Dynamics of convective slippery constraints on hybrid radiative Sutterby nanofluid flow by Galerkin finite element simulation

Abbreviations

A

unsteadiness parameter

b

initial stretching rate

B i

Biot number

B r

Brinkman number

C f

frictional force factor

C p

specific heat ( J kg − 1 K − 1 )

E G

dimensional entropy ( J / K )

E

column vectors of order

E c

Eckert number

h

heat transport factor

κ

thermal conductivity ( W m − 1 K − 1 )

k 0

thermal conductivity of the surface

k ⁎

absorbent factor

n

power-law index

N r

radiation parameter

N G

dimensionless entropy generation

Nu x

local Nusselt number

P r

Prandtl number ( ν / α )

q r

radiative heat flux

q w

wall heat flux

Q

heat source

R e

Reynolds number

S

suction/injection parameter

v 1 , v 2

velocity component ( m s − 1 )

U w

velocity of the stretching sheet

V w

vertical velocity

x , y

dimensional space coordinates ( m )

1 Introduction

Nanofluid which has unusual characteristics and is well-known as the suspending colloidal fluid with metallic or nonmetallic nanosolid-particles was studied and discussed by several investigators. Buongiorno [1] said the absolute swiftness of the nanoparticles can be regarded as the total velocities of basefluid and the relative velocity. Seven slip processes were then analyzed: inertia, Brownian, thermophoretic diffusivity, diffusio-phoresis, Magnus impact, drainage of fluids, and gravitational change. Then, he announced that the Brownian and thermophoretic diffusions can play an imperative role in the non-attendance of turbulent impacts. Increased thermal conduction of conventional thermal transmission fluids, for example, H₂O, ethylene-glycol, and mineral oil, is achieved when nanoparticles are applied to a basic fluid. Nanofluids can thus be used for electrical cooling, cooling engines, solar water heating, nuclear reactor, oscillating heat tubing, and as transmission fluids for heat transmission [2]. Garoosi et al. [3] have studied numerically the nanofluids flowing in which using Buongiorno's model. Eiamsa-ard et al. [4] surveyed the improvement in heat transport of TiO₂–H₂O nanofluid into a heat exchanger tube fitted with superlative double-twisted strips. In different physical situations, recent additions are provided for the treatment of nanofluid with heat and mass transport [5,6,7].

Accurate simulation of nanofluid flowing with the macroscopic model is a prerequisite to building nanofluid-operating equipment [8]. Researchers have generally analyzed individual and two-phase macroscopic models. Provided that the relative speeds of nanoparticle and normal fluid are insignificant, the homogeneity monotonic model may be regarded as a single spectrum with its efficiencies. A homogeneous single-phase model has been used by Maiga et al. [9] for investigating forced convective flow of Al₂O₃–H₂O and Al₂O₃–EG nanoliquids in a homogeneously heated tube at completely established laminar and turbulent flowing schemes. Temperature-dependent characteristics for the convection of nano-fluid models have been shown to predict that those heat transfer improvements are greater than that of nanofluid-dependent model used by Palm et al. [10]. Xuan and Roetzel [11] have suggested the dispersion approach for nanofluids. Heris et al. [12] used a dispersive model for the simulation of laminar nanofluid convection in a rotating duct. Mokmeli and Avval [13] utilized both homogeneity monotonic and dispersive schemes for the study of convection nanofluid heat transport. When comparing numerical findings with experimental values, the dispersive model has been seen to be more correct [8]. A recent survey on single phase nanofluid flows is presented in refs. [14,15,16].

The trend in the numerical investigations of heat transport of nanoliquid is to consider it to be a Newtonian fluid; however, it is well documented that numerous nanofluids display a non-Newtonian behavior so it is important to consider it. Consequently, in this analysis, we take account of the significance of non-Newtonian liquid as a Sutterby liquid. Fluid model in the type of Sutterby explains the flux of pseudoplastic behavior. Non-Newtonian Sutterby fluid has been many uses, including those used to spin the fluid in the full process of drilling operations. These liquids are too utilized in fabricated lubricant processing. Refs [17–29] provide more detailed information.

The Sutterby fluid model exhibits highly refined polymer solutions and is one of the non-Newton-based liquids used to analyze the rheological properties of various materials [30]. We are using the transition process to get the ODE system indirect. This program is solved numerically with a built-in shooting method. The leading values of the different parameters involved in speed, temperature, and concentration are defined. Speed, concentration, and temperature are calculated by numbers. The results obtained show that the speed is reduced by the parameter of the objects. Temperature and concentricity are augmented by the parameter [31]. The study of the behavior of non-Newtonian nanofluids is complex and difficult because of the indirect relationship between stress and stress level, and this is because many of the things that happen in the real world are not real and are not the same. It is very easy to solve a line problem but finding solutions to indirect problems is still very difficult. However, the results obtained by numerical methods give unstable points when sorted; otherwise, gaining a complete understanding of the indirect problem is also difficult. If an indirect problem involves unity or has many solutions this adds to the complexity of the numbers. Although numerical and mathematical methods for solving indirect problems are limited, they also have their advantages. Therefore, we cannot ignore any of these two methods, but it is usually fun to solve an indirect problem through analysis. In addition, perforated media is used to transport and conserve energy in many industrial systems, such as heat pipes, solid matrix heat exchangers, electric cooling, and chemical reactors [32]. Various solutions were reported by Hashim and Hamid [33] on Williamson nanofluid flow about suction variable in the current decade. There have been more recent debates [34–36] in regards to non-Newtonian nanofluids. Refs [37–48] provide recent information that considers nanofluids with flowing and heat transmission in a variety of physical descriptions.

Hybrid nanofluids were introduced by Suresh et al. [49] to extra improve the optimistic characteristics of nanoliquids. Hybrid nanofluids are built by the combination of two different kinds of nanosolid-particles. Recent research in the area of nanofluids has focused on hybrid nanofluid thermal systems. The effect of various kinds of nanoparticles, nanoparticle sizes, the volume of nanosolid-particles in the basis fluid, and thermophysical characteristics of nanofluids is nonetheless crucial. The three-dimensional flux of Cu–Al₂O₃/H₂O hybrid nanofluid using the RK-Fehlberg integrated process is analyzed by Devi and Devi [50]. This is caused by Lorentz’s force by the unidirectional linear expansion of the plate. The findings showed that the Cu–Al₂O₃/H₂O hybrid nanofluid heat exchange rate is higher than that of Cu–H₂O nanofluid. Afrand et al. [51] examinated the influence on the rheology conduct of magnetite hybrid nanofluid-silver/ethylene glycol (Fe₃O₄–Ag/EG) of temperature distribution and nanoparticulate concentration. Hayat and Nadeem [52] took the hybrid Brinkman nanoliquid 3D flow, for an assessment of heat transfer characteristics on a linear extending and circular surface with radiative and homogeneity (heterogeneity) reactive flowing of CuO/H₂O and Ag–CuO/H₂O. Ghadikolaei et al. [53] scrutinized the thermo-physical characteristics of magnetohydrodynamics (MHD) mixture TiO₂–Cu/H₂O nanofluid with common geometrical shapes factor for nanoparticles. Hussain et al. [54] considered a hybrid nanofluid containing Al₂O₃–Cu/H₂O flowing into an open cavity with an adiabatic quadrangular obstruction within the hollow. For numerical solution, the finite element method was utilized and the influences of various potential factors on hybrid nanofluid were addressed. Information on mixture nanoliquid flowing and heat transmission properties can be found in refs [55–59].

Recently, the thermodynamics second-law was employed [60] in order to minimize entropy production to find the optimum engineering scheme. The level of irresponsibilities accumulating during a process has been decided with entropy generation [61]. This can be used to test the efficiency of engineering devices with entropy analysis [62]. For example, Oztop and Salem [63] carried out an analysis of the entropy production for free and mixed convective heat transport models. A final volume approach was taken by Shahzad et al. [64] to examine wavy channel entropy and thermohydraulic efficiency with three-corrugation outlines, which are trapezoidal, sinusoidal, and triangular. Iron salt, tannin, and graphic oxide were assembled into GO–Fe₃O₄ hybrid nanofluid by Mehrali et al. [65] due to the stability and abundance. The overall thermal conductance of the system can be increased by 11% with the usage of hybrid solid nanoparticles. Under the effect of a magnetic field, the efficacy of heat transfer of GO–Fe₃O₄ mixed nanofluid enhances, while entropy is decreased with graphene usage instead of purified H₂O by 41%. Using a hot stretched plate, Shahzad et al. [66] made an analysis on chemical reactions and entropy of the 2nd-grade nanofluid under the effect of nonlinear heat radioactivities and MHD. Increment in Reynolds, Brinkmann, and Hartmann numbers results in system entropy elevating, while reduction is observed for higher temperature. Similar analyses were conducted in refs [67–71] for entropy production of nanofluid with the expandable surfaces using diverse geometry.

The finite element technique is commonly used in mathematics and engineering models to numerically solve differential equations. Other applications include sophisticated geometry testing and different forms of material modeling. Khan et al. [72] developed a design-based model to investigate heat increase and thermal management in a separate lid-driven square cavity. They got numerical results using the Galerkin finite element approach (G-FEM). Mourad et al. [73] investigated uniform MHD free convection and heat transmission in a penetrable cavity using a hybrid Fe₃O₄-MWCNT/water nanofluid filled inner elliptic cylinder. To validate the governing equations, they employed the G-FEM. Rana et al. [74] presented an investigation on the electromagnetism aspect of nanofluid on an expandable cylinder under the suspension influence of gyrotactic algae using the Cattaneo–Christov mass and heat flux concept. They used the finite element approach to acquire the mathematical conclusions of the model’s governing equations. Shah et al. [75] investigated the magnetized non-Darcy flow of free convective hybrid nanofluid on a penetrable cylinder using a stable and ambient magnetic field as well as a constant heat source. To numerically resolve the governing set of equations, the control volume finite element approach was used. Nasrin et al. [76] probed the heat transmission behavior of mixture nanoliquids using sinusoidal and lid-driven cavities with trapezoidal shapes. They used the finite element approach to solve the controlling set of partial differential equations regulating fluid temperature and velocity.

Studies on entropy production of SBHNF are infrequent, and none of the papers available addressed the impacts of a penetrable material, viscous dissipation, and radiation flux, shaped-factor via the extended sheet utilizing the nanoliquid Tiwari–Das scheme in detail [77]. The fluid, swiftness, and temperature in Tiwari-Das (monotonic model) are equivalent. The advantages of the single-phase approach are that the slip processes are ignored so that the model is simpler and numerically easy to solve. But the downside of the approach is that the numerical effects of certain situations vary from the results of experiments. Concentrations range from 3 to 20% in this model volume of nanoparticles. The effects of Cu–EO, GO–EO hybrid, and standard nanofluid were only approximated by numerical findings. Therefore, the present investigation focuses on the powerful liquid feature effects and entropy in a numbered system based on the Keller-box process (KBM) of SBHNF in a boundary-layer in order to bridge a gap.

The structure of the current article is as follows: governing mathematical equations are given in Section 2. Solution to the problem is established in Section 3. Galerkin finite element method employed during the numbering procedure is explained in Section 4. Section 5 tells about the validation of the code. In Section 6, we analyzed the development of entropy. Section 7 summarizes the findings as well as the debate. Outcomes along with future guidelines are provided in Section 8.

2 Flow model formulations

The mathematical flow equations show the moved horizontal plate with the irregular expanding velocity [78]:

where b is an original expanding rate. ¥ w ( x , t ) = ¥ ∞ + b ⁎ x 1 − ξ t is isolated surface temperature, for appropriateness, it is considered as constant at x = 0 . Here b ⁎ , ¥ w , and ¥ ∞ represent the rate of temperature variation, the temperature of the surface, and temperature of surroundings, respectively. According to the assumptions, the plate is slippery and temperature variation is subjected to the surface. The hybrid nanofluid is formulated at first with the addition of Cu solid nanoparticles in EO as basefluid at a fractional size ( ϕ R ) and it is constant at 0.09 throughout the investigation. Graphene oxide GO nanomolecules have been expanded in the mixture to achieve a hybrid nanofluid at the concentricity volume ( ϕ H ) .

2.3 Geometric model

Figure 1 depicts the geometrical flow concept.

Figure 1

Scheme of the flow paradigm.

Figure 1 depicts the diagram of the flowing model. The flowing in this scheme is caused by the unidirectional extending of the surface. The flow slippery phenomenon occurs at the plate, and thermal jumping is used to transmit heat from the lamina to the liquid.

3 Nondimensional transformed system

According to similarity transformation converting the controlling partial differential equations into ordinary differential equations (ODEs), equations (2.7) and (2.8) are boundary-value problem. Stream function ψ is specified as:

The next specified similarity quantities are applied

in equations (2.7) and (2.8). We get

with

where χ and θ signify the similarity variable and nondimensional temperature, respectively. In equations (3.3)–(3.4), the consecutive thermo-physical structures ϕ _a, ϕ _b, ϕ _c, and ϕ _d for SBHNF are specified as

It is noted that equation (2.8) is satisfied directly. In all formulas, ′ symbols differentiate w.r.t χ . A = ξ b , λ = B 2 b 2 , and K = ν f ( 1 − ξ t ) b k , respectively, represent unsteadiness parameter, Deborah number, and porousness of material. P r = ν f α f is the Prandtl amount. Thermal diffusivity parameter, thermal radiative flow, and mass transport are specified as α f = κ f ( ρ C p ) f , N r = 16 3 σ ⁎ ¥ ∞ 3 κ ⁎ ν f ( ρ C p ) f , and S = − V w 1 − ξ t ν f   b , respectively. Λ = b ν f ( 1 − ξ t ) N w is the slippery velocity parameter. E c = U w 2 ( C p ) f ( T w − T ∞ ) and B i = h f k 0 ν f ( 1 − ξ t ) b are the Eckert number and Biot number, respectively. Similarity parameter ( ξ ) and time ( t ) are the parameters on which other variables depend. Hence, a non-similar solution for the model is obtained by determining the final results for locally related parameters.

4 G-FEM: A numerical method

Galerkin finite element technique [88] is used to do numerical computations for governing equations model. The following is a summary of FEM:

1. On the ordinary term [χ _e , χ _e+1], the second order nonlinear terms are incorporated to give the powerless integral formulas with weighted remains.

2. To determine the term of stiffness parameters, powerless residual terms are replaced by finite element formulas.

3. To compute the rigidity factors, a normal term [χ _e , χ _e+1] is used, which is then applied to the cluster technique of equations.

4. The collection of algebraic expressions obtained is linearized. To solve them further, an iterative technique is used with a 10⁻⁶ computational tolerance.

5. A calculational domain is generated by modifying the interval [0, ∞) after identifying the greatest variable value

6. Convergence is noticeable. To achieve decisive results, computations are done using a step size of Δ _χ = 0.001.

Figure 2 depicts the G-FEM flow diagram.

Figure 2

Flow chart of G-FEM.

5 Code verification

Rate of heat transfer is measured, and the obtained results from the current method are compared with the results from the literature [89,90]. In this way, it can be seen that the method employed is valid. Comparison of some validities among a few analyses is presented in Table 4; however, results obtained during the current study are very accurate.

Unsteadiness of controlling equations was solved by Das et al. [89] with the help of RK-Fehlberg method. Jamshed et al. [90] used KBM to find the solution of the current model. KBM provides better and more accurate solutions compared with other methods.

6 Entropy analysis

The entropy of the system is generally increased by porous media. Entropy formation of the nanofluids is described as [90]:

Entropy analysis has the following non-dimensional formula [91]:

By formula (3.2), the non-dimensional entropy formula is:

where R e represents the Reynolds number, Brinkmann number is represented by B r , and Ω symbolizes the dimensionless temperature gradient.

7 Results and discussion

The discussion is based on the numerical outcomes attained from the considered model that are itemized in the previous part. The implied parameters in this section are λ , A , K , ϕ , Λ , N r , B i , E c , Q , S , R e , and B r . Physical performance of dimensionless parameters including entropy, swiftness, and temperature is shown in Figures 3–22. Outcomes for Cu–EO conventional SBNF and GO–Cu/EO non-Newtonian SBHNF are attained. Coefficients of drag force along with temperature variations are described in Table 5. Following are default values λ = 0.1 , A = 0.2 , K = 0.1 , ϕ = 0.18 , ϕ H = 0.09 , Λ = 0.3 , n = 0.2 , Q = 0.3 , P r = 7.38 , N r = 0.3 , B i = 0.2 , E c = 0.2 , R e = 5 , B r = 5 , and S = 0.1 . The effects of Deborah number λ on temperature, entropy outlines, and swiftness of Cu–EO mono SBNF and GO–Cu/EO SBHNF are displayed in Figures 3–5. Calculations are accomplished for λ = 0.1 ,   0.2 ,   0.3 at orderly nanoparticles concentration of ϕ R = 0.09 , ϕ H = 0.09 , and ϕ = 0.18 . The swiftness is enhanced when values of λ are raised as shown in Figure 3. So the thickness of the impetus boundary layer is increased. Increment in λ resulted in reducing the velocity profile which caused the momentum of the border sheet to reduce. Motion of the fluid decreases due to resistance. At λ = 0.1 , GO–Cu/EO hybrid nanoliquid has a comparatively higher thickness of boundary-layer than the Cu–EO nanoliquid. The growing value of the resilience-stress variable resulted in increasing the heat in the boundary layer as shown in Figure 4. The rate of heat transfer, as depicted by Nusselt number, decreases for Cu–EO as well as for GO–Cu/EO. Increment in system entropy is observed with increase in the values of λ , as displayed in Figure 5. The behavioral shifts in the velocity and temperature of hybrid nanoliquid are seen in Figures 6 and 7. The amplitude of the resistive body force decreases with the increase in permeability, so a constant reduction in drag is confronted with fluid, and the flow decreases so that the speed in that borderline tends to zero. The vector K affects the nanofluid density directly by reducing the fluid temperature inside the border layer, leading to a reduction in the fluid permeability of the medium. Entropy of system likewise upsurges, in this case, Figure 8. Increment in variable K increases the heat transfer rate. The influence of nanoparticles size ϕ of mono-nanofluids and ϕ hnf of hybrid-nanofluids are depicted in Figures 9 and 10. Here ϕ R = 0.09 , so analysis is done for varied values of ϕ H . Increment in the values of ϕ and ϕ hnf resulted in the velocity profile of nanofluids decreasing. Meanwhile, a profile of temperature is observed to be elevated. The variable ϕ and ϕ hnf are representing solid nanoparticles volume inside basefluid. As there is a higher thermal conductance in solid-molecules than in traditional liquids, increment in the values of ϕ and ϕ hnf resulted in decreasing the velocity profile of the fluid as shown in Figure 9. Temperature profile in the boundary layer also increases. Figure 12 displays the thickness of the thermal boundary layer to increase with increment in the net thermal conductance of nanofluids. Temperature profile of hybrid nanofluids rises at a higher rate as compared with conventional nanofluids. Figure 11 illustrates that entropy is improved for larger solid nanomolecule sizes. For SBNF and SBHNF, Figures 12–14 show the effect of slippery parameter on the flowing, temperature, and entropy producing outlines, respectively. For the slippery behavior Λ , decreasing behavior in velocity is evident, as the slippery effect decelerated the fluid flowing. The thermal energy of the fluid has an opposite effect as nanofluid temperatures rise with an increase in the variable energy Figure 13. Owing to the increased slip parameter Λ , there is a reduction in the drag force between the fluid and its borders. From Figure 14, the entropy is decreased with rising values of Λ , it can be quickly remarked. Larger Λ values mean that the boundary-layer has a reduced velocity. This decreases the velocity in exchange for reducing the frictional forces within fluid friction, which also reduces frictional irreversibility. Thus, its relation to entropy is reduced by entropy, which reduces the entropy of the system. Figure 15 illustrates how the thermal radiation variable impacts the thermal outlines of SBNF and SBHNF, which shows that the nanofluid temperatures are increased for increasing amounts of N r = 0.1 , 0.3 , 0.5 . The increased heating rate behavior as shown in Table 5 would also contribute to an improvement in thermal rendering and efficacy. If the temperature is boosted, the thickness of the thermal boundary-layer is enhanced. This is because there is a higher heat fluxing. Moreover, heat dissipative transfer E c is the kinetic energy to the gap ratio to the boundary-layer enthalpy. Higher value of E c indicates greater kinetic energy, meanwhile, the friction causes heating on the sheet, which resulted in raising the temperature of the liquid. Figure 17 shows the improvement in temperature with increasing values of E c . The increment in N r and E c parameters can be seen in Figures 16 and 18, respectively on the entropy distribution. According to Table 5, the rate of heat transfer increases for N r and E c in both nanofluids (GO–Cu/EO and Cu–EO), while the gradient of the velocity profile is constant at the plate. Thermal efficiency of the system has a vital role in the shape of nanoparticles. According to the researchers, the surface area of nanoparticles has a major role in increasing or decreasing the heat transfer rate. Analysis is done on the effect of solid nanoparticles on the temperature and system entropy. Figures 19 and 20 show the findings for five different shapes of the nanoparticle. Figure 19 shows the values of the shape-factor while using a different shape. The temperature is boosted for an increased amount of m as seen in Figure 20. In addition, for m = 3 , i.e., spherical nanoparticles, the lowest interface temperature is noticed. It is the result of the net surface area of the sphere along with the highest heat transfer rate of fluid into the inner fluid from the surface. This is also verified by the surface heat transmission coefficient in Table 5. In Figure 22, entropy is boosted with increasing values of m . In the spherical-shaped particles, it is obvious that entropy has the smallest rate. Lastly, Reynolds R e and Brinkmann B r numbers’ impacts are provided on entropy production. The R e value raises the entropy to depend on the outcomes. The interpretation: the inertial forces overpower viscid effects when the R e values are high. As seen in Figure 21, the thermal scheme production of entropy thus boosts. The effect of B r on entropy is shown in Figure 22, where the increase in B r increased the entropy production. The fact is that the rise in B r means more heat is dissipated than heat is carried on the surface and entropy is augmented.

Figure 3

Velocity variation vs λ .

Figure 4

Temperature variation vs λ .

Figure 5

Entropy variation vs λ .

Figure 6

Velocity change with K .

Figure 7

Temperature change with K .

Figure 8

Entropy variations against K .

Figure 9

Velocity variation vs ϕ and ϕ hnf .

Figure 10

Temperature variations with ϕ and ϕ hnf .

Figure 11

Entropy variations with ϕ and ϕ hnf .

Figure 12

Velocity variations with Λ .

Figure 13

Temperature variations with Λ .

Figure 14

Entropy variations with Λ .

Figure 15

Temperature variations vs N r .

Figure 16

Entropy variations vs N r .

Figure 17

Temperature variations vs E c .

Figure 18

Entropy variations vs E c .

Figure 19

Temperature variations with m .

Figure 20

Entropy variations with variable m .

Figure 21

Entropy variations vs R e .

Figure 22

Entropy variations vs B r .

8 Final results and future guidance

In the current analysis, a permeable stretched surface was carried out to calculate the boundary-layer flow of non-Newtonian EO based on Cu and GO hybrid nanofluids. The KBM was employed in an investigation in the presence of porous regime, viscidness dissipative flowing, form, entropy, and thermal radiative impact. The above procedure has the following summary:

1. The temperature profile is raised with λ , K , ϕ , ϕ hnf , E c , B i , and N r parameters, while it is declined with Λ .

2. Velocity is diminished with a swelling impact of λ , ϕ , and ϕ hnf .

3. Entropy is raised with augmentation in λ , A , K , ϕ , E c B i , N r , S > 0 , B r , and R e , while it is diminished with the increment in Λ , hence the thermal efficiency of the model is enhanced.

4. For sphere-shaped nanosolid-particles, the utmost heat transport is seen, while for lamina-shaped nanosolid-particles, a lower heat transport is noticed.

5. According to the current study, Cu-GO/EO hybrid nanofluid is more eminent than Cu/EO in heat transmission.

The results of this analysis can be used as guidance for future research, in which thermal efficacy can be calculated by using various kinds of mixture of non-Newtonian nanoliquids (i.e,. Casson, Falkner–Skan, micropolar, Eyring–Powell, etc.). Furthermore, equations may be generalized to include the effects of the viscidity based on temperature, porousness depending on temperatures, and magneto multi-dimensional slippery flowing.