Abstract
In solar heating, ventilation, and air conditioning (HVAC), communications are designed to create new 3D mathematical models that address the flow of rotating Sutterby hybrid nanofluids exposed to slippery and expandable seats. The heat transmission investigation included effects such as copper and graphene oxide nanoparticles, as well as thermal radiative fluxing. The activation energy effect was used to investigate mass transfer with fluid concentration. The boundary constraints utilized were Maxwell speed and Smoluchowksi temperature slippage. With the utilization of fitting changes, partial differential equations (PDEs) for impetus, energy, and concentricity can be decreased to ordinary differential equations (ODEs). To address dimensionless ODEs, MATLAB’s Keller box numerical technique was employed. Graphene oxide Copper/engine oil (GOCu/EO) is taken into consideration to address the performance analysis of the current study. Physical attributes, for example, surface drag coefficient, heat move, and mass exchange are mathematically processed and shown as tables and figures when numerous diverse factors are varied. The temperature field is enhanced by an increase in the volume fraction of copper and graphene oxide nanoparticles, while the mass fraction field is enhanced by an increase in activation energy.
Introduction
Researchers have concentrated on new energy measuring to meet the requirements and needs of companies in this period. Researchers are interested in developing a few devices with the highest rate of heating and cooling. These might save and maintain optimal energy efficiency. Furthermore, poor heat transmission and flowing base liquid conducting have an impact on the performance and operation of solar collectors. Many efforts have been made in this respect to improve the thermal characteristics of base liquids. Solar energy is the renewable energy source from the sun for industrial applications such as electricity generation^{1,2,3}, heating^{4,5,6}, cooling^{7,8,9}, and desalination^{10,11,12}. The benefits of solar energy technology are that this type of energy is limitless, clean, and has no fuel to burn. The most common types of solar energy are photovoltaic (PV) systems^{13,14,15}, thinfilm solar cells^{16,17,18}, solar power plants^{19,20}, and passive solar heating^{21,22}. The Photovoltaic applications were reported in the field of telecommunications^{23}, agriculture^{24}, used with livestock/cattle^{25}, street lighting^{26}, and rural electrification^{27}. The usage of thinfilm solar cells was in rooftops at the institutional and commercial buildings^{28}, solar farms^{29}, power traffics^{30}, and solar steam generation^{31}. Passive solar heating is implemented in circulation spaces such as lobbies, hallways, and break rooms that allow occupants to avoid the sun.
HVAC stands for heating, ventilation, and air conditioning, whereas AC is defined as conditioning. AC is designed to cool the air and control humidity in the house and was invented by Willis Carrier in 1902^{32}. Besides, the primary purpose of HVAC system for residential^{33,34} and commercial buildings^{35,36} is to provide a heating mode in the winter and cooling mode in the summer. This system also filters smoke, odors, dust, airborne bacteria, carbon dioxide, and other harmful gases to improve air indoors^{37,38}. In addition, HVAC system acts as a humidity controller of air indoors^{39,40}. Meanwhile, the HVAC system powered by solar energy is known as solarHVAC (SHVAC), where it is installed by PV panels to capture the sunlight and convert it into electricity. John Hollick is one of SHVAC innovators, and he patented the method and apparatus for cooling ventilation air for a building^{41}. The solar PV panel is connected to the HVAC to convert the solar energy into electricity to power all the parts responsible for the heating or cooling mode in the HVAC. The benefits of the SHVAC system, instead of traditional HVAC, are lower utility bills, preserve the environment, and ease of installation. HVAC systems have moving parts such as fans and vibrating coils that often break, whereas SHVAC have fewer moving parts and these systems have fewer breakage risks.
Among the several renewable resources that may be put practically anywhere in the globe, solar power promises to be the major technology for the transition to a decarbonized energy supply. The efficacy of a photovoltaic (PV) system is directly proportional to the amount of solar energy available. Many governments see renewables and energy conservation measures as a viable method to reduce coal consumption. The primary solar devices that can convert sunlight into electricity are PV system and concentrated solar power (CSP). CSP concentrates sun radiation to increase the temperature of a working fluid, and this fluid drives a heat engine and electric generator. CSP generates alternating current (AC), which has a high distribution rate on the power network. Besides, PV collects sunlight through the photoelectric effect to generate electricity in the form of a direct electric current (DC). The DC generated by the PV system is then transformed to AC through the inverters to ensure that the electricity is distributed on the power network. CSP stores energy by using Thermal Energy Storage technologies (TES), and it is not subjected to weather restrictions: This means that CSP can be used at all times (cloudy day, overnight, low sunlight, etc.) to generate electricity. On the other hand, PV system only stores low thermal energy compared to CSP, since it only uses a battery instead of the storage technology like TES. Therefore, CSP has more qualities over PV by performing more noteworthy efficiencies, lower speculation costs, gives warm capacity limit, and a superior mixture activity ability with different energizes to satisfy baseload need around evening time^{42}.
Parabolic trough solar collector (PTSC) is one type of CSP system that has been used proficiently in water heating^{43,44}, airconditioning^{45,46}, and solaraircraft^{47,48,49,50,51}. PTSC consists of a reflector with a reflecting surface (parabolicshaped mirror) and a receiver. The reflector collects the incident solar radiation and reflects it onto a receiver located in the focal line of the parabola. The working fluid inside the receiver absorbs the heat from the solar radiation, causing the fluid temperature to increase. Finally, highpressure superheated steam is generated from this working fluid in a conventional reheat steam turbinegenerator to produce electricity. The running fluid in PTSC should have those features: (a) excessive thermal potential and thermal conductivity, (b) low thermal growth and occasional viscosity, (c) strong charge of thermal and chemical properties, (d) minimal charge of corrosive interest and (e) low toxicity^{52}. One of the simplest operating fluids in PTSC is innovated nanofluid referred to as hybrid nanofluid and is ready via way of means of submerging specific nanoparticles withinside the equal base fluid. Therefore, there are recent studies regarding the hybrid nanofluid as a working fluid in PTSC installed in solar aircraft^{47,48,49,50,51}, and when PTSC is equipped with turbulators^{53,54,55,56,57,58}. The following types of hybridizing nanofluid were implemented in the PTSC solar aircraft: Casson hybrid nanofluid^{47}, Reiner Philippoff hybrid nanofluid^{48,49}, and tangent hyperbolic hybrid nanofluid^{50,51}. Meanwhile, A turbulator is a tool that transforms a laminar boundary layer right into a turbulent boundary layer to optimize heat transfer. Hence, various patterns of turbulators inserted in PTSC were reported, such as single twisted turbulator^{53}, obstacles act as turbulator^{54}, finned rod turbulator^{55}, two twisted tape acts as turbulator^{56}, inner helical axial fins as turbulator^{57}, and conical turbulator^{58}.
When it comes to thermodynamic rules, the second law of thermodynamics is far more dependable than the first law due to its limits of efficiency in heat transmission in industrial applications. This second law is applied to reduce the irreversibility of thermal constructions. Irreversibility is observed in a variety of thermofluidic apparatuses, including thermal solar, air separators, and reactors, and that competence loss is entirely interrelated with it. This generated irreversibility is determined by the rate of entropy production. The extinction of functional energy is measured by entropy generating. Any system's generated irreversibility creates continuous entropy, which eviscerates the functional energy required to execute the job. Such energy loss might be produced by heat transport by convective, conductive, and radiative fluxing. Furthermore, magnetic fields, buoyancy, and fluid friction all contribute to the generation of entropy. As a result, entropy generation minimization is required for diverse thermal equipment to acquire an optimal quantity of energy. The degree of entropy generating in crossbreed nanofluid is impacted by the expansion of twofold nanomaterials into the base liquid. The nonNewtonian cross breed nanofluid heavily influenced by entropy age have been examined, where this type of nanofluid contains the following double nanomaterials and basefluid: CuAl_{2}O_{3}/H_{2}O^{59,60,61,62,63,64,65}, CuAl_{2}O_{3}/EG^{66}, CuAg/EG^{67,68}, CuTiO_{2}/H_{2}O^{69,70}, CuAg/H_{2}O^{71}, CuGo/H_{2}O^{72}, CuTi/H_{2}O, CuOTiO_{2}/H_{2}O and C71500Ti_{6}Al_{4}V/H_{2}O^{73}, CuFe_{3}O_{4}/EG^{74}, CuCuO/blood^{75}, AgMgO/H_{2}O^{76}, AgGr/H_{2}O^{77}, CuOTiO_{2}/EG^{78}, Fe_{3}O_{4}–Co/kerosene^{79}, MWCNTFe_{3}O_{4}/H_{2}O^{80}, and MWCNTMgO/H_{2}O^{81}. The thermal properties of hybrid nanofluid over an elastic curved surface^{59}, stretching sheet^{61,63,70,74,78}, disk^{64}, stretching disk^{62}, and wedge^{79} were reported. In addition, the flow of a hybrid nanofluid in a cavity was investigated under the following conditions: square cavity^{68}, porous open cavity^{69}, and vented complex shape cavity^{81}. The investigation of a hybrid nanofluid flow through a channel^{66} and microchannel^{73,77} have been performed, where these channels are rotating^{66}, placed vertically^{73}, and recharging^{77}. The flow of a hybrid nanofluid in an enclosure was studied by Alsabery et al.^{60}, Ghalambaz et al.^{65}, and AbuLibdeh et al.^{76}. Alsabery et al.^{60} implemented the wavy enclosure containing the inner solid blocks, whereas Ghalambaz et al.^{65} considered an enclosed cavity with vertical and horizontal parts in their fluid model. On the other hand, AbuLibdeh et al.^{76} selected a porous enclosure with a trapezoid geometry where this type of geometry is used for cooling purposes on the hybrid nanofluid. Meanwhile, Xia et al.^{67} and Khan et al.^{72} developed the fluid flow model bounded by two rotating parallel frames. The heat analysis of the peristaltic flow of hybrid nanofluid internal a duct become studied through McCash et al.^{71}. The electroosmotic pump is involved in the hybrid nanofluid flow studied by Munawar and Saleem^{75}, with ohmic heating. Shah et al.^{80} chose a porous annulus to study the characteristics of a hybrid nanofluid model.
NonNewtonian fluid models are much more different than those of Newtonianism fluids. The stress values for nonNewtonian fluid are nonlinear functions against strain, yield stress, or timedependent viscosity. Examples of this type of fluid are Casson fluid^{82,83,84,85,86}, Maxwell fluid^{87,88,89,90,91}, nanofluid (also including hybrid case)^{47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81}, etc. Sutterby fluid model is one type of nonNewtonianism fluid^{92}, and it describes the viscosity of dilute polymer solutions^{93}. Polymer solutions have been applied in related industrial phenomena or products, such as turbulent pipe flows^{94,95}, stability of polymer jets^{96,97}, and oil recovery enhancement^{98,99}. The heat and mass transfer withinside the flow of magnetohydrodynamics (MHD) Sutterby nanofluid over a stretching cylinder, with the impact of temperaturestructured thermal conductivity have been explored by Sohail et al.^{100} and Raza et al.^{101}. The bioconvection of Sutterby fluid flow was reported when this fluid flows across the wedge^{102} and between two rotating disks^{103}. Gowda et al.^{104}, Yahya et al.^{105}, and Khan et al.^{106} incorporated the CattaneoChristov heat flux model in their mathematical Sutterby fluid model to archive effective thermal properties. The CattaneoChristov heat flux model was developed when the fluid was bounded by a rotating disk^{104}, flat surface^{105}, and wedge^{106}. The effect of entropy generation and activation energy were considered by Hayat et al.^{107}. In contrast, ElDabe et al.^{108} incorporated the boundaries of the attractive field, compound response, permeable media, heat radiation, gooey dissemination, and couple pressure. Parveen et al.^{109}, Arif et al.^{110}, Jayadevamurthy et al.^{111}, Nawaz^{112}, and Waqas et al.^{113} investigated the thermal performance of the Sutterby fluid model with the presence of various hybrid nanoparticles. The base fluid that has become selected was blood^{109,110}, water^{111}, and ethylene glycol^{112,113}. These researchers^{109,110,111,112,113} implemented the dual nanoparticles in their Sutterby hybrid nanofluid, namely as: (i) Au and Al_{2}O_{3}^{109}, (ii) CuO and Al_{2}O_{3}^{110}, (iii) Cu and SiO_{2}^{111}, (iv) MoS_{2} and SiO_{2}^{112}, and (v) first fluid contained SiO_{2} and SWCNT, and second fluid used MoS_{2} and MWCNT^{113}.
Motivation
The goal of this study is to look at a Sutterby hybrid fluid traveling along a stretchy surface with copper and graphene oxide nanoparticles. The following are the main points of the current study:

The effect of ultrafine strong nanoparticles (copper and graphene oxide) at the Sutterby hybrid fluid has yet to be contemplated.

In the extant literature, no 3D kind of Sutterby nanofluid has been built and explored.

The results of Maxwell speed slippery and Smoluchowski heat slippery bounder situations on hybrid nanofluid impacting on an extensible floor are but to be investigated.
The paper's structure
The following is a summary of the paper's structure.

The governing model was created on the premise of a boundary layer.

The controlling PDEs are converted into ODEs using appropriate similarity transformation.

The ODEs are adapted to 1stordered and resolved a usage of the Keller container numerical method included in MATLAB.

Physical portions along with the pores and drag force factor and Nusselt number are mathematically decided and demonstrated in tables.

Mathematical model's velocity, temperature, and awareness elements are numerically calculated and represented withinside the shape of figures.
Proposed mathematical model
The graphical model is presented in Fig. 1, and the characteristics of the proposed mathematical model are as below:

3D model (as in Fig. 2), where \(x\) and \(y\) axes contain planes, where \(z\)axis fluid flow region is at the third axis \(z\ge 0\).

The fluid rotates along \(z\)axis, showing that this axis acts as the axis of rotation for the rotating fluid. This fluid has an angular velocity \(\Omega\).

The involved fluid in this model is incompressible Sutterby fluid, flowing on an extendable surface. This surface is located at \(xy\)plane.

The Maxwell velocity slip^{114} effect is investigated, by adding the component of stretching \({u}_{w}=dx\), together with the slip length \(\frac{2{\sigma }_{v}}{{\sigma }_{v}}{\lambda }_{0}{U}_{z}\).

The Smoluchowski temperature slip^{115} is added, by implementing the term \(\frac{2{\sigma }_{T}}{{\sigma }_{T}}\left(\frac{2r}{r+1}\right)\frac{{\lambda }_{0}}{{P}_{r}}{T}_{z}\).

Surface temperature and concentration are denoted by \({T}_{w}\) and \({C}_{w}\), respectively. Meanwhile, \({T}_{\infty }\) and \({C}_{\infty }\) represent the ambient temperature as well as concentration.
The physical properties of Sutterby hybrid nanofluid are presented in Eq. (1). The dynamics viscosity, density, precise heat and thermal conductivity of hybrid nanofluid are indicated by \({\mu }_{hnf}\) \({\rho }_{hnf}\), \({\alpha }_{hnf}\), \((\rho {C}_{p}{)}_{hnf}\) and \({k}_{hnf}\), respectively.
A Cauchy tensor of tension for Sutterby liquid is presented as^{116}
in which \(p\), \(I\) and S constitute pressure, identification tensor, and further strain tensor, respectively. Subsequently, S in Eq. (2) is given as
where in \({\mu }_{0}\) is 0 shear fee viscosity, and \(E\) is a material time constant. In Eq. (3), the second one invariant stress tensor \(\dot{\gamma }\) and primary order RivilianErikson tensor \({A}_{1}\) were interpreted in Eqs. (4) and (5), respectively.
The \(m\) values determine the fluid categories, where Newtonian fluid when \(m=0\), pseudoplastic (shear thinning) when \(m>0\), and dilatant (shear thickening) when \(m<0\). In addition, the velocity field of the fluid is taken as \(V=[u(x,y,z),v(x,y,z),w(x,y,z)]\).
Under the restriction as stated above, the modeled equations are premeditated by^{117}:
Equations (6)–(10) are controlled by the following boundary conditions:
In Eq. (9), Rosseland approximation^{118} is added:
where in \({\sigma }^{*}\) and \({\kappa }^{*}\) stand for StefanBoltzmann consistent and imply absorption coefficient, respectively.
The appropriate transformations^{119} have been selected, as shown in (13):
The transformations (13) are implemented to dimensionless the early mathematical model (6)–(10), together with (12). As a result, the following forms have occurred:
After implementing (13) in (11), the dimensionless BCs are:
The final dimensionless governing parameters in (14)–(17) have been derived as
where \({B}_{1}\), \({B}_{2}\), \({B}_{3}\) and \({B}_{4}\) are constants^{120} as below:
Thermophysical properties of copper and graphene oxide nanoparticles^{120,121} have been tabulated in Table 1.
The skin friction coefficients in horizontal \(x\) and vertical axes \(y\) are shown in Eq. (21). From Eq. (21) also, \({\tau }_{xz}\) and \({\tau }_{yz}\)^{122} are expressed in Eq. (22).
Finally, surface drag coefficients are derived as:
The dimensional heat transfer coefficient^{122} is expressed in Eq. (24), where the heat flux \({q}_{w}\) is shown in Eq. (25).
From Eqs. (24), (25), the dimensionless Nusselt number is obtained:
The Sherwood number and the mass flux are given in Eqs. (27) and (28), respectively.
After manipulation of Eq. (28) into Eq. (27), The dimensionless shape of the mass transfer coefficient is
Numerical scheme
Keller box method (KBM)^{123} is selected as the current numerical technique to perform the solutions for the ODEs (14)–(17), together with BCs (18). The coding of KBM is built in MATLAB software, wherein the flow chart of KBM technique is depicted in Fig. 2. The presentday numerical method applies a finite distinction scheme, which is a collocation technique of order 4 and it runs in the back of KBM MATLAB. The abovementioned nonlinear differential problem, i.e., Eqs. (14)–(17) followed by the end point condition supplied by Eq. (18) is solved using the Keller box approach.
 Step 1:

Conversion of ODEs
The aforementioned equations are fairly turned into a new sophisticated first order coupled system:
$$\left.\begin{array}{ll} {y}_{1}^{{\prime}}={y}_{2}, & \quad {y}_{1}(0)=0,\\ {y}_{2}^{{\prime}}={y}_{3}, & \quad {y}_{2}(0)=1+{\Gamma }_{1}s,\\ {y}_{3}^{{\prime}}=\frac{2{B}_{1}{B}_{2}[{y}_{2}2\lambda {y}_{4}{y}_{1}{y}_{3}]}{1\frac{N}{2}{R}_{\eta }{D}_{\eta }{y}_{3}^{2}}, & \quad { y}_{3}(0)=s,\\ {y}_{4}^{{\prime}}={y}_{5}, & \quad {y}_{4}(0)=0,\\ {y}_{5}^{{\prime}}=\frac{2{B}_{1}{B}_{2}[2\lambda {y}_{2}{y}_{1}{y}_{5}+{y}_{4}{y}_{2}]}{1\frac{N}{2}{R}_{\eta }{D}_{\eta }{y}_{5}^{2}}, & \quad {y}_{5}(0)=t,\\ {y}_{6}^{{\prime}}={y}_{7}, & \quad {y}_{6}(0)=1+{\Gamma }_{2}u,\\ {y}_{7}^{{\prime}}=\frac{{P}_{r}f {b}_{4} {y}_{1} {y}_{7}}{\left({B}_{3}+\frac{4}{3}{R}_{\delta }\right)}, & \quad {y}_{7}(0)=u,\\ {y}_{8}^{{\prime}}={y}_{9}, & \quad {y}_{8}(0)=1,\\ {y}_{9}^{{\prime}}=\left(\sigma {S}_{\delta } (1+\Gamma {y}_{6}{)}^{n}exp\left(\frac{E}{1+\Gamma {y}_{6}}\right)\right){y}_{9}{S}_{\delta } {y}_{1} {y}_{9}, & \quad {y}_{9}(0)=v.\end{array}\right\}$$(30)  Step 2:

Domain discretization & difference equations
Likewise, domain discretization in \(x\beta\) plane is signified. In view of this web, net points are \({\beta }_{0}=0,{\beta }_{j}={\beta }_{j1}+{h}_{j}, j=\mathrm{0,1},\mathrm{2,3}...,J,{\beta }_{J}=1\) where, \({h}_{j}\) is the stepsize. Relating central difference formulation at midpoint \({\beta }_{j1/2}\)
$$\left(\frac{({y}_{1}{)}_{j}({y}_{1}{)}_{j1}}{{h}_{j}}\right)=\left(\frac{({y}_{2}{)}_{j}+({y}_{2}{)}_{j1}}{2}\right),$$(31)$$\left(\frac{({y}_{2}{)}_{j}({y}_{2}{)}_{j1}}{{h}_{j}}\right)=\left(\frac{({y}_{3}{)}_{j}+({y}_{3}{)}_{j1}}{2}\right),$$(32)$$\left(\frac{({y}_{4}{)}_{j}({y}_{4}{)}_{j1}}{{h}_{j}}\right)=\left(\frac{({y}_{5}{)}_{j}+({y}_{5}{)}_{j1}}{2}\right),$$(33)$$\left(\frac{({y}_{6}{)}_{j}({y}_{6}{)}_{j1}}{{h}_{j}}\right)=\left(\frac{({y}_{7}{)}_{j}+({y}_{7}{)}_{j1}}{2}\right),$$(34)$$\left(\frac{({y}_{8}{)}_{j}({y}_{8}{)}_{j1}}{{h}_{j}}\right)=\left(\frac{({y}_{9}{)}_{j}+({y}_{9}{)}_{j1}}{2}\right),$$(35)$$\left.\begin{array}{c}\left(1\frac{N}{2}{R}_{\eta }{D}_{\eta }{\left(\frac{({y}_{3}{)}_{j}+({y}_{3}{)}_{j1}}{2}\right)}^{2}\right)\left(\frac{({y}_{3}{)}_{j}({y}_{3}{)}_{j1}}{{h}_{j}}\right)2{B}_{1}{B}_{2}\left(\frac{({y}_{2}{)}_{j}+({y}_{2}{)}_{j1}}{2}\right)\\ 2{B}_{1}{B}_{2}\left[2\lambda \left(\frac{({y}_{4}{)}_{j}+({y}_{4}{)}_{j1}}{2}\right)\left(\frac{({y}_{1}{)}_{j}+({y}_{1}{)}_{j1}}{2}\right)\left(\frac{({y}_{3}{)}_{j}+({y}_{3}{)}_{j1}}{2}\right)\right]=0\end{array}\right\}$$(36)$$\left.\begin{array}{c}\left(1\frac{N}{2}{R}_{\eta }{D}_{\eta }{\left(\frac{({y}_{5}{)}_{j}+({y}_{5}{)}_{j1}}{2}\right)}^{2}\right)\left(\frac{({y}_{5}{)}_{j}({y}_{5}{)}_{j1}}{{h}_{j}}\right)4{B}_{1}{B}_{2}\lambda \left(\frac{({y}_{2}{)}_{j}+({y}_{2}{)}_{j1}}{2}\right)\\ 2{B}_{1}{B}_{2}\left[\left(\frac{({y}_{1}{)}_{j}+({y}_{1}{)}_{j1}}{2}\right)\left(\frac{({y}_{5}{)}_{j}+({y}_{5}{)}_{j1}}{2}\right)+\left(\frac{({y}_{4}{)}_{j}+({y}_{4}{)}_{j1}}{2}\right)\left(\frac{({y}_{2}{)}_{j}+({y}_{2}{)}_{j1}}{2}\right)\right]=0\end{array}\right\}$$(37)$$\left.\left({B}_{3}+\frac{4}{3}{R}_{\delta }\right)\left(\frac{({y}_{7}{)}_{j}({y}_{7}{)}_{j1}}{{h}_{j}}\right){P}_{r}f {b}_{4}\left(\frac{({y}_{1}{)}_{j}+({y}_{1}{)}_{j1}}{2}\right)\left(\frac{({y}_{7}{)}_{j}+({y}_{7}{)}_{j1}}{2}\right)=0\right\}$$(38)$$\left.\begin{array}{c}\left(\frac{({y}_{9}{)}_{j}({y}_{9}{)}_{j1}}{{h}_{j}}\right)+{S}_{\delta }\left(\frac{({y}_{1}{)}_{j}+({y}_{1}{)}_{j1}}{2}\right)\left(\frac{({y}_{9}{)}_{j}+({y}_{9}{)}_{j1}}{2}\right)\\ \sigma {S}_{\delta }{\left(1+\Gamma \left(\frac{({y}_{6}{)}_{j}+({y}_{6}{)}_{j1}}{2}\right)\right)}^{n} \left(1E\left(1\Gamma \left(\frac{(6{)}_{j}+({y}_{6}{)}_{j1}}{2}\right)\right)\right)\left(\frac{({y}_{9}{)}_{j}+({y}_{9}{)}_{j1}}{2}\right)=0\end{array}\right\}.$$(39)  Step 3:

Newton method
Equations (29) through (37) are linearized using Newton's linearization technique
$$\left.\begin{array}{c}\begin{array}{ll}& ({y}_{1}{)}_{j}^{n+1}=({y}_{1}{)}_{j}^{n}+(\delta {y}_{1}{)}_{j}^{n},({y}_{2}{)}_{j}^{n+1}=({y}_{2}{)}_{j}^{n}+(\delta {y}_{2}{)}_{j}^{n},\\ & ({y}_{3}{)}_{j}^{n+1}=({y}_{3}{)}_{j}^{n}+(\delta {y}_{3}{)}_{j}^{n},({y}_{4}{)}_{j}^{n+1}=({y}_{4}{)}_{j}^{n}+(\delta {y}_{4}{)}_{j}^{n},\\ & ({y}_{5}{)}_{j}^{n+1}=({y}_{5}{)}_{j}^{n}+(\delta {y}_{5}{)}_{j}^{n},({y}_{6}{)}_{j}^{n+1}=({y}_{6}{)}_{j}^{n}+(\delta {y}_{6}{)}_{j}^{n},\\ & ({y}_{7}{)}_{j}^{n+1}=({y}_{7}{)}_{j}^{n}+(\delta {y}_{7}{)}_{j}^{n},({y}_{8}{)}_{j}^{n+1}=({y}_{8}{)}_{j}^{n}+(\delta {y}_{8}{)}_{j}^{n},\end{array}\\ ({y}_{9}{)}_{j}^{n+1}=({y}_{9}{)}_{j}^{n}+(\delta {y}_{9}{)}_{j}^{n}.\end{array}\right\}$$(40)  Step 4:

Block tridiagonal structure
The linear mathematical model now has the block tridiagonal shape, written
$$A\Delta =S,$$(41)where
$$A = \left[ {\begin{array}{*{20}l} {[L_{1} ]} & {[N_{1} ]} & {} & {} & {} & {} & {} \\ {} & {[L_{2} ]} & {[N_{2} ]} & {} & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} & {} \\ {} & {} & {} & \ddots & {} & {} & {} \\ {} & {} & {} & {} & {[M_{{J  1}} ]} & {[L_{{J  1}} ]} & {[N_{{J  1}} ]} \\ {} & {} & {} & {} & {} & {[M_{J} ]} & {[L_{J} ]} \\ \end{array} } \right],\quad \Delta = \left[ {\begin{array}{*{20}l} {[\Delta _{1} ]} \\ {} \\ \vdots \\ \vdots \\ \vdots \\ {[\Delta _{{J  1}} ]} \\ {[\Delta _{J} ]} \\ \end{array} } \right]\;{\text{and}}\quad S = \left[ {\begin{array}{*{20}l} {[S_{1} ]} \\ {} \\ \vdots \\ \vdots \\ \vdots \\ {[S_{{J  1}} ]} \\ {[S_{J} ]} \\ \end{array} } \right].$$(42)where the overall size of the blocktriangle matrix A is J × J and the supervector's block size is 9 × 9. LU decomposition method implementation for solving Δ. A mesh size of h j = 0.01 is regarded adequate for mathematical assessment, and the difference between the current and previous iterations for the needed precision has been set at \(1{0}^{6}\).
Result’s verification
The comparative analysis of the numerical values skin friction coefficient values \({f}^{{{\prime}}{{\prime}}}(0)\), are tabulated in Table 2. The comparison is made with the previous researchers^{117,124}, with the various values of rotating parameter \(\lambda\). However, other parameters have remained zero such as consistency parameter, Reynolds, Deborah numbers, and speed slippery (\(N={R}_{\eta }={D}_{\eta }={\Gamma }_{1}=\) 0). Besides, \({B}_{1}={B}_{2}\) is fixed to obtain this comparative analysis. From Table 2, it is clear that the accuracy of the current results is quite high. Therefore, the current numerical scheme KBS is quite reliable, authentic, and acceptable for subsequent calculations.
Result and discussion
This segment shows and discusses the impact of diverse parameters at the floor frictional factor, Nusselt value, speed, energy, and concentricity outlines with the use of tables and figures. In the case of separated boundaries, Table 3 is intended to mirror the effect of wall frictional factors \(C{f}_{x}\) and \(C{f}_{y}\) consistent with the table, changes inside the powerregulation conduct list \(N\), Reynolds number \({R}_{\eta }\), Deborah \({D}_{\eta }\), pivot boundary and speed slippage cause a decline inside the surface coefficient of drag along the \(x\) orientation, however an expansion when the speed slippage boundary \({\delta }_{1}\) is gotten to the next level. This is physically since both the Reynolds number \({R}_{\eta }=\frac{d{x}^{2}}{\nu }\) and the Deborah number \({D}_{\eta }=\frac{{a}^{2}{d}^{2}}{\nu }\) depend on the viscosity of the nanofluid and follows the frictional force is diminished. \(C{f}_{y}\) ascends because of expansions in \(N\), and \({\Gamma }_{1}\) but falls in light of an increment in its values. This is because increasing the rapidity slippage \({\Gamma }_{1}=\frac{2{\sigma }_{v}}{{\sigma }_{v}}{\lambda }_{0}\sqrt{\frac{d}{v}}\) increases the reaction rate, and this effect occurs. Table 4 is expected to examine hotness and mass exchange rates for dimensionless various variables. It is found that when the radiation boundary \({R}_{\delta }\) and Prandtl number \({P}_{r}\) are changed, Nusselt number improves, however, devalues as the temperature slips \({\Gamma }_{2}\). This is because the presence of heat radiation boosts the stored thermal energy and then begins to release it through the nanofluid molecules, which improves the rate of mutual rate of heat transfer, which in turn grows the number of Nusselt. The mass exchange rate increments when \({R}_{\delta }\), substance response rate, Schmidt number \(Sc\), temperature contrast boundary, and fixed value steady \(n\) increment, yet reduces as \({P}_{r}\), heat slippery \({\Gamma }_{2}\), and enactment energy \(E\) decline.
The impact of \({R}_{\eta }\) on \({f}^{{\prime}}(\eta )\) is portrayed in Fig. 3. \({R}_{\eta }\) decides if the conduct is laminar or tempestuous at the actual level. The Reynolds number is the ratio of inertial power to gooey power. It is worth noting that the higher the Reynolds number, the greater the inertial power over the gooey power, the thicker the consistency, and the smaller the motion field. Indeed, increasing the volume fraction of nanoparticles reduces liquid fixation, diminishes liquid thickness, and boosts idleness. Finally, a significant component in the lowering of the rapidity field. Figure 4 shows the impact of \({D}_{\eta }\) on \({f}^{{\prime}}(\beta )\). Physically, smaller Deborah values make the material to operate more freely, resulting in a flow of Newtonian viscidity. With increasing Deborah quantities, the effectual enters the nonNewtonianism zone, with increased elasticity ratings and solidlike behavior. The bigger the Deborah quantity, the stronger the viscidness effect. Deborah values distinguish amongst liquid solids and fluid properties on a physical level. As \({D}_{\eta }\) increases, the fluid changes from a fluid to a solid. The substance behaves like a liquid for lesser \({D}_{\eta }\) and such as a solid for greater \({D}_{\eta }\). As \({D}_{\eta }\) increases, fluid behaviour such as shear thickening becomes more difficult to flow through the surface, lowering \({f}^{{\prime}}\left(\beta \right)\). The behavior of the power law exponent \(M\) at \({f}^{{\prime}}\left(\beta \right)\) (Fig. 5). When shear force is applied, \(N\) affects the viscidity of the nanofluid. The letters \(N\) stand for fluid shear thinning and Newtonianism. Positive variations in \(N\) boost viscidness (shearing thicker) and decrease the velocity of fluid flowing through a ductile surface, thus use caution. Physically, shearing thicker occurs as a result of a larger volume fraction of nanomolecules, a rise in fluid viscosity, and a reduction in fluid rapidity \({f}^{{\prime}}\left(\beta \right)\). The relationship between rotational parameter and \({f}^{{\prime}}\left(\beta \right)\) is shown in Fig. 6. The fractional size of gold nanomolecules is magnified, which reduces \({f}^{{\prime}}\left(\beta \right)\) and the thickness of the momentum boundary layer. An alteration in \({f}^{{\prime}}\left(\beta \right)\), acts like shear thickening. When the torque increases, this cause to incremental changes in the viscosity of the fluid to develop, the nanofluid rapidity decreases. The effect of \({R}_{\eta }\) on \(g\left(\beta \right)\) is depicted in Fig. 7. In opposite of the viscidness influence, \({R}_{\eta }\) emphasizes the relevance of the inertia effect. The consistency of the liquid is decreased, and the liquid speed \(g\left(\beta \right)\) is diminished when \({R}_{\eta }\) is expanded. The motivation behind Fig. 8 is to stress the feature of \({D}_{\eta }\) on \(g(\beta )\). Higher thick powers that lull the liquid speed led to an expansion in \({D}_{\eta }\). The liquid behaves exactly like shearing dilatation due to a consistent change in \({D}_{\eta }\). It's intriguing to see how increasing the quantity of nanomolecules influences liquid thickness while lowering it. Physically, boosting the amount of nanostructure particles enhances liquid consistency, lowering liquid speed and \(g(\beta )\). Figure 9 shows the impact of \({\Gamma }_{1}\) on \({f}^{{\prime}}\left(\beta \right)\). An amplification of \({\Gamma }_{1}\) lessens the worth of \({f}^{{\prime}}\left(\beta \right)\). In the status of slippery limit restrictions, the speed of the plate and the liquid are not equivalent at the plate, bringing about a decrease in liquid speed and a diminishing speed. Figure 10 shows a portrayal of \(g(\beta )\). This is physically because the liquid near the boundary layer is more viscous due to the accumulation of particles close to the surface, which reduces the velocity and increases the further away from the boundary layer. Another important concept is that as the percentage of nanoparticles in the base liquid grows, the thickness of the liquid reduces, making it simpler to travel across an extensible plate. Magnification in the volume part of nanomolecules builds a liquid and diminishes the liquid speed and \(g(\beta )\).
Figure 11 is intended to depict \({R}_{\delta }\) performing on \(\theta (\beta )\). \({R}_{\delta }\) is the most thing of a heat transfer rules in terms of physics. It is commonly known that amplification in \({R}_{\delta }\) causes the heat transfer rate to increase. It is because of an improvement in \({R}_{\delta }\) lowers the average absorbing factor, resulting in amplification in \(\theta (\beta )\). Practically, an increase in the size of the nanomolecules paired with \({R}_{\delta }\) enhances the thermal conducting of the fluid, boosting \(\theta (\beta )\). The effect of \({P}_{r}\) on \(\theta (\beta )\) is depicted in Fig. 12. When \({P}_{r}\) is small, heat diffuses quickly in comparison to velocity (momentum), and vice versa when \({P}_{r}\) is large. Furthermore, because of amplification in \({P}_{r}\), the thickness of the thermal boundary layer declines \(\theta (\beta )\). This is physically due to the inverse relationship between the Prandtl number and the thermal diffusivity, as the lack of thermal diffusivity occurs as a result of the low thermal conducting and thus enhances the Prandtl number, which works to increase the temperature inside the nanoliquid. The link between \({\Gamma }_{1}\) and temperature is seen in Fig. 13. A magnification of \({\Gamma }_{1}\) reduces the space among the surface and surrounding heat, transporting less temperature from a plate to a liquid and, due to the lowering a fluid heat.
Figure 14 emphasizes the effect of chemically response charge \(\sigma\) at the awareness area \(\phi (\beta )\). The physical interpretation refers to the amount \(\sigma (1+\delta \theta {)}^{n}exp\left(\frac{E}{1+\delta \theta }\right)\) magnifies at the likewise of improvement in \(\sigma\) or \(n\) which inspires the destructive chemically reactive action which diminishes the mass size range. The exponential part in the formula means that when the active energy diminishes, the rate constant of a reaction grows exponentially. Because the rate of a reaction is directly proportionate to its rate constant, the rate also grows exponentially^{125}. The impact of \({S}_{\delta }\) at the mass area \(\phi (\beta )\) is defined in Fig. 15. The Schmidt quantity is the ratio of momentum to mass diffusivity. It's well worth noting that a highquality alternate in \({S}_{\delta }\) reduces mass diffusivity. Physically, the fluid viscidness falls because of a growth in \({S}_{\delta }=\frac{\nu }{D}\), which reduces mass diffusion and will increase momentum diffusivity. The presence of \({S}_{\delta }\) maximum possibly reduces the fluid viscidness and \(\phi (\beta )\).
Conclusions
3D rotating Sutterby hybrid fluid with copperGraphene oxide nanomolecules, active energy, impetus, heat slippery bounder constraints, and radiative heat flow is defined in this paper. The numerical solution to the simulated problem was achieved using the MATLAB KBM builtin technique. The following are some of the most important aspects of the results:

The profile \({f}^{{\prime}}(\eta )\) denigrates at the behalf of extension in \({R}_{\eta }\), \({D}_{\eta }\), and \(N\).

Magnification withinside the factors \(\lambda\) and \(N\) monitors to an extension in \(g(\beta )\).

Intensification in \({\theta }_{w}\) boosts \(\theta \left(\beta \right)\) however a decline in \(\theta \left(\beta \right)\) occurs due to an enhancement in \({R}_{\delta }\).

The value of the Nusselt wide variety decreases below amplification in \({\Gamma }_{1}.\)

It is essential that \(\phi \left(\beta \right)\) will increase withinside the case of extension in \(\xi .\)

A positive variant in \({\Gamma }_{2}\) will increase \(\phi \left(\beta \right).\)

The mass fractional size discipline outline reduces for the chemically response factor \(\Gamma .\)
The Kellerbox method could be applied to a variety of physical and technical challenges in the future^{126,127,128,129,130,131,132,133,134,135,136,137,138,139}.
Data availability
The results of this study are available only within the paper to support the data.
Change history
09 May 2023
This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1038/s41598023346284
Abbreviations
 \({T}_{\infty }\) :

Ambient temperature (K)
 \(Re\) :

Reynold’s number
 \(\Omega\) :

Angular velocity
 \({C}_{\infty }\) :

Ambient concentration (\(\frac{\text{mol}}{{\text{m}}^{3}}\))
 \(\dot{\gamma }\) :

Second invariant strain tensor
 e :

Consistency index
 \({R}_{\delta }\) :

Thermal radiation
 \({U}_{w}\) :

Stretching velocity along \(x\)axis (\(\frac{{\text{m}}}{{\text{s}}}\))
 \(Pr\) :

Prandtl number
 \(\sigma\) :

Reaction rate constant (\(\frac{\text{mol}}{\text{lits}}\))
 \({\sigma }_{v}\) :

Velocity accommodation coefficient
 \({\Gamma }_{1}\) :

Temperature slip (K)
 \(n\) :

Fitted rate constant
 \({T}_{w}\) :

Temperature at the wall (K)
 \({\mu }_{0}\) :

Zero shear fee viscosity
 \({D}_{\eta }\) :

Deborah number
 \(\lambda\) :

Rotation parameter
 \(E\) :

Material time constant
 \(T\) :

Temperature
 \(S\) :

Extra stress tensor
 \(N\) :

Powerlaw behavior index
 \({\Gamma }_{1}\) :

Velocity slip (\(\frac{{\text{m}}}{{\text{s}}}\))
 \({q}_{r}\) :

Radiative heat flux ( \(\frac{\text{W}}{{\text{m}}^{2}}\))
 \(E\) :

Activation energy (\(\frac{{\text{J}}}{{{\text{mol}}}}\))
 \({\sigma }_{T}\) :

Temperature accommodation coefficient
 \(Sc\) :

Schmidt number
 \({A}_{1}\) :

RivilianErikson tensor
 \({C}_{w}\) :

Concentration at the wall (\(\frac{\text{mol}}{{\text{m}}^{3}}\))
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W.J. formulated the problem. W.J. and M.R.E. solved the problem. W.J., M.R.E., R.S., A.A.P., M.A., Z.R., S.S.P.M.I., and W.W. computed and scrutinized the results. All the authors equally contributed in writing and proof reading of the paper. All authors reviewed the manuscript.
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Jamshed, W., Eid, M.R., Safdar, R. et al. RETRACTED ARTICLE: Solar energy optimization in solarHVAC using Sutterby hybrid nanofluid with Smoluchowski temperature conditions: a solar thermal application. Sci Rep 12, 11484 (2022). https://doi.org/10.1038/s41598022156857
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DOI: https://doi.org/10.1038/s41598022156857
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