Study of Triangular Fuzzy Hybrid Nanofluids on the Natural Convection Flow and Heat Transfer between Two Vertical Plates

The prime objective of the current study is to examine the effects of third-grade hybrid nanofluid with natural convection utilizing the ferro-particle (Fe3O4) and titanium dioxide (TiO2) and sodium alginate (SA) as a host fluid, flowing through vertical parallel plates, under the fuzzy atmosphere. The dimensionless highly nonlinear coupled ordinary differential equations are computed adopting the bvp4c numerical approach. This is an extremely effective technique with a low computational cost. For validation, it is found that as the volume fraction of (Fe3O4+TiO2) hybrid nanoparticles rises, so does the heat transfer rate. The current and existing results with their comparisons are shown in the form of the tables. The present findings are in good agreement with their previous numerical and analytical results in a crisp atmosphere. The nanoparticles volume fraction of Fe3O4 and TiO2 is taken as uncertain parameters in terms of triangular fuzzy numbers (TFNs) [0, 0.05, 0.1]. The TFNs are controlled by α − cut and the variability of the uncertainty is studied through triangular membership function (MF).


Introduction
Researchers have been attracted by natural convection (NC) flow because of its numerous uses in engineering and scientific problems like heat exchangers, building ventilation, insulation, solar energy collection, refrigeration, nuclear waste repositories, petroleum reservoirs geothermal systems, and chemical catalytic reactors. Convection is used significantly in the manufacturing of solar panels, microstructures during the cooling of molten metals, and free air cooling without the need for fans in real-world applications. Various researchers have looked into the NC-based flow of non-Newtonian and Newtonian fluids between two infinite parallel vertical plates such as Bruce and Na [1] who investigated the heat transfer of NC between vertical flat plates using non-Newtonian Powell-Eyring fluids. Later on, Rajagopal and Na [2] studied the extensive thermodynamic analysis on fundamental functions. e influences of the third-grade non-Newtonian fluid on heat transfer (HT) were examined by Ziabakhsh and Domairry [3] through the homotopy analysis method (HAM). Using the least square method (LSM), Maghsoudi et al. [4] inspected the NC flow of third-grade fluid between two infinite vertical flat plates with a porous media. Mansoor et al. [5] studied the natural convective flow between two vertical plates with the help of the volume of parameter method (VPM) and Runge-Kutta method (RKM). ey show that VPM is better than RKM. Some researchers have explored different flows of fluids between vertical parallel plates analytically and numerically [6,7].
Because of the escalation in energy prices, HT management is extremely important in energy systems. So the nanofluids are the mixtures of liquid and nanoparticles which are used to improve the rate of heat transfer. e performance of nanoparticles in a heat transfer mechanism is excellent as compared to regular fluid. is is due to the dispersed ultrafine particles boosting the thermal conductivity of the fluid and therefore increasing their energy transfer competency. In this way, convective HTof nanofluid is a challenging problem. When a non-Newtonian fluid is moving in a structure, natural convection analysis is a difficult task. e nanoparticle adds to the base fluid and then heat transfer may be increased and this added nanoparticle in the base fluid is named nanofluid [8,9]. Numerous literatures [10,11] disclose the low volume fractions (1-4 volume %) for better performance of thermal conductivity of the fluids. We can utilise nanoparticle concentrations of greater than 20% [12]. When two or more distinct nanoparticles are added to the base fluid, it is referred to as "hybrid nanofluid," also the thermal conductivity is greater than as compared to nanofluid and regular fluid. In the field of heat transfer, hybrid nanofluids have received a lot of attention such as nuclear system cooling, drug reduction, automobile radiators, thermal storage, welding, electronic cooling, solar heating, lubrication, the coolant in machining, generator cooling, defence, biomedical, heating, and refrigeration, etc. Saqib et al. [13] studied the NC flow problem on Jaffry hybrid nanofluid using CNTs (single-and multiwall carbon nanotubes) with carboxy-methyl-cellulose (CMS) as a base fluid between two vertical parallel plates. Hatami and Ganji [14] applied the differential transform method (DTM) to investigate the NC flow of sodium alginate (SA) as a host fluid and silver (Ag) and copper (Cu) as nanofluids between two vertical parallel plates. Maghsoudi et al. [15] investigated natural convective, thermal radiation, HT, and magnetic field of the non-Newtonian nanofluid flow between two infinite vertical flat plates utilizing the Galerkin method (GM). Using the HAM, Rahmani et al. [16] explored the NC flow of non-Newtonian nanofluids between two vertical plates. ey observed that HAM is better than the numerical RK method. e NC flow of non-Newtonian nanofluids between two vertical plates using the generalized decomposition method (GDM) was also studied by Kezzar et al. [17]. ey observed that GDM is better than the numerical RK method. Biswal et al. [18,19] used the HPM in an uncertain environment to examine the NC of nanofluid flow between two parallel plates. e volume fraction of nanoparticle was considered as TFN and also shows the fuzzy result is better than a crisp result. Gabli et al. [20] studied the NC flow of non-Newtonian ferroparticle (Fe 3 O 4 ) nanofluids between two vertical plates with thermal radiation using the Adomian decomposition method (ADM). ey observed that ADM is better than the RK-Feldberg-based shooting method. Devi and Devi [21] inspected the HT and flow problems of hydro-magnetic hybrid nanofluids (Al 2 O 3 + (Cu/H 2 O)) through a stretched sheet.
Fluid flow with heat transfer is essential in science and engineering. Because of extensive physical properties such as chemical diffusion, magnetic effect, and heat transfer, governing fluid equations are converted into linear or nonlinear DEs. After controlling these physical issues, they are transformed into linear or nonlinear DEs. e solution of DEs is strongly affected by the physical problems with associated parameters and initial, geometry, coefficient, and boundary conditions. en, these are not crisp due to the mechanical defect, experimental error, and measurement error, etc. In this scenario, fuzzy sets theory (FST) is a more accurate instrument than assuming genuine physical problems for getting a better understanding of the facts under investigation. To be more specific, FDEs are useful for decreasing uncertainty and determining the best way to define a physical problem with unknown parameters and initial and boundary conditions. e FST was first presented by Zadeh [21] in 1965. FST is a fantastic approach for describing circumstances when information is unclear, imprecise, or uncertain. Later on, Dubois and Prade [22] developed arithmetic procedures on fuzzy numbers (FNs). e trapezoidal, triangular, and Gaussian FNs are three forms of FNs that may be classified. For thoroughness, we will look at TFNs now. e FN is a variable that has a range from 0 to 1. Each numerical value in the range is given a membership grade, with 0 being the lowest grade and 1 being the strongest possible grade. e information contained in crisp partial or ordinary differential equations models of dynamical systems is sometimes incomplete, imprecise, or ambiguous. FDEs are a useful approach for modelling dynamical systems with ambiguity or uncertainty. FNs or TFNs can be used to define this impreciseness or vagueness mathematically. Many researches have been conducted in recent years around the notion of FDEs. e fuzzy differentiability idea was established by Seikala [23] and Kaleva [24] and then they discussed fuzzy integration and differentiation. e FDEs were first presented in 1987 by Kandel and Byatt [25]. For the solution of FDEs, Buckley et al. [26] employed two methods: the extension principle and FNs. For continuous FDEs, Nieto [27] investigated the Cauchy problem. In [28], Lakshmikantham and Mohapatra investigated the initial value problems for FDEs. For the existence and uniqueness solution of FDE, Park and Hyo [29] employed the successive approximation approach. e geometric approach for solving a system of FDEs was devised by Gasilov et al. [30]. e system of FDEs with TFNs was investigated by Nizami et al. [31]. Salahsour et al. [32] used FDE and TFNs to investigate the fuzzy alley effect and the fuzzy logistic equation.
Inspired by the earlier investigations, the goal of this paper is to use the numerical scheme bvp4c to analyze the hybrid nanofluid flow between two vertical parallel plates in a fuzzy environment. e sodium alginate (SA) is the host fluid, while the hybrid nanoparticles are Fe 3 O 4 and TiO 2 .
e impacts of the Eckert number, Prandtl number, and nanoparticle volume friction on velocity and temperature profiles are studied. It has been detected that hybrid nanofluid enhanced the thermal efficiency of the base fluid rapidly as compared to the other fluid and nanofluids. Besides, after checking the accuracy of bvp4c so compare the results of existing work in the literature. e nanoparticle volume fraction has also been treated as an uncertain parameter in this investigation, with fuzzy numbers or triangular fuzzy numbers being used. e FDEs with α − cut approach are used to tackle the natural convection problem in a fuzzy environment. e paper is arranged as follows. We discussed some fundamental preliminaries on FDEs in Section 2. e formation of the crisp problem is described in Section 3. e crisp problem is transformed into FDEs in Section 4. Section 5 contains an explanation of graphs and tables. Section 6 concludes with some closing remarks.

Preliminaries
Some fundamental definitions are given in this section.
Definition 1 (see [21,52]). Fuzzy set is defined as a set of ordered pairs such that U � (y, μ U (y)): y ∈ X, μ U (y) ∈ [0, 1]}, where μ U (y) is the membership function of U, X is the universal set, and mapping is defined as Definition 2 (see [21,52]). α-cut or α-level of a fuzzy set U is a crisp set U α and defined by Definition 3 (see [22,52] e TFN with peak (or center) a 2 , left width a 2 − a 1 > 0, right width a 3 − a 2 > 0, and these TFNs are transformed into interval numbers through α-cut approach, is written as where α ∈ [0, 1]. e membership function is the building block of FST and it is defined by its membership function. TFNs U � (a 1 , a 2 , a 3 ) and α-cut of membership function are shown in Figure 1. An arbitrary TFN satisfies the following conditions: Definition 4 (see [23,25,52]). Let I be a real interval. A mapping u(y; α): I ⟶ F is called a fuzzy process, defined Definition 5 (see [23,25,52]). Let I⊆R, u(y; α) be a fuzzy valued function defined on Similarly, we can define higher-order ordinary derivatives in the same way. An FN by an ordered pair of functions [du(y; α)/dy] α and they satisfy the following conditions: (i)

Problem Formulation
In this proposed problem, Figure 2 portrays the main theme schematically. It consists of two vertical parallel flat plates separated by a distance 2h apart, in which there is a non-Newtonian fluid, which is flowing due to the free convection. e walls at x � h and x � − h are held at constant temperatures T 1 and T 2 , respectively, with (T 1 > T 2 ). is difference of temperature causes the fluid near the walls at x � − h to rise and the fluid near the wall x � b to fall. e fluid is a non-Newtonian sodium alginate-based nanofluid containing Fe 3 O 4 and TiO 2 hybrid nanoparticles. e base fluid and the hybrid nanoparticles are considered to be in thermal equilibrium, with no-slip between them. Some physical properties of the hybrid nanofluid are arranged in Table 1.
Using the above assumptions and Boussinesq approximation [14], the momentum and energy equations of the Computational Intelligence and Neuroscience natural convection flow of an incompressible third-grade nanofluid are as follows [2,3,5,14,20]. e equation of motion is and the equation of energy is as follow: with the following boundary conditions: e dimensionless variables [2] u (y;α)   [14,20].
After removing the bar, we have And dynamic and thermal boundary conditions are Pr where the dimensionless Grashof number (Gr), the Eckert number (Ec), Prandtl number (Pr), and the non-Newtonian viscosity (β).

Formulation of the Crisp Problem into a Fuzzy Problem Using FDEs
e velocity and temperature of nanoparticles are affected by small changes in their volume fraction. Some researchers take the nanoparticles volume fraction in this range [0.01-0.04], implying that fluid flow is solely dependent on these values. en, due to the fixed crisp values of the volume fraction of nanoparticles, uncertainty develops.
Since ϕ 1 representing the volume fraction of Fe 3 O 4 and ϕ 2 represents the volume fraction of TiO 2 , so, in a fuzzy environment, it is preferable to address a complex situation by accepting both volume fractions as FN.
For fuzzy solutions, equations (6)-(8) can be converted into FDE using α − cut approach. So, according to Definitions 4 and 5, we have Computational Intelligence and Neuroscience where 0 ≤ α ≤ 1 u 1 (y, α) is the lower bound and u 2 (y, α) is the upper bound of fuzzy velocity profiles. Similarly, the fuzzy temperature profiles are θ(y, α) � [θ 1 (y, α), θ 2 (y, α)], 0 ≤ α ≤ 1. Table 2 presents the crisp values and TFNs of these FNs. e TFN defined the variation of FN at each α − cut. e TFNs are used to define the triangular MFs of the FNs which is ranging from 0 to 1, see Figure 1. is investigated range is commonly used to develop the aforementioned problem. Now, we present a boundary value problem solver numerical procedure for controlling crisp differential equations (equations (6)-(8)) and FDEs (equations (11)-(13)) with boundary conditions, which are called bvp4c. It is a Lobatto IIIa formula with three stages based on the finite-difference algorithm. It has a collocation polynomial, and in [a, b], the collocation formula yields a sixth-order accurate uniform C1 continuous solution. For error control and mesh selection, the continuous solutions residual is employed. e aforementioned ODEs are transformed into a first-order system as follows: Let and boundary conditions are For the required solution, equations (14) to (18) are coded in MATLAB software.  Tables 3 and 4 show the comparison of velocity and temperature fields at ϕ 1 � ϕ 2 � 0, β � 0.5, Gr � Pr � Ec � 1, with studies by Ziabakhsh and Domairry [3], Manshoor et al. [5], and Biswal et al. [18,19]. For the validation, the current study findings were found to be in excellent agreement. Figure 3 displays the influence of the Prandtl number (Pr) on the velocity and temperature fields while other physical parameters are fixed. e velocity and temperature fields of the hybrid nanofluid rise as Pr increases due to upsurges in the thickness of the boundary layer. e impact of the viscous dissipation parameter (Ec) on the velocity and temperature fields is demonstrated in Figure 4. It can be observed that the velocity and temperature of the hybrid nanofluid enhance with growing the values of Ec. When Ec increases, the dissipation of heat on the boundary layer region increases and also the heat transfer rate increases. e impact of third-grade fluid parameter (β) on the velocity and temperature field 6 Computational Intelligence and Neuroscience  Computational Intelligence and Neuroscience of the hybrid nanofluid is examined in Figure 5. In Figure 5(a), when β increases, then the velocity declines in the region − 0.9 < y < − 0.1 and it is increasing in the region 0.4 < y < 0.8. e reason for this is that when the viscosity of the hybrid nanofluid increases, the boundary layer thickens and the velocity declines. In Figure 5(b), when the value of β increases, the temperature falls. e temperature profile shows very small variations on large values of β because the rate of shear increases and decreases in the boundary layer thickness. e impact of buoyancy forces (Gr) on the velocity profile is portrayed in Figure 6. It can be seen that when Gr is amplified, then the velocity profile displays an increasing trend. Physically, large values of Gr boost the buoyancy force, resulting in a higher thermal force through the use of the viscous force and hence there is an upsurge in hybrid nanofluid velocity. Figures 7 and 8 Figure 7(b) shows the temperature profile increases because of increased heat transfer at ϕ 2 � 0.01. e reason is that the friction of solid particles decreases the host fluid viscosity. Similarly, Figure 8(a) shows that the velocity of hybrid nanofluids increases with an increase in ϕ 2 and Figure 8(b) shows that the temperature profile increases because of increased heat transfer at ϕ 1 � 0.01. Physically, the intermolecular forces between the particles of hybrid nanofluids become weaker, and consequently, the hybrid nanofluid velocity accelerates. Further, it is detected that the thermal boundary layer thickness increases because the temperature profile increases due to higher values of hybrid nanoparticle's volume friction. Figure 9 represents the comparison of nanofluids      Now, we discuss the nanoparticles volume fraction of Fe 3 O 4 (ϕ 1 ) and TiO 2 (ϕ 2 ) in a fuzzy environment. e nanoparticles volume fraction ϕ 1 and ϕ 2 are said to be TFN, as shown in Table 1, and analyzed by α-cut approach (0 ≤ α ≤ 1), as discussed in Section 3.4 in detail. Figures 10 and 11 show the nanoparticles volume fraction of Fe 3 O 4 (ϕ 1 ) and TiO 2 (ϕ 2 ) considered as TFNs (see in Table 2) and then the u(y, α) and θ(y, α) are controlled by α − cut for some particular values of α − cut (α � 0, 0.3, 0.7, 1). In Figure 10, when ϕ 1 is a TFNs, then the u(y, α) and θ(y, α) convert into lower and upper bounds of the velocity and temperature fields. When α − cut � 0, u 1 (y, α) and θ 1 (y, α) represent the nanofluid while u 2 (y, α) θ 2 (y, α) represent hybrid nanofluid at ϕ 2 � 0.04. When α − cut increases the width between u 1 (y, α) and u 2 (y, α) decreases and at α − cut � 1, they coherent with one another. It is noted that the width between u 1 (y, α) and u 2 (y, α) is very less, so the vagueness is less. Similarly, in the case of the θ(y, α), as α rises, the width between θ 1 (y, α) and θ 2 (y, α) reduces, and α � 1, they are coherent with one another. It is  vital to keep in mind that the width between θ 1 (y, α) and θ 2 (y, α) is quite narrow, indicating that the uncertainty is very less. Consequently, in Figure 11, when ϕ 2 is a TFN, then the u(y, α) and θ(y, α) convert into u 1 (y, α), u 2 (y, α), θ 1 (y, α), and θ 2 (y, α). It is essential to note that the width between the lower and upper bounds of velocity and temperature fields is very narrow, which indicates that the uncertainty is minimal. Figures 12 and 13 show the triangular membership functions of the u(y, α) and θ(y, α) for various values of y.

Results and Discussion
In these diagrams, we investigated two different cases. e black lines represent the case where ϕ 1 is used as the TFN and ϕ 2 � 0.04. e green and red dashed lines indicate the representation of ϕ 2 as TFN, whereas ϕ 1 � 0.04. e horizontal axis displays the u(y, α) and θ(y, α) for varying y, while the vertical axis displays the membership values of the u(y, α) and θ(y, α) for varying α − cut. From Figure 12, it can be seen that the width between u 1 (y, α) and u 2 (y, α) is less, therefore the uncertainty is less for numerous values of y. e width between θ 1 (y, α) and θ 2 (y, α) is moderately slight in Figure 13, demonstrating that the impreciseness is neglectable for various values of y. As a result, the uncertain parameters are controlled through TFNs.

Conclusion
e current study focused on the natural convection flow of third-grade (Fe 3 O 4 + TiO 2 )/SA hybrid nanofluid across vertical parallel plates in a fuzzy environment. e impacts of the Eckert number (Ec), the non-Newtonian viscosity (β), Prandtl number (Pr), Grashof number (Gr), and nanoparticles volume fraction (ϕ 1 , ϕ 2 ) on the temperature and velocity profiles have been studied for (Fe 3 O 4 + TiO 2 )/SA hybrid nanofluid. e volume fractions of nanoparticles of Fe 3 O 4 (ϕ 1 ) and TiO 2 (ϕ 2 ) are considered as TFNs with the help of α − cut (0 ≤ α ≤ 1) which control fuzziness. For various values of y, triangular membership plots of fuzzy velocity and temperature profiles were also examined. e following significant finding comes from this investigation: e velocity and temperature profiles rise as the values of Pr, Gr, and Ec increase, whereas the velocity and temperature profiles decrease when the value β increases. e rate of heat transfer upsurges by growing volume fractions of nanoparticles ϕ 1 and ϕ 2 . e present results obtained from numerical technique via bvp4c are found to be in excellent agreement as compared to existing results. e hybrid nanofluid (Fe 3 O 4 + TiO 2 )/SA shows a higher heat transfer rate as compared to nanofluids Fe 3 O 4 /SA and TiO 2 /SA. e results indicate that the crisp solution is always inbetween the upper and lower solutions when α − cut to increase from 0 to 1. e sensitivity of the assumed TFN is held influenced by the unfluctuating width of the fuzzy velocity or temperature. According to the triangular membership plots, the uncertain width of the fuzzy velocity and temperature is less, so the assumed TFNs are less sensitive. Finally, the TFN is represented visually for better understanding. As a result, the TFNs may be used to different heat transfer problems.

Data Availability
No data were used in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.