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

1.   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 and Abbas ^[17] with the slip effect. Yan et al. ^[18] inspected the rheological behaviour of a non-Newtonian hybrid nanofluid in the context of a driven pump. The viscosity of the hybrid nanofluid with the largest volume percentage was lowered by up to 21%, according to this study. Rahman et al. ^[19] used a moving edge to investigate the effects of internal heating, SWNCT and MWNCT on engine oil numerically. Aladdin et al. ^[20] proposed a hybrid nanofluid ${\text{Cu}} + {\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/}}{{\text{H}}_2}{\text{O}}$ flow above a moving permeable sheet. Several researchers are studying the heat transmission of hybrid nanofluids under various physical norms, as stated in ^{[21,22,23,24,25,26,27,28,29,30,31,32,33,34]}.

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. Basha et al. ^[36] investigated the stability of a hybrid (${\text{Ag - MgO/}}{{\text{H}}_{\text{2}}}{\text{O}}$) nanofluid flow over a stretching/shrinking Riga wedge with a stagnation point. Rasool and Wakif ^[37] used a modified Buongiorno's model to study numerically EHMD free convection of second-grade nanofluid on a vertical Riga plate. The EHMD flow of Maxwell nanofluid on the Riga plate was numerically explored by Ramesh et al. ^[38]. Shafiq et al. ^[39] evaluated the Marangoni impact across a Riga plate in the presence of CNTs. Ahmed et al. ^[40] examine the impact of thermal radiation and viscous dissipation on CNTs/water nanofluid over a Riga plate. Nanofluid with EMHD slip characteristics is analytically computed by Ayub et al. ^[41] using parallel Riga plates. The Riga plate was used by Zaib et al. ^[42] to consider the slip effect and mixed convective hybrid nanofluid flow. Abbas et al. ^[43] inspected entropy generation through a Riga plate. Bhatti et al. ^[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 $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}} \right)$ and copper $\left({{\text{Cu}}} \right)$ nanoparticles with engine oil (EO) as base fluid is considered.

• Thermal radiation is considered in nonlinear form.

• The volume fraction of ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ and ${\text{Cu}}$ nanoparticles is taken as a triangular fuzzy number.

• Thermal characteristics of nanofluids $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}} \right),$ $\left({{\text{Cu/EO}}} \right),$ and hybrid $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}} \right)$ 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?

2.   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 $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}} \right)$ and copper $\left({{\text{Cu}}} \right)$ 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. ${u_{sv}} = {U_{sv}}{x^m}$ is the free stream velocity and ${U_{sv}}$ 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 $u = {u_v} + R{u_y}\left(0 \right).$ Here, ${u_v} = {U_v}{x^m}$ indicates the wedge's stretched/shrunk velocity, with ${U_v} > 0$ identifying the stretching wedge, ${U_v} < 0$ representing the shrinking wedge, and ${U_v} = 0$ expressing the static wedge, and $R$ signifies the slip coefficient. Further, $m = {\psi \mathord{\left/ {\vphantom {\psi {\left({2 - \psi } \right)}}} \right. } {\left({2 - \psi } \right)}},$ where $m$ is the Hartree pressure gradient, wedge angle is $\psi,$ and $\Gamma = \psi \pi$ 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 point flow $\left({\Gamma = \pi } \right)$ if $m = 1$ $\left({\psi = 1} \right)$, the flow past a horizontal flat surface $\left({\Gamma = 0} \right)$ if $m = 0$ $\left({\psi = 0} \right).$ In this study, we consider the wedge flow problem so that the value of $m$ must be in the range of $0 < m < 1.$ Also, ${T_f} > {T_\infty },$ where ${T_f}$ is the surface temperature and ${T_\infty }$ 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 $v\, \, {\text{and}}\, \, u$, respectively. The thermal properties of nanofluids and hybrid nanofluids are summarised in Table 1. The volume percentages of ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ and Cu nanomaterials, respectively, are ${\phi _1}$ and ${\phi _2}.$ The hybrid nanofluid is converted to a second-grade fluid by putting ${\phi _1} = 0 = {\phi _2}.$ Here, the liquid density ${\rho _{hnf}},$ dynamic viscosity ${\mu _{hnf}},$ heat capacity $\left({\rho C} \right){p_{hnf}},$ electrical conductivity ${\sigma _{hnf}},$ and thermal conductivity of hybrid nanofluid ${k_{hnf}}.$ For ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ and Cu nanoparticles, the subscripts $f,$ $nf,$ $hnf,$ ${s_1}$ and ${s_2}$ signify the fluid, nanofluid, hybrid nanofluid, and solid components, accordingly. Eq (9) also shows the physical characteristics of engine oil, ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ nanoparticles, and Cu nanoparticles.

The following similarity transformations are illustrated for the governing Eqs (1)-(3) with the constraints (4) in a much simple way ^[36,50]. Where the stream function $\omega$ can be specified as $v = - {{\partial \omega } \mathord{\left/ {\vphantom {{\partial \omega } {\partial x, \, {\text{and}}\, \, u = {{\partial \omega } \mathord{\left/ {\vphantom {{\partial \omega } {\partial y, }}} \right. } {\partial y, }}\, \, }}} \right. } {\partial x, \, {\text{and}}\, \, u = {{\partial \omega } \mathord{\left/ {\vphantom {{\partial \omega } {\partial y, }}} \right. } {\partial y, }}\, \, }}$ while the similarity variable is $\eta :$

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

where Modified Hartmann number ${M_h} = {{\pi {j_0}{M_0}} \mathord{\left/ {\vphantom {{\pi {j_0}{M_0}} {4u_{sv}^2}}} \right. } {4u_{sv}^2}},$ second-grade fluid parameter $\alpha = {{2{\alpha _1}b} \mathord{\left/ {\vphantom {{2{\alpha _1}b} {{\mu _f}}}} \right. } {{\mu _f}}},$ the stretching/shrinking parameter $S = {{{U_v}} \mathord{\left/ {\vphantom {{{U_v}} {{U_{sv}}}}} \right. } {{U_{sv}}}},$ non-dimensional parameter ${a_h} = \left({{\pi \mathord{\left/ {\vphantom {\pi d}} \right. } d}} \right)\sqrt {{{2{\nu _f}} \mathord{\left/ {\vphantom {{2{\nu _f}} {\left({m + 1} \right)}}} \right. } {\left({m + 1} \right)}}},$ slip parameter $\lambda = R{\mu _f}\sqrt {{{{U_{sv}}} \mathord{\left/ {\vphantom {{{U_{sv}}} {{v_f}}}} \right. } {{v_f}}}},$ Prandtl number $\Pr = {{{\alpha _f}} \mathord{\left/ {\vphantom {{{\alpha _f}} {{v_f}}}} \right. } {{v_f}}},$ heat source parameter $H = {{{Q_0}} \mathord{\left/ {\vphantom {{{Q_0}} {{{\left({\rho Cp} \right)}_f}}}} \right. } {{{\left({\rho Cp} \right)}_f}}},$ thermal radiation $Nr = {{ - 16{\delta ^*}T_\infty ^3a} \mathord{\left/ {\vphantom {{ - 16{\delta ^*}T_\infty ^3a} {3{k^*}{k_f}}}} \right. } {3{k^*}{k_f}}},$ Biot number ${B_i} = \left({{{{h_f}} \mathord{\left/ {\vphantom {{{h_f}} {{k_f}}}} \right. } {{k_f}}}} \right)\sqrt {{{2x{v_f}} \mathord{\left/ {\vphantom {{2x{v_f}} {{U_{sv}}\left({m + 1} \right)}}} \right. } {{U_{sv}}\left({m + 1} \right)}}},$ and temperature difference ${\theta _r} = {{{T_f}} \mathord{\left/ {\vphantom {{{T_f}} {{T_\infty }}}} \right. } {{T_\infty }}}$ ^[6,11,12].

The thermophysical properties of the hybrid nanofluids are ^[11,25]:

2.1.   Physical interest

The quantities of physical interest are given by.

(i): The skin friction coefficient $C{f_x}$ and nusslt number $N{u_x}$ are defined by

Then employ (5) into (10) and (11), yielding the following relationship:

where $R{e_x} = {{{u_e}x} \mathord{\left/ {\vphantom {{{u_e}x} {{\nu _f}}}} \right. } {{\nu _f}}}$ is the x-axis local Reynolds number.

2.2.   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 $\sigma {\text{ - cut}}$ technique. Also, $\sigma {\text{ - cut}}$ range from 0 to 1 and controls the fuzziness of the TFN. For further information about this topic see the literature ^{[59,60,61,62,63,64,65,66,67,68,69,70]}.

Let $\phi = \left[{{\text{ }}0, 0.05, 0.1} \right]$ be a TFN defined entirely by three quantities: $0$ (lower bound), $0.05$ (most belief value), and $0.1$ (upper bound) are shown in Figure 2. By the TFN, the Membership function $M\left(\phi \right)$ can be expressed as:

TFNs are converted into interval numbers using $\sigma {\text{ - cut}}$ the approach, which is written as $\bar \theta \left({\eta, \, \sigma } \right) = \left[{{\theta _1}(\eta; \, \sigma), {\theta _2}(\eta; \, \sigma)} \right] = \left[{0 + \sigma (0.05 - 0), \, \, 0.1 - \sigma (0.1 - 0.05)} \right],$ where $0 \leqslant \sigma {\text{ - cut}} \leqslant 1.$

To handle such TFNs use FDEs with the help of the $\sigma {\text{ - cut}}$ technique. The FDEs are converted into lower ${\theta _1}(\eta; \, \sigma)$ and upper bounds ${\theta _2}(\eta; \, \sigma).$

2.3.   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

with boundary conditions are

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 ${10^{ - 6}}.$

3.   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 $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}} \right)$ nanofluid flow over a shrinking/stretching Riga wedge. The bvp4c technique which builds in MATLAB is utilized to acquire the outcomes. The numerical results of flow rate, local Nusselt number, skin friction, and temperature were determined for various values of parameters $\Pr = 2.1,$ $Nr = 0.7,$ ${M_h} = 0.2,$ $H = 0.1,$ $\alpha = 0.7,$ $m = 0.3,$ ${a_h} = 0.3,$ ${\theta _r} = 1.1,$ ${B_i} = 0.8,$ $\lambda = 0.1,$ and $S = 0.6.$ Also, a comparison of second-grade fluid and hybrid nanofluid discussion in Figures 3-12.

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 3 exposes the influence of the modified Hartmann number $\left({{M_h}} \right)$ on the flow rate. It is perceived, that hybrid nanofluid velocity rises more rapidly than that of a second-grade fluid. The momentum boundary layers thin slightly as the ${M_h}$ strength grows. Physically, a higher ${M_h}$ determines the intensity of the external electric field, which promotes fluid flow and, consequently, Lorentz force produces. 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 $\left({{a_h}} \right)$ on flow rate and temperature profiles is portrayed in Figure 5. With swelling ${a_h}$ 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 ${a_h}$ 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 ${\phi _1}$ and ${\phi _2},$ the velocity reduces whereas the thermal field is improved. Physically, the thermal and momentum boundary layers become denser for the higher volume fraction ${\phi _1}$ $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}} \right)$ due to the presence of $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}} \right)$ 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 ${\phi _2}$ $\left({{\text{Cu/EO}}} \right)$, 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 $\left(\alpha \right),$ on the velocity profile is shown in Figure 8. The velocity decreases meaningfully as the value of $\alpha$ 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 $\left(\lambda \right)$ 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 $\left({{B_i}} \right)$ on the heat flux are reviewed in Figure 10. Larger ${B_i}$ 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 $\left({{\theta _r}} \right)$ and the radiation parameter (Nr) on the heat flux is demonstrated in Figure 12. It is evident that the temperature field upsurges when ${\theta _r}$ and Nr are raised. Physically, a larger ${\theta _r}$ 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 ${M_h}, \, \, {\phi _1}, \, \, \, \lambda, \, \, \, \alpha, \, \, {\phi _2}\, \, {\text{and}}\, \, \Pr$ on the local skin friction and Nusselt number are demonstrated in Figures 13-15. Variations of $C{f_x}$, with ${M_h}$ for several values of $\alpha$ are plotted in Figure 13. It is observed that when $\alpha$ is improved the skin-friction coefficient cultivates while ${M_h}$ shows the opposite behaviour. Lorentz's effect raises the skin fraction on the Riga wedge's surface while reducing the distance away from it. Figure 14 exhibits $C{f_x}$ fluctuations with ${M_h}$ for numerous $\lambda$ values. The drag force declined when ${M_h}$ and $\lambda$ upsurged. The $N{u_x}$ lowers by way of ${\phi _1}$ increase while heat transfer rate improved when ${\phi _2}$ increasing as shown in Figure 15. Physically, the heat emitted from the wedge surface when enhancing ${\phi _1}.$

The performance of various significantly flow parameters by the local coefficient of skin friction and Nusselt number is explored in Table 4. For larger values of ${M_h},$ m, ${\phi _1},$ ${\phi _2}$ and $\lambda$ progress the skin friction whereas ${a_h},$ and $\alpha$ decrease the local skin friction factor. The nanoparticles are dragged when a perpendicular magnetic force is applied to a hybrid nanofluid moving across a Riga wedge, boosting the skin fraction. The outcome of the same engineering parameters on the Nusselt number is shown in the next column of Table 4. When the ${a_h},$ $\alpha,$ m, Nr, $\lambda,$ and ${\phi _1}$ are improved, then the heat transfer rate is boosted while it drops for the larger values of the Pr, ${M_h},$ and ${\phi _2}.$ The heat transmission rate of hybrid nanofluids is higher than the regular fluids.

Figure 16 illustrate the fuzzy temperature profile $\left({\theta \left({\eta, \, \, \sigma } \right)} \right)$ of ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}$ by considering the volume fraction as a TFN, i.e., ${\phi _1},$ and ${\phi _2}$ = [0.0, 0.05, 0.1]. For triangular MFs, four sub-plots illustrate the fuzzy temperatures for varying values of $\eta,$ such as 0.25, 0.5, 0.75, and 1. On the vertical axis is the MF of the fuzzy temperature profile for $\sigma {\text{ - cut}}\, \left({0 \leqslant \sigma {\text{ - cut}} \leqslant 1} \right),$ while on the horizontal axis is $\theta \left({\eta, \, \, \sigma } \right)$ for different values of $\eta.$ The obtained fuzzy temperatures are TFN but 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 ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}$ $\left({{\phi _1}} \right),$ ${\text{Cu/EO}}$ $\left({{\phi _2}} \right),$ and ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}$ hybrid nanofluids over MF for several values of $\eta.$ We focus on three possible outcomes in these graphics. The case when ${\phi _1}$ is treated as TFN and ${\phi _2} = 0$ is signified by purple dotted lines. The case when ${\phi _2}$ is treated as TFN and ${\phi _1} = 0$ is denoted by red dotted lines. The hybrid nanofluid with both ${\phi _1}$ and ${\phi _2}$ 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 ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ and ${\text{Cu}}$ are added in a hybrid nanofluid to provide the maximum transfer of heat. When a comparison of ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}$ and ${\text{Cu/EO}}$ nanofluids is analyzed, ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}$ has a higher heat transfer rate because ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}$ has a higher thermal conductivity than ${\text{Cu}}.$

4.   Conclusions

A numerical investigation of second-grade hybrid $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}} \right)$ nanofluid flow across a stretching/shrinking Riga wedge is discussed in this article. Fuzzy nanoparticle volume fraction, 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 $\left({{\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}} \right)$ is found to be a better thermal conductor.

• With higher ${a_h},$ m, $\alpha,$ ${\phi _1}$ and ${\phi _2}$ the hybrid nanofluid and second-grade fluid velocity decline.

• Both fluid velocities are intensified by boosting the credit of ${M_h},$ S, ${a_h},$ m and $\lambda.$

• Thermal efficiency is improved when Nr, ${\theta _r},$ ${a_h},$ m, H and ${B_i}$ are amplified.

• The Nusselt number is reduced with the upsurge in ${M_h},$ and Pr while swelling with the rise in the ${a_h},$ m, $\alpha,$ and $\lambda.$

• For fuzzy analysis through the triangular membership function, the width between lower $\left({{\theta _1}\left({\eta, \, \sigma } \right)} \right)$ and upper $\left({{\theta _2}\left({\eta, \, \sigma } \right)} \right)$ bounds of fuzzy temperature profiles of hybrid nanofluids is maximum so the fuzziness is maximum as compared to nanofluids.

• For fuzzy heat transfer analysis, the ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{ + Cu/EO}}$ hybrid nanofluids are extremely proficient of boosting the heat transfer rate when compared to ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}$ and ${\text{Cu/EO}}$ nanofluids, as showed by triangular fuzzy membership functions. Also, the ${\text{Cu/EO}}$ nanofluid is better than ${\text{A}}{{\text{l}}_{\text{2}}}{{\text{O}}_{\text{3}}}{\text{/EO}}$ the nanofluid when they are compared.

• The behaviour of the drag force is improved with the heightening of ${M_h},$ m, $\lambda,$ ${\phi _1}$ and ${\phi _2}$ whereas the reverse trend can be seen for the ${a_h},$ and $\alpha.$

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.).

Acknowledgement

The authors extend their appreciation to the Deanship of Scientific Research, University of Hafr Al Batin for funding this work through the research group project no. (0033-1443-S).

Competing interests

The authors declare no competing interests.