Entropy production with the flow of nanomaterials through the permeable stretched surface with heterogeneous–homogenous chemical reaction

In various thermodynamic procedures and the optimisation of thermal manipulation, nanofluids flowing through porous media represent an emerging perspective. The main objective of this study, from the perspective of thermal applications, was the investigation of the flow of nanofluid over a horizontal stretched surface embedded in a porous medium. The effects of the chemical reactions on the surface, magnetic field, and thermal radiations were invoked in the mathematical formulation. The non-Darcy model examines the fluid flow in the porous media. The principles of thermodynamics were employed to integrate entropy optimisation methods with the established theoretical approach to analyse the thermal behaviour of nanomaterials in the chemical reactive diffusion processes. Using the Tiwari-Das nanofluid model, the volume fraction of the nanomaterials was merged in the mathematical equation for the flow model. Water was taken as a base fluid and nanoparticles composed of aluminium oxide (Al2O3) and silver (Ag) were used. The significance of radiation, heat production, and ohmic heating were included in the energy equation. Furthermore, an innovative mathematical model for the diffusion of the autocatalytic reactive species in the boundary layer flow was developed for a linear horizontally stretched surface embedded in a homogeneous non-Darcy porous medium saturated with the nanofluid. The computer-based built-in bvp5c method was used to compute numerically these equations for varied parameters. It is clear that the magnetic parameter has a reversal influence on the entropy rate and velocity. Temperature and velocity are affected in the opposite direction from a higher volume fraction estimate. Thermal field and entropy were increased when the radiation action intensified. The inclusion of nanoparticle fraction by the volume fraction of nanoparticles and Brinkman number also enhances the system entropy. Entropy production can be minimized with the involvement of the porosity factor within the surface.


Introduction
Under diverse physical conditions, thermal energy is transported primarily through conduction, convection, and radiation.Radiation has a number of advantages over the other two modes of heat transfer, conduction and convection. 1 Electromagnetic waves are capable of moving through a vacuum because of the radiative heat ux.For heat transport, both conduction and convection require a medium.Convective or conductive heat transport depends linearly on a temperature difference, while radiant heat transport is proportional to the difference in the fourth power of temperature.Radiative heat transfer is increasingly essential at higher temperatures and is used in a variety of engineering applications, including combustion, 2 nuclear power technology, 3 atmospheric reentry, 4 and solar energy. 5Convection and conduction are manifestations of short-range phenomena that are governed by the conservation of energy across any small volume.In contrast to the long-range phenomenon of radiation, the emission, absorption, and reection from the wall will immediately alter the total radiative heat transfer.Every object with a temperature greater than 0 K naturally emits thermal radiation, which has been extensively employed in energy harvesting, thermal transfer, and management. 6A specic method of transferring energy between two objects via thermal radiation is known as radiative heat transfer.Radiant heat (RH) is referred to as far-eld radiative heat transfer when the distance between the heat source and the heat sink is signicantly greater than the characteristic thermal wavelength. 7The Stefan-Boltzmann law limits the radiative heat ux, which is primarily determined by propagating waves.Coupled models have signicant problems with radiative heat transfer for combustion applications.This heat transport mechanism is the most common and dictates accurate modelling in environments with high temperatures and concentrations of radiative species.
The heat energy generated while using various equipment and devices in both small and large sizes must be released into the environment in order to prevent overheating and improve efficiency.Most thermal exchangers use uids to remove the heat generated while a machine is operational.The use of common uids in the process of heat exchange is less acceptable due to their poor thermal conductivity.Recently, researchers created a groundbreaking method to enhance the thermal characteristics and thermal energy exchange rate of common uids.Due to its importance in a variety of scientic elds and the industrial sector, nanotechnology has recently become one of the most signicant research areas.Researchers are closely examining the varied thermal characteristics of such nano-sized particles due to the importance of nanotechnology in thermal design and multifaceted sciences.In order to meet the rising demand for energy resources, resulting from technological appliances, modern nanotechnology innovations combine nanoparticle communications to optimize the heat/mass transfer mechanism.They mixed ordinary uid with metal oxides and nanoparticles of several metals.Typically, nanoparticles are divided into two phases: a liquid phase made up of the base uid and a solid phase made up of the nanomaterials.Nanouids are more sophisticated than ordinary uids since they have unique properties and precise specications.A variety of unique characteristics of nanouids include increased viscosity, thermal conductivity, and transfers of heat.Many metallic and non-metallic nanouid models have been studied because of their improved thermal performance.The rst pioneering study to introduce nanouids was performed by Choi. 8Aerward, Eastman et al. 9 studied the thermal conductivity and heat transmission of nanouids.Shaq et al. 10 studied the bioconvective tangent hyperbolic magnetohydrodynamics (MHD) nanouid ow with Newtonian heating.In order to study the effects of chemical radiation on a spherical-shaped cylinder encased in a porous medium, Tlili et al. 11 used multiple slip effects on an MHD non-Newtonian nanouid.Haider et al. 12 theoretically and numerically explored the impact of the nanoparticle shape on a waterbased aluminium oxide nanouid moving towards a solid cylinder with the consequences of heat transfer.Khamliche et al. 13 used Cu nanoparticles to create high thermal conductivity nano-uids.Gangadhar et al. 14 examined the two-dimensional boundary ow layer of a nanouid over a stretched surface using the spectrum relaxation method.They concluded that when Prandtl numbers increase, the temperature and the nanoparticle volumetric fraction decrease.Furthermore, they concluded that with increased values of the heat absorption parameter, the local Nusselt number increases.Furthermore, Kotha et al. 15 and Gangadhar et al. 16 reported that gyrotactic microorganisms are contained on the boundary layer surface by a nanouid ow.][19][20] Entropy generation is a well-known indicator of the degree of irreversibilities present throughout any heating process.Cooling and heating are signicant events in many engineering procedures and industrial sectors, particularly in the elds of energy and electronics.Entropy coproduction must be optimized to avoid irreversible losses that could impair system performance.Bejan 21 initially established the Bejan number (Be) as the ratio of heat irreversibility to the total loss of heat due to uid frictional forces in order to gure out the generation of entropy.The second rule of thermodynamics to estimate the system irreversibilities allowed us to access the most practicable design of thermal systems. 22,23he presence of irreversibilities reduces the performance of an engineering device, and the level of the availability of this factor in a process is gauged by the entropy generation function.For the energy systems to be designed with the best possible efficiency, the entropy formation must be reduced because it serves as the measurement criterion for the available work destruction of the systems. 24Moreover, the formation of entropy results in a reduction in the useable power cycle outputs of a power production device, and an increase in the power input requires the cycle to produce power.It is crucial to highlight that the investigations based on the second rule of thermodynamics are more trustworthy than those analyses based on the rst law due to the rst law's efficiency limitations in heat transfer engineering systems. 25he analysis of entropy establishment is carried out to boost system performance.Moreover, entropy formation can occur by mass transfer, heat transfer, viscous dissipation, and nite temperature gradients. 26,27ith the broad spectrum of applications in chemical engineering, the study of heat transport in the stretched ow is quite signicant.In the current study, we considered the convective heat transfer coating ow through the boundary layer of a magnetic nanouid over an expanding slit dipped in a porous medium under the magnetic eld.The inquiry is unique in the sense that it involves multiple phenomena, including radiative ux, energy source, porosity characteristics, and dual chemical reactions.The ow pattern is curvilinear when the medium porosity is approximately one, and the inertial effect is brought on by the curvature of the path.The streamlined bend and drag rise quickly with velocity.A Tiwari-Das mechanism, which accounts for the volume fraction and is less complex than the KKL nanoscale model, 28 was used to study the nanoparticles and analyze their nanoscale behavior.Additionally, the Tiwari-Das model enables us to examine the nanoparticle material types, in contrast to the Buongiorno model. 29Both nanoparticles are of homogenous size and shape.The analysis of entropy production for the MHD ow in the presence of a non-Darcy-Forchheimer medium with radiative heat ux and dual chemical reaction on the sheet surface has not yet been discussed.Therefore, the consideration of irreversibility in the ow of radiative heat ux, non-Darcy porous medium, and Lorentz force towards a porous surface is the novelty of this research.Two different nanoparticles silver Ag and aluminum oxide Al 2 O 3 were suspended in water.The current study incorporates viscous heating and ohmic dissipation to create realistic temperature distribution estimations for chemical processes.The conservation relations for mass, energy, momentum, and diffusion (for species X and Ŷ ) are converted into a system of non-linear coupled higher-ordered differential equations using the proper scaling variables.The MATLAB bvp5c procedure uses four-point Gauss-Lobotto formulas to solve the developing nonlinear ordinary differential boundary value problem. 30The dimensionless quantities exhibit an extensive

Paper
Nanoscale Advances display of velocity, temperature, species X and Ŷ ) local Sherwood numbers, frictional coefficient, heat transport coefficient, and concentrations of X and Ŷ species.The study is pertinent to the industrial uid dynamics of stretching surface engineering components with smart magnetic nano-coating at high temperatures.According to our understanding, the current study is the rst to address smart nanouid dynamics.

Quantification and thermophysical features of nanoparticles
(1) The volumetric heat capacities of solid nanoparticles and base uids are indicated in the aforementioned expression by (rC p ) nf and (rC p ) f .The subscripts s, f, and p stand for solid particles, base uid, and nanoparticles, respectively.4 stands for the volumetric percentage of nanoparticles.m stands for the form factor of nanoparticles.
This inquiry takes into account the solid nanoparticles of silver (Ag) and aluminum oxide (Al 2 O 3 ), while water is taken as a base uid.The numerical values of the nanoparticles are shown in Table 1.

Modelling and mathematical formulation
We considered the motion of a viscous nanouid through a stretched surface corresponding to the plane y = 0 with steady and laminar uid layers.As can be seen in Fig. 1, the ow is restricted to the region y > 0, where y is the coordinate measured perpendicular to the stretched surface.The rst-order homogeneous and heterogeneous chemical species reaction processes are included in this study.The particles, i.e., silver (Ag) and aluminum oxide (Al 2 O 3 ), are considered to be nanomaterials.The plate was stretched along the x-axis while experiencing the linear velocity u = V w = ax, where a is a positive constant.Along the y-axis, a magnetic eld B was delivered at a constant strength (i.e., normal to the ow direction).According to general belief, the magnetic Reynolds number is low.As a result, in comparison to the magnetic eld that is provided externally, the induced magnetic eld is rather moderate.
Herein, we use the straightforward boundary layer ow model proposed by Chaudhary and Merkin 31,32 for the interface of homogenous-heterogeneous processes involving the dual chemical species X and Ŷ : Herein, K 1 and K s are the corresponding homogeneous and heterogeneous reaction rate constants, while d and c are the concentrations of the chemical species X and Ŷ .The thickness of the nanoliquid boundary layers is considered to be high.Radiative heat transport is approximated using the Rosseland diffusion algebraic approximation.Radiation is emitted at the stretched surface.Viscous heating due to uid molecule and nanoparticle interaction is added to the formulation.The resulting conservation equations using a Cartesian (x, y) coordinate system, with the involvement of ohmic dissipation (Joule heating) affects porosity factor and heat source, are as follows: [33][34][35]   Nanoscale Advances Paper Since the reaction is irreversible, the reaction rate declines for external ow as well as for the boundary layer.Thus, the mathematical expression for species X and Ŷ occurs as. 36,37 with the following boundary conditions Using similarity transformations We have We anticipate that the diffusion coefficients of the chemical species X and Ŷ are approximately equivalent in size in the majority of applications.Therefore, the ratio of diffusion coef-cients was supposed to be one (3 = 1), leading to the assumption that the diffusion coefficients DX and DŶ are of identical size, i.e., DX = DŶ.Following this relation, we have Eqn ( 18) and (19) gives

Physical quantities
Skin friction and Nusselt number Where the momentum and energy ux at the interface are dened as: Using eqn (15), the dimensionless form of physical quantities are: The apparent mass ux for species X and Ŷ is demarcated as:

Paper Nanoscale Advances
The Sherwood numbers dene the mass uxes for mass species X and Ŷ , i.e., the normalized mass transfer rates at the stretched sheet surface, as follows: The dimensionless forms of eqn (33) and (34) are The present study generalizes analyses from previous studies because they did not take into account Sherwood number behavior in their research.

Entropy modelling
The irreversible process leads to entropy generation optimization.In the entropy production analysis, the viscous dissipation, Joule heating, radiant heat ux, and heterogeneous-homogeneous chemical reaction effects are incorporated.One of the primary goals of the entropy generation is to gauge the irreversibility of any system.The ve factors that cause entropy creation are the temperature differential, magnetic eld, porosity of the medium, viscous dissipation, and species diffusion.Under these circumstances, the dimensional form entropy is provided in ref. 39-41.Here, the rst to h terms on the right hand side of eqn (37) are the entropy generation due to heat transfer, viscous dissipation, Joule heating (magnetic eld), porosity factor, and concentration of the X and Ŷ species, respectively.Furthermore, The overall entropy, abbreviated in the text for global thermodynamics, was obtained by integrating these local values over the entire volume.The following equations can be used to express them: 42 In the normalized notation, we obtain

Dimensionless parameters
The following are dimensionless numbers, i.e., porosity parameter, radiant parameter, magnetic parameter, Prandtl, Eckert, heat-generating parameter, Brinkman number, diffusion number for species and X, and Ŷ temperature ratio parameter.

Numerical method and validation
For the dual diffusive ow specied by eqn ( 16), (17), and ( 23) under the wall and free stream boundary conditions ( 20), (21), and ( 24), the converted nonlinear ODE boundary value problem is a coupled multi-degree system of the seventh order.Because of the signicant nonlinearity, a numerical solution is required.
In order to solve this problem, MATLAB bvp5c solver was used.
Providing adequate initial estimations for the meshes so that the solution converges to the target outcome is the most challenging aspect of solving a BVP.It directly controls the error in the calculation, whereas the more common bvp4c solver is unable to do so.This makes it a superior algorithm.This distinction between the solutions is less obvious at tighter error tolerances than that of Shampine and Kierzenka. 43The fourstage Lobatto IIIa collocation formula was implemented using "bvp5c", a nite difference code, and the associated polynomial yields a C 1 reliable solution that is uniformly precise for the 5thorder in [0, N].The Runge-Kutta implicit formula was used to implement the procedure.The quantitative condensation found in the bvp4c was not used in the bvp5c.Bvp5c tickle with simple differential equations for the anonymous parameters as opposed to bvp4c, which directly deals with unknown parameters.The novel solver is based on the residual control for robustness.The residual was scaled to match the genuine error up to the h order of convergence.If the scaling residual was lower than a specied patience, the asymptotic error was likewise smaller than the tolerance, according to bvp5c, 43 which was proven for a wide range of approaches.The value of X MID at the midpoint and the slopes at both ends of the subinterval are interpolated with the bvp5c.The stepping formula described by Russel and Christansen 44 was employed as follows: X 1 is the initial guess in this case, and K 1 , K 2 , K 3 , and K 4 are estimations with a stepping interval of z = 0.01.The algorithm has quick compilation times, is extremely effective, and delivers good accuracy.For each set of parameter values, the boundary layer region was reached while maintaining an iteration error of less than 10 −6 .Furthermore, Tables 2 and 3  The effects of various physical factors on the velocity prole, temperature and concentration prole, and entropy generation are discussed in detail in "Results and discussion".

Results and discussion
This section elaborates on the consequences of various emerging parameters, such as velocity, temperature, concentration, and entropy, providing physical justication.The structure of ow in the physical sense is illustrated through numerical simulations of velocities, temperatures, and concentration distributions against changes in physical limitations.The nanouid velocity f ′ (h) prole is affected by the magnetic eld parameter M, as shown in Fig. 2. The Lorentz force increases when an electric-conductive uid is subjected to a magnetic eld.The uid rate decreases, since the uid particles face an opposing force, and because of this opposed force, the uid momentum slows down.Physically, the uid particles travelling closer to the boundary layer under the inuence of the Lorentz force slow down.The stretching sheet   The friction between the uid layers causes the temperature of nanouids to rise as the E c number rises.Physically, with viscous heating, the kinetic energy is converted into internal energy as a result of the increase in the temperature of nanouid.Compared to Ag, the nanoparticles have a better temperature prole.The temperature prole of the aggregated nanoparticles is superior as compared to that of the ordinary uid.Higher levels of friction result in the conversion of mechanical energy into thermal energy inside the uid.From Fig. 6, it is evident that the radiation parameter R d signicantly increases the thermal boundary layer and the temperature prole.Physically, a higher radiation parameter results in additional heating as a result more heat is transferred to the corresponding uid, increasing the thickness of the boundary layer and the temperature prole.The Rosseland radiative heat ux model is straightforward and limited to optically dense media capturing the effects of radiant heat transfer in systems for processing materials.The existing simulations offer a solid foundation for the development of more complex radiative ux models using the current model.

Nanoscale Advances Paper
a reduction in the temperature prole and boundary layer thickness.The energy transferred by molecular conduction is inhibited, which results in a reduction in the thermal prole and a reduction in the thickness of the thermal boundary layer as Pr is increased.The stretching sheet regime cooling is accomplished with greater Prandtl numbers.From a physical perspective, the Prandtl number links the heat diffusion rate and the momentum diffusion rate.As the Prandtl number rises (beyond unity), thermal diffusion is suppressed because the momentum diffusion rate is far greater than the thermal diffusion.The effect of the magnetic factor M on Q(h)for the nanouid is shown in Fig. 8.As the magnetic eld intensity increases, the temperature prole increases.The thermal boundary layer thickness increases progressively as well.Physically, as the rate of the Lorentz force increases, the resistance to the motion of the nanoparticles increases, producing additional heat in the ow of the nanouid.Fig. 9 depicts the relationship between the nanouid thermal conductivity and temperature variation with regard to various values of volumetric fractions.
The results show that as the thermal conductivity increases with increasing nanoparticle volumetric friction 4, the temperature increases.One of the elements inuencing the enhancement of the thermal eld of nanouids is the intrinsic thermal conductivity of the nanouid.Actually, by increasing the volumetric friction, the uid molecular movement and the movement of the base uid and nanoparticles are both accelerated.
Energy is transferred from one layer of the nanouid to another, increasing the thermal distribution of nanouids.
As shown in Fig. 10, an increase in the homogeneous reaction parameter K is accompanied by a signicant decrease in the concentration of species X.In the chemical reaction X / Ŷ   12).However, these concept polarities are diametrically opposite.The reaction is destructive (negative) when species X is used, whereas it is constructive when species Ŷ (positive) is used.As species X concentration magnitudes increase with increasing the distance from the stretched surface, a rapid ascent in species X concentration is observed with increasing transverse coordinates, as previously discussed.The impact of Sc on J(h) is observed from the data in Fig. 11.The growth in Sc dwindles J(h) in both scenarios.The smallest Sc < 0.1 represents the highest nanoparticle concentration.As Sc > 0.1 escalations begin, a reduction in concentration happens.Moreover, the NP aggregated uid exhibits better mass transfer than ordinary uid ows.Herein, we nd that uid ow without NP aggregation results in lower mass transfer.It is clear that rising Schmidt numbers signicantly increase the concentration magnitudes, and this trend is consistent across the boundary layer of the stretched surface.In polymers, gaseous diffusion occurs when Sc > 1, therefore, the species diffusion rate exceeds the diffusion rate of momentum, leading to greater concentration magnitudes.In contrast, when Sc < 1, the species diffusion rate is exceeded by the momentum diffusion rate, reducing the concentration values.As the heterogeneous reaction parameter (K C ) is increased, there is a noticeable suppression in the concentration magnitudes, as shown in Fig. 12.It appears in the conditions along the wall edge J ′ (z,0) = K C J(z,0).According to the autocatalytic reaction eqn (2), which states that X / Ŷ , rate K s c, it is connected to the rst order, isothermal catalytic reaction at the substrate (stretched) surface.Physically, the highest conversion of species X to a new species occurs near the stretched surface resulting in minimizing the concentrations.The concentration of species X increases from the wall to the free stream, where it naturally reaches its maximum far away from the stretched surface.Since more and more species X are destroyed as K s increases, species X concentration magnitudes (molecular diffusion) are inhibited.All proles asymptotically converge to the free stream, supporting the simulations of an innite boundary condition.The prominence of thermodynamics inquiry is to assess the temperature and heat variation losses, which detract the available energy.The concept of entropy is explained by the second law of thermodynamics.Concisely, entropy generation explains the dissipated energy within the thermal system, which consequently increases the level of disorder in the system.Entropy proles E T for various values of Brinkman number Br are shown in Fig. 13.It should be noted that the Brinkman number determines the relative signicance of the viscous effect.It can be observed that the E T increases with increasing Br values.A higher entropy is generated by uid friction due to higher levels

Nanoscale Advances Paper
of Br.Furthermore, the impacts of entropy formation are stronger close to the sheet surface.When the uid moves far away from the sheet surface, these effects become less noticeable.Since the Brinkman number is inversely proportional to the square of the sheet stretching velocity, the dimensionless entropy generation number is greater for the bigger values of Br on the sheet surface (Br $ 0.4).The stretching velocity of the sheet affects the uid in such a way that at the surface of the sheet, the uid accelerates and as a result entropy generation rate rises.For Br < 0.4, it is evident that E T drops with increasing distance from the sheet.The inuence of radiation on the entropy rate is shown in Fig. 14.Physically, a higher (R d ) enhances the radiative emission, which intensies the collision of uid particles.From Fig. 15, it can be seen that the system rate of producing entropy decreases as the porosity parameter d increases.As the non-linear drag increases, the system's internal entropy intensies because of frictional heating.There is a considerable impact of uid friction and heat transfer within the entropy production system for different values of d.A lower temperature differential between the uid and the stretched surface is obtained because of the concentrated frictional heating intensications that drastically lower the uid temperature.The increased temperature difference is achieved by d, which increases the surface thermal transport rate from the uid to the sheet walls.The entropy rate for magnetic parameters is illustrated in Fig. 16.Lorentz force increases when magnetic eld intensity increases.Over the boundary layer region, signicant heat generation occurs, and the system's overall disorderliness increases because of the Fig. 13 Total entropy E T sketch with the upshot in Br.
Fig. 14 Total entropy E T sketch with the upshot in R d .
Fig. 15 Total entropy E T sketch with the upshot in d.
Fig. 16 Total entropy E T sketch with the upshot in M. strength of the magnetic eld.For various xed parameters, Fig. 17 shows the variation of frictional entropy generation in the laminar ow regime inside the heat sink with regard to the particle volumetric fraction.EG grows when the particle volume fraction is increased.However, there is a denite difference in the order of this increase.For instance, Al 2 O 3 particles respond sharply to heat transport, which means that the EG produced by changes in the volume percentage of these particles rises because of large friction.Fig. 18 shows the corresponding entropy-generating eld for various diffusivity variable inputs g a and g a .This gure concludes that increasing the diffusivity coefficient inputs leads to an improvement in the entropy creation.The diffusion of the uid particles increases with increasing levels of the diffusivity coefficient.In addition, there is a sharp increase in the prole as the value is increased.

Conclusion
In this study, the irreversibility analysis for the ow of a waterbased nanouid over a stretched surface was examined while considering the dual chemical species features and other physical factors.A complicated geometric stretched surface with many real-world engineering applications was used to model the ow.By dispersing the nanomaterial on the slit surface, the boundary layer separation from the surface was managed.The ow was subjected to external radiation, a heat source, and magnetic effects.In addition, the volumetric friction strategy described in the literature as a modied Tiwari-Das model (TDM) was used to execute this project.1.An increment in velocity is noticed for both nanoparticles against the porosity factor, while the opposite consequence holds against the magnetic parameter.
2. Both, alumina (Al 2 O 3 ) and silver (Ag) nanoparticles have nearly similar effects on thermal and uid ow with an augmented volume fraction.
3. The Eckert number and radiation for both silver (Ag) and alumina (Al 2 O 3 ) nanoparticles show an increase in temperature.
4. The aqueous concentration prole declines as the chemical reaction parameter and Schmidt increase.
5. Entropy generation grows as the Brinkman number, volumetric friction, diffusion parameter, and magnetic parameter, whereas it decreases with increasing porosity parameter.
6.The concentration of species decreases with homogenous and heterogeneous reaction rates.
7. By lowering the porosity within the sheet surface, the formation of entropy can be minimized.
8. The temperature gradient increases as the porosity parameter increases, but the velocity gradient shows the opposite pattern.The concentration gradient is reduced by increasing the values of the Schmidt number and the destructive reaction rate parameter, whereas a positive trend for the constructive parameter is revealed.Wall heat ux (Wm ) Q Heat source q M( X ) , q M( Ŷ ) Wall mass ux (kg m −2 s −1 ) S 00 0 G Entropy generation (Wm
are created to compare the skin friction coefficient C f Re ffiffix of Agand Al 2 O 3 , with the existing literature, in a limited sense, and was found to be in great agreement.

Fig. 2
Fig.2Velocity sketch f ′ (h) with the upshot in M.

Fig. 7
displays how the temperature prole varies for different Prandtl numbers.The thermal boundary layer and the temperature prole tend to be diminished by large values of the Prandtl number Pr. Physically, the temperature of uids with lower thermal diffusivity is lower, as a result, the uid exhibits

Fig. 3
Fig. 3 Velocity sketch f ′ (h) with the upshot in d.Fig. 4 Velocity sketch f ′ (h) with the upshot in 4.

Fig. 5
Fig. 5 Temperature sketch Q(h) with the upshot in E c .

Fig. 6 Fig. 7
Fig. 6 Temperature sketch Q(h) with the upshot in R d .

Fig. 8 Fig. 9
Fig. 8 Temperature sketch Q(h) with the upshot in M.

Fig. 10
Fig. 10 Species concentration J(h) sketch with the upshot in K.

Fig. 11
Fig.11Species concentration J(h) sketch with the upshot in Sc.Fig.12Speciesconcentration J(h) sketch with the upshot in K C .

Fig. 17 Fig. 18
Fig.17Total entropy E T sketch with the upshot in 4.

Table 1
Thermophysical properties of the nanomaterials

Table 3
Evaluation of Nusselt number values, whenC 4 = R d = E c = Q = 0 Applying computational uid dynamics (CFD), a thorough examination and comparison of the thermohydraulic and entropy performance of a water-based nanouid evacuated across a stretched surface was performed.H 2 O/Ag-Al 2 O 3 nanouid was used in the analysis.The numerical model accounts for the changes in operational parameters, including thermal radiation and mass ow rates.The model highlights the rate of entropy development due to the global and local forms of heat transmission, heat loss, and viscous effects.Using sets of sophisticated numerical techniques, including Bvp5c based on nite difference techniques, the governing PDEs (non-linear and coupled) were solved.The following concluding observations are made from the preceding thorough analysis: −3 K −1 ) k* Mean absorption coefficient (cm −1 ) s Electrical conductivity (S m −1 ) J Current density (A m −2 ) s* Stephan Boltzmann constant(Wm −2 K −4 ) B 0 Magnetic eld (Wm 2 ) 4 hf = 4 Ag + 4 Al 2 O 3Volume friction of 4 1 and 4 2