1 Introduction

Nanotechnology has fascinated inventors and investigators for its substantial use in industrial disciplines. For instance, in the pharmacological field, cancer patients are cured by nano-liquid-based machinists who embrace diverse energies and medications. Cooling and heating methods such as lessening receivable heat from electronic processing units, regulating the temperature of atomic reactors, regulating vehicle radiators, and managing heat flow in thermal valves employ nanofluids. These vital properties, together with numerous manufacturing and domestic applications, have captivated researchers and engineers in recent days. Nanofluids engrossed in pure fluids have a propensity to uplift their heat efficiency. A nanofluid comprises elements with nano-scale-sized particles. This impression originated with Choi [1] when he induced nanoelements in a pure fluid, which was called “nanofluid.” The nanoparticles comprise metals, metallic oxides, carbon nanotubes, and carbides. Nanoparticles have noteworthy industrial applications like sunblock for cars that are extra repellant to radiant emissions, car bumpers that have lesser weight, artificial bones that are stronger than nauseating sartorial, and improved sturdiness of several sports items such as balls [2-4].

A Casson fluid is a non-Newtonian fluid in which shear stress is nonlinearly related to velocity gradients. The stress–strain relationship in the Casson fluid is guided by a nonlinear Casson constitutive equation originally formulated by Casson [5]. This fluid has many practical applications and is a critical non-Newtonian fluid that is widely used in fields such as biomechanics, plastics, and metallurgy. Its significance spans food processing, metallurgy, drilling, and biotechnology, which has gathered momentous curiosity from investigators [6-8]. The Casson fluid model is predominantly effective in mathematically representing the property of fluids with non-zero plastic dynamic viscidness, relating it to a near estimation of blood. This model is also pertinent in the exploration of fluids encompassing plasma and proteins and in numerous products such as coal-water assortments, coats, artificial greases, and biotic fluids like tomato sauce, honey, potage, gelatine, and body fluid. For its highly practical applications, studies [9-15] explored further on the properties of MHD Casson fluids, reporting significant results. Reddy et al. [16] analytically analyzed the entropy study of two-phase radiative nanofluid transport in a bioconvective non-Newtonian model via HAM. Shamshuddin et al. [17] explored the impact of ternary nanofluid integration under thermal physical influences with the Cattaneo-Christov model.

The fluid movement past/over a cone has established much courtesy owing to countless applications concerning thermal transmission. It comes across several industrial uses, in addition to countless natural conditions such as medication and biomedical systems, energy preservation systems, aerospace sciences, geological mechanisms, cosmological physics, micro-inverter chips, interstellar technology, the regulatory mechanism of machine oil, and atomic energy control systems. In recent explorations, natural convective flows over a cone are the most exciting disciplines deliberated by investigators under the dynamics of fluid motion. Merk and Prins [18, 19] exhibited the overall connections for analogous results on a procedure occurring at an immobile temperature and explained the results obtained. Bilal et al. [20] considered the influence of numerous physical conditions on Maxwell nanofluid transport through a gyrating cone. Hussain et al. [21] outlined the thermal efficiency of multiple-based nanofluids across a modified cone. Hussain et al. [22] examined the enhancement of heat transport on a 3D gyrating cone of a hybrid nanofluid. Sayed and Fathy [23] reported the thermal transport consequences of a nanofluid past an upright cone, where the solution of the flow was obtained numerically. Additional relevant results on the MHD fluid flow via a cone with different flow geometries can be captured in references [24-27].

The order reactive species is termed to be one and consistent when its rate is directly correlated with reactant concentration and it proceeds entirely within a single-phase volume. Furthermore, the diffusion of species generated or consumed as a result of the reaction may be proportionate to the change in the nth-order concentration amid the interface and surrounding fluid, where n is a positive integer. Meteorological phenomena, haystack burning, milk spray drying, suspended catalyst reactors, and thermal cooling structures are all impacted by convective thermal and solutal transport with chemical reactions [28-30]. With such a significant impact on engineering and industrial applications, Swain et al. [31] outlined the outcomes of higher-order reacting species and slip frontier settings on hydro-magnetic nanofluid transport via numerical solutions. Seth et al. [32] simulated the model of conjugate heating effects on hydro-magnetic free convective Casson fluid movement with nth-order reactive species in permeable media. The upshots of MHD radiative flow nanofluid with inconstant chemical reaction past an upright plate were reported by Rahman and Uddin [33]. Studies [34-39] included the consequences of chemical reactions on MHD nanofluid transport.

Mass and energy fluxes arise as direct consequences of gradients in temperature and concentration, respectively. Thermo-diffusion, commonly referred to as the Soret effect, is driven by temperature gradients, whereas the diffusion-thermo phenomenon, known as the Dufour effect, is instigated by concentration gradients. These effects gain prominence under flow conditions where density variations are present. Moreover, convection plays a pivotal role in heat transfer processes, significantly impacting practical applications like regulating systems, metallurgical mechanisms in the steel factory, and the operation of heat exchangers. Vijay and Sharma [40] explained the cross-diffusive upshots on the hydromagnetic transport of a hybrid nanofluid across a slowing spinning disk, where solutions of the flow were numerically obtained using the BVP Midrich scheme. Ullah et al. [41] performed a comprehensive analysis of entropy generation in the context of double-diffusive dissipative magnetohydrodynamic (MHD) Jeffrey nanofluid transport, incorporating the effects of non-linear slip conditions and activation energy. Sathyanarayana and Goud [42] examined the influence of double-diffusive aspects on reacting hydromagnetic nanofluid transport across an upright cone entrenched in an absorbent material via the Runge–Kutta approach allied with the shooting technique. Ilango and Lakshminarayana [43] discussed the consequences of cross-diffusive effects on dissipative hydro-magnetic Casson nanofluid past a widening surface in a permeable material subjected to an induced magnetic domain. Further detailed scrutiny on cross-diffusive effects can be found in references [44-47].

The influence of viscous dissipation is paramount in areas characterized by high current densities and flows involving either high velocities or elevated viscosities. Viscous dissipation becomes increasingly significant in such contexts, as internal friction produces considerable heat, particularly in regions subjected to intense shear. The effects of dissipation are crucial in the broad scope of technological and engineering practices, embracing aerospace machines, jet propulsion structures, space dynamics, automotive engines, and other advanced mechanical systems. Zainal et al. [48] provided a thorough of dissipative consequences on the transport behavior of MHD hybrid nanofluids past an exponentially widening or contracting sheet. Hazarika et al. [49] scrutinized the thermophoresis upshot on a dissipative motion of a hydromagnetic fluid across a continuous surface in a permeable domain employing $$$ Cu,{Fe}_3{O}_4,\mathrm{and}\ Ag $$$ nanoparticles. Muntazir et al. [50] discussed the radiative and dissipative hydromagnetic nanofluid transport across an absorbent elongating sheet considering $$$ Cu\ \mathrm{and}\ {Al}_2{O}_3 $$$ nanoparticles. Where the transformed equations of the problem were cracked through a semi-analytical tactic utilizing the differential transformation method. Prabhakar Reddy and Sademaki [9] conducted a study of dissipative and radiative consequences on a hydromagnetic Casson fluid transport past an upright porous plate via finite difference method (FDM). The studies [51-55] merit recognition for their significant contributions to the understanding and explanation of dissipation effects in MHD fluid streams.

The investigation of inconsistent heat source and absorption in hydro-magnetic fluid transport is a fundamental insight into the thermal behavior of conducting fluids under the influence of magnetic domains. Non-uniform heat source or absorption pertains to spatial distinctions in thermal generation or consumption within the fluid, a phenomenon that is critically important across numerous manufacturing and engineering applications. This variation significantly enhances temperature distribution and transport properties, playing a pivotal role in systems where precise thermal regulation is essential. Applications span from fission reactors and astrophysical phenomena to electronic device regulators, metal processing, polymer synthesis, and other technologies reliant on controlled thermal transmission. Mythili et al. [56] presented the upshots of higher-order reactive species and nonuniform heat source/absorption on hydromagnetic Casson fluid transport across an upright cone via the FDM. Prabhavathi et al. [57] scrutinized the heat and solutal transport features of a hydro-magnetic nanofluid past an upright cone entrenched in a permeable membrane, incorporating variable heat source and absorption via a numerical approach. Konda et al. [58] explored the upshot of a non-uniform heat generation and consumption on the hydro-magnetic transport of Williamson nanofluids with thermal effects in a penetrable medium. Shi et al. [59] inspected the radiative characteristics of hydro-magnetic Maxwell-nanofluid transport, considering non-uniform heat generation and consumption. Additional significant findings regarding the consequences of non-uniform heat source/sink on nanofluid flow are documented in studies [60-62].

2 Model Description

According to Ramadhyani et al. [64], the heat flux $$$ {q}^{{\prime\prime\prime} } $$$ can be computed from the electrical power $$$ P $$$ and thermal transport surface area $$$ {S}^{\bullet }:{q}^{{\prime\prime\prime} }=P/{S}^{\bullet } $$$ It should be pointed out that the instance $$$ {A}^{\ast }>0;{B}^{\ast }>0 $$$ matches to inner heat source and the occasion $$$ {A}^{\ast }<0;{B}^{\ast }<0 $$$ links to inner thermal consumption.

Where $$$ {u}^{\prime } $$$ is the fluid velocity, $$$ {T}^{\prime } $$$ is the temperature of the fluid, $$$ {C}^{\prime } $$$ is the nanofluid concentration of nanofluid, $$$ {\rho}_{nf} $$$ is the nanofluid density, $$$ {\mu}_{nf} $$$ is the dynamic viscosity of nanofluid, $$$ {\sigma}_{nf} $$$ is the electrical conductivity of nanofluids, $$$ {\beta}_{nf}^{\theta } $$$ is the heat expansivity coefficient, $$$ {\beta}_{nf}^C $$$ is the coefficient of volumetric expansivity, $$$ g $$$ is the gravitational acceleration, $$$ {\left(\rho {c}_p\right)}_{nf} $$$ is the specific heat capacity of the nanofluid, $$$ {k}_{nf} $$$ is the thermal conductivity of nanofluid, $$$ {D}_{nf} $$$ is the mass diffusivity, $$$ {\gamma_k}^{\prime } $$$ is the chemical reaction parameter, $$$ {D}_m $$$ is the chemical molecular diffusion, $$$ {T}_m $$$ is the fluid mean temperature, $$$ {k}_T $$$ is the thermal diffusion rate, $$$ \beta $$$ is the cone apex semi-angle.

And the variables appearing in Equations (10-13) are demarcated as follows in Table 1.

With considerable dimensionless engineering curiosity analogous to heat and mass flow potentials encompass the wall shear stress $$$ {C}_f $$$, heat and mass transfer rates in terms of Nusselt number $$$ {N}_u $$$, and Sherwood number $$$ {S}_h $$$, which are defined as:

3 Finite Element Computation Analysis

To use the GFEM for structures of coupled equations, the leading equations are multiplied by weight functions, and the integral over the element field is performed, resulting in the setting of the results to zero. This process leads to the following variational equations:

It must be emphasized that the demand for continuity in field variables is significantly stricter in its current robust manifestations as depicted in Equations (16-18). To address these challenges, there is a preference for weak formulations. Undoubtedly, these weak formulations assist in lowering the continuity requirements for selected elements, thereby simplifying the approximation functions and allowing the utilization of easily constructed and implemented polynomials. Additionally, weak formulations inherently enforce natural frontier settings. Consequently, the chosen weak variations of the variational formulations are derived by substituting the unknowns in the weighted residual approach in Equations (16-18) with approximate trial solutions, represented by polynomial relationships in Equation (19), and performing integrations by parts over the element domain. The incorporation of borderline settings directly into the weak formulations follows the integrations by parts over the element space. The finite element model of the equations in matrix form is presented as:

Here, $$$ {u}^{\bullet }=\sum \limits_{j=1}^3{W}_j{u^{\bullet}}_j $$$ and $$$ {v}^{\bullet }=\sum \limits_{j=1}^3{W}_j{v^{\bullet}}_j $$$ are the presumed relations that are introduced to simplify the structure of the equations. The Crank–Nicolson technique is used to approximate the time derivatives. Subsequently, the computational domain is split into 448 triangular components, each consisting of three nodes. As a result, there are 285 nodes in the entire domain. Meanwhile, 3 functions are computed at their respective node, then, the order of apiece component matrix is $$$ 9\times 9. $$$ Upon assembling the fundamental equations, an initial arrangement of 855 nonlinear coupled equations is formed. As mentioned before, and are used to linearize the structure of nonlinear coupled equations. An arrangement of 660 linear coupled equations of order is formed after the frontier settings are executed on the structure of linearized coupled equations. The Gauss–Seidel iteration scheme is employed to compute the linearized structure of equations. When the relative error for each variable amid successive iterations is calculated under the convergence requirement in a way that indicates $$$ \sum \mid {\zeta}_i^n-{\zeta}_i^{n-1}\mid \le {10}^{-6}, $$$ where $$$ \zeta $$$ is the overall reliant variable $$$ u,\theta\ \mathrm{and}\ \phi $$$ and $$$ n $$$ is the number of successive iterations.

4 Validation of the Computation Procedures

To display the authentication and precision of the current arithmetical scheme, a comparative analysis of the rate of wall shear stress and mass transfer was computed by removing the nanofluid characteristics and letting $$$ M=0\ \mathrm{and}\kern0.5em {E}_c=0 $$$ in this present study with the preceding work of Mythili et al. [56] attained by FDM is depicted in Table 3. It is professed that there is an excellent pact between the outcomes. This describes the precision of the current numerical method.

5 Results and Discussion

To comprehend the insight of the problem at hand more thoroughly, we performed numerical computations for non-dimensional embedded flow profiles and the engineering quantities, using pertinent parameter values for two distinct Casson nanofluids. We presented twofold solutions for $$$ Ag- WEG $$$ and $$$ {Al}_2{O}_3- WEG $$$ nanofluids to achieve this. Shear stress, thermal, and solutal numbers are calculated numerically and exhibited in table format. To accomplish the numerical results of the current model problem, we adopted the following default values throughout the analysis: $$$ {E}_c=0.5,{K}_A=1,M=2,{S}_r=0.2,\alpha =0.5,{A}^{\ast }=-2,{B}^{\ast }=-2.1,{P}_r=0.71,{D}_u=0.3,t=1,\beta =\pi /4,{N}_1=1,\phi =0.01,n=2,{S}_c=0.2,\mathrm{and}\ {k}_1=0.2. $$$

Figure 2 exemplifies the impact of magnetic field intensity $$$ M $$$ on the velocity of the nanofluid. It is observed that the velocity profile of both nanofluids exhibits a declining trend with increasing $$$ M $$$. This reduction arises from the imposition of a transverse magnetic field, which persuades a Lorentz force acting on the electrically conductive fluid. The resultant retarding force impedes the fluid's motion within the boundary layer, thereby decreasing its velocity. This scenario holds significant importance in the domain of motor rewinding technology, particularly in the precise regulation and control of machine processes across a diverse range of applications. Figure 3 showcases the impact of the absorbent parameter $$$ {K}_A $$$ on nanofluid velocity. It is perceived that nanofluid velocity increases as $$$ {K}_A $$$ is elevated. The underlying physics of this observation lies in the fact that the porosity parameter defines the proportion of the medium's volume composed of void spaces, devoid of solid material. As $$$ {K}_A $$$ increases, the impediment imposed by the absorbent matrix diminishes, owing to the presence of more void spaces through which the nanofluid can traverse. This reduction in resistance facilitates a freer flow of the nanofluid, thereby increasing fluid velocity. In manufacturing, controlled porosity enhances filtration efficiency, thermal insulation, and catalytic reactions, making it essential in sectors such as aerospace, biomedical engineering, and metallurgy. Figure 4 elucidates the variation in fluid velocity for diverse parameters of the Soret number $$$ {S}_r $$$. It is seen that, as $$$ {S}_r $$$ increases, fluid velocity correspondingly rises. Physically, this behavior is attributed to enhanced solute diffusion within the boundary layer, driven by an intensified thermal gradient. The resulting increase in diffusion accelerates the fluid flow, thereby elevating its velocity. Figure 5 illustrates the variation in velocity distribution as a function of the viscous dissipation parameter $$$ {E}_c $$$. It is detected that the fluid's momentum upsurges with the rise in $$$ {E}_c $$$. Physically, the enhancement of $$$ {E}_c $$$ converts kinetic energy into internal energy, which, in turn, augments the fluid's velocity. Considerating and enhancing viscous dissipation is crucial for advancing energy efficiency, thermal regulation, and material processing in industrial systems. Figure 6 presents the influence of the Casson parameter $$$ \alpha $$$ on fluid velocity. It is evident that increasing $$$ \alpha $$$ reduces the fluid's velocity. This effect is attributed to the rise in the fluid's yield stress, which introduces greater opposition to the fluid movement. As $$$ \alpha $$$ increases, the fluid requires a higher force to initiate movement, resulting to a thickening of the borderline stratum. This, in turn, diminishes the shear rate, ultimately resulting in a decrease in velocity. The stimulus of space and temperature-dependent thermal consumption $$$ \left({A}^{\ast }<0;{B}^{\ast }<0\right) $$$ and heat generation $$$ \left({A}^{\ast }>0;{B}^{\ast }>0\right) $$$ on the velocity curves are exhibited in Figures 7 and 8, respectively. It is evident from these depictions that snowballing the parameters of heat generation augment the velocity curves by generating heat within the fluid while lessening the values of heat absorption diminishes the fluid pace by making the fluid cooler. Figure 9 depicts the stimulus of the Prandtl number $$$ {P}_r $$$ on the velocity of the nanofluid. It is noted that as the $$$ {P}_r $$$ surges, the fluid's pace shrinks. This behavior can be explained by the reason that a higher $$$ {P}_r $$$ corresponds to greater fluid viscosity, which effectively thickens the fluid. The augmented viscosity dissipates kinetic energy, thereby reducing the fluid's speed. The Prandtl number balances momentum and thermal conductivity to regulate heat distribution in the MHD nanofluid flow. It is important for energy systems and cooling technologies. Figure 10 demonstrates the upshot of the semi-apex angle $$$ \beta $$$ on fluid velocity. It is evident that increasing $$$ \beta $$$ results in a reduction in fluid momentum. This behavior arises because the inclination angle modifies the influence of gravitational forces, increasing frictional resistance and reducing buoyancy-driven convection. These combined mechanisms slow down the fluid, yielding to a noticeable decrease in velocity as the inclination angle grows. The fluctuation of the ratio mass Grashoff number to the thermal Grashoff number on fluid velocity is illustrated in Figure 11. It is well concluded that fluid speed ascends when $$$ {N}_1 $$$ is raised. Physically, $$$ {N}_1 $$$ impacts fluid velocity by controlling the relative intensity of buoyancy forces due to concentration and temperature gradients. These buoyancy forces drive convection currents within the fluid, hence augmenting fluid velocity. Figure 12 explicates the fluid velocity variability for the sprouting radiation parameter $$$ {N}_r. $$$ There is an enlargement noted in the distributions of the velocity when $$$ {N}_r. $$$ values are progressed. This is due to kinetic energy within the fluid being enhanced causing the fluid to flow quickly.

Figure 13 depicts the temperature behavior in response to the Eckert number $$$ {E}_c $$$. A growing $$$ {E}_c $$$ contributes to the heightened activity of the material particles as a result of the substantial energy stored in the boundary stratum, resulting in an elevation of the fluid temperature. Figure 14 elucidates the impact of the Prandtl number $$$ {P}_r $$$ on fluid temperature, showing that the fluid temperature shrinks with mounting $$$ {P}_r $$$. The underlying reason for this behavior is that higher $$$ {P}_r $$$ indicate that thermal diffusion is comparatively lower likened to momentum diffusion, meaning that thermal energy is transmitted more slowly through the fluid. As a result, in fluids with elevated $$$ {P}_r $$$, heat propagation occurs at a slower rate, leading to a more gradual change in temperature. This phenomenon is of paramount importance in maintaining thermal equilibrium within cooling systems, particularly in engines and high-performance machinery operating under extreme conditions. Figure 15 showcases the impact of nanoparticle volume fraction $$$ \phi $$$ on fluid temperature. It is detected that, the growth of $$$ \phi $$$ demotes the fluid temperature. This fallout is the result of the reduction of the width of the thermal boundary stratum in the flow domain. The distinction of Soret number $$$ {S}_r $$$ on fluid temperature is exposed in Figure 16. It is distinguished that the escalation of $$$ {S}_r $$$ enhances fluid temperature. Physically, an increase in the Soret parameter signifies that temperature variations exert a significant influence on solute diffusion. When a thermal gradient is established, solutes exhibit a propensity to thrust from areas of lower temperature to those of higher temperature due to this coupling effect. This propulsion can result in localized heating, as the solute particles transport thermal energy with them, thereby contributing to an elevation in the overall fluid temperature. Figures 17 and 18 explain that, the heat boundary stratum produces the energy for snowballing the heat generating values, that is, A > 0 and B > 0 which improve the thermal transport, and the energy is engrossed by lessening the heat consumption values, that is, A < 0 and B < 0 yielding to temperature decline pointedly adjacent the frontier layer. It is to be distinguished that the extent of thermal transference for temperature-dependent heat generation is more noticeably associated with surface-dependent heat generation. The performance of Dufour number $$$ {D}_u $$$ on fluid temperature is illustrated in Figure 19. It is perceived that fluid temperature grows when is $$$ {D}_u $$$ enhanced. The physics behind this process is, that in mass diffusion, kinetic energy is transported toward the fluid due to molecular exchanges, resulting in energy imbalance. The energy transmission enhances heat, which boosts the local temperature. The higher the concentration gradient, the greater the energy flow, resulting in a perceptible temperature intensification in improved diffusion processes. The upshot of radiative property $$$ {N}_r $$$ is exemplified in Figure 20. It is detected that the fluid temperature rises with a higher value of $$$ {N}_r $$$. Practically, when fluid is subjected to thermal radiation it engrosses radiant energy which is rehabilitated into heat energy inside the fluid yielding to an escalation of fluid temperature.

The performance of order of chemical reaction $$$ n $$$ toward fluid concentration is professed in Figure 21. It highlighted that $$$ n $$$ is proportional to the concentration of the fluid. The physics behind this scenario is higher $$$ n $$$ values tend to intensify in fluid concentration due to improved reaction rates, more operative exchanges among reactants, and response processes that endorse further reaction, eventually contributing to an advanced concentration of reactants or products in the fluid. Figure 22 elucidates the impact of the Schmidt number $$$ {S}_c $$$ on concentration. species. It is detected that concentration of the fluid diminutions progressively with the elevation of $$$ {S}_c $$$ From a physical standpoint, an expansion in the $$$ {S}_c $$$ signifies a lessening in molecular diffusion. Consequently, species concentration is augmented at lower Schmidt number, while it experiences a marked decline at higher values. Figure 23 reveals the influence of the thermo-diffusion $$$ {S}_r $$$ on fluid concentration. It is noted that snowballing the Soret number enhances fluid concentration. Physically, the $$$ {S}_r $$$ quantifies the relative strength of thermal diffusion compared to molecular diffusion. Optimizing the Soret consequence can improve efficiency in energy systems, chemical processing, biomedical applications, and advanced material technologies. An increase in this parameter amplifies the effect of temperature gradients on concentration distribution, leading to more pronounced variations in fluid reactions. The impression of the chemical reaction parameter $$$ {k}_1 $$$ on fluid concentration is portrayed in Figure 24. A pronounced decline in species concentration is observed with the intensification of $$$ {k}_1 $$$. This reduction is attributable to the chemical reaction's repressive effect, which diminishes the molecular propulsion within the fluid, consequently leading to a significant attenuation in fluid concentration. Figure 25 describes the concentration behavior with time $$$ t $$$ variation. It is perceived that fluid concentration enlarges as time progresses.

5.1 Wall Shear Stress, Rate of Thermal Flow, and Rate of Solutal Flow

The distinction of the wall shear stress for both nanofluids with dissimilar parameters is visible in Table 4. As indicated in this table, the wall shear stress is elevated with the upsurge in $$$ M,{K}_A,\alpha, {D}_u,\mathrm{and}\ {S}_r $$$ but it lessens with an enhancement in $$$ \phi $$$. The rate of thermal transport (Nusselt number) variation with several physical flow parameters is disclosed in Table 5. It is stated from this table that the Nusselt number for both nanofluids upsurges with an increment in $$$ \phi $$$ but a contrary trend is perceived with a surge of $$$ M,{A}^{\ast },{B}^{\ast },\mathrm{and}\ {P}_r. $$$ The deviation of the Sherwood number for both nanofluids is accessible in Table 6. It is perceived that the Sherwood number heightens by intensifying $$$ {S}_c\ \mathrm{and}\ {S}_r $$$ but exhibits a converse trend for a surge in $$$ {k}_1,n,\mathrm{and}\ \phi $$$ values.

TABLE 4. Numerical solutions of the wall shear stress $$$ {C}_f $$$ for distinct parameters.

						$$$ {C}_f $$$
$$$ M $$$	$$$ {K}_A $$$	$$$ \alpha $$$	$$$ {D}_u $$$	$$$ {S}_r $$$	$$$ \phi $$$	$$$ {Al}_2{O}_3- WEG $$$	$$$ Ag- WEG $$$
2.0	1.0	0.5	0.3	0.2	0.01	1.345245	1.923434
4.0						1.357643	1.932111
6.0						1.363215	1.945223
	2.0					1.312364	2.082168
	3.0					1.321345	2.125452
	4.0					1.334542	2.172371
		1.0				1.439871	2.345757
		1.5				1.442152	2.412359
		2.0				1.447654	2.490121
			0.4			1.213476	1.823189
			0.6			1.217652	1.829874
			0.9			1.345615	1.835528
				0.4		1.233476	1.863189
				0.6		1.247652	1.899874
				0.8		1.325615	1.905528
					0.03	1.453216	1.976525
					0.05	1.432672	1.964326
					0.07	1.423447	1.936540

TABLE 5. Numerical calculations of the thermal transport rate $$$ {N}_u $$$ for distinct parameters.

					$$$ {N}_u $$$
$$$ M $$$	$$$ {A}^{\ast } $$$	$$$ {B}^{\ast } $$$	$$$ {P}_r $$$	$$$ \phi $$$	$$$ {Al}_2{O}_3- WEG $$$	$$$ Ag- WEG $$$
2.0	−2.0	−2.1	0.71	0.01	1.295258	1.723471
4.0					1.257643	1.632745
6.0					1.163215	1.545639
	−1.0				−1.682187	−1.212364
	0.0				−1.725438	−1.425438
	2.0				−1.972376	−1.572376
		−1.5			1.239871	1.845761
		2.0			1.042152	1.412352
		3.0			0.747654	1.090125
			1.0		1.313476	1.923188
			1.5		1.217652	1.829871
			2.0		1.145615	1.735523
				0.03	1.233476	1.863189
				0.05	1.347652	1.999864
				0.07	1.425615	2.105628

TABLE 6. Numerical solutions of the mass transfer rate $$$ {N}_u $$$ for distinct parameters.

					$$$ {S}_h $$$
$$$ {S}_c $$$	$$$ {k}_1 $$$	$$$ n $$$	$$$ {S}_r $$$	$$$ \phi $$$	$$$ {Al}_2{O}_3- WEG $$$	$$$ Cu- WEG $$$
0.2	0.2	2.0	0.71	0.01	1.395258	1.823463
0.4					1.457643	1.932741
0.6					1.563215	2.145656
	0.4				1.382181	1.712361
	0.6				1.225433	1.625432
	0.8				1.172375	1.572371
		3.0			1.539871	1.715761
		4.0			1.442152	1.642352
		5.0			0.247654	1.500125
			1.0		1.313465	1.823188
			1.5		1.417651	1.929851
			2.0		1.545613	2.135522
				0.03	1.433476	1.863145
				0.05	1.347652	1.799821
				0.07	1.225615	1.505676

Figures 26-31 present 3D plots that elucidate the response of various parameters to engineering quantities. These visualizations offer a comprehensive insight into the multifaceted factors affecting frictional drag, as well as thermal and solutal transport rates. By revealing the complex interdependencies between these parameters, the diagrams serve as essential tools for rigorous engineering analysis, design refinement, and process optimization. Their capacity to highlight intricate relationships makes them invaluable assets in advancing the understanding of these critical transport phenomena. Figures 26 and 27 exemplify the impression of parameters $$$ {K}_A $$$ and $$$ \phi $$$ on the wall shear stress on both nanofluids, respectively. It is detected that a rise in both $$$ {K}_A $$$ and $$$ \phi $$$ results in an elevation in drug friction. Figures 28 and 29 demonstrate the consequence of parameters $$$ {B}^{\ast } $$$ and $$$ n $$$ on the rate of thermal transport $$$ {N}_u $$$. It is detected that an expansion in $$$ {B}^{\ast } $$$ results in a lessening in $$$ {N}_u $$$. Conversely, when $$$ n $$$ values are amplified, $$$ {N}_u $$$ exhibits an upward trend. Figures 30 and 31 portray the performance of $$$ \alpha $$$ and $$$ {A}^{\ast } $$$ on the Sherwood number $$$ {S}_h $$$. It is perceived that the upgrading $$$ \alpha $$$ lessens the Sherwood number and enlarges when $$$ {A}^{\ast } $$$ values are augmented.

6 Conclusions

Nomenclature