MHD Stagnation Point Radiative Flow of Hybrid Casson Nanofluid across a Stretching Surface

: The current investigation explores the hybrid Casson nanofluid stagnation point flow on a transient stretching surface under the impact of thermal radiation. The Joule heating effect is also considered in this study. Copper and aluminium hybrid nanoparticles are used. The guiding partial differential equations are broken down into nonlinear ordinary differential equations using adequate affinity transmutations. The subsequent equations are worked out by employing the Keller box scheme. The numerical findings for the study are represented by plotting velocity


Introduction
Nanofluids are novel fluids constructed by dissolving nanometer-sized components such as nanosheets, nanotubes, and nanofibers in base fluids.In another manner, nanofluids are nanoscale colloidal solutions that include condensed nanosubstances.Nanofluids have improved thermo physical characteristics and convective heat transfer coefficient, similar to viscosity, thermal diffusivity and thermal conductivity when compared to base fluids like water or oil.It offers a wide range of possible uses in a variety of sectors in heat transfer applications such as industrial cooling systems, nuclear reactors, automotive applications such as coolants, electronic applications such as cooling of microchips, biomedical applications such as nano drug delivery, cancer therapy, nanosurgery, sensing and imaging (Wei Yu & Huaqing Xie [1] , Wong & De Leon [2]).
From the previous works, it is noted that in a few cases of nanofluid, it was witnessed that specific heat is very low when compared to base fluid, also thermo physical properties of nanofluid depend on the dimensions of the nanomaterials used.Due to some lack of clarity in nanofluids, a new type of fluid is created by suspending various nanoparticles as a mixture or as a composite form called hybrid nanofluids.Hybrid nanofluids represent a novel class of fluids prepared as per blend of non-metallic or metallic nano-contented particles with base fluid acquainted to develop heat transfer rate when compared with base fluids.As demonstrated by several applications in cooling systems, heat turbines in industries, heat generators, drug delivery, and biological problems caused by nanofluids, this class of nanofluid model effectively enhances heat transfer phenomena.The goal of integrating two distinct nanoparticles will augment thermal conductivity.These liquids have distinct implementations in the medical sciences field, most medications are generated in the form of hybrid nanofluids, and blood is utilized as base fluid to monitor the chemical reactions of components in the blood.To the improvement of the thermal properties of such liquid, nanoparticles dispersed in the base fluid will enhance thermo physical properties.In the present paper a two-dimensional boundary layer flow of a heat transfer non-Newtonian Casson hybrid Nanofluid (Al 2 + Cu+ Water) across a stretched surface as its stagnation point is considered.Choi and Eastman [3] investigated that by delaying metallic nano particulates trendy base fluid resulting fluid exhibit higher thermal conductivity and advancement in heat transfer was observed.Also, practical applications of Nanofluid are analyzed by Dash et al. [4] intended Casson liquid flow in porous pipe and observed that increasing yield stress of fluid decrease in permeability is observed.Later Lee et al. [5].Empirically measured about thermal conduction of nanofluid.After that Xuan and Li [6] studied heat transfer improvement in the nanofluids case and observed that suspending nanoparticles increases thermal conductivity.Mernone et al. [7] analyzed the peristaltic transport of Casson fluid using the perturbation series method.Later Hong et al. [8] studied Fe nano liquids and observed that the thermal conductivity of the nanofluid rises, and the volume fraction of nanoparticles improves.Later Khan and Pop [9] analyzed Nanofluid flow on a stretched sheet for higher values of Pr, Sherwood number increases.Warrier and Teja [10] discussed that particle size will affect the enhancement of the thermal conduction of Nano liquid.Saidur et al. [11] reviewed the applications of nanofluids.After that, Suresh et al. [12] experimentally studied Cu-Al 2 O 3 -H 2 O hybrid nano-liquid flow and observed that the heat transfer coefficient increases with increasing Reynolds number.K. Bhattacharyya [13] studied "Boundary layer stagnation-point flow of Casson fluid and heat transfer towards a shrinking/stretching sheet.Haq et al. [14] studied Convective heat transfer and MHD effects on Casson Nanofluid flow over a shrinking sheet.Mansur and Ishak [15] studied boundary layer Nanofluid flow over a stretched sheet utilizing the shooting method; observed the hat heat transfer rate reduced with higher numerals of unsteadiness factor.Later esfe et al. [16] aimed experimentally about the conductivity and viscosity of hybrid nanofluid and observed that increasing the thermal conductivity of nanoparticles and dynamic fluid viscosity enhances.Later Harandi et al. [17] studied f-MWCNT Fe 3 O 4 -EG hybrid nano liquids' thermal conductivity; noted thereto thermal conductivity enhances, an increase in solid volume fraction and temperature.Thereafter, Hymavathi and Sridhar [18] studied Casson MHD liquid flow using the Keller box method; observed that an increase in Schmidt number causes a decrease in concentration profiles.Later Ibrahim and Makinde [19] studied Nano fluid Casson flow over a stretched sheet using RK-Fehlberg; observed that Sherwood decreases with a rise in the Casson parameter.Later Devi and Devi [20] studied cu-Al 2 O 3 hybrid nano liquid flow across a stretched sheet; hence the conclusion is the usage of hybrid nanofluid under magnetic field heat transfer rate enhances.Esfe et al. [21] analyzed about thermal conduction of SWCNT-MgO hybrid nanofluid.Ghadikolaei et al. [22] studied MHD Nano liquid Casson flow as well as radiation impact across an inclined stretched sheet RK Fehlberg method; hence the conclusion is increasing radiation parameter heat transfer rate increases.Later Aman et al. [23] presented an influence of the magnetic field flow of a Casson hybrid Nano liquid using the Laplace transform technique.Later Sajid and Ali [24] reviewed the thermal conductivity of nanofluids.Akhgar and Toghraie [25] studied H 2 O-ethylene glycol, SWCNT-TiO 2 hybrid nano liquid flow, which was seen to rise as the temperature and volume fraction of the Nanofluid raised.Balaji Prakash et al. [26] reviewed the radiation impact of MHD Casson Nano liquid on the vertical plate and solved using the closed analytical method velocity of fluid falls with an increment in radiation.Narender et al. [27] explored the flow of stagnation point a Casson nanofluid flow on the radially stretched sheet using the shooting technique; for higher values of Prandtl number, temperature falls were concluded.After that Sridhar et al. [28] explored combined convective Casson fluid flow on an inclined stretched sheet; the increase in permeability parameter velocity decreases.Later Krishna et al. [29] studied Casson hybrid nanofluid using the Laplace transform technique; the Nusselt number rises with increment in heat absorption rate.After that Sridhar and Ganesh [30] studied the MHD flow of Casson fluid across a stretched surface; an increasing radiation parameter will enhance the temperature.C. Y. Wang [31], Free convection on a vertical stretching surface, Thereafter, Ganesh and Sridhar [32] studied MHD Casson fluid through Darcy for chimer porous media and observed that for progressive values of magnetic velocity decreases.Agamid et al. [33] reported Casson-Cu+Al 2 hybrid Nanofluid; an increase in Casson factor velocity of the fluid declines.Xu et al. [34,35] investigates gas passage behaviour in nanopores and discussed that the packing fraction of the gas equation of state offers the smallest limits for the Knudsen number and flow system.Also, with a Knudsen number and a pore diameter, one can easily estimate the apparent permeability of gas for flow-regime conditions from a typical gas apparent permeability curve.Atashafrooz [36] studied hybrid nanofluid flow inside a trapezoidal enclosure with the influence of radiative heat transfer and magnetic field and concluded that in the absence of a magnetic field highest magnitude of heat transfer rate is obtained at the bottom of the enclosure.Sajjadi et al. [37] investigated the influence of magnetic field on particle suspension in a modelled room and concludes that for higher values of magnetic field influence the number of suspended particles on the walls increases also gravity force has a notable influence on particle suspension and their presence on the floor of the modelled room.Delouei et al. [38] investigated experimentally the influence of concentration of the nanofluid on ultrasonic vibrating finned tube heat exchanger and noted that increasing concentration of the nanofluid will influence ultrasonic vibration effect.Also decreasing the surrounding temperature and velocity of the air heat transfer enhances and the positive impact of ultrasonic vibrations decreases.

Formulation of the problem
In the Current investigation, an incompressible, unsteady, two-dimensional stagnation point MHD boundary layer flow of non-Newtonian hybrid Nano liquid over a stretched surface with an impact of radiation is considered.From the Figure 1, the rheological equation of the Casson flow model is .
Here µ B represents plastic dynamic viscosity, p y represents yield stress.Under these assumptions, the governing flow partial differential equations are: where u and v represents velocity components in the x and y directions.k represents thermal conductivity.c p represents specific heat.ρ represents fluid density.U s represents stagnation point flow velocity.υrepresents kinematic viscosity.
Relative boundary conditions: Using the similarity transformations the equation ( 1) which represents the equation of continuity is satisfied by using the relations u 2)-( 3) are transformed to Boundary conditions in (4) are transformed to the form

Method of solution
Equations ( 5)-( 6) along with given BCs are resolved using the Keller box scheme as reported by Cebecci and Bradshaw (33) the steps to follows: 1. Deduce the specified system into 1 st order non-linear equation.

Employ finite difference and then linearized.
3. Deduce the specified system to a tridiagonal system.4. Using block tridiagonal elimination we solve the tri-diagonal system.

Results and discussion
A stagnation point flow of Casson hybrid nanofluid on a stretching surface is considered.The flow is influenced by the applied magnetic field and Joule heating influences.Hybrid nanofluid is prepared by suspending nanoparticles of Copper (Cu) and Aluminium (Al 2 ) in the base fluid.The equations governing the flow model are solved using the Keller Box method.The flow is influenced by non-dimensional parameters Casson (β ), Magnetic (M), nanoparticle volume fraction (φ 1 , φ 2 ), unsteady parameter (S) Prandtl number (Pr), Radiation parameter (Rd), Eckert number (Ec).The Impact of these parameters is observed by scheming temperature and velocity profiles utilizing MATLAB.The Nusselt number is calculated for progressive values of the Prandtl number it is found that on increasing Prandtl number enhancement in Nusselt number is witnesses this phenomenon coincides with previous literature mentioned in Table 1.Also, the numerical calculation of local parameters skin friction coefficient was calculated and observed, skin friction value rises for greater values of Casson parameter, Magnetic parameter depicted in Table 2 and incorporated thermal physical properties of the fluid nano particles in Table 3.An impact effect of the Casson parameter in velocity characterization is portrayed in Figure 2 both Casson+Cu fluid and Casson+Al 2 +Cu hybrid nanofluid.Clearly, for progressive values of Casson parameter velocity declines.An improving Casson factor value fluid elasticity of hybrid fluid increases, so the viscosity of the hybrid nanofluid increases.Figure 3 displays velocity profiles of Magnetic parameter.On improving magnetic parameters, fluid velocity lessens in both cases of nanofluid as well as hybrid nanofluid due to the opposing force generated called Lorentz force.Figure 4 represents velocity profiles corresponding to nanoparticle volume friction factors φ 1 , φ 2 .Improving values of φ 1 , φ 2 velocity profiles exhibits decreasing tendency for both Casson nano and Casson hybrid nanofluids.Because rising values of φ 2 cause a reduction in the thickness of the moment boundary layer.Velocity profiles of the unsteady parameter are portrayed in Figure 5. On increasing the unsteady parameter, the velocity of the fluid decreases owing to improving values on the unsteady parameter it generates a resistive force.Figure 6 demonstrates the temperature profiles of the Casson parameter.Casson factor progressive values on the temperature of the fluid increase.The fluid temperature rises for greater values of the Magnetic factor which is displayed in Figure 7. Figure 8 represents the temperature distribution of the Prandtl number.With higher values of Prandtl number, the decreasing trend is observed in temperature profiles for both Casson nano and hybrid nanofluids cases due to the reason for higher values of Prandtl number, thermal conduction of fluid lessens.Temperature profiles of radiation parameter are portrayed in Figure 9.For higher values of radiation factor, the temperature of fluid increases because on increasing thermal radiation parameter TBL thickness decreases more heat will be generated so temperature profile increases.Figure 10 displays temperature profiles for nanoparticle volume friction parameters.It is visible that the increase in values of φ 1 , φ 2 temperature of the fluid increases for both Casson nano and Casson and nanofluid cases.Figure 11 portrays the temperature profile for various values of the Eckert number.It was noticed, temperature turned out to increase for progressive values of Eckert number.Eckert number arises from the application of the Joule heating effect along with the magnetic field.An Increase in Eckert number drives a rise in kinetic energies resulting increase in the collision of fluid molecules causing a rise in the temperature profile.To validate the numerical method, present results are compared with existing literature by calculating −θ (0) taking φ 1 = φ 2 = 0.By observing the Table 1, the results are found to be good in agreement with the existing literature.Increasing the Prandtl number, the Nusselt number increases.
Also, Numerical observations of the skin friction coefficient were calculated and observed that C f -value.
Improves for greater values of Casson parameter, and Magnetic parameter.

Conclusions
In the present study, Casson hybrid nanofluid flow over a stretching surface is considered with the MHD, Radiation and Joule heating influences.The study is carried out by solving the governing equations using the Keller Box method.Also a comparative study between Cu-Casson and Cu+Al 2 -Casson hybrid nanofluid Cases.Also, the impact of various parameters is analysed by constructing graphs of velocity and temperature comparison of nanofluid (Cu+caason) with hybrid nanofluid (Cu+Al 2 +Casson) using MATLAB.Also, numerical values of local parameters skin friction coefficient and Nusselt number are calculated.The following findings are observed.
• Greater values of Casson parameter, declination is observed in velocity profile and increasing trend is observed in temperature profiles.• Progressive values of Eckert number, increment in temperature profiles observed.

Figure 1 .
Figure 1.Physical geometry of the Problem.

Figure 11 .
Figure 11.Temperature profiles for Eckert number volume fraction.

•
Stronger magnetic parameter impact, fluid velocity reduces, and fluid temperature rises.• An improving value of nanoparticle volume friction factors φ 1 , φ 2 , fluid velocity enhancement, and fluid temperature reduces.• An escalating value of Prandtl number, fluid temperature reduces.• Progressive values of radiation factor, fluid temperature rises.Volume 5 Issue 2|2024| 1659 Contemporary Mathematics

Table 2 .
Skin friction coefficient values for β and M.

Table 3 .
Thermo-physical of fluid and nanoparticles.