Numerical study of unsteady tangent hyperbolic fuzzy hybrid nanofluid over an exponentially stretching surface

The significance of fuzzy volume percentage on the unsteady flow of MHD tangent hyperbolic fuzzy hybrid nanofluid towards an exponentially stretched surface is scrutinized. The heat transport mechanism is classified by Joule heating, nonlinear thermal radiation, boundary slippage, and convective circumstances. Ethylene glycol (EG) as a host fluid along with the nanomaterial’s Cu and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Al}}_{{2}} {\text{O}}_{{3}}$$\end{document}Al2O3 are used for heat transfer analysis is also considered in this investigation. The nonlinear governing PDEs are meant to be converted into ODEs employing appropriate renovations. Then, a built-in MATLAB program bvp4c is employed to acquire the outcome of the given problem. The variation of flow rate, thermal heat, drag force and Nusselt number and their influence on fluid flow with heat transfer have been scrutinized through graphs. An increase in thermal radiation, power law index and nanoparticle volume friction heightens the heat transmission rate. Skin friction is diminished by swelling the power-law index, Weissenberg number, and ratio parameters, whereas it is increased by enhancing the magnetic parameter. The heat transfer rate upsurges with an increase in Weissenberg number and nanoparticle volume fraction. Also, the nanoparticle volume percentage is expressed as a triangular fuzzy number (TFN). The triangular membership function (MF) and TFN are regulated by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi - {\text{cut}}$$\end{document}χ-cut parameter, which has a range of 0 to 1. In comparison to nanofluids, hybrid nanofluids have a higher heat transmission rate, according to the fuzzy analysis. This investigation has applications in the areas of paper manufacturing, metal sheet cooling and crystal growth.

The researchers have studied different types of non-Newtonian fluid (NNF) models for better flow behaviour in the area of engineering and science.A fluid that deviates from Newton's law of viscosity is recognized as a NNF.When forced, viscosities in NNF can change to become more liquid.In a NNF, the shear amount and shear stress correlate.The tangent hyperbolic liquid model is dynamic due to it can identify the shear-thinning and thickening phenomenon 1 .Also, tangent hyperbolic fluids are a form of NNF that belongs to the rate type fluids group and has equations in both high and mild tensile forces that are accommodated.Scientists suggest the Tangent hyperbolic fluid model due to its application in industrial and laboratory trials such as sauces, melted cheese, whipped cream, nail varnish, ketchup, and blood.Naseer et al. 2 also evaluated the flow through the boundaries of tangent hyperbolic fluid across a vertically stretched cylinder.Using the homotopy analysis method (HAM), tangent hyperbolic motile microorganisms' nanomaterial flow was appraised by Shafiq et al. 3 .Hayat et al. 4 reviewed MHD Tangent hyperbolic flow behaviour with variable thickness.Ibrahim and Gizewu 5 detected tangent-hyperbolic fluid flow with dual dispersal mechanisms and a second-order slip barrier at the edge.Using an exponentially extending surface, Siddique et al. 6 assessed the impact of Dufour and Soret on second-grade nanofluid flow in an unsteady MHD system.A boundary layer (BL) flow of MHD tangent hyperbolic fluid completed by an exponentially extending sheet has been developed by several philosophers [7][8][9][10][11][12][13] .phenomenon rather than presumptuous physical complications.To be more particular, FDEs are useful for reducing ambivalence and explaining bodily problems that arise when heat transmission constraints, beginning circumstances, and initial or boundary conditions are fuzzy.In 1965, Zadeh 50 was the first to express the FST.FST is a beneficial tool for sharing circumstances where information is vague, confusing, or imprecise.The objective of membership or belongingness perceives FST.Each component of the discourse universe is given a number from the [0, 1] range by the MF in the FST.Fuzzy differentiability was initially brought up by Seikala 51 .Kaleva debated derivative and integration with fuzziness in 52 .Buckley and Feuring 53 exploited FDEs to solve the nth-order DE using fuzzy initial conditions.Moreover, various scholars have executed FST to create well-known scientific and technical results [54][55][56][57] .
The investigations stated above show no struggle has been made to examine the unsteady tangent hyperbolic hybrid nanofluid flow across an exponentially stretched sheet.Through the use of suitable variables, nonlinear ODEs are produced.To create convergent solutions, the Bvp4c scheme is used.The originality of the work is as follows: • The injection of magnetic flux into the region's flow is critical for managing the dynamic behaviour of the manufacturing process.• The heat equation comprises nonlinear thermal radiation, joule heating, dissipation effects and heat source.
• Fuzzy solutions are designed to assess the empirical uncertain dispersion the nanoparticles volume fractions are regarded as triangle fuzzy numbers using the χ -cut technique, and the fuzzy triangular MFs are used to define χ -cut.• Fuzzy triangular MFs were utilized to compare the nanofluids and hybrid nanofluids.

Formulation
The flow of the equilibrium area of an unsteady tangent hyperbolic fuzzy hybrid nanofluid approaching a convectively warm exponentially stretchy sheet is pursued in two dimensions (2D).The x-axis is perpendicular to the stretching plane, whereas the y-axis is becoming longer upright to the x-axis.A viscous fluid's flow is constrained by y > 0. The benefit of a non-uniform magnetic field, i.e.B(x) = B • e x/2L this is carried out in the opposite direc- tion of the flow field shown in Fig. 1.The lack of a magnetic force that is triggered is due to the modest Reynolds number.It is presumed that the electric field is zero.The Joule heating ramifications are preserved.After the following assumptions, provided equations underlie BL flow and heat transport are given as 3,4  The x-axis and y-axis velocity coefficients are symbolized by u and v, respectively.σ hnf is the electrical conductivity, σ * the Stefan-Boltzmann constant, T ∞ is the reference temperature, k hnf is the thermal conductivity, T is the fluid temperature, Ŵ the material constant, ρ hnf represents the density, ν hnf kinematic viscosity, U • is the reference velocity, c p hnf represents the specific heat of hybrid nanofluid, T w the temperature of the convective fluid under the sheet, K * the ratio of mean assimilation and h f = he x / 2L the index of convective heat transfer.
Table 1 lists the thermophysical features of a hybrid nanofluid.
The following set of similarity transformations is used in Eqs.
(1)-( 3) and their related BCs, we must first transform them into non-dimensional ODEs by adding the following analogous variables: By applying Eq. ( 5), the continuity equation is satisfied and Eqs.(1)-( 3) are here β is unsteady parameter, We is wessingbrg number, B i is Biot number, M is magnetic parameter, Ec is Eckert number, α is power law index, γ is slip parameter, the radiation parameter N r and θ w is a temperature difference.The thermophysical features of hybrid nanofluids are illustrated by Eq. ( 13).
The skin friction coefficient Cf x and local Nusselt number (Nu x ) are offered by where ( 3) Table 1.The Cu thermo-physical properties along with Al 2 O 3 and EG [41][42][43] .

Fuzzification.
A change in the volume concentration value can affect the velocity and temperature profiles of a nanofluid and hybrid nanofluid in practice.Moreover, to observe the present situation, the nanoparticle volume fraction is regarded as a fuzzified parameter in aspects of a TFN (see Table 2).The χ -cut technique is utilized to transfer the constitutional ODEs into FDEs.See the literature for more details about this topic [53][54][55][56] .Let φ =[0, 0.05, 0.1] be a TFN explained essentially by three measures: 0 (lower bound), 0.05 (most perception value), and 0.1 (upper bound) as shown in Fig. 2. By the TFN, the Membership function M(φ) can be expressed as TFNs are transformed into interval numbers employing the χ-cut method, which is indicated as The FDEs are changed into lower θ 1 (η, χ) and upper bounds θ 2 (η, χ).
Solution methodology.The differential system given above has nonlinear equations, making it impossible to determine an exact solution.So we shall use a numerical technique namely the bvp4c method.To use (10) Table 2. TFN of ϕ1 and ϕ2 [53][54][55][56][57] .www.nature.com/scientificreports/this algorithm, convert the set of non-linear ODEs and their BCs to a system of first-order ODEs and beginning assumptions.

Fuzzy numbers
For our specific problem, we are taking 10 −5 as tolerance of error.

Results and discussion
We employed two distinct types of Al 2 O 3 and Cu nanoparticles in base fluid EG that flow across an expo- nentially expanding surface for this research work.Also, this section's big priority is to discuss the habits of numerous parameters, such as the radiation parameter (N r = 0.6) , unsteady parameter (β = 0.5), the Prandtl number (P r = 7) , magnetic parameter (M = 0.3), the Biot number (Bi = 0.2) , the power law index the (α = 0.3), (Ec = 0.7) Eckert number, the temperature ratio (θ w = 1.2) and nanoparticle consentration (φ 1 = φ 2 = 0.02).
The comparison is shown in Table 3 with an impressive accuracy level.As a result, it is expected that the results illustrated by the current numerical approach are extremely precise.
Figure 3a,b outlines the consequence of unsteady parameter (β) on the velocity f ′ (η) and temperature (θ(η)) gradients.We encountered that when β approaches, the velocity actively diminishes while temperature erodes.At the sheet surface, the velocity mindset is due to slip conditions.As we jack up ductility slows down allowing for resistance to fluid motion.Temperature and the grain size of the thermal wave are both seen to be falling functions of β.In fact, as β increases, so does thermal diffusivity.The involvements of the force of slip parameter (γ ) on f ′ (η) and θ(η) is validated in Fig. 4a,b.We intensify the value of, the momentum BL gains, while the velocity is declined.This consequence happens because the disturbance caused by the fluid's velocity being reduced because of frictional smothering between the fluid's droplets and the bottom is partially transferred to the stretching velocity.It appears that the significance of temperature is increasing at a rapid rate.This arises as a byproduct of a change in velocity slip, which slows down the working fluids and increases the thermal conductivity.The involvements of the magnetic parameter (M) on the f ′ (η) and θ (η) are visualized in Fig. 5a,b.Doubling the M certainly lessens both the velocity profile and the cross-sectional area of the BL while temperature increases.The magnetic field exerts a force on Lorentz to evoke resistance to the travel of fluid particles which causes the fluid's velocity minimizes.Magnetic variables, in reality, fixate on the Lorentz force.Higher values M have a high Lorentz force, which inhibits density and raises the fluid temperature.When M = 0, the flow is hydrodynamic, however when M > 0, the flow is magnetohydrodynamic. Figure 6a,b exhibits how the temperature profile shapes as the N r and P r are active for a set of values.The method will ensure that as the N r is rose, the temperature went up see Fig. 6a.It comes as a result of an increase in surface heat flow prompted by N r , which benchmarks an upswing in the temperature profile.Figure 6b showcases the suitability of P r on the θ(η) .The temperature profile prevents as P r thaws, we said.Temperature and the thermal layer's breadth are both depleting P r functions.In reality, when   www.nature.com/scientificreports/P r fluctuates, the thermal diffusivity lessens.This leads to a reduction in the energy transfer threshold, which eventually predisposes to a diminution in the area of the thermal layer.The characteristics of the Weissenberg number (We) are responsible for the f ′ (η) and θ (η) are illustrated in Fig. 7a,b.When We is expanded, the velocity is determined to be decreasing while the fluid's temperature is visible to contribute to the high.The liquid's relaxation time and the specific time process ratio are known as the Weissenberg number.The relaxation speed was adjusted as We kept rising, adding more resistance to the motion of the liquid and, consequently, raising the fluid temperature.
Plots of Eckert number (Ec) and power law index (α) on θ (η) is specified in Fig. 8a,b.It is found that the heat and thermal BL thickness upsurge for larger Ec.The heat transfer rate is enhanced as a consequence of the heat energy being stored in the fluid when Ec climbs owing to friction forces.Variation of θ(η) against α is plotted in Fig. 8b Temperature is a steadily rising function of α in this case.Thermal BL thickness is the swelling function of α.
Figure 9a serves to indicate the temperature variation as a factor of Biot number (Bi).In general, Bi is dictated by the surface's characteristic length, thermal conductivity, and convective heat transfer of the hot fluid under the surface.The constant wall temperature at the surface is regarded by a stronger Bi.A higher Bi causes the thermal BL to thicken and the heating rate to elevate. Figure 9b demonstrates the influence of θ w on the θ(η).For the higher values of θ w is the increment in thermal layer thickness.Figure 10a,b highlights the impression of the volume fraction of nanoparticles (φ 1 and φ 2 ) on the f ′ (η) and θ(η) .It is noticeable that the velocity field falls as φ 1 and φ 2 grow while the hybrid nanofluid's temperature field expands.Physically, the momentum and thermal BL get denser at the larger volume fraction  φ 1 and φ 2 .The fractional nanoparticle size in the base fluid appears to have an impact on the effectiveness of the nanofluid and hybrid variants.Because of the increased load, nanoparticles with a higher good fractional range have a lower flowability.Due to the shear-thinning characteristic's dependence on temperature, this phenomenon occurs.Nanoparticles have a high thermal conductivity and transfer rate under a variety of physical conditions, which is a well-known property.Additionally, it is mentioned that the particles' viscosity will decline at higher temperatures.
The impact of nanofluids Al 2 O 3 /EG(φ 1 ) , Cu/EG(φ 2 ) and hybrid nanofluid (Al 2 O 3 + Cu/EG) on fuzzy tem- perature profile (θ(η, χ )) for distinct values of η are exposed in Fig. 11.We looked at three possible scenarios in these diagrams.Blue-dashed lines signify the situation when φ 1 is engaged as TFN then φ 2 = 0. Black lines  display the deviation of φ 2 whereas φ 1 = 0.In the third scenario, hybrid nanofluid exemplifies through both φ 1 and φ 2 being non-zero.Additionally, the θ (η, χ ) for varied η is displayed on the horizontal axis, and the MF of the θ(η, χ ) for varying χ -cut is displayed on the vertical axis.When especially in to nanofluids Al 2 O 3 /EG and Cu/EG, the hybrid nanofluid (Al 2 O 3 + Cu/EG) is found to be better according to fuzzy analysis.A hybrid nanofluid exhibits a more significant heat transfer rate than the other two cases.Physically, hybrid nanofluid is constructed by combining the combined thermal conductivities of Al 2 O 3 and Cu to allow for the fastest possible heat transfer.Since Cu/EG thermal conductivity is higher than Al 2 O 3 /EG , it exhibits a faster rate of heat transfer when compared to Al 2 O 3 /EG nanofluid.
The numerical variations of the drag coefficient and Nusselt number are depicted in Tables 4 and 5. Table 4 demonstrates the magnitude of growth in relationship to escalating Ec,β and M estimations while higher We, α and γ are associated with the opposite tendency.In Table 5, it is clear that the amplitude of declines in response to higher Ec, M, H, and θ w whereas it grows through growing levels of Bi, N r and We.The thermal efficiency of the liquid can be boosted more efficiently by using a dynamically hybrid nanofluid.Due to the steady dispersion of these two different types of nanoparticles in a single base fluid, the thermal performance improves.

Conclusions
Unsteady MHD tangent hyperbolic fuzzy hybrid nanofluid identified with convective surface boundary conditions and nonlinear thermal radiation is explored in this work.With the help of the Bvp4c scheme, the boundary value problem is addressed numerically.The key points from the present analysis are addressed below: • On fluid velocity, the performances of Weissenberg number is show the increasing behaviour and the mag- netic parameter has the opposite effect on fluid velocity.• The fluid temperature is eroded by the Prandtl number, while the Lorentz force and Biot numbers produce a more fluid temperature.• Skin friction decreases with bigger of α and We but rises with an upsurge in M.
• The rate of heat transfer is higher when the Eckert number, Biot number, Prandtl number and thermal radia- tion are larger.• The fluid temperature rise as P r and Ec escalation, while the fluid velocity decline when β the rise.
• The present findings are validated and found to be in excellent accord with the previous finding.
• Via triangular fuzzy MFs, it is highlighted that hybrid nanofluids are significantly more capable of accelerat- ing thermal efficiency than Al 2 O 3 /EG and Cu/EG nanofluids.• From this research and the figures, we can determine that the hybrid nanofluid has a significantly higher heat transfer rate than the Tangent hyperbolic fluid.• The temperature rises as thermal conductivity improves.Because copper nanoparticles have a higher thermal conductivity than other metallic nanoparticles, their insertion produces more heat than that of any other type of nanoparticle.• The heat transfer rate is enhanced for temperature ratio and nanoparticle volume fractions.

Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
1In respect of dimensionless from one's point of view, where Re x = xU w ν symbolizes the local Reynolds number.

Table 4 .
The values of Cf x for various parameters.

Table 5 .
The values of Nu x for various parameters.