Impact of fuzzy volume fraction on unsteady stagnation-point flow and heat transfer of a third-grade fuzzy hybrid nanofluid over a permeable shrinking/stretching sheet

Abstract

In current work, the unsteady stagnation point’s flow on a special third-grade fuzzy hybrid $\left(\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}\right)$ nanofluid (HNF) through a permeable convective shrinking/stretching sheet has been scrutinized. In addition, the adverse consequences of heat source, viscous dissipation, nonlinear thermal radiation, and fuzzy nanoparticle volume fraction are likewise taken into consideration. Non-linear coupled partial differential equations (PDEs) get transformed into ordinary differential equations (ODEs) using an effective similarity transformation. After that, the ODEs are numerically solved using the bvp4c algorithm. Regarding validation, the present results align with earlier published research. The effects of heat distribution, flow rate, Nusselt number, and skin friction coefficient on hybrid nanofluid dynamics are explored using graphical and tabular forms. The nanoparticle volume fraction is considered a triangular fuzzy number (TFN) [0, 5%, 10%]. With the use of TFNs, ODEs are transformed into fuzzy differential equations (FDEs). The TFNs are controlled using a widely used $\zeta \text{ - cut}$ technique and $\zeta \text{ - cut}\in \left[0,1\right],$ which requires minimal computational effort to examine their dynamical performance. Also, the comparison of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA,}$ $\text{Cu/SA}$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ through the fuzzy membership functions (MFs). The fuzzy MFs show that the hybrid nanofluid ($\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$) in terms of rate of heat transfer is better than both $\text{Cu/SA}$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ nanofluids.

KEYWORDS:

• HNFs
• Third-grade fluid
• Nonlinear-thermal radiation
• Shrinking/stretching sheet
• TFN

Nomenclature

$x,y$	=	Cartesian coordinates
$u,v$	=	Velocity components
$B_0$	=	Uniform Magnetic field
$T$	=	Temperature
$T_w,T_{\mathrm{\infty }}$	=	Reference and ambient temperature
$Q_0$	=	Heat absorption/ generation coefficient
$q_r$	=	Radiatively heat source
${\rho }_{\mathrm{hnf}}$	=	HNF density
${\rho }_f$	=	Density of fluid
${\mu }_{\mathrm{hnf}}$	=	HNF dynamic viscosity
${\mu }_f$	=	Fluid Dynamic viscosity
${\sigma }_f$	=	Fluid Electrical conductivity
$\eta$	=	Similarity variable
$({\text{β}}_T{)}_{\mathrm{hnf}}$	=	HNF thermal-expansion coefficient
$H$	=	Dimensionless heat source
$\mathrm{Nr}$	=	Thermal radiation parameter
$\lambda$	=	Rate of mass transfer parameter
$f\left(\eta \right)$	=	Normal component of the flow
$\theta \left(\eta \right)$	=	Dimensionless temperature
K	=	Third-grade fluid parameter
$R{e}_x$	=	Local Reynold number
$(\rho c_p{)}_{\mathrm{hnf}}$	=	HNF heat capacity
$s_1$, $s_2$	=	Solid nanoparticles of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ and Cu
Pr	=	Prandtl number
$\psi$	=	Stream function
$\mathrm{Ec}$	=	Eckert number
${\theta }_w$	=	Temperature ratio parameter
$M$	=	Magnetic parameter
$\alpha$	=	Shrinking/stretching rate parameter
$\text{β}$	=	Unsteady parameter
$\mathrm{Gr}$	=	Grashof number
${\sigma }_{\mathrm{hnf}}$	=	Electrical conductivity of HNF
$\zeta$	=	cut technique
${\mu }_{\overline{U}}\left(y\right)$	=	Membership function (MF)
$f^{\mathrm{\prime }}(\eta ,\gamma )$	=	Fuzzy velocity
$\overline{\theta }(\eta ,\gamma )$	=	Fuzzy temperature
${\varphi }_2$	=	The volume fraction of copper
${\varphi }_1$	=	Volume fraction of alumina
${\nu }_f$	=	Fluid kinematic-viscosity
$k_f$	=	Fluid thermal conductivity
$k_{\mathrm{hnf}}$	=	HNF thermal conductivity
$N{u}_x$	=	Nusselt numberi
$C{f}_x$	=	Skin-friction coefficient
$(\rho c_p{)}_f$	=	Fluid heat capacity

1. Introduction

Non-Newtonian fluids (NNFs) have played a dominant position in engineering and industrial processes in recent years due to their wide uses. Here, clay coatings, blood, saliva, polymer solutions, certain oils, paints, molten plastics, ketchup, artificial fibers, and lubricants are some instances of NNFs. As dynamic elastic characteristics, these liquids do not obey the well-known Newton's law of viscosity. Some liquids are enormously viscous which does interpret their significant properties of elasticity such as boiling, polymer depolarization, bubble absorption, composite processing, etc. There are three foremost classes of NNFs such as integral type, rate type, and differential type. Third-grade fluid is a well-known subclass of the differential type fluid that displays the normal stress effect but does not show shear thickening or thinning phenomena (Fosdick & Rajagopal, 1979). On the other way, the third-grade-fluid model can visualize both the normal stresses along shear-thinning or thickening phenomena even though the basic equations have numerous complexities (Dunn & Rajagopal, 1995; Pakdemirli, 1994). Various scholars have investigated the flows of a third-grade fluid model from different perspectives such as Keçebas and Yürüsoy (Keçeba & Yürüsoy, 2006) studied the unsteady flow of special third-grade fluid and calculated the solution using the shooting technique. Ellahi and Riaz (Ellahi & Riaz, 2010) examined the flow of a third-grade fluid using magneto-hydro-dynamic (MHD) and erratic viscosity through a pipe with the analytical technique homotopy analysis method (HAM). Sahoo and Do (Sahoo & Do, 2010) examined the effect of slip and magnetic effect of a third-grade fluid via a stretched surface. Later on, Sahoo and Poncet (Sahoo & Poncet, 2011) inspected the slip effect on a third-grade fluid through an exponentially stretched moving sheet. The unsteady flow of a particular third-grade fluid across a moving permeable surface was investigated by Abbasbandy and Hayat (Abbasbandy & Hayat, 2011). Naganthran et al. (Naganthran et al., 2016) inspected the double solution of unsteady flow of a special type of third-grade liquid over a porous shrinking/stretched sheet. Reddy et al. (Reddy et al., 2018) scrutinized the unsteady flow of a third-grade fluid over a cylinder. Also, they determined that the NNF has a substantial impact as compared to NF. Zaib et al. (Zaib et al., 2017; Zaib et al., 2020) examined the convective and nonlinear thermal radiation effect on a special third-grade fluid through a permeable shrinking surface. Several scholars have examined this model's flows from a variety of aspects (Bhattacharyya, 2011; Kumar et al., 2019; Mahmood et al., 2024).

The energy moving from one place to another in the waveform is called thermal radiation. The waves of radiation are moving in all directions. Nonlinear radiation is greatly appreciated in modern science and advanced technologies because its effects and intensity help in the construction of nano-scale devices. Due to their small sizes, nanosized particles in nanofluid absorb a large amount of nonlinear radiative radiation. Consequently, nanoparticles magnify the radiative features of fluids, causing to increase in sufficient absorption of solar collectors. Numerous scholars (Moayedi et al., 2024; Nadeem et al., 2023c; Nasir & Berrouk, 2023; Siddique et al., 2023; Zulqarnain et al., 2023c) are attracted to studying the impact of quadratic thermal radiation.

The MHD is an electrically conductive field, which itself adjusts the magnetic field and creates a current that constructs forces to control the boundary layer flow (BLF). The magnetic field provides a drag force which is named the Lorentz force in a flowing liquid, which holds the fluid flow and therefore raises the fluid temperature. Consequently, MHD has a great reputation in production and manufacturing progressions, such as stretching of plastic sheets, metal casting, plasma studies, turbulent pumps, geothermal energy extractions, optical grafting, MHD generators, metallurgical process, nuclear reactor safety, polymer industry, and furnace structure (Jawad et al., 2021; Krishna et al., 2019b; Krishna & Chamkha, 2019a; Nadeem et al., 2021b; Siddique et al., 2021; Waini et al., 2021).

Classical fluids such as engine oils, ethylene glycol, and water have lower thermal performance that restricts their usage in advanced cooling applications. The nanofluids (NFs) (Choi & Eastman, 1995; Huminic & Huminic, 2018; Shah & Ali, 2019; Xian et al., 2018) are a great invention that effort to raise the heat transmission rate and thermal conductivity. The size of nanoparticles consists of (1-100 nm) in the base fluids. There are different nanoparticles used in the base fluids such as carbides, copper, nitrides, carbon nanotubes metals, alumina, graphite, and metal oxides that enhance the host fluid's thermal conductivity. Mostly NFs are frequently used in solar panels, hybrid-powered vehicles, the latest fuel generation, medicine, cancer treatment, drug delivery, and modern heating, and cooling systems.

HNFs consist of two solid nanosized particles such as $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu,}$ $\text{MWCNTs + F}{\text{e}}_{\text{3}}{\text{O}}_{\text{4}}\text{,}$ $\text{SWCNTs + F}{\text{e}}_{\text{3}}{\text{O}}_{\text{4}}\text{,}$ $\text{Cu + Ti}{\text{O}}_{\text{2}},$ $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Ag,}$ $\text{CuO + Cu,}$ etc. in base fluids (sodium alginate, kerosene, water, engine oil or ethylene glycol). The prime goal of HNFs is to maximize heat transmission and thermal expansion. HNFs are now used in several heat transmission applications such as air conditioning systems, microchannels, mini channel heat sinks, tubular, plate heat exchangers, and helical coil heat exchangers. The literature review reveals a thorough examination of HNFs such as Roy and Pop (Roy & Pop, 2020) examined the upshot of MHD second-grade $\left(\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/}{\text{H}}_{\text{2}}\text{O}\right)$ HNF flow through a permeable shrinking/stretching surface. Devi and Devi (Devi & Devi, 2016; Devi & Devi, 2017) studied the heat transfer with mathematical analysis of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/}{\text{H}}_{\text{2}}\text{O}$ through a shrinking/stretching surface. The impact of heat generation and MHD flow of HNFs flowing over a rotating channel was inspected by Chamkha et al. (Chamkha et al., 2019). Hussanan et al. (Hussanan et al., 2020) considered the heat improvement properties of NNF $\text{F}{\text{e}}_{\text{3}}{\text{O}}_{\text{4}}\text{ + Cu/SA}$ HNF flow through a shrinking/stretching sheet. The behaviors of $\text{Cu + Ti}{\text{O}}_{\text{2}}\text{/}{\text{H}}_{\text{2}}\text{O}$ HNF past a stretching sheet were deliberated by Subhani and Nadeem (Manjunatha et al., 2022). Nadeem et al. (Manjunatha et al., 2019) analyzed the theoretic investigation of the $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/}{\text{H}}_{\text{2}}\text{O}$ HNF through an exponentially curved stretching sheet. Jawad et al. (Nasir et al., 2024) scrutinized the impression of MHD flow and thermal radiation on HNF ($\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/}{\text{H}}_{\text{2}}\text{O}$) second-grade nanofluid over a non-linear stretching/shrinking sheet. The heat transfer of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/}{\text{H}}_{\text{2}}\text{O}$ HNF through a stretching sheet was curious by Yahaya et al. (Krishna et al., 2021). Numerous theoretical and experimental studies have been reported on different HNFs in the modern literature, e.g. (Nadeem et al., 2021a; Nadeem et al., 2022; Nadeem et al., 2023a; Nadeem et al., 2023b; Zulqarnain et al., 2023a; Zulqarnain et al., 2023b).

The relevant variables and parameters are intended to be precise or well-defined in the prevalent research of physical systems expressed through FDEs. However, as errors in experiments and observations are possible, these values may be regarded as ambiguous and unclear. As a result, the majority of physical systems lack sufficient knowledge of the relevant parameters and variables, which could cause uncertainty (Priyadarshini & Nayak, 2023; Yang et al., 2024). Regarding, the fuzzy theory and FDE approaches are presented here as well as the uncertain circumstance in particular. To address the uncertainty brought on by inadequate information that arises in several mathematical models of various real-world processes, FDEs fuzzy analysis has recently been presented. There are thus a few contributions here that deal with fuzzy theory and modeling. The outcomes of a stretched surface on an MHD tangent Casson fuzzy $\left(\text{F}{\text{e}}_{\text{3}}{\text{O}}_{\text{4}}\text{ - A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/}{\text{C}}_{\text{2}}{\text{H}}_{\text{6}}{\text{O}}_{\text{2}}\right)$ HNF were inspected by Shanmugapriya et al. (Shanmugapriya et al., 2024). By using a fuzzy membership function, they identify that HNFs have superior heat transmission than nanofluids. The influence of Cross HNF tiny liquids across a vertical duct with a fuzzy volume percentage was scrutinized by Ayub et al. (Ayub et al., 2024).

Unsteady, heat transfer, and stagnation-point flow of a special third-grade fuzzy HNF $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ through a permeable convective shrinking/stretching surface under the outcome of the nonlinear thermal radiation, viscous dissipation, and heat source/sink have not yet been examined in the literature. The significant numerical technique bvp4c (finite difference technique) is employed to solve dimensionless coupled highly non-linear ODEs. The impacts of the nanoparticle's volume fraction and physical parameters on the velocity and temperature fields are elaborated graphically. In addition, this work compares nanofluid with hybrid nanofluid via a triangle membership function. The proposed problem is modified into fuzzy differential equations and then again applied to the numerical scheme bvp4c. A triangular fuzzy number is used to describe the volume percentage of nanoparticles. The $\zeta \text{ - cut}$ technique is used to handle the triangular fuzzy number.

2. Problem formulation

The 2D unsteady MHD, nonlinear radiation, and stagnation-point flow induced by permeable shrinking/stretching sheet of a third-grade $\left(\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}\right)$ HNF are examined in this investigation. The x-axis of the Cartesian coordinates is parallel to the sheet, and the y-axis is normally to the sheet as shown in Figure 1. The sheet begins to move at time t = 0 with the velocity $U_w\left(x,t\right)=\alpha u_w\left(x,t\right)$ where $\alpha$ is a non-dimensional constant, through $\alpha >0$ representing a stretched sheet and $\alpha <0$ indicating a shrinking sheet, respectively. An external free stream velocity is $u_e\left(x,t\right).$ The velocity of the shrinking/stretching sheet is $\mathrm{bx}/\left(1-\mathrm{ct}\right)+u_w\left(x,t\right)$ with b > 0 and c showing the unsteadiness. Further T_w > T_∞ (heated sheet), T_w is the uniform temperature, T_∞ is the ambient temperature, and $v_w\left(t\right)$ is the mass flux velocity of the sheet.

Figure 1. Flow Geometry.

The foremost equations of the time-dependent flow under investigation as a result of these assumptions are as follows (Dunn & Rajagopal, 1995; Nadeem et al., 2021b; Naganthran et al., 2016; Pakdemirli, 1994; Zaib et al., 2017; Zaib et al., 2020): (1) $\begin{array}{rl}& \frac{\mathrm{\partial }v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=0,\end{array}$(1) (2) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\\ & =\frac{\mathrm{\partial }{u}_e}{\mathrm{\partial }t}+u_e\frac{\mathrm{\partial }{u}_e}{\mathrm{\partial }x}+\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\\ & +\frac{2{\text{β}}_3}{{\rho }_{\mathrm{hnf}}}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}-\frac{{\sigma }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}{B}_0^2\left(u-u_e\right)\\ & +g({\text{β}}_T{)}_{\mathrm{hnf}}\left(T-T_{\mathrm{\infty }}\right),\end{array}$(2) (3) $\begin{array}{rl}& u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}+\frac{\mathrm{\partial }T}{\mathrm{\partial }t}\\ & ={\alpha }_{\mathrm{hnf}}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+\frac{Q_0\left(T-T_{\mathrm{\infty }}\right)}{{\left(\rho c_P\right)}_{\mathrm{hnf}}}\\ & -\frac{1}{{\left(\rho c_P\right)}_{\mathrm{hnf}}}\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}+\frac{{\mu }_{\mathrm{hnf}}}{{\left(\rho c_P\right)}_{\mathrm{hnf}}}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\\ & +\frac{2{\text{β}}_3}{{\left(\rho c_P\right)}_{\mathrm{hnf}}}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^4,\end{array}$(3)

the boundary conditions are: (4) $\begin{array}{l}t<0:v\left(x,t\right)=0,u\left(x,t\right)=0,\\ T\left(x,t\right)=T_{\mathrm{\infty }},\mathrm{\forall }x,y,\\ t\ge 0:u\left(x,t\right)=U_w\left(x,t\right)=\lambda u_w\left(x,t\right),\\ v\left(x,t\right)=v_w,T\left(x,t\right)=T_w\text{at}y=0,\\ u\to u_e\left(x,t\right),T\to T_{\mathrm{\infty }}\left(x,t\right)\text{as}y\to \mathrm{\infty },\end{array}\}$(4) here the free-stream velocity is $u_e\left(x,t\right)=\left(\mathrm{ax}/1-\mathrm{tc}\right)$ with $a$ as a constant and $B_0$ is the magnetic field externally imposed in the y-direction. It should also be observed that ${\text{β}}_3=0,$ simplifies the above case to a Newtonian fluid.

The thermophysical properties of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ are shown in Eq. (5(5) $\begin{array}{l}{\displaystyle \frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}}={\rho }_r=\left[\right(1-{\varphi }_2)\left\{\left(1-{\varphi }_1\right)+{\varphi }_1{\displaystyle \frac{{\rho }_{s_1}}{{\rho }_f}}\right\}\\ +{\varphi }_2{\displaystyle \frac{{\rho }_{s_2}}{{\rho }_f}}],\\ {\displaystyle \frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}}={\mu }_r={\displaystyle \frac{1}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}},\\ {\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{\mathrm{hnf}}}{{\left(\rho C_{\rho }\right)}_f}}=(\rho C_{\rho }{)}_r={\varphi }_2{\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{s_2}}{{\left(\rho C_{\rho }\right)}_f}}+\left(1-{\varphi }_2\right)\\ \left[\left(1-{\varphi }_1\right)+{\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{s_1}}{{\left(\rho C_{\rho }\right)}_f}}{\varphi }_1\right],{\alpha }_r={\displaystyle \frac{k_r}{{\left(\rho c_P\right)}_r}},\\ {\displaystyle \frac{k_r}{k_{\mathrm{nf}}}}={\displaystyle \frac{2k_{\mathrm{nf}}-2{\varphi }_1\left(k_{s_1}-k_{\mathrm{nf}}\right)+k_{s_1}}{2k_{\mathrm{nf}}+{\varphi }_1\left(k_{s_1}-k_{\mathrm{nf}}\right)+k_{s_1}}},\\ {\displaystyle \frac{k_{\mathrm{nf}}}{k_f}}={\displaystyle \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}}},k_r={\displaystyle \frac{k_{\mathrm{hnf}}}{k_f}},\\ {\displaystyle \frac{{\left({\text{β}}_T\right)}_{\mathrm{hnf}}}{{\left({\text{β}}_T\right)}_f}}=({\text{β}}_T{)}_r={\varphi }_2{\displaystyle \frac{{\left({\text{β}}_T\right)}_{s_2}}{{\left({\text{β}}_T\right)}_f}}+\left(1-{\varphi }_2\right)\\ \left[\left(1-{\varphi }_1\right)+{\varphi }_1{\displaystyle \frac{{\left({\text{β}}_T\right)}_{s_1}}{{\left({\text{β}}_T\right)}_f}}\right],{\delta }_r={\displaystyle \frac{{\delta }_{\mathrm{hnf}}}{{\delta }_f}},\\ {\sigma }_r=\left[{\displaystyle \frac{{\sigma }_{s_2}\left(1+2{\varphi }_2\right)+2{\sigma }_{\mathrm{bf}}\left(1-{\varphi }_2\right)}{{\sigma }_{s_2}\left(1-{\varphi }_2\right)+{\sigma }_{\mathrm{bf}}\left(2+{\varphi }_2\right)}}\right]{\sigma }_{\mathrm{bf}},\\ {\sigma }_{\mathrm{bf}}=\left[{\displaystyle \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}\}$(5) ) (Krishna et al., 2021; Nadeem et al., 2023a; Nadeem et al., 2023b). (5) $\begin{array}{l}{\displaystyle \frac{{\rho }_{\mathrm{hnf}}}{{\rho }_f}}={\rho }_r=\left[\right(1-{\varphi }_2)\left\{\left(1-{\varphi }_1\right)+{\varphi }_1{\displaystyle \frac{{\rho }_{s_1}}{{\rho }_f}}\right\}\\ +{\varphi }_2{\displaystyle \frac{{\rho }_{s_2}}{{\rho }_f}}],\\ {\displaystyle \frac{{\mu }_{\mathrm{hnf}}}{{\mu }_f}}={\mu }_r={\displaystyle \frac{1}{{\left(1-{\varphi }_1\right)}^{2.5}{\left(1-{\varphi }_2\right)}^{2.5}}},\\ {\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{\mathrm{hnf}}}{{\left(\rho C_{\rho }\right)}_f}}=(\rho C_{\rho }{)}_r={\varphi }_2{\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{s_2}}{{\left(\rho C_{\rho }\right)}_f}}+\left(1-{\varphi }_2\right)\\ \left[\left(1-{\varphi }_1\right)+{\displaystyle \frac{{\left(\rho C_{\rho }\right)}_{s_1}}{{\left(\rho C_{\rho }\right)}_f}}{\varphi }_1\right],{\alpha }_r={\displaystyle \frac{k_r}{{\left(\rho c_P\right)}_r}},\\ {\displaystyle \frac{k_r}{k_{\mathrm{nf}}}}={\displaystyle \frac{2k_{\mathrm{nf}}-2{\varphi }_1\left(k_{s_1}-k_{\mathrm{nf}}\right)+k_{s_1}}{2k_{\mathrm{nf}}+{\varphi }_1\left(k_{s_1}-k_{\mathrm{nf}}\right)+k_{s_1}}},\\ {\displaystyle \frac{k_{\mathrm{nf}}}{k_f}}={\displaystyle \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}}},k_r={\displaystyle \frac{k_{\mathrm{hnf}}}{k_f}},\\ {\displaystyle \frac{{\left({\text{β}}_T\right)}_{\mathrm{hnf}}}{{\left({\text{β}}_T\right)}_f}}=({\text{β}}_T{)}_r={\varphi }_2{\displaystyle \frac{{\left({\text{β}}_T\right)}_{s_2}}{{\left({\text{β}}_T\right)}_f}}+\left(1-{\varphi }_2\right)\\ \left[\left(1-{\varphi }_1\right)+{\varphi }_1{\displaystyle \frac{{\left({\text{β}}_T\right)}_{s_1}}{{\left({\text{β}}_T\right)}_f}}\right],{\delta }_r={\displaystyle \frac{{\delta }_{\mathrm{hnf}}}{{\delta }_f}},\\ {\sigma }_r=\left[{\displaystyle \frac{{\sigma }_{s_2}\left(1+2{\varphi }_2\right)+2{\sigma }_{\mathrm{bf}}\left(1-{\varphi }_2\right)}{{\sigma }_{s_2}\left(1-{\varphi }_2\right)+{\sigma }_{\mathrm{bf}}\left(2+{\varphi }_2\right)}}\right]{\sigma }_{\mathrm{bf}},\\ {\sigma }_{\mathrm{bf}}=\left[{\displaystyle \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}\}$(5) where $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ and $\text{Cu}$ are nanoparticles having the volume fractions ${\varphi }_1$ and ${\varphi }_2$ respectively. Also $s_1$ and $s_2$ denote the $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ and $\text{Cu}$ solid nanoparticles, respectively. The thermophysical properties of Copper $\left(\text{Cu}\right)$, Sodium Alginate (SA), and Aluminum oxide $\left(\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\right)\text{,}$ are shown in Table 1.

Table 1. The thermophysical characteristics of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$, as well as $\text{Cu}$ and SA. (Krishna et al., 2021; Nadeem et al., 2023a; Nadeem et al., 2023b).

Physical properties	$\rho \left(\mathrm{kg}{m}^{-3}\right)$	$\rho c_p\left(J{\left(\mathrm{kgK}\right)}^{-1}\right)$	$k\left(W{\left(\mathrm{mK}\right)}^{-1}\right)$	${\text{β}}_T\times {10}^{-5}\left(K^{-1}\right)$	$\sigma (m/\mathrm{\Omega }{)}^1$
SA	989	4175	0.6376	99	$2.6\times {10}^{-4}$
$\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$	3970	765	40	0.85	$3.69\times {10}^7$
$\text{Cu}$	8933	385	401	1.67	$5.96\times {10}^7$

Now, for non-dimensionalizing governing equations, we introduce the appropriate similarity variables (Zaib et al., 2017; Zaib et al., 2020). (6) $\begin{array}{rl}\eta & =\sqrt{\frac{a}{\left(1-\mathrm{ct}\right)v_f}}y,\omega =\sqrt{\frac{a{v}_f}{1-\mathrm{ct}}}\mathrm{xf}\left(\eta \right),\\ T& =T_{\mathrm{\infty }}+\left(T_w-T_{\mathrm{\infty }}\right)\theta \left(\eta \right),\end{array}$(6) where $\eta$ similarity variable and $\omega$ stream function are defined as $u=\frac{\mathrm{\partial }\omega }{\mathrm{\partial }y}\text{and}v=-\frac{\mathrm{\partial }\omega }{\mathrm{\partial }x}.$ (7) $\begin{array}{r}\mathrm{Thus}{v}_w=-\sqrt{\frac{a{v}_f}{1-\mathrm{ct}}}\lambda ,\end{array}$(7) where $\lambda$ is the mass transfer parameter through the permeable sheet, with $\lambda$< 0 indicating suction and $\lambda$ > 0 indicating injection, and $\alpha =a/\left(1-\mathrm{ct}\right)$ is the velocity rati o parameter. Here for similarity solution, we assumed that $k_1=k_0/x,$ where $k_1>0$ being a constant (Zaib et al., 2017; Zaib et al., 2020). The following ordinary (similarity) equations can be constructed by substituting (6) into equations (2), (3), and (4), we have. (8) $\begin{array}{rl}& \left(\frac{6K{f}^{\mathrm{\prime \prime }}2+{\mu }_r}{{\rho }_r}\right)f^{\mathrm{\prime \prime \prime }}+1-(f^{\mathrm{\prime }}{)}^2+\text{β}\left(1-\frac{\eta }{2}{f}^{\mathrm{\prime \prime }}-f^{\mathrm{\prime }}\right)\\ & +f{f}^{\mathrm{\prime \prime }}+\frac{{\sigma }_r}{{\rho }_r}M\left(1-f^{\mathrm{\prime }}\right)+({\text{β}}_T{)}_r\mathrm{Gr\theta }=0,\end{array}$(8) (9) $\begin{array}{rl}& {\alpha }_r{\theta }^{\mathrm{\prime \prime }}+Prf{\theta }^{\mathrm{\prime }}-\frac{1}{2}Pr\eta {\theta }^{\mathrm{\prime }}+\frac{\mathrm{PrH\theta }}{{\left(\rho c_P\right)}_r}\\ & +\frac{\mathrm{Nr}{\theta }^{\mathrm{\prime \prime }}}{{\left(\rho c_P\right)}_r}\{1+\theta \left({\theta }_w-1\right){\}}^3\\ & +\frac{3\mathrm{Nr}{\left({\theta }^{\mathrm{\prime }}\right)}^2}{{\left(\rho c_P\right)}_r}\left({\theta }_w-1\right)\{1+\theta \left({\theta }_w-1\right){\}}^2\\ & +\frac{2\mathrm{KPrEc}}{{\left(\rho c_P\right)}_r}(f^{\mathrm{\prime \prime }}{)}^4+\frac{{\mu }_r\mathrm{PrEc}}{{\left(\rho c_P\right)}_r}(f^{\mathrm{\prime \prime }}{)}^2=0,\end{array}$(9) in addition to the constraints (10) $\begin{array}{r}\begin{array}{l}f\left(\eta \right)=\lambda ,f^{\mathrm{\prime }}\left(\eta \right)=\alpha ,\theta \left(\eta \right)=1\text{at}\eta =0,\\ f^{\mathrm{\prime }}\left(\eta \right)\to 1,\theta \left(\eta \right)\to 0\text{as}\eta \to \mathrm{\infty }\end{array}\}\end{array}$(10) The dimensionless parameters are defined as follows:

$K=\frac{k_1{a}^2{\text{β}}_3}{{\left(1-\mathrm{ct}\right)}^2{\rho }_f{v}_f^2},$ $\mathrm{Gr}=\frac{g{\left({\text{β}}_T\right)}_f{\left(1-\mathrm{ct}\right)}^2}{a^2{\rho }_f\left(T_w-T_{\mathrm{\infty }}\right)},$ $M=\frac{{\sigma }_f{B}_0^2\left(1-\mathrm{ct}\right)}{a{\rho }_f},$ $Pr=\frac{v_f}{{\alpha }_f},$ $H=\frac{\left(1-\mathrm{ct}\right)Q_o}{a{\left(\rho c_P\right)}_f},$ $\mathrm{Nr}=\frac{4{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{k_f{k}^{\ast }{v}_f},$ ${\theta }_w=\frac{T_w}{T_{\mathrm{\infty }}}$ $R{e}_x=x{u}_w/v_f,$ $\mathrm{Ec}=\frac{a^2}{{\left(\rho c_p\right)}_f\left(T_w-T_{\mathrm{\infty }}\right){\left(1-\mathrm{ct}\right)}^2},$ $\text{β}=\frac{c}{a},$ here β > 0 or β < 0 indicates an accelerating and decelerating flow respectively.

Now we will discuss some significant engineering quantities like the drag force and the heat transfer rate, which are both defined as (Zaib et al., 2017; Zaib et al., 2020) (11) $\begin{array}{rl}{C}_{\mathrm{fx}}& =\frac{{\mu }_{\mathrm{hnf}}}{u_w^2{\rho }_f}{\left[\frac{2{\text{β}}_3}{{\mu }_f}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^3+\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right]}_{y=0},\\ N{u}_x& =\frac{-x}{k_f\left(T_w-T_{\mathrm{\infty }}\right)}{\left[-q+k_{\mathrm{hnf}}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_r\right]}_{y=0}\end{array}$(11) Using Eq. (6(6) $\begin{array}{rl}\eta & =\sqrt{\frac{a}{\left(1-\mathrm{ct}\right)v_f}}y,\omega =\sqrt{\frac{a{v}_f}{1-\mathrm{ct}}}\mathrm{xf}\left(\eta \right),\\ T& =T_{\mathrm{\infty }}+\left(T_w-T_{\mathrm{\infty }}\right)\theta \left(\eta \right),\end{array}$(6) ) in Eq. (11(11) $\begin{array}{rl}{C}_{\mathrm{fx}}& =\frac{{\mu }_{\mathrm{hnf}}}{u_w^2{\rho }_f}{\left[\frac{2{\text{β}}_3}{{\mu }_f}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^3+\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right]}_{y=0},\\ N{u}_x& =\frac{-x}{k_f\left(T_w-T_{\mathrm{\infty }}\right)}{\left[-q+k_{\mathrm{hnf}}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_r\right]}_{y=0}\end{array}$(11) ), we have, (12) $\begin{array}{rl}\sqrt{{\mathrm{Re}}_x}C{f}_x& ={\mu }_r[f^{\mathrm{\prime \prime }}+2K{\left(f^{\mathrm{\prime \prime }}\right)}^3{]}_{\eta =0},({\mathrm{Re}}_x{)}^{-1/2}N{u}_x\\ & =-{\theta }^{\mathrm{\prime }}\left(\eta \right)[k_r+N_r{\left\{1+\left({\theta }_w-1\right)\theta \left(\eta \right)\right\}}^3{]}_{\eta =0}.\end{array}$(12)

3. Fuzzy formulation

A fuzzy set $\tilde{U}$ is defined as $\tilde{U}=\left\{\right(y,{\mu }_{\tilde{U}}\left(y\right)):y\in X,{\mu }_{\tilde{U}}(y)\in [0,1\left]\right\},$ where X is the universal set, ${\mu }_{\tilde{U}}\left(y\right)$ is the MF or grade of membership of $\tilde{U}$ and mapping is defined as a ${\mu }_{\tilde{U}}\left(y\right):X\to [0,1].$ value of ${\mu }_{\tilde{U}}\left(y\right)$ ranges from 0 to 1. The value ${\mu }_{\tilde{U}}\left(y\right)=0$ indicates that $y$ is not part of the fuzzy set and if ${\mu }_{\tilde{U}}\left(y\right)$ = 1, $y$ is a member of the fuzzy set. When 0 <${\mu }_{\tilde{U}}\left(y\right)$< 1, the MF of $y$ the fuzzy set is imprecise. A set $X_{\zeta }=\{y\in X,\ge \zeta \}$ is called a $\zeta \text{ - cut}$ of $\tilde{U}.$ A crisp set is formed by a bounded $\zeta \text{ - cut}$ $X_{\zeta }$ containing $y,y_{\zeta ,min}\le y\le y_{\zeta ,max},$ becomes a crisp set and is commonly used in fuzzy structural analysis. TFNs are frequently used in the modeling of fuzzy structural parameters due to their mathematical simplicity. $X=\left[{\chi }_1,{\chi }_2,{\chi }_3\right]$ is a TFN defined entirely by three quantities: ${\chi }_1$ (lower bound), ${\chi }_2$ (most belief value), and ${\chi }_3$ (upper bound) are shown in Figure 2.

Figure 2. Membership functions of a TFN.

By the TFN, the MF ${\mu }_{\tilde{U}}\left(y\right)$ can be expressed as. $\begin{array}{r}{\mu }_{\tilde{U}}\left(y\right)=\{\begin{array}{ll}-{\displaystyle \frac{{\chi }_1-y}{-{\chi }_2+{\chi }_1}}& \text{for}y\in [{\chi }_1,{\chi }_2],\\ -{\displaystyle \frac{y-{\chi }_3}{-{\chi }_3+{\chi }_2}}& \text{for}y\in [{\chi }_2,{\chi }_3],\\ & \text{otherwise}\text{.}\end{array}\end{array}$TFNs are converted to interval numbers using a technique that is stated as $\zeta \text{-cut,}$ which is written as $\tilde{U}=\left[u_1(y;\zeta ),u_2(y;\zeta )\right]=[{\chi }_1+\zeta ({\chi }_2-{\chi }_1),{\chi }_3-\zeta ({\chi }_3-{\chi }_2\left)\right]$, where $0\le \zeta \le 1.$

A slight change in the nanoparticle's volume fraction has an impact on temperature and velocity. The flow velocity and heat transfer are entirely dependent on such values, which use a nanoparticle volume in the range of [1% – 4%]. Therefore, ambiguity develops as a result of the static crisp values of nanoparticle volume fractions. The volume fraction of nanoparticles can be treated as TFN due to insufficient information. Because ${\varphi }_1$ and ${\varphi }_2$ indicate the volume fractions of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ and $\text{Cu,}$ it is easier to deal with a complex problem in a fuzzy atmosphere by treating both ${\varphi }_1$ and ${\varphi }_2$ as FN. With TFNs being converted into $\zeta \text{ - cut}$ procedures as shown in Table 2.

Table 2. TFNs of fuzzy nanoparticles of volume fraction.

Fuzzy Numbers	Crisp values	TFN	$\zeta \text{ - cut}$ approach
${\varphi }_1$ $\left(\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\right)$	[1%−4%]	[0%, 5%, 10%]	$\left[5\mathrm{\%\zeta },10\mathrm{\%}-5\mathrm{\%\zeta }\right],\zeta \in \left[0,1\right]$
${\varphi }_2$ $\left(\text{Cu}\right)$	[1%−4%]	[0%, 5%, 10%]	$\left[5\mathrm{\%\zeta },10\mathrm{\%}-5\mathrm{\%\zeta }\right],\zeta \in \left[0,1\right]$

TFNs are being used to express triangular MFs, which range from 0 to 1, as can be seen in Figure 2. For the fuzzy solution, non-linear DEs (7) to (9) can be renovated into FDEs (Priyadarshini & Nayak, 2023; Shanmugapriya et al., 2024; Yang et al., 2024). (13) $\begin{array}{rl}& f^{\mathrm{\prime \prime \prime }}\left(\eta ,\zeta \right)\left(\frac{6K+{\mu }_r}{{\rho }_r}\right)-\text{β}\left(f^{\mathrm{\prime }}\left(\eta ,\zeta \right)+\frac{\eta }{2}{f}^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right)-1\right)\\ & +1+f\left(\eta ,\zeta \right)f^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right)-(f^{\mathrm{\prime }}\left(\eta ,\zeta \right){)}^2\\ & +\frac{{\sigma }_r}{{\rho }_r}M\left(f^{\mathrm{\prime }}\left(\eta ,\zeta \right)-1\right)=-({\text{β}}_T{)}_r\mathrm{Gr\theta }\left(\eta ,\zeta \right),\end{array}$(13) (14) $\begin{array}{rl}& {\alpha }_r{\theta }^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right)-\frac{1}{2}Pr\eta {\theta }^{\mathrm{\prime }}\left(\eta ,\zeta \right)\\ & +\frac{3\mathrm{Nr}{\left({\theta }^{\mathrm{\prime }}\left(\eta ,\zeta \right)\right)}^2}{{\left(\rho c_P\right)}_r}\left({\theta }_w-1\right)\{1+\theta \left(\eta ,\zeta \right)\left({\theta }_w-1\right){\}}^2\\ & +\frac{\mathrm{Nr}{\theta }^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right)}{{\left(\rho c_P\right)}_r}\{1+\theta \left(\eta ,\delta \right)\left({\theta }_w-1\right){\}}^3\\ & +\frac{{\mu }_r\mathrm{PrEc}}{{\left(\rho c_P\right)}_r}(f^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right){)}^2\\ & =-Prf\left(\eta ,\zeta \right){\theta }^{\mathrm{\prime }}\left(\eta ,\zeta \right)-\frac{\mathrm{PrH\theta }\left(\eta ,\zeta \right)}{{\left(\rho c_P\right)}_r}\\ & -\frac{2\mathrm{KPrEc}}{{\left(\rho c_P\right)}_r}(f^{\mathrm{\prime \prime }}\left(\eta ,\zeta \right){)}^4,\end{array}$(14) in addition to the constraints (15) $\begin{array}{ll}f\left(\eta ,\zeta \right)=\lambda ,f^{\mathrm{\prime }}\left(\eta ,\zeta \right)=\alpha ,\theta \left(\eta ,\zeta \right)=1& \text{at}\eta =0,\\ f^{\mathrm{\prime }}\left(\eta ,\zeta \right)\to 1,\theta \left(\eta ,\zeta \right)\to 0& \text{as}\eta \to \mathrm{\infty },\end{array}\}$(15) where the fuzzy velocity $\left(f^{\mathrm{\prime }}(y,\zeta )\right)$ and temperature $\left(\overline{\theta }(y,\zeta )\right)$ fields can be written as $f^{\mathrm{\prime }}(y,\zeta )=\left[{f^{\mathrm{\prime }}}_1\right(y,\zeta ),{f^{\mathrm{\prime }}}_2(y,\zeta \left)\right],$ and $\overline{\theta }(y,\zeta )=\left[{\theta }_1(y,\zeta ),{\theta }_2(y,\zeta )\right],0\le \zeta \le 1.$ Here, $\left({f^{\mathrm{\prime }}}_1\right(y,\zeta ),{\theta }_1(y,\zeta \left)\right)$ is lower bound and $\left({f^{\mathrm{\prime }}}_2\right(y,\zeta ),{\theta }_2(y,\zeta \left)\right)$ is an upper bound of fuzzy velocity and temperature profiles respectively.

3.1. For validation

The numerical outcomes emitted by the above-described technique are validated via the comparison of $-f^{\mathrm{\prime \prime }}\left(0\right)$ with the previously published results of Bhattacharyya (Bhattacharyya, 2011) and Naganthran et al. (Naganthran et al., 2016) as shown in Table 3. This suggests that the computational tools used in this study are accurate and dependable because the results of the numerical simulations express a high degree of agreement with the body of existing literature.

Table 3. Comparison of $-f^{\mathrm{\prime \prime }}\left(0\right)$ with previously published works when $\text{β}=0,$ $M=0,$ $K=0,$ $\lambda =0,{\varphi }_1={\varphi }_2=0,$ and $Pr=\mathrm{0.73.}$

$\alpha$	(Bhattacharyya, 2011)	(Naganthran et al., 2016)	Present result
−0.25	1.4022405	1.4022407	1.4022407
−0.50	1.4956697	1.4956697	1.4956697
−0.75	1.4892981	1.4892982	1.4892982
−1	1.3288169	1.3288168	1.3288168
−1.15	1.0822316	1.0822311	1.0822311
−1.20	0.9324738	0.9324733	0.9324733

4. Results and discussion

The boundary conditions (10) and the ODEs (8) – (9) are numerically determined using the bvp4c algorithm built-in Matlab program. Different dimensionless parameters with standard values such as $M=0.3,$ $K=0.1,$ ${\varphi }_1={\varphi }_2=0.04$, $\lambda =0.2,$ $\alpha =-0.2,$ $Pr=0.7,$ ${\theta }_w=1.2,$ $\mathrm{Ec}=0.2,$ $\mathrm{Gr}=0.2,$ $\mathrm{Nr}=0.3,$ $\text{β}=0.5,$ and $H=0.1$ effect on velocity $\left(f^{\mathrm{\prime }}\left(\eta \right)\right)$ and temperature $\left(\theta \left(\eta \right)\right)$ fields are illustrated in Figures 3–14 for fluid (Third-grade) and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ hybrid nanofluid. Table 3 replicates the performance of numerous parameters as discussed above on Nusselt number as well as skin friction. Also, HNF profiles are shown in all of the figures as red solid lines, whereas the third-grade fluid's profiles are indicated by blue dashed lines.

Figure 3. Effect of Gr on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 4. Effect of K on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 5. Effect of $\lambda$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 6. Effect of $\text{β}$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 7. Effect of Nr on $\theta \left(\eta \right).$

Figure 8. Effect of ${\theta }_w$ on $\theta \left(\eta \right).$

Figure 9. Effect of Ec on $\theta \left(\eta \right).$

Figure 10. Effect of Pr on $\theta \left(\eta \right).$

Figure 11. Effect of M on $f^{\mathrm{\prime }}\left(\eta \right).$

Figure 12. Effect of H on $\theta \left(\eta \right).$

Figure 13. Effect of ${\varphi }_1$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 14. Effect of ${\varphi }_2$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right).$

Figure 3 indicates the impression of buoyancy forces (Gr) on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$ for both liquids. Due to the rise in Gr, the $f^{\mathrm{\prime }}\left(\eta \right)$ of the fluid and HNF upsurges while the $\theta \left(\eta \right)$ decreases. Physically, when the buoyancy force works on the shirking sheet then stimulates particles to move toward the sheet; as a result, $f^{\mathrm{\prime }}\left(\eta \right)$ enhances, and the $\theta \left(\eta \right)$ field declines. The upshot of the shear-thinning/thickening parameter (K) on the fluid/HNF $f^{\mathrm{\prime }}\left(\eta \right)$ (a) and $\theta \left(\eta \right)$ (b) have been examined in Figure 4. The $f^{\mathrm{\prime }}\left(\eta \right)$ of the fluid and HNF declined as K is improved due to the thickness of the momentum BL. Therefore, the $\theta \left(\eta \right)$ of fluid and HNF amplified with an upsurge in K. The influence of the suction parameter $\left(\lambda \right)$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$ for both liquids is exposed in Figure 5. It is detected that when $\lambda$ is amplified the $f^{\mathrm{\prime }}\left(\eta \right)$ of hybrid nanofluid grows while a diminish in the $\theta \left(\eta \right)$ of the fluid and HNF is noted. The rise in flow rate is due to the creation of a vacuum by the suction of fluid and hybrid nanofluid. For decreasing the temperature, the heat is discharged from the sheet due to the fluid moving faster. Figure 6 displays the time-dependent parameter $\left(\text{β}\right)$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$ for both liquids. It is evident that $f^{\mathrm{\prime }}\left(\eta \right)$ upsurges for enlarging $\text{β}$ and, consequently, the velocity BL thickness reduces. The influence of $\text{β}$ amplifies the speed of liquid flow so, increasing the both fluid velocity. The temperature rises due to an increase in $\text{β}.$ Physically, with a rise in $\text{β}$ the thermal BL thickness upsurges consequently the heat transmission upsurges. Figure 7 demonstrates the impression of thermal radiation (Nr) on $\theta \left(\eta \right)$ for both liquids. It is apparent that when Nr is augmented, the temperature rises so the thermal BL is enhanced as the surface temperature grows. Physically, radiation improves the Brownian motion of tiny particles so they hit each other and induced frictional energy changes into thermal energy. Also, the temperature of regular fluids is lower than HNFs.

Figure 8 describes the outcome of the temperature ratio parameter $\left({\theta }_w\right)$ on the $\theta \left(\eta \right)$ for both liquids. It is observed that the temperature of fluids enhance with an upsurge of ${\theta }_w.$ Physically when increasing ${\theta }_w,$ the strength of non-linear thermal radiation will magnify so, the temperature of both fluids increases. HNFs acquire higher temperatures than regular fluids. The thermodynamical variation of the Eckert number (Ec) observed in $\theta \left(\eta \right)$ is designated in Figure 9. As can be observed, when Ec is increased, $\theta \left(\eta \right)$ of both liquids increases. Physically, the Ec indicates the ratio of kinetic energy to enthalpy, and the total work is done against the viscosity and the kinetic energy is changed into internal energy. So, viscous dissipation increases the fluid temperature. The role of Prandtl number (Pr) in $\theta \left(\eta \right)$ for both fluids is examined in Figure 10. The upsurge in Pr leads to a decline in the thermal BL, thus dropping the heat flux. The rise in Pr makes the density weaker than kinematic viscosity, which produces resistant strength against the fluid flow. This spectacle makes the fluid and HNF thicker, thus, a failure in the temperature is observed. Figure 11 displays the inspiration of the magnetic parameter (M) on the $f^{\mathrm{\prime }}\left(\eta \right)$ for both liquids. It can be seen that the fluid velocity falls with the larger values of M. Physically, the variations in M cause accrue Lorentz forces which reduce the fluid velocity. Figure 12 illustrates the consequence of the heat absorption/generation parameter (H) on $\theta \left(\eta \right).$ The $\theta \left(\eta \right)$ upsurges when the (H > 0) rises. Physically, during an rise in heat generation, the convection current decreases the density of the fluid which causes an increase in $\theta \left(\eta \right).$ The upshot of the nanoparticles volume fraction $\left({\varphi }_1\right)$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$ is depicted in Figure 13. Due to the rise ${\varphi }_1,$ the $f^{\mathrm{\prime }}\left(\eta \right)$ declines, but the temperature profile is found to escalates. As a result, the momentum and thermal BL become thicker for large value of ${\varphi }_1.$ Physically, the thermophysical characteristics of the HNF are changing due to the presence of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ nanoparticles in the liquid, and thus the flow velocity reduces by the viscous force and the heat diffusion increase by larger thermal conductivity. Figure 14 describes the effect of the volume fraction of Cu nanoparticles $\left({\varphi }_2\right)$ on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$. It is detected that when ${\varphi }_2$ is increased, on $f^{\mathrm{\prime }}\left(\eta \right)$ and $\theta \left(\eta \right)$ increase gradually. Physically, the density of HNF falls as ${\varphi }_2$ climbs, boosting velocity and temperature. As a consequence, the intermolecular contacts between HNF particles weaken, and the HNFs velocity increases.

This article also covers the triangular MF comparison of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ $\text{Cu/SA}$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}\text{.}$ Using the $\zeta \text{ - cut}$ approach $\left(0\le \zeta \le 1\right)$, the volume fraction of nanoparticles ${\varphi }_1$ and ${\varphi }_2$ is taken to be TFNs (see Table 2). FDEs and the $\zeta \text{ - cut}$ technique are used to convert the fuzzy velocity and temperature equations into lower and upper bounds.

Figure 15 illustrates the MFs of $\overline{\theta }(y,\zeta )$ for different values of $\eta$ to compare the nanofluids $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ $\left({\varphi }_1\right),$ $\text{Cu/SA}$ $\left({\varphi }_2\right),$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ HNFs. We examined three possible cases in these figures. Black-dashed lines suggest the case when ${\varphi }_1$ is taken as TFN and ${\varphi }_2=0$. Blue lines represent ${\varphi }_2$ which is taken as TFN whereas ${\varphi }_1=0$. In the third case, red lines expression the $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ with both ${\varphi }_1$ and ${\varphi }_2$ non-zero. The $\overline{\theta }(y,\zeta )$ for varying $\eta ,$ is presented on the horizontal axis, while the MF of the $\overline{\theta }(y,\zeta )$ for varying $\zeta \text{ - cut}$ are shown on the vertical axis. It can be realized that when comparing of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ and $\text{Cu/SA}$ with $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$, because of its more noticeable temperature difference than the other two, the HNF performs better. The combined thermal-conductivities of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$ and $\text{Cu}$ provide the extreme heat transfer. When a evaluation of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ and $\text{Cu/SA}$ nanofluids is analyzed, $\text{Cu/SA}$ has a higher heat transfer rate because Cu has a higher thermal-conductivity than $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}$. The $f^{\mathrm{\prime }}(y,\zeta )$ against $\eta$ is discussed in Figure 16 using the same three scenarios as in Figure 15. It's also worth noting that the $f^{\mathrm{\prime }}(y,\zeta )$ of $\text{Cu/SA}$ is greater than that of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ or $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$.

Figure 15. Comparison of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA,}$ $\text{Cu/SA}$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ for varying of $\eta .$

Figure 16. Comparison of $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA,}$ $\text{Cu/SA}$ and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ for varying of $\eta .$

Table 4 summarizes the implications of numerical values for a variety of physical parameters on the surface drag and heat transfer rate of a $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$. Surface drag is amplified over the surface with parameters such as magnetic parameter, mass transfer rate, third-grade fluid parameter, heat source, ${\varphi }_1$ and ${\varphi }_2$ while it is concentrated for the Pr. The upshot of the constraints on $N{u}_x$ is given in the next column of Table 4. When the Pr, mass transfer rate, thermal radiation parameter, velocity ratio parameter, temperature ratio parameter, magnetic parameter, heat source parameter, ${\varphi }_1$ and ${\varphi }_2$ are boosted, the heat transfer rate progresses, however, it lowers for bigger values of the Ec and the third-grade fluid parameter. Additionally, compared to traditional fluids, hybrid nanofluids have a faster heat transmission rate.

Table 4. Numerical values of f ″(0) and θ′(0) for dissimilar values of control constraints.

Gr	K	$\lambda$	$\text{β}$	Nr	${\theta }_w$	Ec	Pr	M	H	$\alpha$	${\varphi }_1$	${\varphi }_2$	$-(\mathrm{Rex}{)}^{0.5}C{f}_x$	$(\mathrm{Rex}{)}^{-0.5}N{u}_x$
0	0.1	−0.2	0.5	0.3	1.1	0.2	0.7	0.3	0.1	0.2	0.04	0.04	2.3184960	0.1237478
0.2													2.5219463	0.1200648
0.4													2.7251868	0.1145654
	0												2.1662727	0.1467936
	0.1												2.5219463	0.1200648
	0.2												2.8957643	0.0891000
		0											2.7459778	0.2161342
		0.1											2.8632128	0.1826756
		0.2											2.9838140	0.2509072
			0										2.7040176	0.3890744
			0.4										2.9285064	0.2804895
			0.8										3.1479800	0.1543014
				0.1									2.9803989	0.2050381
				0.2									2.9821912	0.2283689
				0.3									2.9838140	0.2509072
					1								2.9824729	0.2398319
					1.1								2.9838140	0.2509072
					1.2								2.9853002	0.2633956
						0.1							2.9801049	0.3647286
						0.2							2.9838140	0.2509072
						0.3							2.9875319	0.1366259
							0.1						3.0104394	0.1920474
							0.2						3.0034009	0.2187456
							0.3						2.9977260	0.2368757
								0					2.8014263	2.8014163
								0.3					2.8984179	2.8629896
								0.6					3.0043032	2.9237749
									0				2.9781770	0.3355373
									0.1				2.9838140	0.2509072
									0.2				2.9902164	0.1577988
										0			2.5598961	0.3598542
										−0.1			2.7774007	0.3067130
										−0.2			2.9838140	0.2509072
											0.02		2.8272940	0.2362926
											0.04		2.9838140	0.2509072
											0.06		3.1470906	0.2649790
												0.02	2.6964628	0.2426622
												0.04	2.9838140	0.2509072
												0.06	3.2830830	0.2579704

5. Concluding remarks

In this study, the modeled PDEs are rebuilt into ODEs employing appropriate transformation in terms of momentum and thermal energy. The bvp4c scheme is used to solve the converted ODEs. Graphs show the effects of different physical parameters on momentum as well as the thermal profiles of both (third-grade) and $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ hybrid nanofluid. The Nusselt number and surface drag force for the different values of constraints are provided in tabular form. A triangular fuzzy membership function is used to compare the nanofluid with the hybrid nanofluid for a better understanding and rate of heat transfer. Transformed ODEs are converted into FDEs with the help of TFNs then used numerical scheme bvp4c is used. Several noteworthy discoveries from this investigation are listed below.

• ${\varphi }_1$ declines the flow rate while temperature distribution boosts up. The temperature, velocity distribution, and rate of heat transfer increase as ${\varphi }_2$ rising.

• When the comparison of $K$ and HNFs is analyzed, it is noticed that the sheet temperature and the heat transference rate of HNFs are noticeably greater.

• The higher value of the $\text{β}$ causes a rise in the temperature and velocity fields.

• The heat transfer rate boosts for the larger values of the temperature ratio, heat source, Eckert number, and nonlinear thermal radiation parameter.

• The drag force escalates when the heat source, mass transfer rate, $K$, ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ growths and it declines when a magnetic parameter and Pr upsurge.

• The Nusselt number enhances with growing values of the temperature ratio, magnetic parameter, and thermal radiation while diminishing with rising values of $K$, ${\varphi }_1$ and ${\varphi }_2.$

• Triangle fuzzy MFs demonstrated that $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{ + Cu/SA}$ are significantly more effective at increasing the rate of heat transfer than $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ and $\text{Cu/SA}$ nanofluids. When compared, the $\text{Cu/SA}$ nanofluid performs better than the $\text{A}{\text{l}}_{\text{2}}{\text{O}}_{\text{3}}\text{/SA}$ nanofluid as well.

Acknowledgements

Open Access funding provided by the Qatar National Library.

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.

Disclosure statement

No potential conflict of interest was reported by the author(s).