Numerical study of second-grade fuzzy hybrid nanofluid flow over the exponentially permeable stretching/shrinking surface

The study of hybrid nanoliquids can aid in developing numerous advanced features that facilitate heat transmission, such as pharmaceutical processes, hybrid-powered engines, microelectronics, engine cooling, and domestic refrigerators. In the current study, a mathematical model is designed to elaborate the physical inception of an unsteady second-grade hybrid nanofluid with ${\text{Al}}_2{\mathrm{O}}_3-\text{Cu}/\text{SA}$, a combination concentrated over the permeable exponentially heated stretching/shrinking sheet under hydromagnetic, heat source/sink, and viscous dissipation implications. The set of similarity transforms is used to convert underlying partial differential equations into the system of ordinary differential equations. The well-known homotopy analysis method is applied to tackle the formulated differential system in the MATHEMATICA program, which can obtain non-uniqueness outcomes. The imprecision of nanofluid and hybrid nanofluid volume fractions was modeled as a triangular fuzzy number [0%, 5%, 10%] for comparison. The double parametric approach was applied to deal with the fuzziness of the associated fuzzy parameters. The nonlinear ordinary differential equations are converted into fuzzy differential equations, and the homotopy analysis method is used for the fuzzy solution. In terms of code validity, our results are matched to previous findings. The features of several parameters against the velocity, surface-friction coefficient, heat transfer, and Nusselt number are described via graphs. Furthermore, the nanoparticle volume fraction magnifies the fluid temperature and retards the flow profile throughout the domain, according to our findings. Thermal profiles increase with progress in the heat source, nanoparticles volumetric fractions, viscous dissipation, and nonlinear thermal radiation. The percentage increase in the drag force and heat transfer rate are 15.18 and 5.54 when the magnetic parameter takes input in the range 0.1 ≤ M ≤ 0.3 and nanoparticle volume fraction inputs 0.01 ≤ ${\varphi }_1$ ≤ 0.15. From our observation, the hybrid nanofluid displays the maximum heat transfer compared to nanofluids. This important contribution will support industrial growth, particularly in the processing and manufacturing sectors.

1 Introduction

Investigations into non-Newtonian materials have been ongoing since the past century due to their unique characteristics and fascinating rheological properties. These materials are widely used across various industries, including chemical engineering, metal processing, food, and plastics. Non-Newtonian fluids have a range of applications, including biofluids, glassblowing, synthetic fibers, cosmetics, food, pharmaceuticals, shampoo, and metal spinning. These fluids exhibit different behaviors and can be classified as dilatant, shear-thickening, thixotropic, or shear-thinning. Rheologists have identified various fluid models, such as Casson, Maxwell, Burgers, Williamson, Oldroyd-B, third-grade, Jeffrey, micropolar, Sisko, and Sutterby Cross. However, second-grade fluids behave differently under different conditions, which explains the characteristics of shear-thickening, shear-thinning, and Newtonian effects. Second-grade fluids have gained the attention and devotion of intellectuals due to their dynamic properties [1–8]. Stretching a plastic sheet, on the other hand, is not always linear. An exponentially stretched sheet’s heat transport characteristics have a broader range of technical applicability. The heat transfer ratio of the continuously expanded surface increases rapidly with the expansion rate and temperature variations, which regulates the outcome when the copper wire is thinned and diluted. The techniques involved in these methods significantly impact the final product quality due to the effect of stretching kinematics and concurrent heating or cooling. Khan and Sanjayanand [9] analyzed a second-grade fluid’s steady flow and heat conductivity with an exponentially extending surface using the Runge–Kutta fourth-order (RK4) method. Rehman et al. [10] investigated the steady flow of a second-grade fluid over an exponentially stretching sheet using the Keller box and homotopy analysis approaches. Nadeem et al. [11] explored the flow and heat transfer of second-grade (viscoelastic) liquids in thermal radiation. Ramzan and Bilal [12] calculated the mixed convection of a second-grade nanofluid caused by time-dependent MHD, thermal radiation, and diffuse surfaces. Pakdemirli et al. [13] used perturbation analysis to examine the properties of a second-grade fluid. Recently, many researchers have studied second-grade nanofluids over an exponentially stretching surface [14–23].

Professionals like unsteady flow in several engineering organizations since it contributes to better mechanisms over their deeds [24, 25]. Moreover, even in ideal flow conditions, unnecessary destabilizing effects can occur around the system. The behavior of unstable boundary layer (BL) flow is unique compared to steady-state flow because the control equation has additional time-dependent conditions that degrade the structure of BL separation and fluid motion. However, through a healthier consideration of unstable fluid flow presentations in manufacturing dealings, contemporary enterprise techniques that permit improved structure dependability, productivity, and cost saving of multiple dynamical devices are possible [26]. Zaib et al. [27] discussed the computational exploration of a time-dependent flow with heat flux past an exponentially contracting surface.

The spectacle of heat transfer in electromagnetic waves is called thermal radiation. It happens because the two mediums have a significant temperature difference. In manufacturing and physical science, radiative influences are a crucial part. In the polymer manufacturing sectors, where heat-controlling variables influence the ultimate product quality to some extent, thermal radiation impacts are essential in controlling heat transfer. In addition, the radiation effects of missiles, aircraft, solar radiation, gas turbines, liquid metal fluids, spacecraft, nuclear power plants, and MHD accelerators are also prominent. Pantokratoras and Fang [28] were pioneers in examining the effect of nonlinear thermal radiation on Sakiadis flow. Dogonchi and Ganji [29] evaluated the impact of radiant heat on the MHD flow of a water-based nanofluid in a channel that can shrink, stretch, and diverge or converge. Khan et al. [30] studied the radiation flow of hybrid nanofluids through porous surfaces [30]. Many researchers [31–36] are involved in nonlinear thermal radiation.

Recognizing the need for improved thermal conductivity in traditional fluids, a new type of nanofluid called “hybrid nanofluid” is presented to provide highly industrialized heat conductivity. Two or more semiconductor materials are mixed with a base fluid to make a hybrid nanofluid. Different nanomaterials include carbon nanotubes [37], metals, metal oxides, and carbides. Numerous investigators are now interested in hybrid nanofluid due to its significance for the betterment of thermodynamic characteristics in real-world applications [38, 39], as a result of Choi and Eastman’s [40] outstanding findings that gave the unique notion of nanoliquid. Hybrid nanofluids are also used in various applications, including electrical gadget cooling [41], cooling of domestic refrigerators [42], automobile braking fluid, transformers, heat exchangers, and solar water heating [43]. Suresh et al. [44] explored the effects of a hybrid nanofluid $\left({\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{Water}\right)$ in a circular tube that was uniformly heated. Momin [45] investigated the thermal act of a hybrid nanofluid in a spherical tube and demonstrated that the hybrid nanofluid improves thermal conductivity compared to a conventional working liquid. Waini et al. [46] explored the influence of buoyancy on hybrid $\left({\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{Water}\right)$ nanofluid flow toward the stagnation point of an exponentially stretching/shrinking vertical sheet. They determined that the $\left({\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{Water}\right)$ hybrid nanofluid had a greater rate of heat transfer than the Cu/water nanofluid. Khan [47] numerically examined the convection of copper (Cu + Water) and nanoliquid across a spinning disc in a porous media. Cu–water has a faster heat transfer rate than ${\text{Al}}_2{\mathrm{O}}_3/\text{Water}$, and the presence of porous media raised the thickness of the thermal boundary layer. Takabi and Salehi [48] analyzed the heat transfers of $\left({\text{Al}}_2{\mathrm{O}}_3/\text{Water}\right)$ nanofluids and $\left({\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{Water}\right)$ hybrid nanofluids with a heat source. The literature is well stocked with further information on this topic [49–53].

The fuzzy set theory (FST) [54] has proved to be a valuable technique for modeling uncertainties in recent decades, providing models with a more accurate view of reality and allowing them to express themselves with a broader perspective [55–59]. After modeling real-world problems, they convert into partial differential equations (PDEs) or ordinary differential equations (ODEs). Uncertainty issues may arise during the development of a dynamic model. Researchers must deal with inaccurate data, parameters, dynamical variability, and complex relationships. As a result, many scientists use fuzzy models to depict dynamical systems to prevent artificial data accuracy and produce more realistic results. The fuzzy differential equation (FDE) is critical in overcoming these challenges. Initially, Chang and Zadeh [60] proposed the basic idea of fuzzy derivatives. Dubois and Prade [61] proposed the idea of fuzzy numbers (FNs) for solving an FDE. Kaleva [62] introduced the concept of FDEs in a fuzzy environment. Recently, FDEs played a significant role in fluid dynamics, such as the effects of MHD and gravitation on the third-grade fluid through an inclined channel in a fuzzy atmosphere, which were quantitatively explored by Nadeem et al. [63]. They used the triangular fuzzy numbers to analyze ambiguity. The heat transmission of SWCNTs MWCNTS on a third-grade nanofluid along an inclined channel in a fuzzy atmosphere was explored by Siddiqui et al. [64]. For comparison and uncertainty, they used nanoparticle volume fraction as TFN.

A careful review of the previously cited literature reveals several breaks and confines. No preceding studies have examined the unsteady MHD flow of the second-grade hybrid $\left({\text{Al}}_2{\mathrm{O}}_3-\text{Cu}/\text{SA}\right)$ nanofluid over the exponentially stretching/shrinking sheet with heat source/sink and viscous dissipation in their research outline. Also, the nanoparticle volume fraction of nanofluid and hybrid nanofluid are taken as triangular fuzzy numbers using the double parametric concept for comparison and uncertainty. The homotopy analysis technique was used to tackle the problem under consideration. The impact of important parameters on heat and flow field quantities and nanoparticle volume fraction is graphed and briefly discussed. This innovative contribution might help improve industrial manufacturing, predominantly in the processing and industrial areas.

The motivations for performing this analysis inspire the following research questions:

1) How do the thermal characteristics of nanoparticles vary when nonlinear thermal radiation features are used?

2) How do different developing parameters affect heat transfer and flow rates?

3) How does heat transfer improve in heat source/sink and magnetic force implications?

4) Why is the homotopy analysis method (HAM) preferred over the other methods?

5) Ho\w does the Lorentz force affect the velocity of the second-grade hybrid nanofluid by applying the magnetic field?

2 Mathematical formulation

The time-dependent, 2D incompressible, and unsteady flow of the MHD viscoelastic (second-grade) hybrid ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ nanofluid over the exponentially stretching/shrinking surface is engaged into interpretation in this research, as shown in Figure 1. $u_w\left(x,t\right)=\lambda e^{x/L}\left(a/\left(1-ct\right)\right)$ signifies the stretching/shrinking velocity, where lambda represents a constant that relates to stretching $\lambda >0$ . In shrinking $\lambda <0$ cases of the velocity rate, L indicates the characteristic length and c denotes the unsteadiness. $v_w\left(x,t\right)=\left({\nu }_o{e}^{x/2L}/\sqrt{1-ct}\right)$ signifies the mass flux velocity, where $v_o$ is the constant. The ambient and reference temperatures are labeled as $T_w$ and $T_{\infty },$ correspondingly, while $T_w=T_{\infty }+e^{x/2L}\left(T_o/\left(1-ct\right)\right)$ regulates the temperature circulation close to the surface. The magnetic field is expected to be $B\left(x\right)=e^{x/2L}\left(B_o/\sqrt{\left(1-ct\right)}\right)$, with $B_o$ indicating an identical magnetic field. The viscous, source/sink, and nonlinear thermal radiation impacts are also deliberated.

FIGURE 1

FIGURE 1. Flow problem.

When using the BL approximation, the governing equations for continuity, momentum, and heat are established on all of the preceding assumptions [12, 51]:

$\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x}=0,(1)$

$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\alpha }_1}{{\rho }_{hnf}}\left(\frac{{\partial }^{3}u}{\partial t\partial y^3}+u\frac{{\partial }^{3}u}{\partial x\partial y^3}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial y^2}+\frac{\partial u}{\partial y}\frac{{\partial }^{2}v}{\partial y^2}+v\frac{{\partial }^{3}u}{\partial y^3}\right)-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}u,(2)$

$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+\frac{\partial T}{\partial t}={\alpha }_{hnf}\frac{{\partial }^{2}T}{\partial y^2}+\frac{Q_0\left(T-T_{\infty }\right)}{{\left(\rho c_P\right)}_{hnf}}+\frac{16{\delta }^{*}}{3k^{*}{\left(\rho c_P\right)}_f}\left(T^3\frac{{\partial }^{2}T}{\partial y^2}+3T^2{\left(\frac{\partial T}{\partial y}\right)}^2\right)\\ +\frac{{\mu }_{hnf}}{{\left(\rho c_P\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{{\alpha }_1}{{\left(\rho c_P\right)}_{hnf}}\left(\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial y\partial t}+u\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}+v\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial y^2}\right),\end{array}(3)$

and the boundary conditions are

$\left.\begin{array}{l}t<0:T=T_{\infty },v=0,u=0,\forall x,y,\\ t\ge 0:v=v_w,u=U_w\left(x,t\right)=\lambda u_w\left(x,t\right),T=T_w\text{\hspace{1em}at\hspace{0.5em}}y\to 0,\\ u=0,T=T_{\infty }\text{\hspace{1em}as\hspace{1em}}y\to \infty ,\end{array}\right\}(4)$

where $u$ and $v$ indicate the velocity components along the $x-\mathrm{a}\overline{\mathrm{x}}\text{is}$ and $y-\mathrm{a}\overline{\mathrm{x}}\text{is},$ respectively, while the fluid temperature is denoted by T. The dynamic viscosity ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ is ${\mu }_{hnf}$, ${\rho }_{hnf}$ is the density of ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$, ${\left(\rho Cp\right)}_{hnf}$ is the ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ heat capacity, $k_{hnf}$ is the ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ thermal/heat conductivity, and ${\delta }_{hnf}$ is the electrical conductivity ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}.$ The aluminum oxide $\left({\text{Al}}_2{\mathrm{O}}_3\right)$ thermophysical properties, along with copper (Cu) and sodium alginate (SA) nanoparticles, are revealed in Table 1. Equation (5) contains the thermophysical properties of ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$. Here, $A{l}_2{O}_3$ and $Cu$ are nanoparticles having the volume fractions ${\varphi }_1$ and ${\varphi }_2$, respectively.

TABLE 1

TABLE 1. $A{l}_2{O}_3$ thermophysical properties along with $Cu$ and SA [51].

The thermophysical properties of hybrid nanofluids are as follows [51]:

$\left.\begin{array}{l}{\rho }_r=\frac{{\rho }_{hnf}}{{\rho }_f}=\left[\left(1-{\varphi }_2\right)\left\{\left(1-{\varphi }_1\right)+\frac{{\rho }_{s_1}{\varphi }_1}{{\rho }_f}\right\}+\frac{{\rho }_{s_2}{\varphi }_2}{{\rho }_f}\right],{\mu }_r=\frac{{\mu }_{hnf}}{{\mu }_f}={\left(1-{\varphi }_1\right)}^{-2.5}{\left(1-{\varphi }_2\right)}^{-2.5},\\ {\left(\rho C_{\rho }\right)}_r=\frac{{\left(\rho C_{\rho }\right)}_{hnf}}{{\left(\rho C_{\rho }\right)}_f}=\frac{{\varphi }_2{\left(\rho C_{\rho }\right)}_{s_2}}{{\left(\rho C_{\rho }\right)}_f}+\left(1-{\varphi }_2\right)\left[\left(1-{\varphi }_1\right)+\frac{{\left(\rho C_{\rho }\right)}_{s_1}{\varphi }_1}{{\left(\rho C_{\rho }\right)}_f}\right],\\ k_r=\frac{k_{hnf}}{k_{nf}}=\frac{2k_{nf}-2{\varphi }_1\left(k_{s_1}-k_{nf}\right)+k_{s_1}}{2k_{nf}+{\varphi }_1\left(k_{s_1}-k_{nf}\right)+k_{s_1}},\frac{k_{nf}}{k_f}=\frac{2k_f-2{\varphi }_2\left(k_{s_2}-k_f\right)+k_{s_2}}{2k_f+{\varphi }_2\left(k_{s_2}-k_f\right)+k_{s_2}},\\ {\sigma }_r=\frac{{\sigma }_{hnf}}{{\sigma }_{bf}}=\left[\frac{{\sigma }_{s_2}\left(1+2{\varphi }_2\right)+2{\sigma }_{bf}\left(1-{\varphi }_2\right)}{{\sigma }_{s_2}\left(1-{\varphi }_2\right)+{\sigma }_{bf}\left(2+{\varphi }_2\right)}\right],{\sigma }_{bf}=\left[\frac{{\sigma }_{s_1}\left(1+2{\varphi }_1\right)+2{\sigma }_f\left(1-{\varphi }_1\right)}{{\sigma }_{s_1}\left(1-{\varphi }_1\right)+{\sigma }_f\left(2+{\varphi }_1\right)}\right]{\sigma }_f.\end{array}\right\}(5)$

The following similarity transformations are presented in [10] to simplify the governing Eqs 1–3 along with the boundary conditions (4). The stream function $\omega$ can be expressed as a customizable form $v=-\partial \omega /\partial x,u=\partial \omega /\partial y$, and the similarity variable is $\eta :$

$\left.\begin{array}{l}\omega =\sqrt{\frac{2la{\nu }_f}{1-ct}}{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2L$}\right.}f\left(\eta \right),\eta =\sqrt{\frac{a}{2l{\nu }_f\left(1-ct\right)}}{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2L$}\right.}y,\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\\ u=\frac{a{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}}{1-ct}{f}^{\prime }\left(\eta \right),v=-\sqrt{\frac{a{\nu }_f}{2l\left(1-ct\right)}}{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2L$}\right.}\left(f\left(\eta \right)+\eta f^{\prime }\left(\eta \right)\right).\end{array}\right\}(6)$

Using Eq. 6, Eqs (2), (3) can be condensed to the following set of nonlinear ODEs in the context of the abovementioned relations [36]:

$\begin{array}{l}\left(\frac{{\mu }_r}{{\rho }_r}\right)f^{‴}-2{\left(f^{\prime }\right)}^2-\beta \left(\frac{\eta }{2}{f}^{″}+f^{\prime }\right)+f{f}^{″}+\frac{\alpha }{{\rho }_r}\left(\beta f^{‴}+2f^{\prime }{f}^{‴}-{\left(f^{″}\right)}^2-f{f}^{iv}+\frac{\beta \eta }{2}{f}^{iv}\right)\\ -\frac{{\sigma }_r}{{\rho }_r}M{f}^{\prime }=0,\end{array}(7)$

$\begin{array}{l}\left(1+\text{Nr}{\left(1+\theta \left({\theta }_w-1\right)\right)}^3\right){\theta }^{″}+3\mathrm{N}\mathrm{r}{\left({\theta }^{\prime }\right)}^2\left({\theta }_w-1\right){\left(1+\theta \left({\theta }_w-1\right)\right)}^2+\mathrm{Pr}f{\theta }^{\prime }-\mathrm{Pr}\beta \left(\frac{1}{2}\eta {\theta }^{\prime }+\theta \right)\\ -\mathrm{Pr}\theta f^{\prime }+\frac{\text{PrH}\theta }{{\left(\rho c_P\right)}_r}+\frac{{\mu }_r\text{PrEc}}{{\left(\rho c_P\right)}_r}{\left(f^{″}\right)}^2+\frac{\alpha \text{PrEc}}{{\left(\rho c_P\right)}_r}{f}^{″}\left(2\eta f^{″}+2f^{\prime }{f}^{″}-f{f}^{‴}+\eta \beta f^{‴}\right)=0,\end{array}(8)$

with the constraints

$\left.\begin{array}{cc}f\left(\eta \right)=s,f^{\prime }\left(\eta \right)=\lambda ,\theta \left(\eta \right)=1& \text{at\hspace{1em}}\eta =0,\\ f^{\prime }\left(\eta \right)=0,\theta \left(\eta \right)=0& \text{as\hspace{1em}}\eta \to \infty .\end{array}\right\},(9)$

where the unsteadiness parameter is $\beta =2Lc/a{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$, the magnetic parameter is $M=2{\delta }_{f}L{B}_o^2/a{\rho }_f$, the Prandtl number is $\mathrm{Pr}={\nu }_f/{\alpha }_f$, the second-grade fluid parameter is $\alpha =a{e}^{x/L}/2L{\mu }_f\left(1-ct\right)$, the Eckert number is $Ec=a^2/{\left(1-\text{ct}\right)}^2\left(T_w-T_{\infty }\right){\left(C_{\rho }\right)}_f$, the heat generation/absorption parameter is $H=2L\left(1-\text{ct}\right)\left(T_w-T_{\infty }\right){\mathrm{Q}}_o/a$, and the suction parameter is $s=-v_o\sqrt{2L/a{\nu }_f}$.

The stretching/shrinking parameter is $\lambda$. The coefficient of skin friction $\left({\mathrm{C}}_{fx}\right)$ and the local Nusselt number $\left({\text{Nu}}_{\mathrm{x}}\right)$ are, thus, demarcated as follows [12]:

$C_{fx}=\frac{1}{{\rho }_f{u}_e^2}{\left[{\mu }_{hnf}\frac{\partial u}{\partial y}+{\alpha }_1\left\{u\frac{{\partial }^{2}u}{\partial x\partial y}+v\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial t\partial y}+2\frac{\partial u}{\partial y}\frac{\partial u}{\partial x}\right\}\right]}_{y=0},(10)$

$N{u}_x=-\frac{x}{k_f\left(T_w-T_{\infty }\right)}{\left[k_{hnf}\frac{\partial T}{\partial y}+\frac{16{\sigma }^{*}{T}_{\infty }^3}{3k^{*}}\frac{\partial T}{\partial y}\right]}_{y=0}.(11)$

Using Eq. 6 in Eq. 10 and Eq. (11) yields the following relationship:

$\sqrt{{\text{Re}}_x}{C}_{fx}=\left({\mu }_r{f}^{″}\left(0\right)+\alpha \left(7f^{\prime }\left(0\right)f^{″}\left(0\right)-f\left(0\right)f^{″}\left(0\right)+3\beta f^{″}\left(0\right)+\eta f^{″}\left(0\right)f^{″}\left(0\right)+\eta \beta f^{‴}\left(0\right)\right)\right),(12)$

${\left({\text{Re}}_x\right)}^{-0.5}N{u}_x=-\left(k_r+Nr{\left(1+\theta \left(0\right)\left({\theta }_w-1\right)\right)}^3\right){\theta }^{\prime }\left(0\right),(13)$

where $R{e}_x=u_{e}x/{\nu }_f$ is the x-axis-local Reynolds number. Moreover, the graphical explaination of triangular fuzzy number is given in the Figure 2.

FIGURE 2

FIGURE 2. Membership function of TFN [54].

2.1 Homotopy analysis method

The HAM is a multifaceted investigative system that solves nonlinear equations with several variables. Based on Eq. 9, the HAM computes consequential Eqs 7, 8. Linear operators and preliminary approximations are mandatory to surprise the process through this technique. Consequently, we used them as $\left({\mathrm{\Lambda }}_f,{\mathrm{\Lambda }}_{\theta }\right)$ linear operators and $\left(f_0\left(\eta \right),{\theta }_0\left(\eta \right)\right)$ initial assessments to resolve motion and energy transform equations using the abovementioned method. See [9–12] for further facts on this method.

$f_0\left(\eta \right)=s-\lambda \left(1-e^{-\eta }\right),{\theta }_0\left(\eta \right)=e^{-\eta },(14)$

${\mathrm{\Lambda }}_f\left[f\left(\eta \right)\right]=f^{‴}-f^{\prime },{\mathrm{\Lambda }}_{\theta }\left[\theta \left(\eta \right)\right]={\theta }^{″}-\theta .(15)$

The properties of the operator described above are as follows:

$\left.\begin{array}{l}{\mathrm{\Lambda }}_f\left(A_1+A_2{e}^{-\eta }+A_3{e}^{\eta }\right)=0,\\ {\mathrm{\Lambda }}_{\theta }\left(A_4{e}^{-\eta }+A_5{e}^{\eta }\right)=0,\end{array}\right\}(16)$

where $A_{js}$ (j = 1, 2, 5) are arbitrary constants.

$\left.\begin{array}{l}\left(1-q\right){\mathrm{\Lambda }}_f\left[\stackrel{\sim }{F}\left(\eta ;q\right)-{\stackrel{\sim }{f}}_0\left(\eta \right)\right]-q{h}_f{N}_f\left[\stackrel{\sim }{F}\left(\eta ;q\right)\right]=0,\\ \left(1-q\right){\mathrm{\Lambda }}_{\theta }\left[\stackrel{\sim }{\theta }\left(\eta ;q\right)-{\stackrel{\sim }{\theta }}_0\left(\eta \right)\right]-q{h}_{\theta }{N}_{\theta }\left[\stackrel{\sim }{F}\left(\eta ;q\right),\stackrel{\sim }{\theta }\left(\eta ;q\right)\right]=0,\end{array}\right\}(17)$

where $h_f\text{\hspace{0.5em}and\hspace{0.5em}}{h}_{\theta }$ signify non-zero auxiliary parameters, $q\in \left[0,1\right]$ represents an embedding parameter, and $\stackrel{\sim }{F},\text{and\hspace{0.5em}}\stackrel{\sim }{\theta }$ represent the mapping occupations for $f\left(\eta \right),\text{and\hspace{1em}}\theta \left(\eta \right)$, respectively.

The boundary conditions become [12]

$\left.\begin{array}{l}\stackrel{\sim }{F}\left(0;q\right)=s,{\stackrel{\sim }{F}}^{\prime }\left(0;q\right)=\lambda ,\stackrel{\sim }{\theta }\left(0;q\right)=1,\\ {\stackrel{\sim }{F}}^{\prime }\left(\infty ;q\right)=0,\stackrel{\sim }{\theta }\left(\infty ;q\right)=0,\end{array}\right\}(18)$

$\begin{array}{l}{N}_f\left[\stackrel{\sim }{F}\left(\eta ;q\right)\right]=\frac{d{\stackrel{\sim }{F}}^{‴}\left(\eta ;q\right)}{d\eta }-2{\left(\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta }\right)}^2+\frac{d\stackrel{\sim }{F}\left(\eta ;q\right)}{d\eta }\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }-\beta \left(\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta }+\frac{\eta }{2}\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }\right)\\ +\alpha \left(\beta \frac{d{\stackrel{\sim }{F}}^{‴}\left(\eta ;q\right)}{d\eta }-{\left(\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }\right)}^2-\stackrel{\sim }{F}\left(\eta ;q\right)\frac{d{\stackrel{\sim }{F}}^{⁗}\left(\eta ;q\right)}{d\eta }+2\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta }\frac{d{\stackrel{\sim }{F}}^{‴}\left(\eta ;q\right)}{d\eta }+\frac{\beta \eta }{2}\frac{d{\stackrel{\sim }{F}}^{⁗}\left(\eta ;q\right)}{d\eta }\right)-M\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta },\end{array}(19)$

$\begin{array}{l}{N}_{\theta }\left[\stackrel{\sim }{F}\left(\eta ;q\right),\stackrel{\sim }{\theta }\left(\eta ;q\right)\right]=\left(1+\text{Nr}{\left(1+\stackrel{\sim }{\theta }\left(\eta ;q\right)\left({\theta }_w-1\right)\right)}^3\right)\frac{d{\stackrel{\sim }{\theta }}^{″}\left(\eta ;q\right)}{d\eta }-\mathrm{Pr}\stackrel{\sim }{\theta }\left(\eta ;q\right)\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta }\\ +3\text{Nr}{\left(\frac{d{\stackrel{\sim }{\theta }}^{\prime }\left(\eta ;q\right)}{d\eta }\right)}^2\left({\theta }_w-1\right){\left(1+\stackrel{\sim }{\theta }\left(\eta ;q\right)\left({\theta }_w-1\right)\right)}^2+\mathrm{Pr}D\stackrel{\sim }{F}\left(\eta ;q\right)\frac{d{\stackrel{\sim }{\varphi }}^{″}\left(\eta ;q\right)}{d\eta }\\ +\mathrm{Pr}\stackrel{\sim }{F}\left(\eta ;q\right)\frac{d{\stackrel{\sim }{\theta }}^{\prime }\left(\eta ;q\right)}{d\eta }+\text{PrEc}{\left(\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }\right)}^2-\mathrm{Pr}\beta \left(\frac{1}{2}\eta \frac{d{\stackrel{\sim }{\theta }}^{\prime }\left(\eta ;q\right)}{d\eta }+\stackrel{\sim }{\theta }\left(\eta ;q\right)\right)\\ +\alpha \text{PrEc}\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }\left(\begin{array}{l}2\eta \frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }+2\frac{d{\stackrel{\sim }{F}}^{\prime }\left(\eta ;q\right)}{d\eta }\frac{d{\stackrel{\sim }{F}}^{″}\left(\eta ;q\right)}{d\eta }\\ +\eta \beta \frac{d{\stackrel{\sim }{F}}^{‴}\left(\eta ;q\right)}{d\eta }-\stackrel{\sim }{F}\left(\eta ;q\right)\frac{d{\stackrel{\sim }{F}}^{‴}\left(\eta ;q\right)}{d\eta }\end{array}\right)+\mathrm{Pr}\frac{d{\stackrel{\sim }{\theta }}^{\prime }\left(\eta ;q\right)}{d\eta }\left(Nb\frac{d{\stackrel{\sim }{\varphi }}^{\prime }\left(\eta ;q\right)}{d\eta }+Nt\frac{d{\stackrel{\sim }{\theta }}^{\prime }\left(\eta ;q\right)}{d\eta }\right).\end{array}(20)$

Equations (7)–(9) convert into nonlinear operators like Eqs 18–21, and then, the series solution becomes

$\left.\begin{array}{l}f\left(\eta \right)=f_0\left(\eta \right)+{\displaystyle \sum _{m=1}^{\infty }}{f}_m\left(\eta \right),\\ \theta \left(\eta \right)={\theta }_0\left(\eta \right)+{\displaystyle \sum _{m=1}^{\infty }}{\theta }_m\left(\eta \right).\end{array}\right\}(21)$

2.2 Fuzzification

Using fuzzy concepts, comparing nanofluid and hybrid nanofluid is also explored in this study. The nonlinear ODEs convert into FDEs, and the nanoparticle volume percentage is taken as a TFN. The governing FDE is converted into a double parametric form. In this case, Eq. 8 can be converted into an interval form using the $\chi -\text{cut}$ concept. Here, $\chi$ and $\omega$ are parameters that range from 0 to 1, controlling the fuzziness of the uncertain parameters. The aforementioned problem was solved using the HAM as well. The slight variation in the volume percentage of nanoparticles impacts the flow rate and heat. These parameters alone determine the nanofluid’s flow rate and heat transfer because some researchers estimate that the volume percentage of nanoparticles falls within the [1%–4%] range. It is preferable to address a challenging situation in a fuzzy atmosphere by getting volume fractions as a TFN since ${\varphi }_1$ and ${\varphi }_2$ signify the volume fraction of ${\text{Al}}_2{\mathrm{O}}_3/\text{SA}$ and Cu/SA, respectively, as shown in Table 2. The volume fractions of nanoparticles used in this study are classified as TFNs, with the TFNs being transformed into $\chi -\text{cut}$ methods, and the fuzziness of the TFNs is controlled by $\chi -\text{cut}$ [64].

TABLE 2

TABLE 2. ${\varphi }_1\text{\hspace{0.5em}and\hspace{0.5em}}{\varphi }_2$ transform into TFN [54, 55].

Let ${\varphi }_1={\varphi }_2=\left[0,0.05,0.1\right]$ be a TFN that is described by the three values highlighted in Figure 3: 0 (lower bound), 0.05 (most belief value), and 0.1 (upper bound). As the input value moves from position 0 to position 0.05, the value of the membership function climbs linearly from 0 to 1 and then linearly declines from 1 to 0 as the input value moves from position 0.05 to position 0.1. Eq. 22 represents the mathematical form of the triangular fuzzy membership function as follows:

$\text{Membership function}=\left\{\begin{array}{l}\frac{0-\eta }{0.05-0}\text{\hspace{1em}for\hspace{1em}}\eta \in \left[0,0.05\right],\\ \frac{\eta -0.1}{0.1-0.05}\text{\hspace{1em}for\hspace{1em}}\eta \in \left[0.05,0.1\right],\\ 0,\text{otherwise}.\end{array}\right.(22)$

FIGURE 3

FIGURE 3. Impression of $f^{\prime }\left(\eta \right)$ for M.

The $\chi -\text{cut}$ technique is used to convert TFNs into an interval form and is represented as ${\varphi }_1={\varphi }_2=\left[0+\chi \left(0.05-0\right),0.1-\chi \left(0.1-0.05\right)\right]$, where $0\le \chi -\text{cut}\le 1.$

To handle this scenario, the FDEs are renewed into lower ${\theta }_1\left(\eta ,\chi \right)$ and upper ${\theta }_2\left(\eta ,\chi \right)$ bounds.

3 Results and discussion

An unsteady flow analysis was performed on a second-grade hybrid $\left({\text{Al}}_2{\mathrm{O}}_3-\text{Cu}/\text{SA}\right)$ nanofluid above the exponential surface. Because of the viscous and nonlinear radiation heat transfer amalgamation, conductive fluids are studied in this context. The consequence of dynamic parameters on the speed and temperature profile of the system is scrutinized. In addition, an estimated analysis method called the HAM is used to follow the transformation equation generated from the abovementioned model. For the simulation of our model, we absolute the key parameters, such as M = 0.2, $\beta =0.5,\alpha =0.2,$ s = 0.2, Pr = 10, ${\theta }_w=1.2,$ Ec = 0.3, H = 0.1, $\lambda =0.3,{\varphi }_1=0.02,\text{and\hspace{0.5em}}{\varphi }_2=0.02.$ Table 2 is created to confirm the values of ${\theta }^{\prime }\left(0\right)$ [36]. The values obtained by the HAM in the current survey agree well with the literature.

The influence of the magnetic (M) parameter on the velocity field is depicted in Figure 3. For higher values of M, the velocity dropped in both cases. Lorentz pressure is responsible for this phenomenon, which arises from the cooperation of electric and magnetic fields during an electrically conducted fluid flow. So, the fluid velocity in the BL is controlled by the generated Lorentz force. As a result, as M rises, the velocity of the fluid and hybrid nanofluids falls. The interaction of magnetic fields is significant in different technical and industrial applications, such as crude oil extraction, geothermal systems, and groundwater hydrology. The change of the second-grade parameter ($\alpha$) in motion is shown in Figure 4. The rise in $\alpha$ clues to an enrichment in the velocity of liquid and hybrid nanofluids. This is because as $\alpha$ increases, the viscosity and viscous forces of the fluid decrease. The effects of an unsteady parameter ($\beta$) and suction parameter (s) on velocity and temperature fields are shown in Figures 5, 6. The temperature and velocity contours drop when $\beta$ and s are increasing. The increase in $\beta$ and s decreases the momentum and thermal boundry layer. Figure 7 shows the impression of stretching/shrinking parameters $\left(\lambda \right)$ on velocity and temperature dispersals. When $\lambda$ increases, the velocity also increases while the temperature diminishes. Because the stretching parameters are set to higher levels, the temperature and thickness of the BL are reduced. Due to the exposure of the cooler to the ambient fluid, the BL thickness reduces with growing values of stretching parameters. Figure 8 shows the variation of nanoparticle volume fraction $\left({\varphi }_1\right)$ on the velocity and temperature distributions. When ${\varphi }_1$ increases, the velocity declines while the temperature boosts up. The variability of the volume fractional of nanoparticles $\left({\varphi }_2\right)$ on velocity and temperature gradients is shown in Figure 9. When ${\varphi }_2$ progress, speed drops while temperature upsurges. The main reason for the decay in the velocity is that as the values of the volume fractional of nanoparticle grow, the resistive force also increases, reducing the fluid flow speed. Physically, the energy is discharged from the exponential sheet due to the nanoparticle’s resistive force. More energy is generated when more nanoparticles are added, causing the temperature to rise. Furthermore, the optimum temperature may be achieved because a hybrid nanofluid has a higher thermal conductivity than a mono nanofluid. Figure 10 shows the features of radiation parameters (Nr) and liquid temperature. High Nr approximations support the random motion of particles. As a result, more particles collide and produce more heat. As a result, the heat of the fluid increases. Figure 10 shows the thermal profile matures when the temperature ratio parameter $\left({\theta }_w\right)$ rises. These consequences specified that when θw develops, the temperature difference $\left(T_w-T_0\right)$ upsurges, instigating the fluid temperature to increase. Figure 11 pierces the heat generation parameter (H) impressions on the temperature field. It is noticed that as the H > 0 grows, the temperature distribution improves. Physically, higher heat production shows more heat within the boundary layer, increasing the temperature field.

FIGURE 4

FIGURE 4. Impression of $f^{\prime }\left(\eta \right)$ for $\alpha$.

FIGURE 5

FIGURE 5. Impression of $f^{\prime }\left(\eta \right)$ (A) and $\theta \left(\eta \right)$ (B) for $\beta$.

FIGURE 6

FIGURE 6. Impression of $f^{\prime }\left(\eta \right)$ (A) and $\theta \left(\eta \right)$ (B) for s.

FIGURE 7

FIGURE 7. Impression of $f^{\prime }\left(\eta \right)$ (A) and $\theta \left(\eta \right)$ (B) for $\lambda$.

FIGURE 8

FIGURE 8. Impression of $f^{\prime }\left(\eta \right)$ (A) and $\theta \left(\eta \right)$ (B) for ${\varphi }_1$.

FIGURE 9

FIGURE 9. Impression of $f^{\prime }\left(\eta \right)$ (A) and $\theta \left(\eta \right)$ (B) for ${\varphi }_2$.

FIGURE 10

FIGURE 10. Impression of $\theta \left(\eta \right)$ for Nr and ${\theta }_w$.

FIGURE 11

FIGURE 11. Impression of $\theta \left(\eta \right)$ for H.

As shown in Figure 12, f ′′(0) increases with M and decreases as $\alpha$ grows. Due to the Lorentz drag force, an increase in the M value leads to a substantial confrontation to fluid flow, which reduces the fluid velocity and momentum BL thickness, upsurges the velocity, and thus, increases the shear stress of the exponential stretch sheet. The behavior of $f^{″}\left(0\right)$, the unsteady parameter $\left(\beta \right)$, and the suction/injection parameter (s) is revealed in Figure 13. It can be detected that the drag force declines with the rise in $\beta$ and s. Physically, growth in $\beta$ and s results in an augmentation in the fluid density, due to which more friction is observed by the fluid particles. Figure 14 shows the impact of Nr and H on $N{u}_x$. It is observed that $N{u}_x$ reduces with an increase in Nr and H. $N{u}_x$ decreases when ${\varphi }_2$ increases, while $N{u}_x$ increases when ${\varphi }_1$ increases, as shown in Figure 15. Physically, heat is emitted from the exponential sheet when enhancing ${\varphi }_1\text{\hspace{0.5em}and\hspace{0.5em}}{\varphi }_2.$

FIGURE 12

FIGURE 12. Impression of M and $\alpha$ on $C_f$.

FIGURE 13

FIGURE 13. Impression of s and $\beta$ on $C_f$.

FIGURE 14

FIGURE 14. Impression of Nr and $H$ on $Nu$.

FIGURE 15

FIGURE 15. Impression of ${\varphi }_1$ and ${\varphi }_2$ on $Nu$.

3.1 Fuzzy results and discussion

Figure 16 portrays the calculated fuzzy temperature using volume fractions of ${\varphi }_1$ and ${\varphi }_2$ as the TFN [0%, 5%, 10%] for different values of $\eta ,$ 1, 2, 3, and 4; four subplots delineate the fuzzy temperatures for triangular MFs. The vertical axis represents the MF of the fuzzy temperature bend $\chi -\text{cut\hspace{0.17em}}\left(0\le \chi -\text{cut}\le 1\right)$, and the horizontal axis represents the fuzzy temperature curve with varying values of 𝜂. The resulting fuzzy temperature is TFN, but not symmetric, while a portion of the fuzzy volume is symmetric TFN. These variations might be due to the nonlinearity of the governing FDE. It was also revealed that hybrid nanofluids had a wider width than nanofluids. As a result, the hybrid nanofluid is uncertain according to the TFN. On the other hand, Figure 16 shows the comparison of $A{l}_2{O}_3/SA$ $\left({\varphi }_1\right),$ $Cu/SA$ $\left({\varphi }_2\right),$ and $A{l}_2{O}_3+Cu/SA$ hybrid nanofluids through MF for numerous values of $\eta$. In these figures, we evaluated three scenarios. When ${\varphi }_1$ is preserved as TFN and ${\varphi }_2=0$, it is signified by blue shapes. When ${\varphi }_2$ is preserved as TFN and ${\varphi }_1=0$, it is signified by red shapes, and the black lines show that the hybrid nanofluid is non-zero with both ${\varphi }_1$ and ${\varphi }_2$. It is observed that the temperature change in hybrid nanofluids is more noticeable than in two nanofluids; the performance of hybrid nanofluids is better. To deliver the maximum transmission of heat in hybrid nanofluid joined, the thermal conductivities of $A{l}_2{O}_3$ and $Cu$. $A{l}_2{O}_3/SA$ have a higher heat transfer during the comparison of $A{l}_2{O}_3/SA$ and $\text{Cu}/SA$ because the thermal conductivity of $A{l}_2{O}_3$ is higher than that of $Cu.$ The comparative analysis is provided in Table 3 of the proposed technique with prevailing approaches.

FIGURE 16

FIGURE 16. Comparison of ${\text{Al}}_2{\mathrm{O}}_3/\text{SA}$, $\text{Cu}/\text{SA}$, and ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ hybrid nanofluids for $\omega =0.5$ and different values of $\eta$.

TABLE 3

TABLE 3. Comparison of current results of ${\theta }^{\prime }\left(0\right)$ with the work of Haider et al. [36] for variation in Pr and M when H = 0.0, Nr = 0.0, $\beta =0.0$, and Ec = 0.0

4 Conclusion

This study analyzed the unsteady MHD second-grade hybrid $\left({\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}\right)$ nanofluid flow caused by the exponentially stretching/shrinking surface. Viscous dissipation, nonlinear thermal radiation, and heat scores/sink are also considered. An analytical approach, the HAM, is implemented for the outcome of the formulated problem. For validity, extant outcomes were equated with prevailing consequences. The impacts of non-dimensional physical parameters on velocity and temperature profiles for second-grade fluid and hybrid nanofluid are examined and discussed via graphs. Furthermore, ${\varphi }_1\text{\hspace{0.5em}and\hspace{0.5em}}{\varphi }_2$ are said to be TFNs using the $\chi -\text{cut}$ technique. Comparison and uncertainty are studied through triangular fuzzy graphs. The foremost goals of this study are as follows:

• The fluid velocity is dropped with the magnetic parameter, while the fluid velocity is boosted with the second-grade fluid parameter.

• The fluid temperature increases while the fluid velocity declines with the improvement of ${\varphi }_1\text{\hspace{0.5em}and\hspace{0.5em}}{\varphi }_2$.

• The fluid temperature boosts against higher values of ${\theta }_w$, Nr, and H, whereas the reverse holds for the unsteady parameter, suction parameter, and Prandtl number.

• The fluid velocity grows versus the stretching/shirking parameter while the fluid temperature declines.

• The skin friction coefficient is reduced with a rise in unsteady and second-grade parameters while growing with magnetic parameters.

• For higher values of Nr, H, ${\varphi }_1,\text{and\hspace{0.5em}}{\varphi }_2$, the surface heat transfer rate decreases.

• The maximum width of the fuzzy fluid temperature of the hybrid nanofluid was observed during a fuzzy analysis using a triangular MF, indicating that the fuzziness level is higher than that of regular nanofluids.

• The ${\text{Al}}_2{\mathrm{O}}_3+\text{Cu}/\text{SA}$ hybrid nanofluids showed exceptional capability to increase the heat transfer rate in ${\text{Al}}_2{\mathrm{O}}_3/\text{SA}$ and $\text{Cu}/\text{SA}$ during fuzzy heat transfer analysis compared to regular substances. It has also been observed that the performance of $\text{Cu}/\text{SA}$ is far better than that of ${\text{Al}}_2{\mathrm{O}}_3/\text{SA}$.

The findings of this study can be used to drive future progress in which the heating system’s heat outcome is analyzed with nanofluids or hybrid nanofluids of various kinds (Maxwell, third-grade, Casson, Carreau, micropolar fluids, etc).

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.

Author contributions

RZ, MN: conceptualization, methodology, and writing–original draft. MN, MS: data curation, investigation, resources, software, and writing–original draft. IS: formal analysis, supervision, validation, and writing–original draft. IK, AM: funding acquisition, project administration, resources, visualization, and data curation, writing.

Funding

The authors declare financial support was received for the research, authorship, and/or publication of this article. This project was funded by Zhejiang Normal University, Jinhua, Zhejiang, China.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature