusselt number.

Figure 2(a) displays the role of the power law index ( n ) over velocity distribution F ′ ( ζ ) . From the figure, it is observed that the escalating n reduces F ′ ( ζ ) . Actually, the behavior of hybrid nanofluid changes to shear thickening from shear thinning with the increase in n . This effect reduces the momentum boundary layer thickness that causes augmentation in the velocity curve. Figure 2(b) shows the impact of a magnetic factor ( M ) over velocity distribution F ′ ( ζ ) . The greater the value of ( M ) , the greater the decline in F ′ ( ζ ) . The more resistance to the hybrid nanofluid flow is observed for higher ( M ) . Physically, the higher M produces Lorentz force to the hybrid nanofluid flow that results from a fall in the velocity field F ′ ( ζ ) . Figure 2(c) shows the impact of the Weissenberg number ( We ) on velocity distribution F ′ ( ζ ) . It has been noted that the higher ( We ) decays F ′ ( ζ ) . The reason is that the We is directly related to the relaxation time. Also, the more relaxation time means that the flow is caused by more resistance. So, by this fact, the hybrid nanofluid experiences more resistance with the higher ( We ) . Thus, the hybrid nanoliquid flow velocity distribution reduces with the increase in ( We ) . Figure 2(d) reveals the role of a suction factor ( S ) on the velocity curve F ′ ( ζ ) . The higher value of ( S ) reduces the momentum boundary layer thickness that causes a diminution in the velocity field.

Figure 3(a) shows the impact of the Weissenberg number ( We ) on temperature distribution ϑ ( ζ ) . From this figure, we see that increasing the value of ( We ) augments ϑ ( ζ ) . In fact, the higher ( We ) boosts up the thermal boundary layer thickness that eventually heightens the thermal profile. Therefore, the higher ( We ) augments the temperature distribution ϑ ( ζ ) . The impact of thermal Biot number ( Bi ) on temperature distribution ϑ ( ζ ) is depicted in Figure 3(b). It is observed from the figure that the higher ( Bi ) enhances ϑ ( ζ ) . This effect is because the rising ( Bi ) increases the coefficient of heat transfer which is actually in direct relation with the thermal Biot number. The higher coefficient of heat transfer increases the rate of heat transport which results in augmentation in the thermal boundary layer thickness and thermal distribution. Consequently, the higher ( Bi ) increases the temperature distribution ϑ ( ζ ) . The effect of the thermal radiation factor ( Rd ) over the thermal curve ϑ ( ζ ) is perceived in Figure 3(c). It is observed that the higher ( Rd ) augments ϑ ( ζ ) . Physically, the higher ( Rd ) means that we produce more heat to the hybrid nanofluid flow system outcome augmentation in the thermal boundary layer thickness and thermal distribution. So, the higher ( Rd ) enhances the thermal distribution ϑ ( ζ ) significantly.

Figure 4(a) shows the role of (S) on the concentration distribution Φ ( ζ ) . The higher S reduces the concentration distribution Φ ( ζ ) . Figure 4(b) shows the role of Schmidt number ( Sc ) on concentration distribution Φ ( ζ ) . By definition, the Schmit number has opposite phenomena with the mass diffusivity which means that the larger Schmidt number will decline the mass diffusivity of the hybrid nanofluid flow and also declines the concentration boundary layer thickness. Therefore, the higher Schmidt number reduces the concentration distribution Φ ( ζ ) . Figure 4(c) exhibits the significance of the chemical reaction factor ( Kr ) on concentration distribution Φ ( ζ ) . From the Figure, it is observed that the higher ( Kr ) reduces the concentration distribution Φ ( ζ ) significantly. The reason is that the higher values of the chemical reaction factor reduce the concentration boundary layer thickness which causes a reduction in the concentration distribution Φ ( ζ ) .

Any system that has a high entropy generation indicates that the device has a low level of efficiency. This is because the entropy generation represents the overall quantity of entropy that is considered in any entropy process.

Figure 5(a–c) is plotted to view the significance of entropy production for the Weissenberg variable ( We ), magnetic field parameter ( M ) , and radiation parameter ( Rd ) due to the cross-hybrid nanofluid ( GO + Ag / KO ) and single nanofluid ( GO / KO ). These sketches display the comparison of cross-hybrid nanofluid ( GO + Ag / KO ) and nanofluid ( GO / KO ). It is an exploration that the hybrid nanofluid exhibits greater entropy comparison of nanofluid. From Figure 5(a), it is clear that the entropy process Ns ( ζ ) declines with the bigger values of We. Physically, it provides more frictional forces between the particles, which leads to a reduction in the momentum boundary layer. Figure 5(b) displays that the entropy production Ns ( ζ ) boosts due to enhancing values of the magnetic field ( M ) . The nanofluid is smaller compared to the hybrid nanofluid. The features of (M) on Ns ( ζ ) are shown in Figure 5(c). It is seen that Ns ( ζ ) increases when (M) rises in this case. As (M) increases, a stronger Lorentz force is generated, increasing both the resistance to liquid flow and Ns ( ζ ) . Figure 6(a)–(c) shows the influence of the Weissenberg variable ( We ), magnetic field parameter ( M ) , and radiation parameter ( Rd ) on Bejan number Be ( ζ ) . The influence of ( We ) on Be ( ζ ) is seen in Figure 6(a). Here there is a decrease in the value of Be ( ζ ) when ( We ) is enhanced. The features of ( M ) on Be ( ζ ) are depicted in Figure 6(b). In this case, higher ( M ) increases Be ( ζ ) . The influence of ( Rd ) on Be ( ζ ) is seen in Figure 6(c). Be ( ζ ) demonstrates unequivocally an upward trend for ( Rd ).

Figure 7(a)–(c) reveals the role of skin friction, heat, and mass transfer for the numerous values of S , Rd , and κ against We , Bi , and Sc . Figure 7(a) shows the fluctuation of the skin friction coefficient for hybrid nanofluid ( GO + Ag / KO ) and mono nanofluid ( GO / KO ) as a function of increasing Williamson variable We and power law index parameter n. The decrease in the skin friction coefficient that occurs with the increasing levels of ( We ) may easily be observed. When the magnitude of the Williamson variable gets stronger, frictional manipulations occur with tougher fluidity because of the increased resistance, physically. Also surprising is the fact that the hybrid nanofluid (GO + Ag/KO) has decreased skin friction magnitude.The skin friction coefficient increases dramatically for both fluid types as the index value (n) increases. Figure 7(b) reveals the correlation between the thermal radiation parameter ( Rd ) and (Bi) fluid temperature the local Nusselt number for a cross flow hybrid nanofluid (GO + Ag/KO) and a mono nanofluid (GO/KO). Figure 7(b) shows that the local Nusselt number, both for ( Rd ) and (Bi), increases with the distance. Since ( Rd ) and (Bi) are linear in the heat transfer, it is discovered that better values for the parameters for GO/KO and GO + Ag/KO increase the local Nusselt number. When compared to other fluids, the local Nusselt number of the hybrid nanofluid (GO + Ag/KO) is greater. Figure 7(c) is plotted for fluctuations in Sherwood number with numerous values of (Sc) against ( κ ) . The escalation in Re x − 1 / 2 Sh x is considered for larger magnitudes of (Sc) and ( κ ) .

The influence of cross-hybrid nanofluid and nanofluid is exhibited in Figure 8(a) and (b). Finally, the bar plot for the hybrid nanofluid and the nanofluid is shownin Figure 9(a) and (b).

Table 3 shows the impact of (S) for numerous magnitudes of (We), (M), and (n) on the drag friction factors of the nanofluid and hybrid nanofluid. From the Table, we see that the higher ( S ) reduces − F ″ ( 0 ) of the nanofluid and hybrid nanofluid. Comparing the effect of the drag friction factors of the nanofluid and hybrid nanofluid, the hybrid nanofluid experiences more effect of ( S ) than the nanofluid. In other words, one can say that the hybrid nanofluid is strongly affected by the suction factor.

Table 4 exhibits the effects of (Bi) and (Rd) on the heat transfer rates of the nanofluid and hybrid nanofluid. As discussed in the above figures, the thermal Biot number augments the coefficient of heat transfer rate which eventually heightens the rate of heat transfer. Also, the higher thermal radiation means that we have provided more heat to the fluid flow system. This additional heat heightens the gradient heat. Comparing the nanofluid and hybrid nanofluid, the hybrid nanofluid is greatly affected by the (Bi) and (Rd) factors.

Table 5 shows how the Schmidt number (Sc) and chemical reaction parameters (Kr) affect the mass transfer rate. Table 5 shows that as the chemical reactive parameter (Kr) is enhanced, the local Sherwood number boosts in magnitude. Parameters for the chemical reaction and Schmidt number both increase as the size of the local mass transport rate increases.

4 Conclusion

Theoretical descriptions are given for the heat and mass transfer in a cross-hybrid nanofluid flow generated by a linear stretch to a sheet. Ag and GO are mixed with the base fluid KO. The importance of cross-hybrid nanofluid concentration is highlighted, among other things. Below is a list of some noteworthy observations.:

• The hybrid nature of the system allows for precise control of the velocity while simultaneously increasing the rate of heat transfer at the surface.

• As M , We , and S are larger, it becomes clear that the velocity of cross-type hybrids and mono-nanofluids decreases, whereas n exhibits the opposite behavior.

• Cross-hybrid and mono nanofluid temperatures rise as We , Rd , and Bi are amplified.

• Entropy production and Bejan number rise with the increases in the We, Rd, and M.

• The local Nusselt number increases with the increase in the values of Bi and Rd, whereas the skin friction coefficient increases with the increase in the values of We but decreases with the increase in the values of M .

• The rate of mass transfer is enhanced by rising Sc and κ .