Computational analysis of radiative engine oil-based Prandtl–Eyring hybrid nanofluid flow with variable heat transfer using the Cattaneo–Christov heat flux model

In the present analysis, we study the energy transference through engine oil-based Prandtl–Eyring nanofluid flow through a heated stretching surface. The nanofluid is prepared by adding copper (Cu) and titanium dioxide (TiO2) nanoparticles (NPs) to the base fluid engine oil. The flow mechanism and thermal transmission are observed by exposing the nanofluid flow through the heated slippery surface. The influences of permeable surface, radiative flux and heat absorption/generation are also elaborated in this study. The flow of nanofluids has been designed using a PDEs system, which are then transformed into a set of ODEs via resemblance modification. The numerical technique “shooting method” is used to solve the acquired nonlinear set of non – dimensional ODEs. The results are physically exemplified through tables and plots. It has been detected that the accumulation of nanomaterials in the engine oil, reduces the skin friction while accelerating the energy transfer rate. The velocity field significantly decelerates with the encouragement of the porosity factor, and volume fraction of NPs. However, the temperature profile significantly escalates with the encouragement of the porosity factor, and volume fraction of NPs.


Introduction
Base uids (traditional liquids) play a critical role in transferring heat in industrial operations. In general, the heat transferability of these liquids is poor. To overcome this obstacle, nanoparticles (<100 nm) are inserted to enhance abilities of heat transport. Choi and Eastman 1 proposed this idea rst. Solid particles conduct thermal heat more efficiently than liquids, which is a widely accepted fact. Therefore, the addition of particles of nanoscale size to conventional uids signicantly enhanced their thermal conductivity. Nanoparticles is the name given to these solid particles. A nanouid is a uid that consists of both the nanoparticles and base uid. Eastman 2 claimed in an experimental study that a slight quantity of nanoscale solid substantial particles can enhance the thermal conductivity of normal liquids. This research found that adding Cu NPs or carbon nanotubes (CNTs) at 1% enhanced the thermal efficiency of ethylene glycol (the base uid) by 40-50% (volume fraction). This is due to nanouids playing an important part in electromechanical devices, advanced cooling systems, heat exchange, and so on. The stimulus of radiant energy on a hydromagnetic unsteady liquid ow through a leaky stretchable sheet with heat and mass conversion was described by Bilal et al. 3 The micro-rotation characteristic was discovered to be caused by the permeability factor. Ghasemi et al. 4 revealed the inuence of magnetic pitch on nanouid ow across a stretching sheet at the stagnation point. Furthermore, the outcomes revealed that as the Lewis number upsurges, so does the heat ux of the nanouid. The magnetohydrodynamic water-based NF ow containing motile microorganisms and nanotubes across a porous vertical oating substrate was investigated by Algehyne et al. 5 Increasing heat absorption and production rates were thought to increase the rate of energy transference. Investigators 6-11 looked into the thermal properties of nanouids by incorporating multiple types of nanoparticles into the base uid.
Recently, hybrid nanouids (HNF), a more advanced type of nanouid, have been introduced. The hybrid nanouids are made up of an ordinary liquid and two or more types of nanoparticles. In terms of heat transport, hybrid nanouids outperform conventional nanouids. Suresh et al. 12 exercised an experimental two-step method to create a hybrid nanouid composed of Al 2 O 3 -Cu/water. Madhesh et al. 13 conducted experiments on a copper-titania (Cu-TiO 2 ) hybrid nanocomposite and copper-titania (Cu-TiO 2 ) HNF ows with volume fraction ranging from 0.1% to 1.0%. The outcomes discovered that for a volume concentration of up to 1%, the heat ux rate is enhanced by 49%. Toghraie et al. 14 carried out a research on the mixture of a ZnO-TiO 2 /EG HNF to illustrate the inuences of nanoparticle temperature and concentration on the conduction of the HNF. The outcomes were intriguing, showing that at 50 degrees Celsius, heat radiation was 32% with a volume fraction of 3.5%. In addition to these experimental efforts, scientists have concentrated on theoretical research of hybrid nanoliquid ows. Gul et al. 15 compared Yamada-Ota and Hamilton-Crosser HNF models that contained silicon carbide (SiC) and titanium oxide TiO 2 nanoparticles (NPs) in diathermic oil. Aer being stimulated with a magnetic dipole, the ow of the hybrid nanoliquid was predicted to occur over a larger surface. The key ndings indicated that the Yamada-Ota model outperformed the Hamilton-Crosser hybrid nanoliquid ow model in terms of heat transfer efficiency. Arif et al. 16 studied theoretically ternary hybrid nanoliquid ow with base liquid water between two parallel sheets with various nanoparticle shapes such as cylinders, spheres, and platelets of carbon nanotubes, aluminium oxide, and graphene, respectively. The unsteady uid ow and energy transmission of a Cu-Al 2 O 3 / water-based HNF over an axially impermeable contracting and extending substrate were investigated by Khan et al. 17 A hybrid nanouid (Cu-Al 2 O 3 /water) was discovered to accelerate heat transmission when compared to a conventional uid. Elattar et al. 18 explored the ow of a steady electrically charged HNF through an opaque thin exible sheet using the computational technique PCM. The inuence of velocity index and Hall current raises the velocity contour, while changes in particle volume and sheet thickness lower it. Wang et al. 19 inspected the inuence of a biochemical reaction on a HNF unsteady ow along a texture that was expanding. The presence of the unsteadiness variable has been observed to regulate the transition from laminar to turbulent ow. Alharbi et al. 20 characterised the ow of an energy propagating high conductivity ternary HNF that included nanocrystals as well as an extended sheet. When ternary hybrid NPs are varied the base uid thermal conductivity is greatly improved. Ahmad et al. 21 looked into the heat transport properties of engine oil containing nanoparticles like Cu and TiO 2 . They found that the effectiveness of copper and titanium oxide in engine oil is overlooked.
Prandtl-Eyring nanouid is a mixed convection ow of nanoparticles with activation energy that is non-linear. As a result, numerous non-Newtonian uid models have been offered in the literature. The Prandtl-Eyring uid is one of them. Darji et al. 22 described visco-inelastic liquid ow boundary layer similarity solutions. Hayat et al. 23 inspected the consequence of magnetohydrodynamics on the ow of a peristaltic dissipative Prandtl-Eyring liquid. According to the results, Akbar 24 discovered the convective boundary constraints of Prandtl-Eyring liquid ow with peristatic properties. Khan et al. 25 conducted a mathematical computational evaluation of bio convection on PEF. The effects of thermo/phonetic force and Brownian motion on electrically conducting PEF generated by strained shallow were investigated by Abdelmalek et al. 26 Because of the numerous implementations in nanouid mechanics, investigators are developing a wave-based heat transfer methodology rather than a diffusion operation. [27][28][29] Heat transfer is a well-established phenomenon that happens as a consequence of temperature variations between two distinct objects or between components of an identical system.
For decades, Fourier's 30 fundamental law of heat conduction, has been used to measure heat transfer properties. Later, with concern, it was discovered that this model produces a parabolic energy equation with an initial disruption that lasts across the process. A "paradox of heat conduction" denotes this weakness in the Fourier model. Cattaneo 31 addresses this aw by incorporating a relaxation term into the Fourier approach. Aerward, Christov 32 created the Cattaneo-suggested relationship with the Oldroyd upper-convected variant via frame-indifferent alteration. This type of relationship is identied as the Cattaneo-Christov (CC) ux model. Kumar et al. 33 used a CC ux model to investigate the features of Dusty uid of dissolved HNF ows in two phases through an elongated cylinder. The shooting approach with RK-Fehlberg system, were used for numerical results. Ramzan et al. 34 used the CC model and MHD impact with heterogeneous reactions adjacent to a stagnation point to calculate the Williamson uid ow. It should be acknowledged that the uid parameter has a diametrically opposed inuence on velocity and temperature proles. Shah et al. 35 used the CC model to investigate heat transfer in a 2D (two dimensional) ow of Ree-Eyring nanoliquid through a stretching sheet.
The primary goal of this research is to determine how heat absorption and thermal radiation, as well as viscous dissipation and energy transportation, occur in engine oil-based NF ow over a heated elongating surface. Cu and TiO 2 nanoparticles are added to engine oil to create the nanouid. The nanouid ow was modelled as a system of PDEs, which are then transformed into a set of ODEs via resemblance modication. The numerical technique "shooting method" is used to solve the acquired nonlinear set of nondimensional ODEs. Local velocity gradient and Nusselt number statistics are estimated and analysed. Despite the fact that the phenomena described in this manuscript have never been attempted before.

Mathematical formulation
We have assumed the 2D viscous dissipative nanouid ow over an irregular moving horizontal porous plate. The plate is moving with velocity U w ðx; tÞ ¼ bx 1 À zt , where b is a stretching rate as shown in Fig. 1. The ow has been investigated in terms of thermal radiation and viscous dissipation. At b* and T w represents the energy variation, surrounding temperature and surface heat, respectively. The surface of the plate is supposed to be slippery.

Formal model
The uid ow model is displayed in Fig. 1 as:

Governing equations
The constitutive Prandtl-Eyring model [33][34][35] for the nanouid ow in a porous medium and heat equation with variable temperature, thermal radiation, Cattaneo-Christov heat ux model, and heat source/sink utilizing the approximate boundary-layer are: Boundary conditions 21 Here, v, and u are the velocity components, a 1 and C 1 are the uid parameters, m nf is the dynamic viscosity of nanouid, r nf is the density, k nf is the thermal conductivity, Q 0 is the heat source, h f is the heat transition constant and K is the surface permeability, U E is the heat ux for which model equations is The varying thermal conductivity is classied as: 2.3. Thermo-physical material properties of nanouid and base uid Different researchers presented mathematical models that explain the effective characteristics of heat transfer in the nanouids. These models present physical characteristic of the nanouid in term of relevant physical characteristics of the solid nanoparticles and base uid. The density of a ferrouid (r ff ) is related to the density of the uid (r f ) and that of the solid nanoparticle phase (r s ) as follows: 35 Here f is the volume fraction of NPs. Similarly the volume specic heats are correlated as: The dynamic viscosity of the uid and the nanouid are given by; To estimate the efficient thermal conductivity of the nano-uid, the Maxwell-Garnetts (MG) model can be utilized: In the present study heat transfer analysis through copper (Cu) and titanium dioxide (TiO 2 ) nanoparticles (NPs) to the base uid engine oil has been achieved into the account the characteristics using in Table 1.

Dimensionless formulations model
Using all assumptions and velocity led on eqn (1)-(4), eqn (1) hold identically, and the dimensionless process for eqn (2)-(4), the stream functions are expressed as: The similarity variables are: Fig. 1 The physical representation of flowing fluid across a stretching surface.
Using eqn (8) & (9) in eqn (1)-(3), we get the following dimensionless form of a system of ODEs: The transform boundary conditions are: where, Also

Numerical solution
The obtained system of ODEs (eqn (13)- (15)) is further reduced to the 1 st order differential equations through the following variables framework: The transform boundary conditions are:

Result and discussion
This section explained the physical procedure and trend that underpin each plot and table. The physical sketch of the ow problem was elaborated in Fig. 1. Fig. 2-13 show the behaviour of velocity and energy outlines in relation to various physical constraints. the other parameters are held constant. This graph shows that increasing the value of A causes an upsurge in the value of velocity. Because higher values of A tend to reduce viscosity, which overwhelms the resistance offered by the liquid. Fig. 3 depicts how the uid velocity gradient tends to decrease as uid parameter b increases. It is physically true because b varies inverse proportion with momentum diffusivity, resulting in a decrease in velocity gradient. Fig. 4 portrays the impacts of the porosity parameter b 0 on the velocity distribution. The velocity decreases as b 0 increases. Physically, the existence of a porous medium has increased the medium's opposition to uid ow. Fig. 5 shows how the volumetric concentration f affects the velocity prole f ′ (h). When the particle volume f fraction is risen, the velocity prole f ′ (h) reduces. As the volume fraction f of the nanoparticles grows, the uid thickens, and a conicting force develops, leading to deceleration. Fig. 6 depicts a graphical representation of the behaviour of velocity proles f ′ (h) as a function of the velocity slip parameter L. In general, L calculates the amount of slip at the cylinder's surface. Here, we examine how uid velocity decreases as L increases. It is because L primarily reduce speed of uid motion, conrming a reduction in net movement of uid molecules. Because there is less molecular progression, velocity elds decline. Fig. 7 indicates the behaviour of the velocity eld for various values of S. Suction is an efficient method for preventing boundary layer separation, as well as controlling velocity and heat energy. The amount of uid particles is close to the wall aer reaching the maximum value of the suction/blowing parameter. Accordingly, the outline of the associated boundary layer becomes thinner     over time, and the velocity proles decelerate as S strength increases. Fig. 8-13 reported the conduct of the energy prole q(h) versus the thermal relaxation parameter d e , volume fraction parameter f, Biot number B z , heat generation constraint Q, thermal radiation term Rd and Eckert number Ec. Fig. 8 depicts that the variation of Eckert number Ec on temperature prole. The stimulus of Eckert number Ec on nanouid temperature is evident because an increase in Eckert number accelerates advective transport (kinetic energy). As a result, uid particles interact together more frequently, and these collisions convert kinetic energy (KE) into thermal energy. Accordingly, the temperature prole upsurges. Fig. 9 displays the stimulus of the thermal relaxation parameter on temperature distribution. Temperature distribution decreases as the thermal relaxation parameter d e increases. It is also observed that the thickness of the thermal boundary layer diminutions. This is due to the fact that as d e intensies, the material particles necessitate more time to transferal heat to their neighbouring droplets. To put it another way, for advanced values of the d e parameter, the material reveals a non-conducting property, which contributes to a narrower temperature distribution. Fig. 10 exposed that the temperature upsurges with the upshot of the volume fraction parameter f. Because of the collision of tiny nanoparticles in the ow eld produces thermal energy, which advances the temperature of the uid. Therefore, the addition of Cu and titanium NPs enhances the energy propagation q(h). Fig. 11 display the impression of Biot number B z on the energy     prole. The Biot number is related to the surface's convective boundary constraints. As b z boosts, so does the temperature prole close to the surface, raising the temperature close to the surface and, consequently, the thickness of the thermal boundary layer, as realized in Fig. 11. Fig. 12 indicates that the variation of thermal radiation Rd on temperature prole. The temperature prole improves as thermal radiation (Rd) values boost. Larger Rd values have dominant impact on conduction. As a result of the radiation, a signicant quantity of heat is distributed into the system, raising the temperature. Fig. 13 depicts the encouragement of a heat generation factor Q on uid temperature. The temperature prole is seen to boost as Q goes up. Heat is generated in the ow regime as a result of positive modications in the heat generation parameter, causing a rise in uid temperature.  Table 2. It can be noticed from Table 3 that the rising quantity of thermal relaxation parameter, Eckert number and thermal radiation declined the energy transmission rate. In Table 4 the presents results are compared with previous published results, and excellent agreement is noticed in both results.

Conclusions
We analyzed the energy transference and entropy generation through nanouid ow through a heated extending surface. The nanouid is organized by the accumulation of Cu and TiO 2 NPs in the engine oil. The ow of nanouids has been designed using a PDEs system, which are then transformed into a set of ODEs via resemblance modication. The numerical technique "shooting method" is used to solve the acquired nonlinear set of nondimensional ODEs. The outcomes are physically exem-plied through gures and tables. The key ndings are: The velocity f ′ (h) signicantly improves with the inuence of parameters A, while declining with the effects of parameter b, suction parameter S, slip parameter L, porosity factor, and volume fraction of nanoparticles.
The temperature q(h) curve boosts with growing values of the thermal relaxation coefficient d e and volume fraction parameter f.
The temperature q(h) pointedly boosts with the stimulus of Q, Eckert number Ec, and thermal radiation parameter Rd.
The accumulation of nanomaterials in the engine oil, speeding up the skin friction, while reduces the energy transfer rate.
The uid parameter b boost the skin friction, while the variant in porosity parameter dimmish the skin friction.

Data availability
The data that support the ndings of this study are available from the corresponding author upon reasonable request.