Numerical Study of Cattaneo–Christov Heat Flux on Water-Based Carreau Fluid Flow over an Inclined Shrinking Sheet with Ternary Nanoparticles

Abstract

Due to their capacity to create better thermal conductivity than standard nanofluids, hybrid nano-fluids and modified nanofluids have notable applications in aerospace, energy materials, thermal sensors, antifouling, etc. This study aims to the modified and hybrid nanofluid flow with the Carreau fluid over a sloped shrinking sheet. The Cattaneo–Christov heat flux also takes into account. To determine the thermal efficiency of the heat, three different kinds of nanomaterials, copper oxide ($CuO)$, copper $\left(Cu\right),$ and alumina ($A{l}_2{O}_3$), are used. The similarity alteration commutes the insolubility of the model into ODEs. The conclusions are attained by program writing in MATLAB software and dealing with them through the bvp4c solver with the shooting method. The skin-friction amount decreases with the inclined sheet and local Weissenberg parameter for both modified and hybrid nanofluid. An upsurge thermal relaxation parameter declines the skin-friction coefficient for modified nanofluid flow and increases the skin-friction coefficient for hybrid nanofluid flow. The heat transfer rate is upsurged with modified and hybrid nanofluid for thermal relaxation parameter. Furthermore, the presentation includes the development of skin friction coefficient and Nusselt number values for specific parameters. Through benchmarking, numerical solutions are validated using certain limiting situations that were previously published findings, and typically solid correlation is shown.

1. Introduction

Researchers have expressed great interest in studying non-Newtonian fluids because of their numerous uses. The abundance of these fluids in nature is the fundamental cause of this. Non-Newtonian performance is also used in services like lubrication and biomedical flows and in mining, where slurries and mud are frequently curbed. The significance of modeling non-Newtonian fluid flow phenomena for the industry is evident. There has been a significant amount of research in non-Newtonian fluids, but many more studies are required in non-Newtonian fluid models. Many researchers have studied the power-law model to look into non-Newtonian effects because of its relative simplicity. The power-law model, however, has some drawbacks. Here, we consider the Carreau fluid model, another viscous model, due to the shortcomings of the power-law model, particularly at meager and high shear rates. The Carreau viscosity model helps designate the flow performance of fluids in the high shear rate section. Shear thickening (dilatant) and shear thinning (pseudo-plastic) fluid characteristics are defined by the Carreau fluid, a subdivision of the non-Newtonian fluid model.

Additionally, Fourier’s law indicated the heat transmission mechanism in 1822. The idea that the medium under examination instantly recognizes the actual temperature is based on this law. To remedy this issue, Cattaneo introduced a thermal relaxation period to Fourier’s law. This phrase describes the time a medium need to transport heat to the surrounding particles. Christov also enhanced this model. The heat flux model of Cattaneo–Christov is the new model (C-C) name. Ref. [1] discussed the phase dissemination dynamics using a sand column with a changing water table in a capillary fringe. Ref. [2] discussed the MHD power-law fluid flow with prescribed heat flux along with the heat source/sink and the power-law-dependent thermal conductivity over an extending permeable surface. Ref. [3] studied the time-independent 2D stagnation point fluid flow and heat transference to a melting extending/reduction sheet. The viscous dissipation and heat conductivity were analyzed by [4] with the unvarying temperature and unvarying heat flux for improved power law fluids moving between the equivalent plates with single plate stirring. Ref. [5] inspected the peristaltic Carreau fluid flow in a porous rectangular channel. The wall shear stress delivery was studied by [6] in six-well culture plates with planar orbital transformation. Ref. [7] deliberated the 2D Carreau fluid flow passes through a porous extending sheet along with convective boundary circumstance. Ref. [8] researched the melting phenomenon on the time-independent mixed convective flow about a vertical surface fixed in a porous [9] medium. Ref. [10] investigated for coupled flow and heat transmission of an upper convective Maxwell fluid above an extending plate along with velocity slip impact. Ref. [11] deal with nanofluids mixed convective heat transmission past a concentric erect annulus. Ref. [12] deliberated the blood flow or Carreau fluid over a tapering artery with stenosis. Ref. [13] discussed the Blasius and Sakiadis flow of Carreau fluid along with the Deborah number. Ref. [14] inspected the 2D MHD Powell–Eyring fluid flow over an extending sheet. Ref. [9] discussed the Cattaneo–Christov heat flux of the spinning viscoelastic fluid flow over a spreading surface. The heat transmission to a Carreau fluid flow over a nonlinear extending surface was inspected by [15]. Ref. [16] presented an MHD Carreau fluid flow over a convective energized surface in the company of nonlinear thermal radiation. Ref. [17] discussed the Cattaneo–Christov theory and thermal conductivity (depending on temperature) of Jeffrey fluid over a non-linear stretchy surface along with inconstant width and stagnation point flow.

Other applications of magnetohydrodynamic (MHD) viscous incompressible flow of electrically conducting fluid include processing magnetic materials, managing heat transfer, removing impurities from crude oil, Hall generators, etc. Such flows result in the generation of a Lorentzian magnetic body force that is transverse to the direction of the applied magnetic field and helps control energy flux, dampen oscillations, and high-temperature plasmas. Colloidal suspensions of magnetizable nanoparticles with magnetic, fluid, and thermal characteristics make up magnetic nanofluids. Ref. [18] explored the mixed-convective heat transmission of water-Cu nanofluids privileged a 90° angle curvature microtube. Ref. [19] explored the time-dependent convective micropolar fluid flow with Soret, reaction rate, and radiation impacts passing through an upright porous surface with dynamic viscidness (depending on temperature) and unvarying vortex viscidness. Ref. [20] analyzed the 3D forced convective Carreau fluid flow over a bi-directional strained surface. Ref. [21] studied the 2D hydromagnetic Carreau fluid flow passes an inconstant extending sheet. Ref. [22] discussed comparing $Cu-A{l}_2{O}_3$/water hybrid nanofluid and $Cu$/water nanofluid flows past an extending sheet. Ref. [23] explored the non-Fourier heat flux influence on the stagnation-point Carreau fluid flow. Ref. [24] presented the dual explanations of a Carreau fluid flow over a non-linear slopped reduction surface in the company of immeasurable shear rate viscidness. A time-dependent hydromagnetic natural convective flow with radiative impact and reaction rate impact passes through a porous erect plate explored by [25]. Ref. [26] studied the entropy production impression on hydromagnetic hybrid nanofluid ($A{l}_2{O}_3–Cu$/$H_{2}O$) flow because of the porous stretching sheet with inconstant heat flux in the presence of the electric field. Ref. [27] studied a Slippage impact on peristaltic transport hydromagnetic Jeffery hybrid nanofluid ($Ti{O}_2–Cu$/$H_{2}O$) in an asymmetric channel with viscous dissipation and Hall current impressions.

With their low heat resistance and effective thermo-physical characteristics, nanofluids have become one of the most desirable areas of study. Further, maintaining the intended performance of several industrial and the cooling of technical devices, including computers, laptops, power electronics, motors, and high-powered rays, is essential. Results showed a 23.8% improvement at 0.1% solid copper nanoparticle volume fraction-Their greater surface area and thermal conductivity suit copper nanoparticles to this growth. We spoke about more than fifty distinct nanofluids made of water, ethylene glycol, and motor oil with SiO₂, $A{l}_2{O}_3$, $Ti{O}_2$, $Zr{O}_2$, and diamond particles. The range of solid nanoparticles was 0:25 to 8%, with particle sizes restricted to 10 to 150 nm. The classical theory does not specify thermal conductivity for nanofluids. A few analyses have been developed for two types of particles floating in a base fluid called a hybrid nanofluid. To create a hybrid nanofluid, two different kinds of solid nanoparticles have been dispersed throughout the base fluid in several experimental and computational studies. The next generation of nanofluids is hybrid nanofluids, and the next generation of hybrid nanofluids is modified nanofluids. Ref. [28] discussed the thermal features of the ternary hybrid nanomaterials $(CuO-Cu-A{l}_2{O}_3$) between two equivalent walls with entropy creation and nonlinear thermal radiation. Ref. [29] focused on hydromagnetic Carreau nanofluid flow over a paraboloid surface along with the Cattaneo–Christov heat flux. Ref. [30] inspected the entropy creation of radiative Carreau fluid flow in a slopped microchannel in viscous heating. Ref. [31] examined the radiative flow of Oldroyd-B liquid induced by a stretchy sheet with cross diffusion and chemical reaction effects. Ref. [32] analyzed the Xue model and Yamada–Ota model hybrid nanofluid flow with a slopped magnetic field over a dynamic cylinder. Refs. [33,34] discussed a 2D hydromagnetic SWCNTs or MWCNTs/saline water nanofluid flow with mixed convection above an extending/shrinking plate with a melting effect.

Here, we concentrate on copper, aluminium oxide, and copper oxide as the three most common nanoparticles. Due to their thermal solid and chemical durability, antimicrobial capabilities, and antibacterial characteristics, aluminium oxide nanoparticles are an excellent choice for water treatment. Al₂O₃ nanoparticles also exhibit properties useful in biological applications such as medication delivery, biofiltration, and sensors. Significant anti-bacterial properties of copper make it helpful in treating infectious epidemics. It is a well-established antibacterial and antifungal agent. The radiative $F{e}_3{O}_4,Mo{S}_2$-$H_{2}O$ hybrid nanofluid flow was inspected by [35] over a slopped plate with ascent heating and heat source/sink impressions. A study on the second law analysis and mixed convective rheology of the ($A{l}_2{O}_3–Ag/H_{2}O$) hybrid nanofluid flow influenced by magnetic induction properties to a stretching sheet with viscous dissipation and internal heat generation effects studied by [36]. Ref. [37] investigated the thermal radiation and non-uniform heat flux influences on MHD hybrid nanofluid ($CuO-F{e}_2{O}_3$/$H_{2}O$) flow along a stretching cylinder with velocity slip condition. Ref. [38] studied the heat transmission of an engine oil-based fluid flow at an inclination of $45°$ to the plane. Ref. [39] focused on the inclined plate solar collector substituted by the hybrid nanofluid holding $MgO, CuO$ with MWCNTs and water. Ref. [40] explored the ternary nanofluid flow at a non-linear extending Riga plate with variable viscidness. Ref. [41] studied the unsteady 3D water-driven hybrid nanofluid with the consequences of brick-shaped nanocomposites (ceria and zinc oxide) with the thermal link of heat source/sink and variable thermal conditions within the magnetic environment. Ref. [42] investigated the flow of the glycerin-based carbon nanotubes with velocity slip in Darcy–Forchheimer porous medium on a convectively heated Riga plate along with the Cattaneo–Christov theory. Ref. [43] inspected the thermal conductivity, Cattaneo–Christov heat flux, and activation energy in 2D incompressible nanofluid flow with base fluid water over a curved extending sheet looped in a round using the Koo–Kleinstreuer–Li model.

Heat transmission is one of the fundamental and most significant phenomena in several engineering and manufacturing fields. The function of heat transmission is communicated by several technical and manufacturing developments, including extrusion operations, electronic chilling, the refining sector, and premium heated oil. Fluids considered to be in a nano-size configuration are called nanofluids. These liquids blend base liquids and nanoparticles, such as carbon nanotubes or carbides, oxides, and metals (including glycol, ethylene, oil, water, etc.). The creation of semiconductor materials, crystal growth, welding processes, the glass industry, material sanitization, and many other industrial processes have significantly benefited from the research of melting phenomena in recent years. The variable viscidness (depending on temperature) and thermal conductivity were explored by [44] with modified Fourier’s law in a fluid flow with tiny materials over a revolving disk. Ref. [45] inspected the modified Fourier heat flux influence of MHD radiative nanofluid flow privileged a hole occupied with ethylene glycol-multiwalled carbon nanotubes. Ref. [46] discussed the study of the convective flow of Cross fluid ($A{l}_2{O}_3-Cu/CMC$) containing carboxymethyl cellulose water over a stretching sheet with convective heating. Ref. [47] studied the hydromagnetic mixed convective flow of CNTs nanofluid in water past a heated stretchy plate with injection/suction, heat consumption, viscous dissipation, and radiation. Ref. [48] examined the bi-directional Williamson fluid flow in a porous extending sheet along with the thermophoresis, Brownian motion, zero mass flux, and modified Darcy’s law. The hydromagnetic stagnation point flow of $A{l}_2{O}_3$/water nanofluid over a reduction sheet and viscous dissipation, melting, and Ohmic heating impressions were elucidated by [49]. Ref. [50] explicated the $Mo{O}_2$-$Co/$ sodium alginate-based hybrid nanofluid flow with the ramped motion of a limitless inclined plate. Ref. [51] deliberated entropy production investigation for a peristaltic flow in a rotating medium with generalized complaint walls.

The primary goal of this paper is to analyze the performance of a Carreau-modified nanofluid flowing over a surface that is inclined and shrinking in the context of the Cattaneo–Christov heat flux. Till now, nobody studied the Carreau-modified nanofluid flow in the presence of the Cattaneo–Christov heat flux over an inclined shrinking surface. The three different forms of modified nanoparticles—copper oxide $\left(CuO\right),$ copper nanoparticles $\left(Cu\right),$ and aluminium oxide ($A{l}_2{O}_3$)—are used to examine the thermal deal. By using the bvp4c approach along with the shooting technique, the simulated problem solution is calculated. The behavior of the relevant impressions has been investigated through diagrams and tables. Additionally, the findings of earlier analyses by Wahid et al. [49] and Bachok et al. Ref. [3] have been compared with those of the present investigation. We expect the current study to assist other future researchers by using several flow models and varied geometries.

2. Mathematical Design

The next are the situations and managerial ethics that govern the current model given in Figure 1:

• 2D hydromagnetic time-independent stagnation-point flow
• Carreau modified nanofluid flow with $CuO, Cu,$ and $A{l}_2{O}_3$nanoparticles
• Inclined shrinking sheet
• Cattaneo–Christov heat flux
• The motion of the reduction sheet is $u_w\left(x\right),$ and the movement of the inviscid flow is $u_e\left(x\right),$ The melting temperature is $T_m$ and the free stream temperature $T_{\infty }$.

The suitable Navier–Stoke’s equations that control the flow conventions are the equation of continuity, velocity, and energy, which were obtained by (Wahid et al. [49] and Bachok et al. [3], Ahmad and Pop [8], Devi and Devi [22]):

Equation of continuity:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

Equation of motion:

$$\begin{array}{cc}\hfill {\rho }_{mnf} (u\frac{\partial u}{\partial x} + & v\frac{\partial u}{\partial y}) \hfill \\ & ={\rho }_{mnf}{u}_e\frac{\mathrm{d}{u}_e}{\mathrm{d}x}+{\mu }_{mnf}\left(\frac{{\partial }^{2}u}{\partial y^2}\right){\left(1+{\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\right)}^{\left(n-1\right)/2}+{\mu }_{mnf}(n\hfill \\ & -1){\Gamma }^2\left(\frac{{\partial }^{2}u}{\partial y^2}\right){\left(\frac{\partial u}{\partial y}\right)}^2{\left(1+{\Gamma }^2{\left(\frac{\partial u}{\partial y}\right)}^2\right)}^{\left(n-3\right)/2}-{\sigma }_{mnf}{B}_0{}^2(u\hfill \\ & -u_e)+g {\left(\rho {\beta }_T\right)}_{mnf}cos\alpha \left(T-T_m\right)\hfill \end{array}$$

(2)

Equation of temperature:

$$\begin{array}{cc}\hfill {\left(\rho C_p\right)}_{mnf} [u\frac{\partial T}{\partial x} & + v\frac{\partial T}{\partial y}\hfill \\ & +{\lambda }_T\{\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right)\frac{\partial T}{\partial x}+\left(u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}\right)\frac{\partial T}{\partial y}+u^2\frac{{\partial }^{2}T}{\partial x^2}\hfill \\ & +v^2\frac{{\partial }^{2}T}{\partial y^2}+2uv\frac{{\partial }^{2}T}{\partial x\partial y}\left\}\right]\hfill \\ & =k_{mnf}\frac{{\partial }^{2}T}{\partial y^2}+{\sigma }_{mnf}{B}_0{}^2{\left(u-u_e\right)}^2+{\mu }_{mnf}{\left(\frac{\partial u}{\partial y}\right)}^2 \hfill \end{array}$$

(3)

The boundary circumstances for the existing case are as tracks (Jumana et al. [34]):

$$\begin{array}{c}u=u_w\left(x\right); T=T_m; \frac{k_{mnf}}{{\rho }_{mnf}}\left(\frac{\partial T}{\partial y}\right)=\left(L+C_s\left(T_m-T_0\right)\right) v\left(x\right); at y=0\\ u\to u_e\left(x\right); T\to T_{\infty };at y\to \infty \end{array}$$

(4)

The components of motion in the coordinates of $y$ and $x$are denoted by $v$ (meter/second-m/s) and $u$ (meter/second-m/s), respectively, $u_w\left(x\right)=cx$ and $u_e\left(x\right)=ax$ with a and c are the constants, $C_s$ standing for the solid surface specific heat, L stands for the latent heat, $T_0$ depicts the solid surface temperature, $T_m$ depicts the melting point temperature, acute angle ($\mathsf{\alpha }$), λ_T is the thermal relaxation constants, the heat that is transmitted to the melting surface is equivalent to the sum of the sensible energy mandatory to get the temperature $T_0$ up to the melting point and the melting heat temperature at a point $T_m$, $T$ (Kelvin-K) is the fluid temperature and $\Gamma$ is the time constant. Furthermore, $mnf$ stands for modified nanofluid, $hbnf$ stands for hybrid nanofluid, ${\left(\rho C_p\right)}_{mnf}$signifies the heat capacity of the modified nanofluid, $B_0$ (Tesla-T) shows the strength of the magnetic field, the gravitational acceleration is g, ${\sigma }_{mnf}$ depicts the electrical conductivity, ${\left(\rho {\beta }_T\right)}_{mnf}$ stands for the thermal expansion coefficient, $k_{mnf}$ signifies the thermal conductivity of the modified nanofluid, ${\rho }_{mnf}$symbolizes the density of the modified nanofluid, and ${\mu }_{mnf}$ denotes the modified nanofluid dynamic viscosity. Remember that the Newtonian situation is signified by the power law index $n=1$ in Equation (2). Naturally, a power law index value between 0 and 1 characterizes the shear thinning behavior of fluids, whereas $n>1$ results in the shear thickening behavior.

Table 1. The methods and boundaries for the modified and hybrid nanofluid in the scheme above equations are (Abbasi et al. [28]):

Thermal Properties	Modified Nanofluid	Hybrid Nanofluid
Thermal Diffusivity	${\alpha }_T{}_{mnf}=\frac{k_{mnf}}{{\left(\rho C_p\right)}_{mnf}}$	${\alpha }_T{}_{hbnf}=\frac{k_{hbnf}}{{\left(\rho C_p\right)}_{hbnf}}$
Viscosity	$\frac{{\mu }_{mnf}}{{\mu }_{Bf}}=\frac{1}{{\left(1-{\varphi }_{CuO}\right)}^{2.5}{\left(1-{\varphi }_{Cu}\right)}^{2.5}{\left(1-{\varphi }_{A{l}_2{o}_3}\right)}^{2.5}}$	$\frac{{\mu }_{hbnf}}{{\mu }_{Bf}}=\frac{1}{{\left(1-{\varphi }_{Cu}\right)}^{2.5}{\left(1-{\varphi }_{CuO}\right)}^{2.5}}$
Heat Capacity	$\begin{array}{cc}\hfill \frac{{\left(\rho C_p\right)}_{mnf}}{{\left(\rho C_p\right)}_{Bf}}=(1-& {\varphi }_{A{l}_2{o}_3}){\displaystyle (}(1-{\varphi }_{Cu}){\displaystyle (}\left(1-{\varphi }_{CuO}\right)\hfill \\ & +{\varphi }_{CuO}\frac{{\left(\rho C_p\right)}_{CuO}}{{\left(\rho C_p\right)}_{Bf}})+{\varphi }_{Cu}\frac{{\left(\rho C_p\right)}_{Cu}}{{\left(\rho C_p\right)}_{Bf}})\hfill \\ & +{\varphi }_{A{l}_2{o}_3}\frac{{\left(\rho C_p\right)}_{A{l}_2{o}_3}}{{\left(\rho C_p\right)}_{Bf}}\hfill \end{array}$	$\begin{array}{cc}\hfill \frac{{\left(\rho C_p\right)}_{hbnf}}{{\left(\rho C_p\right)}_{Bf}}={\varphi }_{Cu}& \frac{{\left(\rho C_p\right)}_{Cu}}{{\left(\rho C_p\right)}_{Bf}}\hfill \\ & +(1-{\varphi }_{Cu})\left(\left(1-{\varphi }_{CuO}\right)+{\varphi }_{CuO}\frac{{\left(\rho C_p\right)}_{CuO}}{{\left(\rho C_p\right)}_{Bf}}\right)\hfill \end{array}$
Density	$\begin{array}{cc}\hfill \frac{{\left(\rho \right)}_{mnf}}{{\left(\rho \right)}_{Bf}}=(1-{\varphi }_{A{l}_2{o}_3})& {\displaystyle (}(1-{\varphi }_{Cu}){\displaystyle (}\left(1-{\varphi }_{CuO}\right)\hfill \\ & +{\varphi }_{CuO}\frac{{\left(\rho \right)}_{CuO}}{{\left(\rho \right)}_{Bf}})+{\varphi }_{Cu}\frac{{\left(\rho \right)}_{Cu}}{{\left(\rho \right)}_{Bf}})\hfill \\ & +{\varphi }_{A{l}_2{o}_3}\frac{{\left(\rho \right)}_{A{l}_2{o}_3}}{{\left(\rho \right)}_{Bf}}\hfill \end{array}$	$\frac{{\left(\rho \right)}_{hbnf}}{{\left(\rho \right)}_{Bf}}={\varphi }_{Cu}\frac{{\left(\rho \right)}_{Cu}}{{\left(\rho \right)}_{Bf}}+(1-{\varphi }_{Cu})\left(\left(1-{\varphi }_{CuO}\right)+{\varphi }_{CuO}\frac{{\left(\rho \right)}_{CuO}}{{\left(\rho \right)}_{Bf}}\right)$
Thermal expansion	$\begin{array}{cc}\hfill \frac{{\left(\rho {\beta }_T\right)}_{mnf}}{{\left(\rho {\beta }_T\right)}_{Bf}}=(1-& {\varphi }_{A{l}_2{o}_3}){\displaystyle (}(1-{\varphi }_{Cu}){\displaystyle (}(1-{\varphi }_{CuO})\hfill \\ & +{\varphi }_{CuO}\frac{{\left(\rho {\beta }_T\right)}_{CuO}}{{\left(\rho {\beta }_T\right)}_{Bf}})+{\varphi }_{Cu}\frac{{\left(\rho {\beta }_T\right)}_{Cu}}{{\left(\rho {\beta }_T\right)}_{Bf}})\hfill \\ & +{\varphi }_{A{l}_2{o}_3}\frac{{\left(\rho {\beta }_T\right)}_{A{l}_2{o}_3}}{{\left(\rho {\beta }_T\right)}_{Bf}}\hfill \end{array}$	$\begin{array}{cc}\hfill \frac{{\left(\rho {\beta }_T\right)}_{hbnf}}{{\left(\rho {\beta }_T\right)}_{Bf}}={\varphi }_{Cu}& \frac{{\left(\rho {\beta }_T\right)}_{Cu}}{{\left(\rho {\beta }_T\right)}_{Bf}}\hfill \\ & +(1-{\varphi }_{Cu})\left(\left(1-{\varphi }_{CuO}\right)+{\varphi }_{CuO}\frac{{\left(\rho {\beta }_T\right)}_{CuO}}{{\left(\rho {\beta }_T\right)}_{Bf}}\right)\hfill \end{array}$
Thermal Conductivity	$\begin{array}{c}{k}_{nf}=\frac{k_{CuO}+\left(M-1\right)k_{Bf}-\left(M-1\right){\varphi }_{CuO}\left(k_{Bf}-k_{CuO}\right)}{k_{CuO}+\left(M-1\right)k_{Bf}+{\varphi }_{CuO}\left(k_{Bf}-k_{CuO}\right)} k_{Bf},\\ k_{hbnf}=\frac{k_{Cu}+\left(M-1\right)k_{nf}-\left(M-1\right){\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}{k_{Cu}+\left(M-1\right)k_{nf}+{\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)} k_{nf},\\ k_{mnf}=\frac{k_{A{l}_2{o}_3}+\left(M-1\right)k_{hbnf}-\left(M-1\right){\varphi }_{A{l}_2{o}_3}\left(k_{hbnf}-k_{A{l}_2{o}_3}\right)}{k_{A{l}_2{o}_3}+\left(M-1\right)k_{hbnf}+{\varphi }_{A{l}_2{o}_3}\left(k_{hbnf}-k_{A{l}_2{o}_3}\right)} k_{hbnf}\end{array}$	$k_{hbnf}=\frac{k_{Cu}+\left(M-1\right)k_{nf}-\left(M-1\right){\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}{k_{Cu}+\left(M-1\right)k_{nf}+{\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)} k_{nf}$$k_{nf}=\frac{k_{CuO}+\left(M-1\right)k_{Bf}-\left(M-1\right){\varphi }_{CuO}\left(k_{Bf}-k_{CuO}\right)}{k_{CuO}+\left(M-1\right)k_{Bf}+{\varphi }_{CuO}\left(k_{Bf}-k_{CuO}\right)} k_{Bf}$
Electrical Conductivity	$\begin{array}{c}{\sigma }_{nf}=\frac{{\sigma }_{CuO}+\left(M-1\right){\sigma }_{Bf}-\left(M-1\right){\varphi }_{CuO}\left({\sigma }_{Bf}-{\sigma }_{CuO}\right)}{{\sigma }_{CuO}+\left(M-1\right){\sigma }_{Bf}+{\varphi }_{CuO}\left({\sigma }_{Bf}-{\sigma }_{CuO}\right)} {\sigma }_{Bf},\\ {\sigma }_{hbnf}=\frac{{\sigma }_{Cu}+\left(M-1\right){\sigma }_{nf}-\left(M-1\right){\varphi }_{Cu}\left({\sigma }_{nf}-{\sigma }_{Cu}\right)}{{\sigma }_{Cu}+\left(M-1\right){\sigma }_{nf}+{\varphi }_{Cu}\left({\sigma }_{nf}-{\sigma }_{Cu}\right)} {\sigma }_{nf},\\ {\sigma }_{mnf}=\frac{{\sigma }_{A{l}_2{o}_3}+\left(M-1\right){\sigma }_{hbnf}-\left(M-1\right){\varphi }_{A{l}_2{o}_3}\left({\sigma }_{hbnf}-{\sigma }_{A{l}_2{o}_3}\right)}{{\sigma }_{A{l}_2{o}_3}+\left(M-1\right){\sigma }_{hbnf}+{\varphi }_{A{l}_2{o}_3}\left({\sigma }_{hbnf}-{\sigma }_{A{l}_2{o}_3}\right)} {\sigma }_{hbnf}\end{array}$	${\sigma }_{nf}=\frac{{\sigma }_{CuO}+\left(M-1\right){\sigma }_{Bf}-\left(M-1\right){\varphi }_{CuO}\left({\sigma }_{Bf}-{\sigma }_{CuO}\right)}{{\sigma }_{CuO}+\left(M-1\right){\sigma }_{Bf}+{\varphi }_{CuO}\left({\sigma }_{Bf}-{\sigma }_{CuO}\right)} {\sigma }_{Bf}$${\sigma }_{hbnf}=\frac{{\sigma }_{Cu}+\left(M-1\right){\sigma }_{nf}-\left(M-1\right){\varphi }_{Cu}\left({\sigma }_{nf}-{\sigma }_{Cu}\right)}{{\sigma }_{Cu}+\left(M-1\right){\sigma }_{nf}+{\varphi }_{Cu}\left({\sigma }_{nf}-{\sigma }_{Cu}\right)} {\sigma }_{Bf}$

shows the formulas for topographies of nanoparticles.

where $C_p$ is the specific heat at constant pressure, ${\varphi }_{\mathrm{CuO}}$ represents the volume fraction of $CuO$, ${\varphi }_{\mathrm{Cu}}$ describes the volume fraction of $Cu, {\varphi }_{A{l}_2{o}_3}$ describes the volume fraction of $A{l}_2{O}_3$, and $k_{Bf}, {\mu }_{Bf}, {\rho }_{Bf}$ and ${\sigma }_{Bf}$ represent the thermal conductivity, dynamic viscosity, density, and electrical conductivity of the regular fluid, respectively. The subscripts $Bf, hbnf, mnf, CuO,Cu$, and $A{l}_2{O}_3$ symbolize the base fluid, modified nanofluid, hybrid nanofluid, copper oxide, copper, and aluminum oxide nanoparticles, respectively. As an outcome, Table 1 covers data from the working fluid and three different nanomaterials $CuO, Cu$, and $A{l}_2{O}_3$. The physical properties of nanofluids are given in

Table 2. Physical properties of $CuO-Cu$ -$A{l}_2{O}_3$ /water hybrid nanoparticles (Abbasi et al. [28]):

Physical Characteristics	Water	$\mathit{C}\mathit{u}\mathit{O}$	$\mathit{C}\mathit{u}$	$\mathit{A}{\mathit{l}}_2{\mathit{O}}_3$
$\rho$	997.1	6320	8933	3970
$C_p$	4179	531.8	385	765
k	0.613	76.5	401	40
$\sigma$	$5.5\times {10}^{-6}$	$6.9\times {10}^{-2}$	$59.6\times {10}^6$	$35\times {10}^6$
${\beta }_T$	21	1.80	1.67	0.85
$Pr$	6.2	-	-	-

.

In the existing circumstances, we may clarify our investigation by holding the next non-dimensional quantities (Jumana et al. [34]):

$$\eta =\sqrt{\frac{a}{{\nu }_{Bf}}}y, v=-\sqrt{a{\nu }_{Bf}}f, u=ax{f}^{\prime }, \theta \left(\eta \right)=\frac{T-T_m}{T_{\infty }-T_m}$$

(5)

In Equations (1) and (4), the changed ODEs and boundary conditions are obtained by rearranging (6) to (8) as:

$$\begin{array}{cc}\hfill \frac{{\mu }_{mnf}/{\mu }_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}(1+& n{W}^2{f^{″}}^2){\left(1+W^2{f^{″}}^2\right)}^{\left(n-3\right)/2}{f}^{‴}+f{f}^{″}+1+{f^{\prime }}^2\hfill \\ & -\frac{{\sigma }_{mnf}/{\sigma }_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}Mn\left(f^{\prime }-1\right)+\frac{{\left(\rho {\beta }_T\right)}_{mnf}/{\left(\rho {\beta }_T\right)}_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}cos\alpha G_T\theta =0.\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \frac{1}{Pr}(\frac{k_{mnf} /k_{Bf}}{{\left(\rho C_p\right)}_{mnf} /{\left(\rho C_p\right)}_{Bf}}& -{\delta }_T{f}^2){\theta }^{″}+f{\theta }^{\prime }-{\delta }_{T}f{f}^{\prime }{\theta }^{\prime }\hfill \\ & +\frac{{\mu }_{mnf}/{\mu }_{Bf}}{{\left(\rho C_p\right)}_{hbnf} /{\left(\rho C_p\right)}_{Bf} }Ec{f^{″}}^2\hfill \\ & +\frac{{\sigma }_{mnf}/{\sigma }_{Bf}}{{\left(\rho C_p\right)}_{hbnf} /{\left(\rho C_p\right)}_{Bf} }EcMn{\left(f^{\prime }-1\right)}^2=0\hfill \end{array}$$

(7)

With the boundary situations in issue

$$\begin{array}{c}f\left(0\right)+\frac{Me}{Pr}\frac{\frac{k_{mnf}}{k_{Bf}}}{\frac{{\left(\rho \right)}_{mnf}}{{\left(\rho \right)}_{Bf}}}{\theta }^{\prime }\left(0\right)=0, f^{\prime }\left(0\right)=\lambda , \theta \left(0\right)=0 at \eta =0\\ f^{\prime }(\infty )\to 1, \theta (\infty )\to 1 at \eta \to \infty \end{array}$$

(8)

The local Weissenberg number $\left(W\right)$, Hartmann parameter $\left(Mn\right)$, local Grashof temperature number (G_T), $\mathrm{Prandtl}\mathrm{effect}\left(Pr\right),\mathrm{thermal}\mathrm{relaxation}\mathrm{parameter}\left({\delta }_T\right)$ Eckert number $\left(Ec\right)$, Stretching/Shrinking parameter ($\lambda$), and Melting parameter $\left(Me\right)$ are all parameters utilized to designate the nondimensional amounts in Equations (6) to (8). The constant shrinking parameter is $\lambda =\frac{c}{a}<0$, while the static sheet is represented by $\lambda =0$. These quantities are specified by formula as

$$\begin{array}{c}W=\Gamma \sqrt{\frac{x{a}^3}{{\nu }_{Bf}}}\Gamma ,Mn=\frac{{\sigma }_{Bf}{B}_0{}^2}{{\rho }_{Bf}a},G_T=\\ \frac{g{\left({\beta }_T\right)}_{Bf}\left(T_{\infty }-T_m\right)}{a^{2}x},Pr=\frac{{\nu }_{Bf}}{{\propto }_{Bf}},{\delta }_T={\lambda }_{T}aEc=\frac{a^2{x}^2}{{\left(C_p\right)}_{Bf}\left(T_{\infty }-T_m\right)},\\ \mathrm{and}Me=\frac{{\left(C_p\right)}_{Bf}\left(T_{\infty }-T_m\right)}{L+C_s\left(T_{\infty }-T_m\right)},\mathrm{and}\lambda =c/a.\end{array}$$

We note that the Stefan numbers ${\left(C_p\right)}_{Bf}\left(T_{\infty }-T_m\right)/L$ and $C_s\left(T_{\infty }-T_m\right)/L$ for the base fluid and solid surface, respectively, are combined to form the melting parameter Me.

The shear stress and heat transfer rate are physical amounts of practical engineering concern, and they are explained as tracks:

The skin friction is $C_f=\frac{{\tau }_w}{{\rho }_{Bf}{u}_e^2}$, and the Nusselt number is $N{u}_x=\frac{x{q}_w}{k_{Bf}\left(T_{\infty }-T_m\right)}$.

The surface shear stress ${\tau }_w$ is supposed by ${\tau }_w={\mu }_{mnf}{\left(\frac{\partial u}{\partial y}\left(1+\frac{\Gamma }{\sqrt{2}}\frac{\partial u}{\partial y}\right)\right)}_{y=0}$, we get

$$C_{fx}R{e}_x^{1/2}={\mu }_{mnf}/{\mu }_{Bf}[f^{″}\left(0\right)+\left(W/2\right){(f^{″}\left(0\right))}^2],$$

The rate of heat transfer $q_w$ is supposed by $q_w=-k_{hbnf}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}$(here negative sign shows that the temperature is reducing from higher to lower.), we get

$$N{u}_{x}R{e}_x^{-1/2}=-\left(k_{mnf}/k_{Bf}\right){\theta }^{\prime }\left(0\right),$$

where, $R{e}_x=\frac{u_e\left(x\right)}{{\nu }_{Bf}}$ is the Reynolds number.

3. Numerical Structure

Equations are resolved through the bvp4c solver. All numerical amounts and diagrams are established with MATLAB, which is debated in the tables and graphs. Let

$$f=y\left(1\right), f^{\prime }=y\left(2\right), f^{″}=y\left(3\right), \theta =y\left(4\right), {\theta }^{\prime }=y\left(5\right).$$

Equations (6) to (8) reduce into newform as follows:

$$\begin{array}{cc}\hfill \frac{{\mu }_{mnf}/{\mu }_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}(1+& n{W}^{2}y{\left(3\right)}^2){\left(1+W^{2}y{\left(3\right)}^2\right)}^{\left(n-3\right)/2}{f}^{‴}+y\left(1\right)y\left(3\right)+1+y{\left(1\right)}^2\hfill \\ & -\frac{{\sigma }_{mnf}/{\sigma }_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}Mn(y\left(1\right)-1)+\frac{{\left(\rho {\beta }_T\right)}_{mnf}/{\left(\rho {\beta }_T\right)}_{Bf}}{{\rho }_{mnf}/{\rho }_{Bf}}cos\alpha G_{T}y\left(4\right)\hfill \\ & =0\hfill \end{array}$$

$$\begin{array}{cc}\hfill \frac{1}{Pr}(\frac{k_{mnf} /k_{Bf}}{{\left(\rho C_p\right)}_{mnf} /{\left(\rho C_p\right)}_{Bf}}& -{\delta }_{T}y{\left(1\right)}^2){\theta }^{″}+y\left(1\right)y\left(5\right)-{\delta }_{T}y\left(1\right)y\left(2\right)y\left(5\right)\hfill \\ & +\frac{{\mu }_{mnf}/{\mu }_{Bf}}{{\left(\rho C_p\right)}_{hbnf} /{\left(\rho C_p\right)}_{Bf} }Ecy{\left(3\right)}^2\hfill \\ & +\frac{{\sigma }_{mnf}/{\sigma }_{Bf}}{{\left(\rho C_p\right)}_{hbnf} /{\left(\rho C_p\right)}_{Bf} }EcMn{\left(y\left(2\right)-1\right)}^2=0\hfill \end{array}$$

With the boundary conditions in issue:

$$\begin{array}{c}y0\left(1\right)+\frac{Me}{Pr}\frac{\frac{k_{mnf}}{k_{Bf}}}{\frac{{\left(\rho \right)}_{mnf}}{{\left(\rho \right)}_{Bf}}}y0\left(5\right)=0, y0\left(2\right)=\lambda , y0\left(4\right)=0 at \eta =0\\ y\infty \left(2\right)\to 1, y\infty \left(4\right)\to 1 at \eta \to \infty \end{array}$$

The choice $\mathsf{\eta }(\infty )=5\mathrm{or}7$ designates that each numerical outcome accesses asymptotic assets ideally in this method.

4. Code Verification

Justification of existing conclusions is proved in current research.

Table 3. Assessment regarding the amounts of $f^{″}\left(0\right)$ (at $\lambda =2$ ) and $-{\theta }^{\prime }\left(0\right)$  (at $\lambda =1$ ) with Me, when $Pr = 1$, and non-appearance of further rest numbers.

Me	${\mathit{f}}^{″}\left(0\right) \mathit{a}\mathit{t} \mathit{\lambda }=2$			Me	$-{\mathit{\theta }}^{\prime }\left(0\right) \mathit{a}\mathit{t} \mathit{\lambda }=1$
	Bachok et al. [3]	Wahid et al. [49]	Current Work		Bachok et al. [3]	Wahid et al. [49]	Current Work
0	−1.8873066	−1.887306668	−1.887306668	0	−0.7978846	−0.797884573	−0.797884573
1	−1.5804839	−1.580483902	−1.580483902	1	−0.5060545	−0.506054476	−0.506054476
2	−1.4427473	−1.442747275	−1.442747275	2	−0.3826383	−0.382638279	−0.382638279
3	−1.3592105	−1.359210504	−1.359210504	3	−0.3119564	−0.311956405	−0.311956405

endows a juxtaposition of the known research consistencies. However, highly accurate outcomes for the present investigation are searched.

5. Result and Discussion

The properties of separate water-based nanofluid, notably $CuO, Cu$, and $A{l}_2{O}_3$/water (spherical shape), are assessed utilizing a bvp4c solver. The implications of developing flow features are enumerated utilizing a variety of produced graphs and tables. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and

Table 4. The rates of skin-friction and Nusselt with $Mn, G_T, \alpha , n, W, {\delta }_T$ and $Ec$ when $Pr = 6.2, Me = 0.1, \lambda = -0.1, {\mathsf{\varphi }}_{\mathrm{CuO}}=0.20,{\mathsf{\varphi }}_{\mathrm{Cu}}=0.25$ and ${\mathsf{\varphi }}_{{\mathrm{Al}}_2{\mathrm{O}}_3}$ = 0.35.

Mn	GT	α	n	W	δT	Ec	CuO-Cu-Al2O3/H2O		CuO-Cu/H2O
							${\displaystyle {{\displaystyle \mathbf{C}}}_{{{\displaystyle \mathbf{f}}}_{{\displaystyle \mathbf{x}}}}\sqrt{2}{\displaystyle \mathbf{R}}{{\displaystyle \mathbf{e}}}_{{\displaystyle \mathbf{x}}}{}^{1/2}}$	$\mathbf{N}{\mathbf{u}}_{\mathbf{x}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{-1/2}$	${\mathbf{C}}_{{\mathbf{f}}_{\mathbf{x}}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{1/2}$	$\mathbf{N}{\mathbf{u}}_{\mathbf{x}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{-1/2}$
0.1							8.927047972	−6.198828532	4.894402794	−3.813227424
							11.143034493	−8.020221164	5.433157323	−4.288368467
							13.383275047	−9.812182534	6.031744065	−4.809842051
	0.1						8.927047972	−6.198828532	4.893446770	−3.811927735
	1.5						9.039911046	−6.286452677	4.993525133	−3.903802630
0.5	1						9.184889066	−6.399489483	5.119261263	−4.020277174
		$\pi /6$					8.934277295	−6.204740426	4.900228211	−3.818651672
		$\pi /4$					8.927047972	−6.198828532	4.894402794	−3.813227424
		$\pi /3$					8.917629262	−6.191131516	4.886812746	−3.806164135
1			0.1				8.927047972	−6.198828532	4.894402794	−3.813227424
			0.5				8.949440436	−6.201321300	4.926321011	−3.817245819
			0.9				8.971965771	−6.203819563	4.958732360	−3.821286307
				0.1			8.927047972	−6.198828532	4.894402794	−3.813227424
				1			6.336483032	−5.819039998	2.683474467	−3.374176093
				2			4.222481257	−5.312300497	1.626149290	−2.969048910
					0.0005		−2.969048910	−6.198664878	4.894402756	−3.813102206
					0.0010		8.927047972
					0.0015		8.927047703
						0.1	8.927047972	−6.198828532	4.894402794	−3.813227424
						0.5	8.833144216	−17.794316758	4.848093454	−9.939227882
						0.7	8.787808564	−23.467557884	4.825687631	−12.942578233

and

Table 5. The rates of skin-friction and Nusselt with Pr, Me, $\mathsf{\lambda }$ , ${\mathsf{\varphi }}_{\mathrm{CuO}},{\mathsf{\varphi }}_{\mathrm{Cu}},$ and ${\mathsf{\varphi }}_{{\mathrm{Al}}_2{\mathrm{O}}_3}$ when Mn = ${\mathrm{G}}_{\mathrm{T}}$ = n = W = ${\mathsf{\delta }}_{\mathrm{T}}$ = Ec = 0.1 and $\mathsf{\alpha }=\mathsf{\pi }/4$.

Pr	Me	λ	${\mathsf{\varphi }}_{\mathbf{C}\mathbf{u}\mathbf{O}}, {\mathsf{\varphi }}_{\mathbf{C}\mathbf{u}}$	${\mathsf{\varphi }}_{\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3}$	CuO-Cu-Al2O3/H2O		CuO-Cu/H2O
					${\mathbf{C}}_{{\mathbf{f}}_{\mathbf{x}}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{1/2}$	$\mathbf{N}{\mathbf{u}}_{\mathbf{x}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{-1/2}$	${\mathbf{C}}_{{\mathbf{f}}_{\mathbf{x}}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{1/2}$	$\mathbf{N}{\mathbf{u}}_{\mathbf{x}}\sqrt{2}\mathbf{R}{\mathbf{e}}_{\mathbf{x}}{}^{-1/2}$
5.2					8.921092404	−5.561627924	4.890271980	−3.442597882
6.2					8.927047972	−6.198828532	4.894402794	−3.813227424
7.2					8.931869189	−6.805867799	4.897747610	−4.164290966
	0.1				8.927047972	−6.198828532	4.894402794	−3.813227424
	1				8.462962288	−5.372604369	4.616471710	−3.271921106
	2				8.080005569	−4.719097824	4.390985730	−2.851024087
		−0.1			8.927047972	−6.198828532	4.894402794	−3.813227424
		−0.4			10.188466357	−7.053427611	5.507954725	−4.116246768
		−0.7			10.766721071	−7.859191814	5.700393663	−4.309395629
			0.20		7.948470379	−5.660306008	4.297838195	−3.449087491
			0.30		12.479347624	−8.043118957	6.952368528	−5.004649610
			0.40		20.542630767	−11.787848721	11.479921474	−7.332749263
				0.25	7.396199827	−5.360729882
				0.35	9.006005543	−6.207577317
				0.45	11.361562610	−7.324523129

exhibit the interactions of various physical factors on the amounts of the Nusselt, skin friction, temperature, and velocity obtained using MATLAB programming. According to our measurements, the physical factors for the present study are as follows$: Pr=6.2, \alpha =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right., {\varphi }_{CuO}=0.20, {\varphi }_{Cu}=0.25, {\varphi }_{A{l}_2{O}_3}=0.35, {\delta }_T=0.001, W=Mn=GT=n=Me=Ec=0.1,$ and $\mathsf{\lambda }=-0.1$.

Figure 2a, Figure 3a, and Figure 4a depict the falling momentum profiles of the growing Weissenberg parameter, acute angle, and local Grashof temperature number, respectively. The fluid velocity toward the Weissenberg number is shown in Figure 2a. It is detected that with an upsurge Weissenberg number, the motion outline declines because the growing amounts of the Weissenberg number bring improvement in the relaxation time of the fluid particles, and hence viscosity occurs more prominent, which makes resistance to the fluid flow; as an outcome, the fluid motion declines.

In detail, Figure 3a shows the effect of an inclination towards fluid motion. It is observed that there is an opposite connection between an inclination and motion outline; for significant amounts of an inclination, the motion diagram falloffs. When we upsurge an inclination about the x-axis, the impact of gravity falls, which fetches a failure in fluid motion within a boundary layer. The upsurge in local temperature Grashof number declines the velocity profile (Figure 4a) because of the decrease in the thickness of the momentum boundary layer. Figure 2b and Figure 3b depict the declining temperature profiles of the growing Weissenberg parameter and an acute angle, respectively. Figure 4b displays the increasing influence of the temperature of the Grashof number due to the upsurge thickness of the thermal boundary layer for both modified and hybrid nanofluid.

Figure 5a,b illustrate the rising momentum and temperature outlines of the power law index, respectively. The increasing magnetic number declines to the velocity profile (Figure 6a) and grows to the temperature profile (Figure 6b). It is observed that the motion graphs reduce for more significant amounts of the magnetic field parameter. When we improve the magnetic field parameter, a resistive force named Lorentz force actively offers resistance against fluid particles; consequently, horizontal motion reduces. Thermal energy is released due to the additional effort required to pull the nanofluid against the magnetic field’s action. This warms the nanofluid, raises temperatures (Figure 6b), and thickens the thermal boundary layer for a sheet that is a shrinking sheet. Due to the growing melting effect, both velocity and temperature graphs fall off, as seen in Figure 7. It is also detected that the temperature reduces (Figure 7b), and the thermal boundary layer width improves for more significant values of the melting parameter. Physically, growth in the value of the melting parameter causes an upsurge in molecular movement, resulting in dissipation in energy and a decrease in the temperature of the fluid.

Figure 8a depicts the upsurge velocity profile for the thermal relaxation parameter. An upsurge temperature impression of the thermal relaxation, Eckert and Prandtl number is seen in Figure 8b and Figure 9a,b, respectively. The temperature increases with thermal relaxation due to the thickness of the thermal boundary layer. The influence of the Eckert number on temperature outline is depicted in Figure 9a. It is noticed that the temperature profile is enhanced for positive values of the Eckert number. When the Eckert number increases, fluid particles are more dynamic and energetic; consequently, average kinetic energy rises, which yields an augmentation in liquid temperature. From the definition of the Prandtl number, it is explicit that greater $Pr$ has a lesser thermal diffusivity. Due to the result of the melting parameter, the thermal boundary layer thickness increases by increasing $Pr$values and improves the thermal boundary layer width, as exposed in Figure 9b.

Figure 10a shows the decline momentum profile with the shrinking parameter. In the detail of Figure 10b, the temperature profile increases $\eta <2.5$ with the modified nanofluid and $\eta <0.7$ and the hybrid nanofluid. The temperature profile declines $\eta >2.5$ with the modified nanofluid and $\mathsf{\eta }>0.7$ with the hybrid nanofluid. An increasing impression of volume-fraction of tiny particles falloffs to the velocity and temperature profiles for both modified and hybrid nanofluid (as shown in Figure 11 and Figure 12). The effects of solid particles on the velocity field are shown in Figure 11a and Figure 12a. The acceleration of the velocity profile is caused by the growth of solid nanoparticles in the modified nanofluid and hybrid nanofluid. The increased collision with suspended nanoparticles is what drives this phenomenon; and temperature declines due to the falling thermal boundary layer.

Table 4 and Table 5 show the flow and heat transfer rate of various non-dimensional parameters for both modified and hybrid nanofluid flows. The upsurge Grashof, power law index, and volume fractions of nanosized particles grow to the skin-friction coefficient. The skin-fiction rate declines with the increasing acuate angle, Weissenberg parameter, and melting impacts. The Nusselt number rises with the acute angle and melting impacts. A growing impression of the Grashof, power law index, thermal relaxation parameter, and volume fractions of tiny particles falloffs to the heat transfer rate.

6. Conclusions

This ground-breaking work explains the $CuO-Cu$-alumina/water-based Carreau-modified nanofluid flow with Cattaneo–Christov heat flux across an inclined moving shrinking sheet. It uses the bvp4c solver for the validation of the results. The synopsis of the conclusions drawn from this research are as follows:

• The velocity and temperature profiles decline with the rising Weissenberg parameter and inclined surface.
• The velocity declines with hybrid and modified nanofluid flow with the increasing volume fraction of nanoparticles.
• The motion and temperature grow with the upsurge power law index parameter.
• For hybrid and modified nanofluid flow, the power law index declines to heat transfer rate and grows to skin friction rate.
• The Nusselt number decreases with the upsurge thermal relaxation parameter in hybrid and modified nanofluid flow.
• An upsurge volume fraction of tiny particles reduces both velocity and temperature profiles.