Heat and Mass Transfer of the Darcy-Forchheimer Casson Hybrid Nanofluid Flow due to an Extending Curved Surface

Mathematics Department, University of Peshawar, Peshawar, Pakistan Laboratoire Lmdan, Departement De Mathematiques De Decision, Facultíe des Sciences Economiques et Gestion, Universite Cheikh Anta Diop De Dakar, BP 5683 Dakar Fann, Senegal Department of Industrial Engineering, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia Chemistry Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia


Introduction
The flow over an extending surface has received much importance due to its significant role in several sectors of industry and engineering, such as condensation process, spinning of fiber and continuous casting of fiber, plastic sheet extraction, paper production, and many others. Crane [1] was the first to study the flow over an expanding planar surface. Many researchers have since followed the concept of the crane [2][3][4][5], expanding the sheets to investigate various aspects of this form of flow. Sajid et al. [6] addressed boundary layer flow and micropolar fluid, concluding that the curvature effect leads to a reduction in boundary layer size. Gul et al. [7] have investigated the flow of the boundary layer on the stretching surface using the Fractional Order Derivatives Scheme. Imtiaz et al. [8] demonstrated the fluid flow under the upshots of the magnetic field over an extending curved surface. It has been noticed that with the action of curvature coefficient, the energy profile is enhanced. Rosca et al. [9] have analyzed the flow caused by contracting and expanding sheets. Saeed et al. [10] offered a complete investigation of the Darcy hybrid nanoliquid flow through a curved surface that is exponentially expanding. The outcomes signify using SWCNTs, MWCNTs, and Fe3O4 nanomaterials for the increase in the nusselt number. Ali et al. [11] analyzed the hydrological importance of wave propagation of hybrid nanofluid over a warmed extended curved surface with the impacts of a magnetic field using bvp4c. The suspension of carbon nanotubes in a magnetite nanoliquid promotes local surface drag but reduces local heat flow. Kumar et al. [12] have reported the radiation impact on the Casson fluid across the exponentially curved sheet. Hayat et al. [13] explored ferroliquid flow with the mass and heat transition across a curved stretching surface. Hussain et al. [14] reported the findings of their investigation on hybrid nanofluid flow across a curved sheet. The outcomes of the survey revealed that the energy transference efficiency in hybrid nanocrystals is higher than that in nanofluids for large frequencies of the curvature index. Qian et al. [15] and Khan evaluated that how heat transmission and radiation were affected by the conducting flow over a curved extending surface. Their study was found to be in good accord with a previously published finding.
The heat transmission in carbon nanofluids has gotten a lot of interest from researchers in a variety of fields in the last few years. CNTs are carbon nanotubes with a fundamental chemical structure and a carbon atom formation wrapped in a cylindrical shape. CNTs have superior chemical, thermophysical, and mechanical characteristics, making them ideal for usage as a particulate in a base fluid. They offer various advantages over other nanomaterials due to their tiny size, structure, configuration, dimension, and hardness. Haq et al. [16] evaluated the computational findings for conductive fluid using carbon nanomaterials along an extensive surface. Ahmadian et al. [17] addressed a 3D model of an unsustainable hybrid nanofluid flow with fluid and momentum transmission caused by surface accelerating displacement. The use of hybrid nanoparticles is thought to have enhanced the carrier fluid's thermal properties substantially. Because of the C-C link, CNTs are more effective than other forms of nanoparticles in the carrier fluid. CNTs nanofluid may be further functionalized to get the desired result, which may be used in a range of applications through noncovalent and covalent manipulation [18]. Saeed et al. [19] have considered the nanofluid containing CNTs and iron oxide nanomaterials using the flow of fluid over a curved surface. Gul et al. [20,21] studied the flow of nanofluids to enhance heat transfer and thermal applications. Alghamdi et al. [22] have observed the flow of hybrid nanofluid through a blood artery for medications. Using the bvp4c tool, Li et al. [23], and Ding et al. [24] used (MWCNTs) in the base liquid to evaluate heat transmission. Akbar et al. [25] described the influence of a magnetic field on the flow of CNT nanofluids through a moving permeable channel. Gul et al. [26] and Bilal et al. [27] used an inclined extending cylinder to examine the Darcy-Forchhemier hybrid nanoliquid flow. They examined the carbon nanotubes (CNTs) and iron oxide Fe 3 O 4 as two distinct nanomaterials. Ahmed et al. [28] represented temperature propagation in a wavy-wall impermeable enclosure through nanofluids. It was discovered that increasing the waviness of the sheet boosts both the heat transmission rate. Yarmand et al. [29] investigated how graphene nanoplatelets/platinum hybrid nanofluids with diverse properties may improve heat transfer rates. Sajid et al. [30] examined the thermophysical characteristics of hybrid and single-form nanotubes using numerical methods. They determined that the size, type, concentra-tion, and temperature fluctuation of nanoparticles had a significant impact on the thermophysical characteristics of nanofluid. Kumar et al. [31] examined the solar radiation impact on the flow of ferromagnetic hybrid nanofluid. Gowda et al. [32] studied the flow of nanofluid over the stretched and curved surface using (KKL) relation. Kumar et al. [33] have used the concept of the magnetic dipole for the flow of nanofluid over a cylinder. Zeeshan et al. [34] have studied the couple stress nanofluid flow using the paraboloid model.
The curved surface for the fluid flow has many applications in the mechanical and automotive industry. Sanni et al. [35], Jawad et al. [36], and Saeed et al. [37] have studied the fluid flow on a curved surface using various kinds of nanofluids for the heat transfer enhancement. Hayat et al. [9,38], Rosca, and Pop [39] have explained the homogeneous-heterogeneous reaction phenomena using the curved surface for the flow pattern. Okechi et al. [40], Asghar et al. [41], and Hayat et al. [42] have used the nonewtonian fluid flow over the curved surface with various extensions considering Darcy-Forchheimer flow medium. The related work to the proposed model can be seen in the References [9,35,[43][44][45].
The inertia effect is taken into account by incorporating a squared component to the momentum equation, called Forchheimer's modification [46]. Muskat [47] used the term "Forchheimer factor" to describe this new concept. It is critical to include non-Darcy consequences in convective transport analysis to properly represent realworld challenges. The novelty of the model has been presented as

Mathematical Formulation
The Darcy-Forchheimer flow considers CNTs and Fe 3 O 4 nanomaterials across an expanding curved sheet. The viscous fluid flow has been expressed in the permeable space by the Darcy-Forchheimer effect. The flow is assumed across the stretching sheet, with radius R, as depicted in Figure 1. The term ðr, sÞ is taken as the space coordinate and ðu, vÞ is the velocity component. Here, U w ðsÞ = ae s/L is the exponential stretching velocity, T w is the curved surface, and T ∞ is the ambient temperature. Keeping in view, the above superposition, the energy, and momentum equations along with their boundary conditions are expressed as [9,[36][37][38][39]46]

Journal of Nanomaterials
The basic flow conditions are Here, K * and F = C b /sK * 1/2 , are the permeability and iner-tia factors.

Journal of Nanomaterials
The k hnf is the thermal conductivity, ϕ 1 and ϕ 2 are the volume friction parameters, ðC p Þ MS is the specific heat capacity, ρ MS and ρ CNT are specified densities of Fe 3 O 4 and CNTs, and Sc is the Schmidt number, respectively.

Journal of Nanomaterials
By eliminating H from Eqs. (9) and (10), we get The transform conditions are where Fr, λ, and k are the Forchheimer, porosity, and curvature parameters, respectively, which can be rebound as The local Nusselt number, Sherwood Number, and Skin friction are expressed as
The sum of residual error is ε t

Results and Discussion
The goal of this portion is forward to see how the temperature and velocity profiles function under the effect of the predicted factors. The flow configuration is shown in Figure 1. The OHAM technique's progress has been  Figure 2(c). The greater value of the parameter δ improves the temperature distribution. Figure 3 illustrates the effects of the (porosity term) and k on velocity and temperature. This conclusion can be drawn that the velocity f ′ ðηÞ decrease, while the temperature field is increased versus rising values of porosity   Figure 5(d). The thickness of the boundary layer improved with the increasing value of Pr and consequently, the temperature profile reduces. Figures 6(a) and 6(b) illustrate to elaborate the consequences of curvature parameter k and Schmidt number Sc on mass transport ΦðηÞ profile. The mass transition rate reduces with the influence of Schmidt number while enhancing with the positive effects of curvature term, because the fluid mean viscosity becomes thick as the number of carbon and iron oxide particulates continues to increase.
The surface drag force ffiffiffiffiffiffiffiffiffi Re/2 p C f x for carbon nanoliquid and Fe 3 O 4 is declared via Figures 7(a) and 7(b). It is been evidenced that as the curvature and the volumetric parameters are increased, the skin friction drops. Figures 7(c) and 7(d) demonstrate the numerical results for the Nusselt number ðL/SÞðRe/2Þ −1/2 Nu x . It has been discovered that the heat conversion rate accelerated as the number of carbon nanomaterials in the conventional fluids and the Prandtl number expanded. Figure 7(e) indicates that the Sherwood number ðL/SÞðRe/2Þ −1/2 Sh x is an enhancing factor of the Schmidt number. Table 1 displays the thermophysical properties of solid substrates and basic fluids. The OHAM technique's consolidation has been computed up to the 30th iteration and is reported in Table 2. Table 3 offers a comparative analysis of the current study to the existing literature.

Conclusion
We addressed the Darcy-Forchheimer flow of Casson hybrid nanoliquid induced by an extended curved surface in this problem. The momentum and energy equations are included in the flow model, which is set up as a system of partial differential equations. The "Homotopy analysis method" is used to further build up the analytic arrangement of modeled equations. This mathematical model attempts to highlight the dominance of nanofluid in heat and mass transmission in advanced technologies and industries. The following are the core findings:

Data Availability
Data are available in the manuscript.

Conflicts of Interest
No such interest exists.