The importance of this investigation is to examine the heat and mass transportation of magneto nanofluid movement along a heated sheet with exponential temperature-dependent density, entropy optimization, thermal buoyancy, activation energy, and chemical reaction aspects. The influence of these factors in cutting tools by means of machining and nanofluid lubrication is a significant process in cutting zone, chip cleaning, lubricating, and cooling productivity in milling. The corresponding energy activation and chemical process are essential to understand the thermal behavior of nanofluid. The appropriate transformations are used to solve nonlinear partial differential equations within the framework of ordinary differential equations using stream functions and similarity variables. The Keller box method is employed to efficiently solve these equations computationally under the Newton–Raphson approach. Through tables and figures, the fluid velocity, temperature distribution, and concentration consequences are sketched using various controlling parameters. It is seen that the fluid temperature function increases with noticeable amplitude as the Eckert factor, variable density, chemical-reaction, and activation energy increase. It is found that the noticeable enhancement in heat and mass transportation is deduced for maximum Brownian motion and thermophoresis. This work is important in various applications such as cutting fluids, drilling, brake oil, engine oil, minimum quantity lubrication, enhanced oil recovery, and controlled friction between the tool-chip and tool-work during machining operations.

## I. INTRODUCTION

Nanofluid is very important for enhancing the effectiveness of minimum quantity lubrication (MQL) systems to remove excessive heating. Due to MQL, the surface roughness improves with cooling aspects. The nanofluid consists of base fluids with tiny manufactured nanoparticles, which are known as nanomaterials suspended within them. MQL is a prime choice among base fluids for nanotechnology applications due to its impressive cooling efficiency when compared to traditional overflow cooling techniques. The concept of nanofluids being used as lubricants has had a very significant response in the advancements of nanotechnology. It offers precise control over the dimensions, shape, and surface finish of the material waste in the form of cutting zones. The findings of recent nanofluid research have a potential impact on enhanced oil recovery (EOR) processes with the interactions of oil-based materials and microscopic particles. For various machining tools, improved efficiency is expected to be largely driven by the significantly enhanced and temperature-sensitive thermal conductivity offered by nanofluid. Cutting tools are indispensable across a wide range of manufacturing and scientific processes. These tools play a crucial role in various applications such as liquid film vaporization and shaping polymeric materials through aerodynamics to crystal formation, cooling solid materials, designing manufacturing equipment, heat exchange systems, glass, and polymer industries. Significant attention has been directed toward the study of boundary layer movement in this problem. This form of flow plays a vital role in numerous technological processes, including plastic coating bending, continuous molding, the production of fiberglass and paper, food processing, and polymer shaping. This research is explored with temperature-dependent density impact on the momentum-driven convection flow rate of nanofluid along a stretched plate. Researchers have employed assumptions of the Rosseland assumption to describe heat flux, considering variable fluid density variation for temperature fluctuations. Furthermore, this research is based on the heat and mass transfer in electromagnetic nanofluids through heated surfaces. Some studies have focused on natural convection in radiated liquids near parallel spherical surfaces by using the impact of temperature-dependent fluid density. Additionally, researchers have explored the magneto-hydrodynamic (MHD) motion of nanofluid in extensible geometries by incorporating slip conditions. This initial investigation with the consequences of thermo-density over stretched sheet within nanofluid is deduced under thermophoresis and Brownian motion simulations. Many technical processes, like the bending of plastic sheets, continuous molding, the production of paper and fiberglass, food processing, and polymer molding, need this kind of movement. The author^{1} investigated the thermal-dependent density influence on the momentum-driven convection flow rate of a power-law fluid using a moving semi-infinite perpendicular plate. The heat differential between the plate and the surrounding liquid was large. In the presence of radiant heat, researchers^{2} completed the examination of longitudinal momentum-driven flow separation of a visually opaque thick liquid in the incredibly heated perpendicular smooth plate. It was believed that the fluid density would change dramatically with heat. Azzam^{3} evaluated the radiant heat flow rate of electromagnetic liquid through the vertical shape. The density of the liquid was assumed to be perpendicular to the heated surface. The naturally occurring convection motion of radiated liquids through a parallel spherical surface was addressed by the examiner.^{4} It was predicted that the density of water is a constant function of heat. The MHD (magneto-hydrodynamic) motion of a fluid having nanoparticles contained through extensible geometry was examined by Awais *et al.*.^{5} In this investigation, slip circumstances were used. A stretching surface in a nanofluid with a smooth fluid flow problem was discussed in Refs. 6–11 statistically.

In the presence of activation energy and chemical reaction, both initial and subsequent stages of lubricating oil extraction can be improved through the (EOR) method. Chemical reaction and activation energy aspects in EOR have proven to be highly effective in extracting more oil by keeping operational costs low as compared to traditional techniques. This EOR method is very useful for the improvement of lubricating oils to reduce surface roughness, excessive heating in cutting tools, cracking and damaging parts during cutting tools, and cooling behavior in cutting tools. First, these lubrication oils can affect the dampening capabilities of constituents, which involve micromechanics at the surface and particle level. Second, considering the micromechanics of bigger fuel molecules, nanoparticles can affect the stiffness at the water–oil interface. Moreover, nanoparticles can stick to surfaces in the form of oil or lubricating grease to prevent excessive temperature and damage. Mjankwi *et al.*^{12} reviewed the implications of variable fluid qualities on the temperature conductivity and diffusion rate in the field of MHD motion of nanofluid across an inclined stretched sheet with radiant heat and chemical interactions. In the magneto-hydrodynamic phenomenon, Akbar *et al.*^{13} concentrated on the viscous motion of a cu–water/methanol dispersed fluid having nanoparticles over an externally stretched surface. In the circumstance of a magnetized field, Daniel *et al.*^{14} described the hydro-magnetic transport of a nanofluid past a curved elongated surface with variable thickness. Viscous dispersion, Joule heating, and chemical interactions were used in the procedure. The researcher^{15,16} explored the implications of nanofluid dispersion on MHD hybrid convective stagnant phase motion with dynamically stratified water-soluble nanofluid over transparent elastic sheet. Khan *et al.*^{17} evaluated the consequences of ultraviolet rays and chemically reactive materials on a Carreau nanofluid generated from a stretched plate with a shifted diameter. To prevent heat loss in high-temperature networks, a study was investigated for thickness, magneto-hydrodynamics, and radioactivity impact on thermal and mass properties of nanofluid over flexible sheet. They considered the effects of thermophoresis and Brownian motion in their analysis.

The impact of exponential temperature-dependent density on the flow of heat and mass transportation of nanofluid across heated sheets is very significant in lubricating oils, cutting tools, surface roughness, high friction devices, and gayer oils. For high dense fluids, the lubrication will be dropped in a very short time. For less dense fluids, the movement of nanoparticles will improve the motion of lubricating oil in the gayer section of automobiles, high friction devices, and cutting tools with maximum cooling efficiency. Thermodynamic transport and magneto-hydrodynamic (MHD) issues can collaborate to produce magnetic flux and electrical density with specific characteristics. These parameters are essential for metalworking operations, chilling continuous sheets, spinning spindles through a stationary fluid, purification of molten iron, and handling non-metallic impurities in metals. To improve the stability in thermal transfer analysis in thermo-physical heat and mass transportation, magnetic field, activation energy, chemical reaction, and variable density are essential. Enhanced oil recovery (EOR) relies on the implementation of liquid injections, nano-foam engineering, and the production of nano-emulsions. In the framework of nanofluid EOR, the fundamental basis lies in the interaction of nanoparticles with the substances involved. When nanomaterial sticks to a metal surface, the surface will be more wet and oily for maximum energy work. In the field of automotive engineering, there is a growing need for better brake systems that can dissipate heat more effectively. This is important in the automobile industry to reduce drag for the improvement of aerodynamic vehicles. The special focus of this study is how thermal density affects magneto-hydrodynamic flow in nuclear power plants, energy production, and areas of astrophysics and meteorology. In the existence of heating radiation in a thermodynamic equation, the authors^{18,19} evaluated the thermal transfer and flow properties of a sticky nanofluid across a gradually expanding surface. In an electrically connected Maxwell nanofluid, an elastic sheet with thermal efficiency, slip-velocity constraints, radiant heat, and the consequences of heat radiation were explored by Aziz and Shams.^{20} The mathematical examination was conducted by Murugesan and Kumar^{21} to assess the implications of thermal emission on the MHD motion of a radiating nanofluid across a flexed porous sheet, including heat dissipation. The implications of an electrical field and reaction of chemicals on the mixed conduction flow of the boundary layer in nanofluid along a semi-infinite extending surface were explored in Ref. 22. Through stretched surfaces, Haroun *et al.*^{23} probed the transmission of heat and mass in magneto-hydrodynamic nanofluid motion. The simultaneous impacts of MHD, extraction, and radioactivity were addressed in Ref. 24 to see the forced convection flow through the boundary layers of a nanofluid across an enormously flexed surface submerged in a dynamically stratified substrate. With the presence of nanoparticles, Narender *et al.*^{25} concentrated on computational research on chemical reactions and fluid dissipation on the stable flow of MHD boundary layer nanofluid through an extending plate. John and Mansur^{26} researched the influence of dust particles and slipping impacts on the flow of the boundary layer of a dusty nanofluid across an extensible horizontal sheet. In recent work,^{27–31} authors investigated the chemical reaction, entropy generation, and heat-mass transportation of nanofluid over stretched sheet numerically.

The novelty of this research is to use nanofluid for heat reduction and friction reduction during machining operations, especially in work pieces and cutting tools, brake systems in automobiles, high friction devices, drilling, and milling systems. To achieve this objective, this study involves the impact of electromagnetic field, activation energy, entropy generation, chemical reaction, and exponential temperature-dependent density on magnetically driven nanomaterials over an extended heated surface. For this investigation, the aspects of thermophoresis and Brownian motion are used. The impact of exponential temperature-dependent density provides a noticeable improvement in lubricating oils, minimum lubricating oils, and enhanced oil recovery with less or high dense fluid behavior. The impact of magneto-hydrodynamic is also very important to prevent the surface from excessive temperature and cracking. Many previous studies have investigated with constant density in different nanofluids. However, this research is based on the exponential form of temperature-dependent fluid density by using the Keller box method to develop solutions. The developed mathematical model is solved through stream functions and similarity variables with the help of the Newton–Raphson technique. The presented figures and illustrations are examined to provide a comprehensive understanding of the topic. To the best of our knowledge, this marks the first exploration of how entropy generation, activation energy, and variable density collectively influence the behavior of nanofluid under high temperature differences. This work is very important in various applications such as cutting fluids, drilling, brake oil, engine oil, minimum quantity lubrication (MQL), enhanced oil recovery (EOR), and controlled friction between the tool-chip and tool-work during machining operations.

## II. FLOW FORMULATION

The focus of this research is to explore the computational responses of various factors, including thermal density, magnetic field, generated entropy, and buoyancy force, on the characteristics of nanoparticles within a nanofluid. The purpose of this work is to examine how these elements influence the transfer of heat and mass over an extended surface by considering the impact of Brownian motion and thermophoresis. To address this complex issue, a simplified set of partial differential equations with stream-function adjustments is developed. These equations are then converted into ordinary equations using similarity variables. The mixed form of Keller box and Newton–Raphson approach is used to solve this problem. The results of this analysis are presented in a quantitative manner with the material properties and their behavior graphically illustrated in tabulated form for a clear and comprehensive understanding of the system.

*x*and

*y*directions, correspondingly. Here, the x axis aligns with the stretched surface, while the y axis is perpendicular to it. The temperature at the wall is represented by T

_{w}, while the ambient temperature is denoted by T

_{∞}. The liquid thermal conductivity is expressed by κ, and its specific heat is indicated by

*C*

_{p}. The thickness of the material is denoted by δ, the particle density by

*ρ*

_{p}, the fluid kinematic viscosity by ν, the acceleration of gravity by g, and the density of the fluid is indicated by

*ρ*. Equation (1) presents the law of conservation of mass with variable density, and the momentum flow of nanofluid with variable density and magnetic field is presented in Eqs. (2) and (3). The aspects of variable density, entropy optimization, Brownian motion, and thermophoretic in the energy equation are presented in Eq. (4). The impact of Brownian diffusion, chemical reaction, and activation energy in fluid concentration is presented in Eq. (5) by following:

^{1–5}

^{,}

*P*illustrates pressure,

*ρ*symbolizes the base fluid density,

*α*denotes heating diffusion,

*ν*stands for viscosity kinematics,

*a*represents a specific constant number,

*D*

_{B}relates to Brownian distribution,

*D*

_{T}pertains to thermophoretic distribution, and

*τ*signifies the heat proportion of nanoparticles and fluid capacity. In simple terms, these symbols and parameters are used to describe and understand the behavior of temperature and velocity within the context of the study. The stream factors and similarity factors are employed to transform partial equations into ordinary equations, which are listed below in Eqs. (7) and (8),

^{7–12}

*θ*is shown for dimensionless temperature, $Le=vDB$ is determined by Lewis parameter,

*η*is demonstrated by the factor of similarity, Pr $=\nu \alpha $ is denoted by Prandtl number, reaction rate is $\sigma =kr2a$, temperature difference is $\delta =Tw\u2212T\u221eT\u221e$, activation energy is $E=Ea\kappa T\u221e$,

*m*is dimensionless rate constant with rage −1 <

*m*< 1 and $\nu =\mu \rho $ is denoted by kinematical viscosity by following.

^{13–18}The ordinary form of Eq. (6) is written as follows:

## III. NUMERICAL AND COMPUTATIONAL SCHEME

*l*(

*η*), and

*m*(

*η*),

^{18–23}

*η*

_{n−1},

*η*

_{n}with $\eta n\u221212$ by using Eq. (23). The equation for the algebraic representation of the center-difference and average difference system is now described below. Equations (23) and (24) are used in Equations (14)–(23) for the basic form of algebraic equations with mid-points, average difference, and central difference aspects as follows:

^{23–26}

## IV. RESULTS AND DISCUSSION

This paper investigates the interaction of heat density, entropy generation, and activation energy in a two-dimensional laminar mixed convection flow of magneto nanofluid toward a stretched sheet. In contrast to the traditional mathematical model, the fluid density used in this study is described exponentially with a temperature dependent relation. With this novel method, the solution holds true for a broad range of temperature changes and yields more accurate results. The defined boundary conditions are considered to develop the boundary-layer equations uniquely in this mechanism. Notably, a stream function framework is used further to simplify the dimensionless governing mathematical equations. A set of non-similar computations is used to analyze the data. The Keller box method is utilized for the numerical integration. Geometric analysis is performed on the numerical results with this present method and shows good agreement with the predictions. This study explores how a number of crucial flow factors affect several flow characteristics, including velocity, temperature distribution, and concentration distribution. The activation energy, density parameter, thermophoretic parameter, magnetic-force parameter, buoyancy parameter, Eckert number, and Prandtl parameter are utilized. Investigating these factors allows us to gain a deeper understanding of how they influence the behavior of the flow and the resulting physical characteristics. Tables and graphs are carefully drafted to present the results with a good evaluation and significance of the model.

### A. Steady behavior of fluid velocity, temperature function, and fluid concentration profiles

Figures 2(a)–2(c) are presented to show temperature, velocity, and concentration distributions for different values of activation energy. Regarding velocity, it is evident that there is a strong correlation between these variables because the velocity graph noticeably increases as activation energy decreases. However, in terms of temperature distribution, the data show that, particularly when considering entropy generation, temperature rises to its maximum values in tandem with activation energy. When the activation energy decreases, the concentration distribution is contracted. These results demonstrate the important influence of activation energy on these crucial variables in the system being studied. There are clear and noticeable differences in each of the plots. Physically, the activation energy refers to an energy threshold or barrier, which is very important to form reactions for making different products. Heating energy from outside sources is usually the foundation of the activation energy required for developing processes. The variation in a chemical-reaction enthalpy is known as the heat of reactions. The activation energy and chemical reaction are very useful to make lubricating oils. Figures 3(a)–3(c) illustrate the temperature, velocity, and concentration distribution using reaction rate by taking activation energy and entropy optimization effects. First, it is clear that as the reaction rate rises, the water-based fluid velocity increases. On the other hand, velocity tends to decrease in tandem with a decrease in reaction rate. A significant pattern in the temperature distribution appears for higher reaction rates under certain amplitudes. With different reaction rates, a unique behavior in the distribution of temperature throughout the system is deduced. Furthermore, there is a distinct trend in the concentration rate. It rises when the rate of reaction falls, and these shifts are distinguished by noticeable fluctuations. These findings demonstrated the important influence of reaction rate on the temperature, velocity, and concentration distribution in relation to activation energy and entropy optimization. Physically, it is valid because the reaction rates develop energy using activation energy and are used extensively to analyze, identify, and test a wide range of materials. Chemical reactions and activation energy play a vital role in fluid dynamics and heat transmission. It is also important in nuclear reactors, electronic appliances, manufacturing, and automobile industries. With a focus on entropy production, the effects of temperature, velocity, and concentration distribution on the thermal density parameter are shown in Figs. 4(a)–4(c). These graphs show very noticeable differences in the velocity distribution. The fluid consisting of water is found to have a higher velocity as its density decreases. On the other hand, velocity tends to decrease as the density parameter rises. In relation to the temperature distribution, a clear trend becomes apparent. At lower density parameters, the distribution exhibits noticeable amplitude. The decreasing amplitude is found for the high density parameter. Similarly, the concentration sketch shows an increasing trend as the fluid density decreases. When entropy generation is present, the maximum density exhibits the lowest concentration. Physically, this phenomenon can be explained by the fact that at higher densities, buoyant pressure increases and causes more variations in density due to temperature. A less dense fluid will move faster than a denser fluid. For high exponential temperature density, the temperature of the fluid will increase with the maximum concentration of nanoparticles.

Figure 5(a) illustrates the impact of the Eckert number on the velocity distribution with a discernible increase and consistent asymptotic behavior. Figure 5(b) illustrates the significant variations in the water-based fluid temperature as the Eckert number rises. It is seen that the enhancing amplitude in fluid temperature is depicted for high entropy generation. It is noted that the fluid concentration decreases as the Eckert number increases. The concentration distribution is shown in Fig. 5(c), where it decreases with increasing Eckert number and increases for lower entropy generation. Each physical profile is found with unique changes. Physically, it is expected because the Eckert number is a dimensionless quantity that expresses the connection between the kinetic energy movement and boundary-layer enthalpy change. It is also utilized to characterize the heat transport dissipation. Furthermore, the high Eckert number increases the transformation of kinetic energy to internal energy, which enhances the temperature distributions. Figures 6(a)–6(c) illustrate the consequences of magnetic force with the impact variable density and entropy production on fluid velocity, temperature function, and fluid concentration profile of nanofluid over stretched surfaces. It is discovered that the fluid velocity and temperature decrease as magnetic pressure increases with high Prandtl. For lower values of the magnetic parameter, there is a noticeable increment in both temperature and velocity that is depicted with less variation. Conversely, when the magnetic parameter drops, the concentration distribution rises. This behavior is consistent with the theory because the lower frictional force inside the viscosity layers increases the heat conduction with increasing Prandtl value. Physically, it is expected because the maximum 20% rate in heat transmission increases with the laminar movement of magnetically driven fluid by applying the magnetic field, which enhances the heating efficiency under high magnetic intensity. Moreover, the influence of magnetic fields protects our planet from charged particles and cosmic radiation emitted by the Sun. The magnetic field works like a coating material in many devices to protect excessive temperature differences.

### B. Nusselt quantities and Sherwood quantities for Brownian motion and thermophoresis parameters

The effects of thermophoresis on the mass and heat transfer physical processes across the heated surface are drafted physically in Figs. 7(a) and 7(b). For high thermophoretic parameters, the mass and heat transfer rates progressively improve and exhibit notable variations when thermal densities and entropy optimization are used. Conversely, when the thermophoretic parameter decreases to its minimum values, the decreasing rates of heat and mass transmission are deduced, especially with higher Prandtl numbers. This is due to the fact that a liquid with a higher Prandtl number generally has a lower thermal conductivity, which increases heat transfer at the surface and decreases conduction. Physically, it is expected because the thermophoresis is a transportation force that develops in a high temperature gradient due to maximum exponential temperature density. Thermophoresis is very significant in high temperature regions with maximum temperature density variations. It describes the movement of nanoparticles in high temperature differences, which provides the maximum increment in heat and mass transportation. The effects of thermal densities and entropy optimization on mass and heat transmission function are drafted for several values of the Brownian motion parameter in Figs. 8(a) and 8(b). Notably, it is observed that a decrease in dimensionless heat transfer for the highest Brownian motion parameter is found, particularly with Prandtl number 7.0. On the other hand, the dimensionless mass transfer increases for the highest Brownian motion parameter in the presence of buoyancy and magnetic forces. It is also noted that the high Sherwood quantity is deduced with high Brownian effects because Brownian force develops the constant movement of nanoparticles in the fluid flow physically. Brownian motion force prevents the particles from settling down, with improved stability in colloidal solutions. With the increase of Brownian motion, the heat transport and temperature decrease, but mass transmission across the fluid is enhanced. This behavior can be attributed to the combined influence of density and magnetic attraction, which acts as a controlling force. This force can be harnessed for various applications, including the magnetic coating of cables and metallic materials, as well as the development of magneto-hydrodynamic energy.

### C. Numerical results of skin-friction, Nusselt, and Sherwood quantities

The non-dimensional skin friction, heat rate, and mass transmission across a stretching sheet are calculated computationally and numerically in Tables I–II by using a number of parameters, including the temperature-difference, the Eckert parameter, and the effects of thermal density and entropy. These calculations provide valuable perceptions of the behavior of the system under different conditions. One interesting observation is that an increase in mass transfer is evident when the buoyancy parameter reaches its maximum value of 0.9. However, both heat transfer and skin friction show a decreasing pattern with the temperature-difference parameter in Table I. It is seen that the rate of friction decreases as the value of the temperature-difference factor increases. The increased gravitational impact at higher buoyancy values results in enhanced temperature gradients within the small fluid nanoparticles for promoting maximum mass transfer. On the other hand, this intensified temperature-difference reduces the heat transfer and skin friction. The Eckert parameter effects on the numerical characteristics of mass transfer, skin friction, and heat rate are presented in Table II. Here, it is discovered that larger values of the Eckert parameter correspond to the maximum values of each physical attribute. As the Eckert parameter decreases, both the dimensionless heat-mass transportation rates decrease as well. It is seen that the rate of heat transportation and mass rate enhance as Eckert factor enhances with prominent change. The lower response in each numerical data is calculated for the lower Eckert factor. In Table III, the comparative findings for Lewis and Prandtl values with the existing literature^{6} are deduced numerically. The results demonstrate a prominent agreement between our calculations and the prior research. This underlines the reliability and consistency of our findings with established scientific knowledge. It is seen that the Nusselt quantity decreases and the Sherwood quantity increases as the value of the Lewis factor increases because the Lewis number presents the relative importance of heat and mass diffusion in fluid dynamics. It is also seen that the Nusselt quantity and Sherwood quantity increase as the value of the Prandtl index increases because the Prandtl factor presents the kinematic viscosity and thermal diffusivity of fluid.

. δ | ″(0)
. f | −(0)
. θ′ | −ϕ′(0)
. |
---|---|---|---|

0.1 | 1.116 690 859 808 825 | 0.817 039 128 307 756 | 0.892 601 403 110 183 |

0.3 | 0.907 505 841 434 577 | 0.629 538 072 563 911 | 1.222 378 468 495 322 |

0.5 | 0.784 942 522 867 972 | 0.373 890 818 701 673 | 1.362 275 440 358 294 |

0.7 | 0.367 537 306 821 681 | 0.196 891 654 885 512 | 1.470 478 364 789 618 |

0.9 | 0.143 499 336 076 714 | 0.107 297 572 381 414 | 1.562 084 422 804 949 |

. δ | ″(0)
. f | −(0)
. θ′ | −ϕ′(0)
. |
---|---|---|---|

0.1 | 1.116 690 859 808 825 | 0.817 039 128 307 756 | 0.892 601 403 110 183 |

0.3 | 0.907 505 841 434 577 | 0.629 538 072 563 911 | 1.222 378 468 495 322 |

0.5 | 0.784 942 522 867 972 | 0.373 890 818 701 673 | 1.362 275 440 358 294 |

0.7 | 0.367 537 306 821 681 | 0.196 891 654 885 512 | 1.470 478 364 789 618 |

0.9 | 0.143 499 336 076 714 | 0.107 297 572 381 414 | 1.562 084 422 804 949 |

Ec
. | ″(0)
. f | −(0)
. θ′ | −ϕ′(0)
. |
---|---|---|---|

0.1 | 2.183 396 932 316 465 | 1.413 888 055 062 002 | 1.349 043 083 809 308 |

0.4 | 2.410 935 199 782 845 | 1.703 348 871 052 019 | 1.466 341 806 440 679 |

0.7 | 2.632 294 288 980 072 | 2.225 416 928 317 450 | 1.683 769 945 241 027 |

1.0 | 2.847 505 480 184 784 | 2.962 897 609 657 398 | 1.801 183 250 674 839 |

1.5 | 3.756 803 884 505 053 | 4.001 905 836 106 870 | 2.318 483 648 737 034 |

Ec
. | ″(0)
. f | −(0)
. θ′ | −ϕ′(0)
. |
---|---|---|---|

0.1 | 2.183 396 932 316 465 | 1.413 888 055 062 002 | 1.349 043 083 809 308 |

0.4 | 2.410 935 199 782 845 | 1.703 348 871 052 019 | 1.466 341 806 440 679 |

0.7 | 2.632 294 288 980 072 | 2.225 416 928 317 450 | 1.683 769 945 241 027 |

1.0 | 2.847 505 480 184 784 | 2.962 897 609 657 398 | 1.801 183 250 674 839 |

1.5 | 3.756 803 884 505 053 | 4.001 905 836 106 870 | 2.318 483 648 737 034 |

. | . | Khan and Pop^{6}
. | Present outcomes . | ||
---|---|---|---|---|---|

L_{e}
. | Pr . | Nusselt . | Sherwood . | Nusselt . | Sherwood . |

5.0 | 1.067 | 1.094 | 1.055 | 1.090 | |

10.0 | 0.820 | 2.312 | 0.818 | 2.298 | |

15.0 | 0.718 | 3.076 | 0.716 | 3.054 | |

25.0 | 0.618 | 4.124 | 0.616 | 4.116 | |

1.0 | 0.562 | 2.059 | 0.561 | 2.046 | |

2.0 | 0.796 | 1.951 | 0.793 | 1.933 | |

5.0 | 0.970 | 1.984 | 0.967 | 1.970 | |

10.0 | 0.820 | 2.311 | 0.817 | 2.303 |

. | . | Khan and Pop^{6}
. | Present outcomes . | ||
---|---|---|---|---|---|

L_{e}
. | Pr . | Nusselt . | Sherwood . | Nusselt . | Sherwood . |

5.0 | 1.067 | 1.094 | 1.055 | 1.090 | |

10.0 | 0.820 | 2.312 | 0.818 | 2.298 | |

15.0 | 0.718 | 3.076 | 0.716 | 3.054 | |

25.0 | 0.618 | 4.124 | 0.616 | 4.116 | |

1.0 | 0.562 | 2.059 | 0.561 | 2.046 | |

2.0 | 0.796 | 1.951 | 0.793 | 1.933 | |

5.0 | 0.970 | 1.984 | 0.967 | 1.970 | |

10.0 | 0.820 | 2.311 | 0.817 | 2.303 |

## V. CONCLUDING REMARKS

The present study is based on the complex link between heat density, activation energy, and entropy production in magneto nanofluid with two-dimensional mixed convection flow across a stretched sheet. The exponential form of temperature dependent density is applied for the enhancement of temperature differences under Brownian movement and thermophoresis aspects. To tackle this numerical problem, the boundary-layer equations are developed with specific mechanisms for well-defined boundary conditions. Further simplification of dimensionless governing mathematical equations is created using a system of non-similar equations. A stream function structure provides the necessary support for this simplification, and then integrated equations are solved numerically using the Keller box method, allowing us to assess our findings in relation to prior research. The geometric evaluation of the numerical results from this technique revealed a high level of agreement, reaffirming the accuracy and reliability of the approach. The influence of various governing flow factors, including activation energy, density factor, Eckert number, magnetic-field, thermophoretic factor, buoyancy factor, temperature-difference factor, and Prandtl element, on velocity, temperature, and concentration distributions is discussed. These assessments were presented prominently through graphs and tables. In summary, this study has shed light on the intricate interplay of these factors in a complex system. These findings contribute to a deeper understanding of the physics and hold significance in various practical applications. The importance of this study has unveiled several key findings, as follows:

It is observed that the velocity graph exhibits an increment as activation energy decreases, with notable variations. Similarly, in the existence of entropy generation, the temperature reaches its maximum magnitude as activation energy increases.

It is seen that the noticeable temperature distribution enhances with maximum amplitude at lower density parameter values but diminishes with increasing density parameter. For a particular Prandtl number, the concentration distribution correspondingly rises as the temperature-dependent density falls.

It is depicted that the prominent increment in fluid temperature is deduced for increasing the Eckert parameter with noticeable variations. For lower magnetic force, the fluid velocity and temperature function increase, but the fluid concentration is enhanced significantly.

It is found that a gradual enhancement in both heat and mass transfer rates is noticed as the thermophoretic parameter increases, leading to significant changes. The maximum increment in heat transportation is deduced at the small Brownian motion factor. Maximum improvement in mass rate is deduced for high Brownian motion.

It is noticed that the dimensionless heat transfer is decreased for the maximum value of the Prandtl number, specifically with Pr = 7.0 under maximum density variations. On the other hand, the dimensionless mass transfer is increased for the highest value of the thermophoretic parameter, especially in the presence of buoyant and magnetic forces.

It is seen that the noticeable increment in the rate of skin friction is deduced for the high temperature-difference factor. Maximum improvement in heat and mass transmission is found for the small temperature-difference factor in the context of thermal density, entropy optimization, and external forces.

The influence of Soret/Dufour, thermal conductivity, slip effects, variable surface temperature, reduced gravity, and heat source/sink on the steady and unsteady nanofluid flow over a stretching sheet, moving wedge, circular cylinder in a wake, and vertical plate using variable density effects can be investigated for future work.

## ACKNOWLEDGMENTS

This project was supported by Researchers Supporting Project No. RSPD2024R909, King Saud University, Riyadh, Saudi Arabia.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Zia Ullah**: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **Md. Mahbub Alam**: Data curation (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). **Aamir Abbas Khan**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal). **Shalan Alkarni**: Methodology (equal); Resources (equal); Software (equal). **Feyisa Edosa Merga**: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Qaisar Khan**: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.