Coupled Effects of Lorentz Force, Radiation, and Dissipation on the Dynamics of a Hybrid Nanoﬂuid over an Exponential Stretching Sheet

: The ﬂow and heat transfer induced by an exponentially shrinking sheet with hybrid nanoparticles are investigated comprehensively in this paper. Nanoparticles are considered due to their unusual characteristics such as extraordinary thermal conductivity, which is signiﬁcant in advanced nanotechnology, heat exchangers, material sciences, and electronics. The main objective of this research is to enhance heat transportation. The ﬂow model is ﬁrst transformed and simpliﬁed to a system of ordinary differential equations utilizing non-dimensional quantities and similarity functions. Then, the desired system is solved with the help of the Runge–Kutta numerical method and the shooting technique in MATLAB script. The results show that a stronger porosity parameter raises the temperature while diminishing the velocity. Additionally, they emphasize that augmentations in the magnetic parameter, Eckert number, radiation parameter, and the volume fractions of titanium dioxide and silver nanoparticles are all proportional to the temperature proﬁle.


Introduction
Nanofluids are specific types of fluids that have nanoparticles scattered throughout a conventional fluid, usually liquids such as water, engine oil, or kerosene.These conventional heat transfer fluids possess limited heat transfer capabilities due to their relatively low thermal conductivity.This limitation hinders their effectiveness in enhancing the performance and compactness of various engineering electronic devices.As a consequence, there is a great need to develop heat transfer fluids with considerably increased conductive qualities as a means of improving their thermal properties.The presence of nanoparticles in fluids can substantially augment both the effective thermal conductivity and viscosity of the base fluid, leading to a significant enhancement in its heat transfer capabilities.Early research looked into the ways in which nanofluids can improve heat transfer.Choi et al. [1] presented experimental data showing enhanced thermal performance in comparison to base fluid when analyzing the effects of nanoparticle dispersion.This study contributed to the understanding of nanofluid behavior and its potential applications.Khan and Pop [2] were the first to analyze the development of the steady boundary layer flow, heat transfer, and nanoparticle fraction over a stretching surface in a nanofluid.Hayat et al. [3] studied the impact of silver and copper nanoparticles on the entropy of a viscous fluid produced by a rotating disk, finding that the surface drag force and Nusselt number both increase as the volume fraction of the nanoparticles is enhanced.Hsiao [4] presented heat and mass transfer in a hydromagnetic flow featuring magnetic and viscous dissipation effects with micropolar nanofluids near a stretching sheet.It was assumed that magnetic field and Joule heating were induced.References to nanofluids can be found by Akbar et al. [5], Mustafa et al. [6], Ibrahim et al. [7], Mathews and Hymavathi [8], and Javed et al. [9], all of which have demonstrated the importance of the stagnation fluid flow.
In recent years, there has been growing interest in the innovative concept of enhancing heat transfer within boundary layer flow problems through the application of hybrid nanofluids.It is believed that these characteristics can significantly improve heat transfer rates in comparison to regular nanofluids, primarily due to their synergistic effects [10].Devi and Devi [11] conducted a study focusing on the flow past a permeable stretched surface while considering the influence of hydromagnetic effects.They utilized a hybrid nanofluid consisting of different nanoparticles suspended in water, and introduced a novel thermophysical model specific to hybrid nanofluids.Recently, Khashi'ie et al. [12] investigated the flow of a hybrid nanofluid over a shrinking disc, observing that the heat transfer rate decreased as the hybrid nanoparticles concentration increased.This phenomenon was attributed to the greater suction force applied to the shrinking sheet.Various researchers have used the thermophysical approach to examine the influence of physical factors on hybrid nanofluid flows, including Leong et al. [13], Gul and Saeed [14], Basit et al. [15], Mahabaleshwar et al. [16], Kayalvizhi and Kumar [17], and Waqas et al. [18].
The flow over a stretching surface plays an important role in many engineering processes, including extrusion, metal spinning, hot rolling, wire drawing, crystal growing, production of glass fiber, production of plastic and rubber sheets, cooling of filaments, and more.In the historical context of fluid dynamics, the investigation of a steady twodimensional flow over a linearly stretched surface can be attributed to the pioneering research conducted by Crane [19].Magyari and Keller [20] were the first to propose the use of an exponentially stretched surface to investigate the distribution of the wall temperature and its effect on the flow and heat transfer characteristics.They found that for any fixed Prandtl number, raising the temperature distribution parameter reduces the thickness of the thermal boundary layer.Hayat et al. [21] worked on the melting heat transfer and radiation effects within the stagnation point flow of a carbon-water nanofluid.Their findings revealed that higher values of the melting parameter in combination with smaller nanoparticle volume fractions led to a decrease in the skin friction coefficient.Mabood et al. [22] developed an analytical solution for MHD boundary layer flow over an exponentially stretching sheet while considering thermal radiation.They found that the magnetic and radiation parameters significantly influence the flow characteristics, resulting in increased temperature within the thermal boundary layer and friction.Recently, Zaib et al. [23] studied the impact of nanoparticles on flow and heat transfer over an exponentially shrinking sheet, discovering that the thermal boundary layer thickness is enhanced as the nanoparticle volume fraction increases due to the shrinking sheet.Following this innovative study, flow field over extending surfaces has received considerable attention, and a great deal has been written on it.Examples include Aziz [24], Dessie and Kishan [25], Cortell et al. [26], Rafique et al. [27], Alharbi et al. [28], Yirga et al. [29], and Hayat et al. [30].
In our study, we consider the coupling of the Lorentz force, radiation, and dissipation effects together, which is essential in order to provide a comprehensive understanding of nanofluid behavior.These interactions are all interconnected and collectively influence the heat transfer and flow characteristics of nanofluids, especially in scenarios involving magnetic fields and elevated temperatures.Investigating them together can provide a comprehensive and precise representation of nanofluid phenomena [31].
Although many outcomes have been reported in the literature, they have not been properly analyzed as a whole.Therefore, the objective of the present study is to investigate the effects of hybrid nanoparticles on fluid flow and heat transfer induced by an exponentially stretching sheet.A model of titanium dioxide (TiO 2 ) and silver (Ag) nanoparticles with water (H 2 O) as the base fluid is assumed.This approach provides an innovative perspective on modeling the various forces involved in heat transfer for such hybrid nanofluids.For validation purposes, the numerical outcomes of this study are compared with previously published data.The current research has applications in metal spinning, drawing of plastic films, glass blowing, crystal growth, and filament cooling [32].The objective of this study is to provide numerical answers to the following research questions: • What impact do the relevant parameters have on the flow profiles?• What changes occur in the heat transfer rate due to the hybrid nanofluid?• What are the impacts of the critical parameters on the physical quantities?

Mathematical Formulation
A time-independent 2-dimensional TiO 2 -Ag/H 2 O hybrid nanofluid flow is examined via an exponentially stretching sheet.The x-axis lies parallel to the stretching surface and the y-axis is perpendicular to the surface, as shown in Figure 1.The sheet accelerates in the x-direction (x ≥ 0) at an exponential rate Ūw = c 0 e x l , and its velocity profile is vw = v 0 e x 2l .We denote the reference length as l, while c 0 and v 0 are both positive constants.The sheet is kept at an exponentially changing temperature Tw = T∞ + T0 e  Under the above conditions, the governing equations are as follows [33,34].Continuity equation: ūx + vy = 0. ( Momentum equation: Energy equation: (3) Equation ( 1) describes the mass conservation for an incompressible flow, while Equations ( 2) and (3) introduce the rate of convection change on the left-hand side.In Equation ( 2), the right-hand side consists of three distinct terms; the first represents viscous effects, the second signifies body forces, particularly magnetic effects, and the third relates to thermal conductivity.In Equation (3), the right-hand side can be categorized similarly; the first term deals with thermal diffusion, the second involves flux gradients, the third encompasses heat generation and temperature differences, and the fourth accounts for viscous dissipation.
The boundary conditions are provided by For the Rosseland approximation q r [35], Equation (3) can be written as follows: Next, the similarity transformation [36,37] is introduced: In view of the above appropriate relations, Equation ( 1) is satisfied identically; Equations ( 2) and (3) respectively become [33,34] and the boundary conditions can be transformed into The aforementioned system was solved numerically by the Runge-Kutta method with the shooting technique in MATLAB R2023a, Mathworks, Lodz, Poland.The availability of the chosen numerical method was demonstrated by the precise agreement of the surface temperature gradient with previously published results, shown in the remarkable agreement appearing in Table 1.
The associated non-dimensional parameters are provided by The wall shear stress and thermal flux are respectively provided by the relations The skin friction C f x and Nusselt number Nu x in dimensionless form are The thermophysical properties of the base fluid and nanoparticles are shown in Table 2.  [34].

Properties Hybrid Nanofluid
Dynamic Viscosity Thermal Conductivity where

Implementation of the Method
The system of ODEs in Equations ( 7) and ( 8), subject to the boundary conditions in Equations ( 9) and (10), is solved numerically using the shooting method [39].To convert Equations ( 7) and (8) into a system of first-order ODEs, we introduce the new variables f = y 1 , f = y 2 , f = y 3 , θ = y 4 , θ = y 5 to make up the following set of five first-order ordinary differential equations: . The corresponding dimensionless boundary conditions are To solve the above system of first-order ODEs using the shooting method, five initial conditions are required.To do this, we first guess two unknown initial conditions y 3 (0) = p and y 5 (0) = q.Suitable guesses for two unknown missing conditions p and q are chosen such that the two known boundary conditions are approximately satisfied for η → ∞.Newton's iterative scheme is applied to improve the accuracy of the missing initial conditions p and q until the desired approximation is met.Our computations were performed for the various emerging parameters and the appropriate bounded domain [0, η max ] instead of [0, ∞), with the positive real number η max chosen to ensure that no significant variations appear in the results for values greater than η max .The stopping criteria for the iterative process is where ζ is a very small positive real number.

Validation of the Numerical Scheme
For all our computations, ξ was chosen as 10 −7 .To further ensure the reliability and validity of the code, the results of −θ (0) were checked against the rising inputs of M, R d , and Pr as reported by Goud et al. [33], Ishak et al. [36], and Bidin et al. [38], which were successfully reproduced as presented in Table 1.Furthermore, the Matlab-created RK computations showed a high rate of convergence.Thus, our numerical calculations are validated.

Results and Discussion
The system of ODEs in Equations ( 7) and ( 8) was solved numerically by implementing the Runge-Kutta method with the shooting technique in MATLAB.The basic discretization methods were the finite difference method (FDM), finite volume method (FVM), and finite element method (FEM).However, the computational costs and time required by these methods are much higher for calculating unknowns as compared to the Runge-Kutta method.The Runge-Kutta method with shooting technique is a powerful scheme for solving ODEs and finding the solutions of flow problems.The physical influence of suitable parameters on the velocity and temperature profiles is demonstrated in Figures 2-12.
The velocity profiles in Figure 2 demonstrate that the rate of transport decreases significantly when the magnetic parameter M leads to an increase in the Lorentz force.This indicates that the transverse magnetic field opposes transport, likely due to increased resistance caused by changes in the Lorentz force.However, the velocity gradient representing the surface shear stress increases with rising M. Therefore, M acts as a control parameter for the surface shear stress.Figure 3 depicts the variation of the velocity f (η) with η for several values of the porosity parameter K p .This is explained physically by the fact that an increase in the K p parameter reduces fluid velocity, as higher K p values indicate a greater proportion of larger pores within the porous medium.This increased void space offers less resistance to fluid flow, allowing the fluid to move more slowly through the medium.The impact of the suction parameter s upon the velocity field can be seen in Figure 4.It can be observed that increasing the suction parameter s in a fluid flow leads to a significant decrease in velocity.This is consistent with the physical reality that when s increases, the stretched sheet that interrupts the flow momentum remains with the boundary layer of momentum, causing a decrease in the boundary layer thickness and a reduction in the velocity field.
Figure 5 shows the impact of the magnetic parameter M on the temperature profiles.From the profiles, it can be seen that an increase in M leads to the production of a Lorentz force that increases the friction between the fluid layers, in turn causing thicker thermal boundary layers that improve the temperature profile.The influence of the radiation parameter R d on the temperature field θ(η) can be seen in Figure 6.Increasing the radiation parameter R d leads to an increase in temperature, as higher R d values indicate a greater contribution of radiative heat transfer, elevating the overall temperature within the system.Thus, radiative heat transfer becomes more significant, resulting in higher temperatures, especially near radiative heat sources.Figure 7 demonstrates the favorable effect of the Eckert number Ec on the temperature field, with an increase in the Eckert number Ec leading to an increase in temperature.Physically, higher values of Ec signify that kinetic energy dominates over thermal energy in the fluid flow, resulting in increased temperature.Figures 8 and 9 respectively show how the heat source β and porosity parameters K p produce an internal heat source (which can be seen in [40,41]) that boosts the fluid temperature.Figure 10 visualizes the effect of the suction parameter s on the temperature field.It can be observed that increasing the suction parameter s leads to a decrease in temperature.Physically, suction removes fluid from the system, which can result in reduced temperatures, particularly near the suction point.Figures 11 and 12 respectively show the favorable influence of the volume fraction of Ag and TiO 2 nanoparticles on θ(η).Physically, this because the inclusion of Ag and TiO 2 nanoparticles increases particle collision and improves fluid thermal conductivity.
The effects of significant parameters on the skin friction coefficient and Nusselt number are presented in Tables 3 and 4, respectively.With modifications in the values of M, K p , s, φ 1 , and φ 2 , the drag coefficient diminishes; see Table 3.These variables constrict the flow, which in turn reduces the drag coefficient.Because R d enhances the surface heat flux, the thermal transportation rises with R d , and conversely decreases with increased values of M, Ec, β, φ 1 , and φ 2 (see Table 4).Table 5 shows the values of thermophysical features of water and used nanoparticles.

Conclusions
In the paper, we have comprehensively examined the effects of hybrid nanoparticles on the fluid flow and heat transfer induced by an exponentially stretching sheet.For validation purposes, our numerical outcomes were compared with previously published data, with good agreement being achieved.The influences of the different parameters on fluid profiles were numerically solved using the Runge-Kutta method with shooting technique.Our significant findings are summarized below:

•
The velocity profile is adversely affected by the magnetic field, whereas the temperature is positively impacted.

•
The velocity profile is lowered by an increase in the porosity parameter, while the temperature is increased.

•
The temperature profiles are proportional to the changes in the radiation parameter, magnetic field parameter, Eckert number, and volume fractions of TiO 2 and Ag nanoparticles.

•
The magnetic, suction, and porosity parameters are all inversely linked to the drag coefficient, as are the volume fractions of TiO 2 and Ag nanoparticles.

•
The radiation parameter has a positive correlation with the heat transfer rate, while the magnetic parameter, heat source parameter, viscous dissipation parameter, and volume fractions of TiO 2 and Ag nanoparticles all have negative correlations.
In future research, the analysis provided in this paper could be expanded by taking into account other non-Newtonian fluids, as well as favorable effects such as nonlinear thermal radiation and space-dependent heat sources.

x
2l , where T0 is the reference value and T∞ is the ambient value.The variable magnetic field is Bx = B0 e x 2l , where B0 represents the uniform magnetic strength normally applied to the flow region.The effects of porosity Kx = K0 e −x l and changing heat source Qx = Q0 e x l are considered as well.

Figure 1 .
Figure 1.Geometry of the problem.

Table 1 .
Comparison of −θ (0) with different values of R d , M, and Pr.

Table 2 .
The hybrid nanofluid's efficient thermophysical models, as provided by