Multiscale tribology analysis of MHD hybrid nanofluid flow over a curved stretching surface

In this study, we investigate the interactions of a hybrid nanofluid on a curved surface that is being stretched. The magnetic field is perpendicular to the flow and interacts with a mixture of molybdenum disulfide and argentum nanoparticles suspended in pure water, forming a hybrid nanomaterial. Our investigation considers heat transport analysis under different conditions, such as magnetohydrodynamic, Darcy–Forchheimer porous medium flow, Joule heating, and a convective boundary condition. We employ numerical and statistical methods to study the problem's intricacies comprehensively. Our findings indicate that Darcy–Forchheimer flow includes viscous and inertial forces, which results in higher flow rates and reduced skin friction. Additionally, the convective boundary condition leads to uniform temperature distribution within the hybrid material due to rapid internal heat transfer relative to surface resistance, significantly increasing the heat transfer rate.


Introduction
Hybrid nanouids are advanced colloidal suspensions that combine nanoparticles from different materials to create unique and synergistic properties.Incorporating different nanoparticles involves blending these nanoparticles to create a specialized nanouid with enhanced thermal, electrical, and other functional characteristics.Lee et al. 1 reviewed conicting ndings on the thermal conductivity of nanouids.The study highlighted experimental milestones, proposed mechanisms and models, addressed data inconsistencies, and suggested directions for future research.The aim was to optimize nano-uids for enhanced thermal properties in various applications.Lee et al. 2 developed surfactant-free zinc oxide nanouids using a pulsed wire evaporation method in ethylene glycol.Five labs conducted round-robin tests to measure the thermal conductivity of three samples using in-house setups and a commercial device.Choi et al. 3 discussed nanouids' internal forced convective heat transfer attributes with experimental features.Nadeem et al. 4 inspected hybrid nanomaterial distribution by considering two distinct nanoparticles in pure water over a curved surface.Waqas et al. 5 numerically computed melting heat transfer in the nonlinear radiative ow of hybrid nanouids due to permeable stretching curved surface.Wahid et al. 6 explored the ow and heat transfer of a hybrid nanouid induced by an exponentially stretching/shrinking curved surface.More interesting studies regarding nanouids are addressed in ref. 7-22.Regression analysis is a statistical technique used to establish empirical correlations between parameters.It has practical applications in uid mechanics, allowing researchers to understand, predict, and optimize various uid behaviors.These correlations are crucial for characterizing ow patterns, predicting pressure drops, and modeling turbulence in complex uid systems.It also supports the analysis of experimental data and validation of numerical simulations and helps to design efficient systems while making informed decisions in uid mechanics.Moreover, quadratic and multiple quadratic regression analyses are specialized approaches that can accommodate linear and non-linear relationships between variables and establish predictive models that enhance the understanding and manipulation of uid behavior.Najm 23 examined polynomial chaos methods for probabilistic uncertainty quantication in computational uid dynamics (CFD) predictions.The study reviewed various CFD applications and challenges, such as ow in porous media, incompressible and compressible ows, thermouid and reacting ows, and crosscutting challenges related to time unsteadiness and longtime horizons.Zhang et al. 24 presented an improved airfoil design using a modied shape function transformation.The design was validated through computational uid dynamics and experimental testing.Optimization was performed using a modied multi-island genetic algorithm combined with nonlinear programming.The resulting optimized airfoil demonstrated enhanced performance and li-to-drag ratio, providing insights for future airfoil design.Kumar et al. 25 employed spectral relaxation to investigate nanouid ow over a porous stretching sheet, considering slip, mixed convection, dissipations, and nanoparticle control.The study discussed velocity, temperature, and concentration graphs; veried prior ndings; conducted regression analysis for local Nusselt numbers; and highlighted the impact of thermophoretic and Eckert number factors on diffusion.Liu et al. 26 used quadratic regression orthogonal combination (QROC) and a genetic algorithm (GA) to optimize the coal pyrolysis ltration system's performance and extend the lter tube lifespan with compelling predictions and low mean square error.References are provided for studies discussing issues associated with regression analysis (ref.27-33).
The signicance of this study lies in examining the combined effects of molybdenum disulde and argentum nanoparticles suspended in a water solution over a curved surface, inuenced by an inward magnetic eld, magnetohydrodynamics, Darcy-Forchheimer porous medium ow, Joule heating, and a convective boundary condition.This yields insights into intricate heat transport interactions through a hybrid nanomaterial and porous medium.The reason we have combined molybdenum disulde (MoS 2 ) and silver (Ag) nanoparticles is their versatility and compatibility with different nanomaterials (their mixture can be customized to achieve specic results).The investigation employs numerical and statistical methodologies for a comprehensive study of the intricacies of the problem.The ndings of this study hold relevance in various applications such as advanced thermal management systems, nanoparticle-enhanced heat exchangers, and innovative cooling technologies where the combined effects of molybdenum disulde and argentum nanoparticles suspended in a water solution can be harnessed to optimize heat transfer efficiency and system performance.
The article was structured as follows: Section 2 encompassed mathematical modeling, Section 3 presented regression analysis, Section 3 contained a discussion of results, and Section 4 offered a summary of the ndings.

Modeling of the problem
A two-dimensional ow of a hybrid nanouid is considered over a curved surface with a radius "a 0 ," which is undergoing stretching.The problem is formulated in curved coordinates (r,z), and an applied magnetic eld of intensity "B 0 " with an inward perpendicular direction to the ow is present.This eld interacts with a mixture of molybdenum disulde (MoS 2 ) and argentum (Ag) nanoparticles suspended in a pure water (H 2 O) solution, creating the hybrid nanomaterial.The velocity eld is V = [U(r,z), V(r,z), 0].The analysis encompasses heat transfer effects while accounting for magnetohydrodynamics, Darcy-Forchheimer porous medium ow, Joule heating, and a convective boundary condition.The investigation involves both numerical and statistical approaches to study the problem comprehensively.Fig. 1 provides a visual representation of the problem's geometry, while Table 1 offers essential thermophysical values for both the nanoparticles and the base uid.
By incorporating the assumptions mentioned above and considering conditions such as negligible viscous dissipation ), the following set of PDEs can be derived (following ref. 7): with Uðr; zÞ ¼ U w ðzÞ ¼ U 0 z; V ðr; zÞ ¼ 0; À C****k bf vTðr; zÞ vr ¼ g 0 ½T w À Tðr; zÞ; at r ¼ 0; Uðr; zÞ/0; vUðr; zÞ vr /0; Tðr; zÞ/T N ; as r/N: For hybrid nanouid the electrical conductivity is dened by Hybrid nanomaterial density by the Xuan and Li model is Dynamic viscosity via the Brinkman model for the hybrid nanomaterial is The Maxwell model dened thermal conductivity of the hybrid nanomaterial using For the conversion of ow as mentioned above and heat transfer related PDEs and associated BCs into non-dimensional form, we introduce the following dimensionless non-similar variables (following ref. 7).
Aer making use of the associated velocity component in terms of stream function, we get eqn (1) identically satised, and the rest of the equations (eqn (2)-( 5)) take the following form aer using rst-order truncation vðÀÞ vz ¼ 0 :: Eliminating P from eqn ( 13) and ( 14), we get with BCs where Aer using appropriate substitutions in eqn (19)

Discussion
The primary objective of this section is to analyze the responses of ow (F h (x,h)), temperature (q(x,h)), skin friction ðC f ffiffiffiffiffi ffi Re p Þ; and Nusselt number Nu ffiffiffiffiffi ffi Re p with the variation of different physical parameters.

Discussion of methodology
The governing non-dimensional PDEs and their BCs are solved using the NDSolve numerical tool in Mathematica.It simplies the solving process by automatically selecting the most appropriate method and controlling the adaptive step size, making it easier for users to solve complex differential equations with high accuracy.Similarity transformations are applied to simplify the PDEs, treating them as ODEs.NDSolve automatically selects a numerical method suitable for the equations and desired precision.The size of time or space steps is adjusted Nanoscale Advances Paper using NDSolve as the solution progresses, focusing computational effort where it is needed.Starting from the initial conditions and progressing to the desired nal time or spatial domain, NDSolve integrates the equations over the specied range.It returns the numerical solution, oen as interpolating function objects, which can be used for analysis and visualization.Mathematica's built-in functions can create plots and visual representations of the solution.The technique also provides information about the accuracy of the solution, including error estimates.Each graph presented in this section offers a comparative analysis of the behavior of hybrid (MoS 2 + Ag + water) and nanouid (MoS 2 + water) solutions.The nanouid consists of MoS 2 nanoparticles suspended in a water base uid, while the hybrid nanouid incorporates two types of nanoparticles: MoS 2 and Ag, both dispersed in a water base uid.Throughout the comparative study, f np1 = 0.05 = f np2 is kept constant for the MoS 2 + Ag + water case, whereas f np1 = 0.1 and f np2 = 0 are xed for MoS 2 + water.The results are visually depicted using solid lines for hybrid nanouid (MoS 2 + Ag + water) and dashed lines for nanouid (MoS 2 + water) solutions.Moreover, Table 1 is constructed for thermophysical properties of nanoparticles and base uid while Tables 2 and 3 are for regression coefficients of skin friction and Nusselt number respectively.Fig. 2(a) and (b) illustrate the response of M* on F h (x,h) and q(x,h).It is seen that with the variation of M* from 0.1 to 0.4, the opposition force created by B 0 causes the ow eld to decline.Physically, this is because particles experience resistance which  slows their motion and consequently, lowers F h (x,h), see Fig. 2(a).However, due to B 0 , Joule heating promotes the energy transport among the particles.Physically, the system's kinetic energy boosts, which eventually enhances q(x,h), see Fig. 2

(b).
The behavior is the same for hybrid nanouid (MoS 2 + Ag + water) and nanouid (MoS 2 + water) solutions.However, the MoS 2 + Ag + water solution causes a more prominent effect.
Fig. 3(a) and (b) depict the outcome of m* on F h (x,h) and q(x,h).
The variation is recorded from 0.1 to 0.4.This parameter is generated due to the Hall effect.Physically, it causes charged particles in the uid to experience a Lorentz force as the particles move through a magnetic eld.The uid's dynamics is affected by this interaction, which leads to an increase in F h (x,h), see Fig. 3(a).Specically, it amplies existing driving Fig. 5 Impact of g* on F h (x,h) (a) and q(x,h) (b).

Nanoscale Advances Paper
forces for uid motion, resulting in accelerated ow in certain regions.On the other hand, the Hall effect's Lorentz force does not only impact F h (x,h) but can also affect q(x,h).When charged particles move through the magnetic eld, their trajectories can be altered by the force, leading to changes in q(x,h).This effect can even cause a decrease in q(x,h) in some regions of the uid due to the redistribution of thermal energy resulting from the interaction between the Lorentz force and the uid ow, see Fig.However, the opposite trend is visible for Nu ffiffiffiffiffi ffi Re p : Physically, due to the complex interplay between the Hall effect and magnetic eld the energy transport rate is declined, see Fig. 4(b).Fig. 5(a) and (b) describe the effect of g* on F h (x,h) and q(x,h).The impression noted suggests that when uid ows through a curved path, its inertia causes curvature to generate centrifugal forces.These forces are directed away from the center of curvature and can lead to an increase in F h (x,h) on the outer side of the curve, see Fig. 5(a).This increase in velocity oen results in higher kinetic energy, which can elevate q(x,h) due to the conversion of kinetic energy into thermal energy through dissipation, see Fig. .The Darcy-Forchheimer law expands upon Darcy's law to account for inertial effects in porous media.It highlights that as F h (x,h) decreases, see Fig. 6(a), inertial forces accounted for result in lower ow rates compared to predictions made using Darcy's law (which only considers viscous forces).However, the effect of Fr* on C f ffiffiffiffiffi ffi Re p is more complex.C f ffiffiffiffiffi ffi Re p is determined using the s w at the solid-uid interface, and both viscous and inertial affect it.Physically, an increase in inertial forces counteract the increase in C f ffiffiffiffiffi ffi Re p caused by lower velocity, resulting in a net inclination of C f ffiffiffiffiffi ffi Re p ; and see Fig. 7 for the behavior of hybrid nanouid (MoS 2 + Ag + water) and for nanouid (MoS 2 + water) solutions.However, l* suggests that due to porosity, there are more spaces available for uid to ow through.This increased porosity creates additional pathways for uid movement, resulting in higher F h (x,h).The availability of open spaces allows uid to move more freely through the medium, leading to greater F h (x,h), whereas l* results in a lower volume of the solid material within the medium, which reduces the resistance against uid ow.Therefore, there is less interaction between the uid and solid surfaces, resulting in C f ffiffiffiffiffi ffi Re p ; see Fig. 7. Fig. 8(a)-(d) dene the reaction of f np1 and f np2 on F h (x,h) and q(x,h) for hybrid nanouid (MoS 2 + Ag + water) and for nanouid (MoS 2 + water) solutions.The f np1 notation is for MoS 2 and f np2 is for Ag.Due to the increment in f np1 and f np2 , F h (x,h) is augmented.Physically, by increasing f np1 and f np2 , F h (x,h) and q(x,h) are heightened due to the enhanced thermal conductivity of the nanoparticles.This increases heat transfer rates within the uid, creating a temperature gradient that drives uid motion through thermally induced convection.The energy absorption by the nanoparticles and their transfer to the uid contribute to increasing q(x,h).Note that the MoS 2 + Ag + water solution causes a more prominent effect.Thus, an increment in   Nanoscale Advances Paper f np1 and f np2 improves heat transfer properties and thermally driven uid motion, leading to enhanced F h (x,h) and q(x,h), see From Table 2, it is seen that the absolute value of the regression coefficient of M* is greater than the regression coefficient of m*.Hence, the effect of M* is prominent over m* on skin friction ðC f ffiffiffiffiffi ffi Re p Þ: Similarly, from Table 3, it is observed that the absolute regression coefficient of Ec* is more than that of g*.Therefore, it is concluded that Ec* is more effective for Nusselt number Nu ffiffiffiffiffi ffi Re p in comparison with g*.

Final remarks
This study investigated the complex interactions of a hybrid nanouid owing over a curved surface subjected to stretching.A magnetic eld perpendicular to the ow and directed inward engaged with a composite of molybdenum disulde and argentum nanoparticles suspended in a pure water solution, forming a hybrid nanomaterial.The study analyzed heat transport through magnetohydrodynamics, Darcy-Forchheimer porous medium ow, Joule heating, and a convective boundary condition.Using numerical and statistical methods, the study gained valuable insights into the behavior of the hybrid nanouid system under specied conditions.The main key points are: ❖ The MoS 2 + Ag + water solution caused a more prominent effect than MoS 2 + water.
❖ Joule heating promoted the energy transport among the particles, while Lorentz force caused the ow eld to lessen, resulting in an increment in skin friction.❖ The Hall effect affected uid's dynamics by increasing velocity, while decreasing skin friction.However, thermal transport declined but the Nusselt number boosted up.
❖ Curvature augmented velocity and causes conversion of kinetic energy into thermal energy, which inclined thermal transport.
❖ The volume fraction improved heat transfer properties and thermally driven uid motion.
❖ Darcy-Forchheimer ow included viscous and inertial forces which resulted in higher ow rates and diminished skin friction.
❖ Porosity declined ow and enhanced skin friction.❖ The Biot number caused uniform temperature distribution within the material due to rapid internal heat transfer relative to surface resistance.
❖ The biot number augmented the heat transfer rate signicantly.
❖ The Eckert number caused heat capacity to enhance and elevated thermal transport; however, it lessened the Nusselt number.
❖ Regression analysis suggested that the effect of the Eckert number on the Nusselt number was more than that of the Biot number.
❖ The impact of Darcy-Forchheimer ow through regression analysis was prominent over the porosity parameter on skin friction.

Limitations and scope of future work
The study's use of numerical and statistical methods demonstrates its potential for insightful analysis.However, experimental validation is necessary to enhance the model's credibility further and ensure its practical application accuracy.Although the ndings are specic to the investigated parameters, they provide a valuable basis for comprehending the behavior of hybrid nanouids and allow for customized applications in a broad range of scenarios.Real-world complexity, including factors such as turbulence and impurities, emphasizes the importance of future research to expand on this foundation, considering the intricate nature of dynamics of uid in various environments.

Fig. 4
Fig. 4 Impact of M* against m* on C f ffiffiffiffiffiffi Re p (a) and Nu ffiffiffiffiffiffi Re p (b).
3(b).Fig. 4(a) and (b) depict the impact of M* against m* on The variation of M* is from 0.1 to 0.4.The opposing force causes the particles to experience resistance, which is skin drag.Thus, due to variation of M*, C f ffiffiffiffiffi ffi Re p enhances, but m* causes C f ffiffiffiffiffi ffi Re p to lessen, see Fig. 4(a).
5(b).Fig. 6(a) and (b) explain the impact of Fr* and l* on F h (x,h).Fig. 7 identies the inuence of Fr* against l* on C f ffiffiffiffiffi ffi Re p

Fig. 7 Fig. 8
Fig. 7 Impact of l* against Fr* on C f ffiffiffiffiffiffi Re p :

Fig. 9
Fig. 9 Impact of g* and f np2 against f np1 on C f ffiffiffiffiffiffi Re p (a and c) and Nu ffiffiffiffiffiffi Re p (b and d).

Fig. 8 (
a)-(d).Fig. 9(a)-(d) provide physical signicance of g* and f np2 against f np1 on C f ffiffiffiffiffi ffi Re p and Nu ffiffiffiffiffi ffi Re p : The graphs are designed for hybrid nanouid (MoS 2 + Ag + water) and for nanouid (MoS 2 + water) solutions.As g* enhanced F h (x,h), see Fig. 5(a), it causes a reduction in opposition forces, see Fig. 9(a).Therefore, C f ffiffiffiffiffi ffi Re p declines but due to f np1 , C f ffiffiffiffiffi ffi Re p is boosted up.Similarly, the heat transfer rate is reduced due to an augmentation in g*, see Fig. 9(b).The effect of f np1 causes Nu ffiffiffiffiffi ffi Re p as well.The physics behind this is that the better thermal conductive properties of MoS 2 augment Nu ffiffiffiffiffi ffi Re p : Fig. 10(a) and (b) give explanation of the impact of b and Ec* on q(x,h).Fig. 11 identies the inuence of Ec* against b on Nu ffiffiffiffiffi ffi Re p for hybrid nanouid (MoS 2 + Ag + water) and for nanouid (MoS 2 + water) solutions.An increment in b represents a shi from efficient internal heat transfer to surface resistance as the dominant factor in heat transfer processes for both qðx; hÞ and Nu ffiffiffiffiffi ffi Re p : For b ( 1, a uniform temperature distribution increment is seen within the material due to rapid internal heat transfer relative to surface resistance.In contrast, b [ 1 indicates slower internal heat transfer than surface heat transfer, which can result in temperature gradients within the material and notable differences between surface and bulk temperatures, see Fig. 10(a) and 11.Meanwhile, increasing Ec* signies a greater signicance of kinetic energy changes in a uid than heat transfer rates.This can increase q(x,h), while decreasing Nu ffiffiffiffiffi ffi Re p for hybrid nanouid (MoS 2 + Ag + water) and for nanouid (MoS 2 + water) solutions, see Fig. 10(b) and 11.

Fig. 11
Fig. 11 Impact of Ec* against b on Nu ffiffiffiffiffiffi Re p :

Table 1
Nanoparticle (MoS 2 and Ag) and base fluid (H 2 O) thermal features multiple quadratic regression model is a statistical technique for modeling the relationship between a dependent variable and one or more independent variables.For this purpose we have taken 51 values of l* and Fr*, such that l* ˛[0.1,0.5] and Fr* ˛[0.1,0.7].In the case of regression analysis of Nu ffiffiffiffiffi ffi Re p ; we have chosen 51 values of Ec* and b such that b ˛[0.1,0.5] and Ec* ˛[0.1,0.3].During these analyses all other physical parameters are kept xed.The multiple quadratic regression A

Table 2
Regression coefficients of skin friction ðC f