Significance of heat transfer for second-grade fuzzy hybrid nanofluid flow over a stretching/shrinking Riga wedge

: This investigation presents the fuzzy nanoparticle volume fraction on heat transfer of second-grade hybrid 2 3 Al O +Cu/EO nanofluid over a stretching/shrinking Riga wedge under the contribution of heat source, stagnation point, and nonlinear thermal radiation. Also, this inquiry includes flow simulations using modified Hartmann number, boundary wall slip and heat convective boundary condition. Engine oil is used as the host fluid and two distinct nanomaterials ( Cu and 2 3 Al O ) are used as nanoparticles. The associated nonlinear governing PDEs are intended to be reduced into ODEs using suitable transformations. After that `bvp4c,' a MATLAB technique is used to compute the solution of said problem. For validation, the current findings are consistent with those previously published. The temperature of the hybrid nanofluid rises significantly more quickly than the temperature of the second-grade fluid, for larger values of the wedge angle parameter, the volume percentage of nanomaterials. For improvements to the wedge angle and Hartmann parameter, the skin friction factor improves. Also, for the comparison of nanofluids and hybrid nanofluids through membership function (MF),


Introduction
Interest in non-Newtonian fluid flows has expanded dramatically in recent decades due to its widespread use in food, chemical process industries, construction engineering, power engineering, petroleum production, and commercial applications. Furthermore, the boundary layer flow of non-Newtonian fluids is particularly important because of its application to a variety of engineering challenges, including the prospect of reducing frictional drag on ships and submarine hulls. Non-Newtonian fluids research gives engineers, mathematicians, and computer scientists some interesting and challenging challenges for these reasons. Non-Newtonian fluids include materials like ketchup, blood, silly putty, paints, suspensions, toothpaste and lubricants. Because shear rate and shear stress are nonlinearly associated in non-Newtonian fluids, perfect forecasting of all connected features for such fluids is not feasible using a single model. The classic Navier-Stock equation is useful for describing a variety of dynamics and rheological features of these fluids, including retardation, stress differences, memory effects, elongation, relaxation, yield stress, and so on. The class of viscoelastic/second-grade fluids is one category of differential type fluid for which analytic solutions can be reasonably expected. The viscoelasticity of fluids causes the order of differential equations describing the flow to grow. The normal stress effects can also be predicted using a second-grade fluid model. In their article, Rasool et al. [1] demonstrated the flow of second-grade fluid across an upright Riga surface. Abbas et al. [2] presented an entropy analysis of nanofluidic flow over a Riga plate. Shtern and Tsinober [3] investigated the stability of Blasius-type flow over a Riga surface. The mass and heat transfer of tangent hyperbolic nanofluids through a Riga wedge was inspected by Abdal et al. [4]. Over a Riga Plate, Gangadhar et al. [5] evaluated EMHD (Electro-magneto-hydro-dynamic) and Convective heat second-grade nanofluid. Khan and Alzahrani [6] evaluated the melting process and bioconvection on second-grade nanofluid through a wedge. In recent years, there has been a lot of study on second-grade liquid [7][8][9][10][11][12][13].
Several studies have shown that nanofluids have higher heat transfer capabilities than traditional fluids, which is why conventional fluids can be replaced with nanofluids. Nanotechnology has grown in relevance in different disciplines of science and the industrial zoon in recent years, making it one of the most powerful study areas. Researchers are paying particular attention to the varied thermal properties of such nano-sized nanoparticles because of the importance of nanoparticles in the generation of thermal engineering, multidisciplinary sciences and industries. The interaction of nanoparticles is being used in recent nanotechnology developments to improve the thermal transfer mechanism to meet the growing demand for energy resources. Nanoparticles' diverse uses in biomedical sciences include many illness diagnoses, heart surgery, cancer cell extinction, brain tumours, and a variety of medical applications. Thermally enhanced nanoparticles are used in numerous industrial applications such as heat and cooling processes, thermal extrusion systems, power generation and solar systems. Nanofluids are created by adding a small number of solid particles (10-100 nm range) [13]. Hybrid nanofluids are very useful for a variety of applications such as automobile radiators, biomedical, nuclear system cooling, coolant machining, microelectronic cooling, drug reduction, thermal storage and solar heating. On a mechanical level, they have advantages like chemical stability and excellent thermal efficiency, allowing them to perform more efficiently than nanofluids. Suresh et al. [14] proposed the concept of hybrid terms. The hybrid nanofluid's experimental outcomes were also investigated. He demonstrated that hybrid nanofluids increased the heat transfer rate by two times more than nanofluids. Sundar et al. [15] examined the viscous fluid of a hybrid nanofluid at low volume fractions. The stagnation-point flow of hybrid nanostructured materials fluid flow was evaluated by Nadeem et al. [16]. The hybrid nanoparticle fluid flow in a cylinder was addressed by Nadeem  In previous decades, fluids having higher electrical capacitance, such as fluid metals, were preferred. Externally applied (connected or common) magnetic disciplines of moderate strength can control the composition of the boundary layer in those types of liquids. The current created by the external field is insufficient to control the stream if the electrical permeability of the liquid (ocean water) is low, necessitating the use of an electromagnetic actuator to control it. Gailitis and Lielausis [35] invented that the Riga plate is unique in that it contains and imposes electric and magnetic fields strong enough to induce Lorentz forces parallel to the surface, thus restricting the flow of moderately conducting liquid. Also, the Riga-plate is a type of actuator. It could be utilized to prevent flow separation within an effective agent's radiation, skin friction, and subsurface pressure gradients.  [44] numerically explained the viscous nanofluid across a Riga plate. Furthermore, numerous authors recently studied their work mentioned by [45][46][47][48][49][50][51][52][53][54][55] based on Riga plate flow.
Differential equations (DEs) play a very important role in explaining the framework for modelling systems in biology, engineering, physics and other disciplines. On the other side, when a real-world situation is simulated using ODEs, we can't always be sure that the model is working properly since dynamical framework information is sometimes inadequate or confusing. Authors employed a fuzzy environment instead of a fixed value to overcome this type of issue, transforming ODEs into fuzzy differential equations (FDEs). Zadeh [56] established the concept of fuzzy set theory (FST) in 1965. The FST was formulated to overcome uncertainty caused by a lack of information in many mathematical models. This theory has recently been investigated further with a variety of applications being explored. Chang and Zadeh [57] were the first to establish the concept of fuzzy derivatives. After that, Dubois and Prade [58] invented the extension principle. The fuzzy initial value problem was developed by Kaleva and Seikkala in [59][60][61]. In the last decades, researchers used FDEs in fluid dynamics such as Borah et al. [62] used fractional derivatives to study the magnetic flow of second-grade fluid in a fuzzy environment. Later, Barhoi et al. [63] studied a permeable shrinking sheet in a fuzzy environment. Nadeem et al. [64] studied heat transfer analysis using fuzzy volume fraction. Additional information on FDEs and their applications are also offered in [65][66][67][68][69][70][71][72][73].
In the above literature, most of the researchers investigated the flow of different fluid models (Cassan, Maxwell and Newtonian fluid) over the wedge along with different physical impacts. Yet the flow of a second-grade fuzzy hybrid nanofluid past the wedge is not studied. Being inspired by the major features of a magnetic field, boundary slip, stagnation point flow, and heat transfer, the second-grade fuzzy hybrid nanofluid over a stretching/shrinking Riga wedge has been examined in the current study. The following points illustrate the novelty of this article:  The use of aluminia   2 3 Al O and copper   Cu nanoparticles with engine oil (EO) as base fluid is considered.
 Thermal radiation is considered in nonlinear form.
 The volume fraction of 2 3 Al O and Cu nanoparticles is taken as a triangular fuzzy number.
 Thermal characteristics of nanofluids   2 3 Al O /EO ,   Cu/EO , and hybrid   2 3 Al O +Cu/EO nanofluids are compared with the help of the fuzzy membership function.
The subsequent research questions will be addressed as part of the appraisal of this study:  What are the advantages of employing convective boundary conditions in the cooling process rather than a constant wall temperature?
 What effect would have second-grade fluid parameter have on the velocity profile?
 What will be the effect of nonlinear thermal radiation on thermal profile?
 What will be the impact of wedge angle on heat transfer rate?

Mathematical formulation
We suppose that an electrically conductive second-grade hybrid nanofluid flows incompressible past a stretching/shrinking Riga wedge with slip boundary conditions as shown in Figure 1. We have chosen aluminium   2 3 Al O and copper   Cu as nanoparticles with Engine oil (EO) as the base fluid. By examining the heat source, stagnation point, nano-linear thermal radiation, and convective boundary conditions.
is the free stream velocity and sv U is a positive constant. The boundary wall is supposed to be moving, and a velocity slip is allowed, which is expressed on the wall as  and    defines the total angle of the wedge. Furthermore, it is important to note that the value of m is between 0 and 1, for stagnation

 
In this study, we consider the wedge flow problem so that the value of m must be in the where f T is the surface temperature and T  is the ambient temperature. The basic equations for hybrid nanofluid flow are given by [11,12,36,50] main assumption stated above.
the boundary conditions are: The velocities in the y and x directions are represented by and v u, respectively. The thermal properties of nanofluids and hybrid nanofluids are summarised in Al O and Cu nanoparticles, the subscripts , f , nf , hnf 1 s and 2 s signify the fluid, nanofluid, hybrid nanofluid, and solid components, accordingly. Eq (9) also shows the physical characteristics of engine oil, 2 3 Al O nanoparticles, and Cu nanoparticles.
Equations (2)-(4) may be reduced to a set of nonlinear ODEs in the setting of the above-mentioned relations by using the similarity transformations (5): with boundary conditions are The thermophysical properties of the hybrid nanofluids are [11,25]:

Physical interest
The quantities of physical interest are given by. x h n f f e y Then employ (5) into (10) and (11), yielding the following relationship: where is the x-axis local Reynolds number.

Fuzzification
In practice, a change in the volume fraction value can impact the temperature and velocity profiles of a nanofluid and hybrid nanofluid. As a result, the nanoparticle volume fraction is viewed as a fuzzy parameter in terms of a TFN (see Table 2) to examine the current problem. The governing ODEs are converted into FDEs with help of -cut TFNs are converted into interval numbers using -cut

Numerical scheme
The differential type fluid model's governing flow equations are highly nonlinear, and exact solutions to the governing nonlinear problem are impossible to find due to the great complexity. In Fluid dynamics many problems are in non-linear form. The numerical techniques generally can be applied to nonlinear problems in the computation domain. This is an obvious advantage of numerical methods over analytic ones that often handle nonlinear problems in simple domains and it has taken less time to solve. bvp4c is a finite difference code that implements the three-stage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fourth-order accurate uniformly in the interval of integration. Mesh selection and error control are based on the residual of the continuous solution. Consequently, for these sorts of problems, a numerical methodology can be utilised to determine the numerical solution. By converting the present governing problem into an associated of first-order equations, we can achieve this. Here, we are discussing 1 Pr The above set of ODEs (16) and (17) with BCs (18) can be numerically explained by employing the bvp4c technique in MATLAB, which is a finite-difference algorithm with the highest residual error of 6 10 .

Results and discussion
The concentration of the investigation is to establish the significance of fuzzy volume fraction, heat generation and nonlinear radiation, on the heat transport properties of second-grade hybrid   A comparison table is also created to validate our computations with previous results, which is shown in Table 3. The present result is better as compared to the existing results.   Figure 4 shows the velocity profile for varying values of the shrinking/stretching parameter (S). As the magnitude of S grows the fluid and hybrid nanofluid's flow rate upsurges. This is due to the uniform movement of the fluid velocity with the boundary surface. The momentum boundary layer thickens physically, causing the flow rate to rise. As a result, no force opposes the fluid's flow across the sheet's surface. The inspiration of the dimensionless parameter   h a on flow rate and temperature profiles is portrayed in Figure 5. With swelling h a inputs, the velocity of both fluids decreases while the thermal efficiency improves. Lorentz forces cause the flow of fluid to decrease while the heat transfer rate improves when the h a is raised. The velocity profile and temperature distribution for varying wedge angle parameter (m) are plotted in Figure 6. When m is larger, the velocity of the second-grade fluid and hybrid nanofluid declines while the temperature of the second-grade fluid and hybrid nanofluid improves. Physically, the velocity declines due to boundary layer thickness improving whereas temperature raises. Figure 7 shows how the concentration of nanoparticles affects the flow and thermal fields of fluid and hybrid nanofluids. It has been noted that with the growth of 1  and 2 ,  the velocity reduces whereas the thermal field is improved. Physically, the thermal and momentum boundary layers become denser for the higher volume fraction 1    2 3 Al O /EO due to the presence of   2 3 Al O +Cu/EO hybrid nanoparticles in the ordinarily fluid, which generates too much resistance than fluid, and therefore, the velocity diminishes, and the heat of the fluid rises. The density of the hybrid nanofluid and fluid upsurges as the larger values of 2

  
Cu/EO , boost the heat and reduce velocity. As an outcome, the intermolecular interactions between the particles of hybrid nanofluids strengthen, and the hybrid nanofluid and fluid's heat transfer rate rise. The inspiration of the non-Newtonian parameter  ,  on the velocity profile is shown in Figure 8. The velocity decreases meaningfully as the value of  is increased. This feature resulted in a significant rise in the thickness of the momentum boundary layer. The fact is that larger normal stress puts a push on the neighbouring particles, forcing them to move quickly. The velocity slip    imprinted in Figure 9 represents the fluid flow. The velocity slip leads the velocity to grow, as has been shown. The velocity slip is predicted to cause additional disruption and accelerate the fluid motion. Physically, when velocity slip rises, the fluid velocity intensifies, resulting in higher applied forces to push the expanding wedge and energy transfer to the liquid. However, due to their importance in the thermophysical properties of hybrid nanofluid, it is noticed that hybrid nanofluid has the highest velocity when compared to fluid. Features of the Biot number   i B on the heat flux are reviewed in Figure 10. Larger i B indicates that more heat is transported from the surface to the nanoparticles, and as a consequence, the temperature rises. Figure 11 indicates the impact of the heat source parameter (H) on the heat flux. It is recognized that as the H goes up, the heat flow enhances. Also, as compared to a second-grade fluid, the heat transmission rate of a hybrid nanofluid is higher. A considerable amount of heat energy is released from the wedge to the working fluid during the heat generation process, which strengthens the temperature field in the boundary layer region near the stretching/shirking wedge. Furthermore, at a smaller distance from the wedge, the temperature profile decays to zero. The influence of the temperature ratio   r  and the radiation parameter (Nr) on the heat flux is demonstrated in Figure 12. It is evident that the temperature field upsurges when r  and Nr are raised. Physically, a larger r  indicates a significant temperature difference between the wedge wall and the surrounding environment. The boundary layer thickness improves as the temperature varies. Physically, the radiative component enhances small particle mobility, pushing random moving particles to collide and converting frictional energy to heat energy. A hybrid nanofluid's temperature is higher than that of a conventional fluid in both cases. Impressions of various parameters   The heat transmission rate of hybrid nanofluids is higher than the regular fluids.  not triangle-symmetric, while both fuzzy volume percentage is TFN and symmetric. The nonlinearity of the governing differential equation may cause these changes. Hybrid nanofluids were also shown to have a larger width than nanofluids. Consequently, the TFN considers the hybrid nanofluid to be uncertain. On the other way, Figure 16 demonstrates the comparison of nanofluids 2 3 Al O /EO    and 2  non-zero is shown in the third case by blue lines. It can be perceived that the hybrid nanofluid performs better due to the temperature variance in the hybrid nanofluid being more prominent than both nanofluids. Physically, The combined thermal conductivities of 2 3 Al O and Cu are added in a hybrid nanofluid to provide the maximum transfer of heat. When a comparison of 2 3 Al O /EO and Cu/EO nanofluids is analyzed, 2 3 Al O /EO has a higher heat transfer rate because 2 3 Al O has a higher thermal conductivity than Cu.
At h=0.25 At h=0.5 At h=0.75 At h=1 Figure 16. Comparison of nanofluids 2 3 Al O /EO, Cu/EO and 2 3 Al O +Cu/EO hybrid nanofluids for various values of .

Conclusions
A numerical investigation of second-grade hybrid   convective, nonlinear thermal radiation, and slip boundary conditions are also studied. Using some appropriate transformations, the governing PDEs are turned into non-linear ODEs. In Matlab, a numerical method known as the bvp4c strategy is used to solve the problem. In addition, when compared to previous results in the literature, the new numerical results are outstanding. The major findings are as follows:  Improvement of the size of nanomaterials in Engine oil can boost the rate of heat transfer.
When compared to a conventional second-grade fluid, a hybrid nanofluid   2 3 Al O +Cu/EO is found to be a better thermal conductor.  For fuzzy heat transfer analysis, the 2 3 Al O +Cu/EO hybrid nanofluids are extremely proficient of boosting the heat transfer rate when compared to 2 3 Al O /EO and Cu/EO nanofluids, as showed by triangular fuzzy membership functions. Also, the Cu/EO nanofluid is better than 2 3 Al O /EO the nanofluid when they are compared. The findings of this study can be used to drive future progress in which the heating system's heat outcome is analyzed with non-Newtonian nanofluids or hybrid nanofluids of various kinds (i.e., Maxwell, Third-grade, Casson, Carreau, micropolar fluids, etc.).