Magneto-hydrodynamics (MHD) is the study of the movement of an electrically conducting fluid in the presence of a magnetic field. The Joule heating enters the energy equation in the MHD linked to the heat transfer problem, along with the additional body force term, or Lorentz force. Because it has many practical applications in geophysical and astrophysical phenomena, including stellar and planetary processes, the study of MHD of viscous electrically conducting fluids is important. Radiation is the term used to describe the transfer of energy using electromagnetic waves. When charged particles in matter move thermally, electromagnetic radiation is produced. This is known as thermal radiation. Heat transfer in radiation does not require an intermediary medium, whereas conduction and convection require a medium for heat transfer. A "mass transfer" occurs when there is a change in a mixture's species concentration, which results in the transfer of mass. In everyday life, mass transfer is frequently observed in processes like drying and evaporation, water vapour transfer into dry air, and smoke diffusion into the environment through towering chimneys. Convective mass transfer and diffusion mass transfer are the two distinct forms of mass transfer. Forced and natural convection combine to form mixed convection. Where buoyancy forces are present, we infer forced convection. Examples include the circulation of blood in heated animals, the shock waves produced by explosions, and convection ovens. A solid matrix with a network of linked voids is what is known as a porous media. Examples of porous medium domains include soil, sand, fissured rock, karstic limestone, ceramics, bread, lungs, kidneys, aquifers from which groundwater is pumped, reservoirs that produce oil and/or gas, sand filters for purifying water, packed beds in the chemical engineering sector, and the root zone in agriculture.

Crane discovered an accurate solution by examining the flow caused by a stretching plate whose velocity is proportionate to the distance from a slit. Following up on Crane's ground-breaking breakthrough, numerous scientists expanded on it in a variety of ways for viscous fluids on a stretching sheet, such as Takhar examined the flow and mass distortion of chemical species undergoing first and higher-order reactions across a continually stretched sheet under the influence of an applied magnetic field. Hayat and Javed have generalized three-dimensional MHD flow over a porous stretched sheet. Krishnaiah et al. discovered numerical solutions for a class of nonlinear differential equations that arise in the heat and mass transport of a non-Newtonian fluid towards an exponentially extending surface. The behaviour of non-Newtonian fluids is characterized by the Casson fluid model. Konda et al. examined the flow of melting heat transfer through a porous media that contains thermal radiation as a stretching sheet. Gangadhar and Suneetha studied the soret and DuFour effects of a two-dimensional, stable, viscous, incompressible, electrically conducting, and laminar MHD free convection flow in the presence of a porous media and heat generation/absorption. Pal and Mondal investigated the transfer of heat and mass through natural convection over a stretching sheet immersed in a fluid-saturated porous medium under a uniform transverse magnetic field in the presence of variable thermal conductivity, Soret and DuFour effects, thermal radiation, viscous dissipation, non-uniform heat source/sink, and temperature-dependent variable viscosity. Patil examined the combined effects of a magnetic field and cross-diffusion on mixed convection flow across an exponentially stretched sheet. Thumma and Mishra looked into how non-uniform heat sources and sinks, as well as Joule and viscous dissipation under velocity slip, convective surface temperature, and zero mass flux circumstances, affect the formation of the 3D Eyring–Powell fluid system in the field of fluid dynamics. Karim et al. studied the DuFour and Soret effects of thermal radiation and an electrically conducting viscous incompressible fluid interaction on the flow over an inclined linearly stretched sheet in the presence of mass transfer and heat allowed by a transversely applied uniform magnetic field, with heat generation accounting for the Rosseland diffusion approximation. Mondal et al. examined the mass transfer of a semi-infinite permeable inclined flat plate in the presence of a non-uniform heat source/sink and chemical reaction, as well as thermophoresis and Soret-DuFour on MHD mixed convective heat. Kalyani et al. looked into the impacts of thermal diffusion, heat source/sink effects, and magnetohydrodynamics on coupled free-forced convection and mass transfer flow across a semi-infinite vertical porous flat plate embedded in a porous medium. Aastha and Chand examined the heat-generating and chemically reacting nanofluid's free convective flow with mass transfer via a vertically moving porous plate in a conducting field, accounting for the Soret and DuFour effects as well as the viscous dissipation effect. Ramadevi et al. examined using three-dimensional (3D) computer modeling the magnetohydrodynamic (MHD) Carreau fluid flow, controlled by a stretching surface affected by mass and heat transfer. Variable thermal conductivity, Joule heating, irregular heat source/sink, and chemical reaction. Hayat et al. examined the properties of flow and heat transmission through a stretching cylinder submerged in a thermally stratified medium. In addition, the effects of thermal radiation and uneven heat generation are taken into account. Ali et al. investigated Soret effects, Hall and MHD effects, and constant free convective mass transport over a vertical porous plate with heat generation. Seth, Tripathi, & Rashidi investigated the effects of diffusion-thermo and thermodiffusion on MHD natural convection flow of an incompressible, heat-absorbing, viscous fluid that conducts electricity and has variable mass and heat fluxes near an inclined stretching sheet in a non-Darcy porous medium while accounting for chemical reaction. Sheikh discovered the analytical solution for the flow of a viscous MHD fluid across a porous oscillatory stretching sheet when thermophoresis is present. Reddy & Chamkha investigated the properties of heat and mass transfer of nanofluids based on Al₂O₃ and TiO₂ over a stretched sheet through a porous media in the presence of radiation, chemical reaction, magnetic field, thermo-diffusion and diffusion-thermo effects, and heat generation/absorption. Ragupathi et al. looked into the possibility of comparing the Fe₃O₄/Al₂O₃ nanoparticle with H₂O/NaC₆H₉O₇ based nanofluids on the Riga plate with non-uniform heat source-sink effects to get distinct results. Koli and Salunkhe focused on the combined effects of heat radiation and magnetic field on convective nanofluid flow towards a permeable stretched sheet in the presence of suction/injection, thermophoresis, and Brownian motion effects. Lakshmi et al. investigated the effects of uniform transverse magnetic field and non-uniform heat source/sink on Sisko fluid flow across a non-linear stretched sheet. Reddy et al. analyzed the heat and mass transfer of an unsteady magneto-hydrodynamic hybrid nano liquid flow over a stretching/shrinking surface with chemical reaction, suction, slip effects, and thermal radiation. Water is considered the base fluid, and a combination of alumina (Al₂O₃) and titanium oxide (TiO₂) nanoparticles is considered a hybrid nanoparticle.

The current work aims to investigate the effects of thermal diffusion (Soret) and diffusion-thermo (DuFour) on MHD mixed convective flow of a viscous nanofluid across a stretching sheet in the presence of non-uniform heat source/sink and chemical reaction embedded in a porous medium under a magnetic field. Using the appropriate similarity transformations, the set of dimensional partial differential equations is converted into dimensionless ordinary differential equations, which are then solved numerically using the Keller box method. Graphs and tables are used to illustrate and discuss the results that were collected.

This study examines the heat radiation soret and DuFour effect of MHD mixed convection of incompressible viscous nanofluid across a stretching sheet with a non-uniform heat source/sink and chemical reaction on a two-dimensional stable electrically conducting nonlinearly permeable sheet. Figure 1 shows the shape of this flow. This study's investigation takes into account the following suppositions:"

1) The problem's geometry is assumed to be represented by a coordinate system with a perpendicular Y-axis and a horizontal X-axis.

2) We explore the flow through a stretched sheet of a two-dimensional barrier layer.

3) We studied the flow of nanofluids at y ≥ 0 (the observed coordinate normal to the stretched surface) in the presence of a magnetic field.

4) At the boundary, the temperature T and volume concentration C of the nanoparticles are considered to be $T_w$and $C_w$ at the wall, and $T_{\infty }$ and $C_{\infty }$at a distance from the wall.

5) Furthermore, it is hypothesized that when a linear function is utilized, there will be a decrease in temperature gradient in the viscous fluid flow. Using Taylor's technique, the expansion $T^4$of moves towards a temperature of free stream $T_{\infty }$

Governing equations of the problem (Chandu. M. Koli and S. N. Salunkhe )

Continuity Equation

Equation of Momentum

Equation of Energy

Where $q^{\text{'}\text{'}\text{'}}=\frac{K{u}_w}{\mathit{x\; v}} [ A_{1 }\left[T_W-T_{\infty } \right] f^{\text{'}}+ B_{1 }\left[\mathit{ T}- T_{\infty } \right]$

Equation of Species nanoparticle volume concentration

Boundary conditions

$u\to 0,v\to 0,\mathrm{ }T\to \mathrm{ }{T}_{\infty }$, $C\to \mathrm{ }{C}_{\infty }$ as $y\to \infty$

Transformation of similarity is presented

When Eq. (6) is applied, the equation of continuity is fulfilled in the same way, and both Eqs. (2) to (4) and (6) have the same form.

“The corresponding boundary conditions (5) to transform into

Here physical parameters involved are specified

The important physical values are the skin friction coefficients' physical parameters, the local share wood number, and the local Nusselt number.”

Where $\mathrm{R}{e}_x=\mathrm{ }\frac{U_0\mathit{ x }\left(\exp \left(\frac{x}{\mathit{L }}\right)\right)}{v}$ be the local Reynolds number

The Keller box method is an implicit unconditionally stable technique capable of solving a variety of different engineering problems. The Keller box method includes the following steps:

1) To compute the approximate solution, first write the governing equations into a first-order system of equations.

2) The domain is then discretized, allowing for calculation over each subdomain rather than the complete domain. This produces more precise outcomes.

3) To get finite difference equations, central-difference derivatives and the average of function midpoints are utilised.

4) The resultant equations are then linearized using Newton's approach, as Keller described. Also, write them in a tridiagonal matrix.

Finally, the ultimate result is obtained by using LU decomposition.

Equations (7) - (9) subject to boundary conditions (10) are solved numerically using the Keller box method.

We start with introducing new independent variables

With change boundary conditions

To validate programme code for verification, Table 1 compares current Nusselt number results with published results by Magyari and Keller , Bidin and Nazar , El-Aziz and Sharma et al in the absence of Nanofluid, with S = 0, λ = 1.5, and R = 0. This table demonstrates that data generated by the new code and the prior code are comparable in quality (Magyari and Keller , Bidin and Nazar , El-Aziz & Sharma et al ).

The behaviour of the temperature, concentration, and velocity profiles has been carefully examined. Theoretical heat and mass transfer analysis has been carried out using graphical and numerical methods.

As shown in Figures 3-5, the symbol M stands for the magnetic condition and the effect of the magnetic field on temperature, concentration, and velocity. A transverse magnetic field causes the Lorentz force, which in turn produces a force that retards velocity movement. When the temperature and concentration profiles grow, the velocity decreases because the retarding force increases with the value of M. Notably, the solutal concentration profile is lowered farther from the sheet due to the significant magnetic strength. The impact of the suction/injection parameter (S) on the dimensionless stream-wise velocities (V) is shown in Figure 6. The graphs in Figure 6 show that the parameter S has a significant impact on the boundary layer's thickness. As S is raised, the flow seems to drastically slow down. The boundary layer is forced closer to the wall by suction or injection, which lowers velocity. The suction/injection causes a reduction in the thickness of the momentum boundary layer. The ratio λ describes the stretching speeds between the two directions, y, x. The stretching rate ratio parameter increases when the y-direction velocity increases and the x-direction velocity decreases, as shown in Figure 7. The temperature profile seen in Figure 8 is initiated by the Prandtl number. As the Prandtl number continuously increases, the thermal diffusivity weakens and the thermal boundary layer thins. Figure 9 depicts the impact of thermal radiation parameter R on the temperature field. The profile of temperature increases as a function of R. The thickness of the thermal boundary layer is improved by an increase in the thermal radiation parameter. The temperature and concentration graphs fluctuate in response to variations in the Brownian motion parameter Nb, as shown in Figures 10 and 11. The concentration profile graph and temperature are influenced by Brownian motion in that when the parameter values rise, the concentration boundary layer thickness decreases and the temperature rises. The graph also shows that as Nb values rise, the thermal boundary layer thickness does not vary significantly. Figures 12 and 13 illustrate how changes in the thermophoresis parameter Nt cause fluctuations in the temperature and concentration graphs. The temperature drops and the concentration rises as the Nt values rise. The effect of the Lewis number on the volume concentration of dimensionless nanoparticles is seen in Figure 14. We find that for larger Le values, there is a significant decrease in the nanoparticle volume fraction. The ratio of heat to mass diffusivity is the definition of the dimensionless Lewis number. The thickness of the thermal boundary layer increases while the thickness of the boundary layer containing the volume concentration of nanoparticles decreases when Le is increased. in Figures 15-17 Grashof number rises with increasing velocity profile. Increased values of Grashof number induce an augmentation in shrinking forces, which in turn causes an increase in fluid flow. As the Gr numbers rise, the concentration and temperature decrease. From The Grashof Number for mass transfer is illustrated in Figures 18-20. As Gc increases, concentration, temperature, and velocity values decrease.

Figure 21 shows the impact of various permeability parameter values on velocity fields. It is evident that a porous media increases the restriction on the fluid flow, slowing the fluid's velocity. when a result, when the permeability parameter increases, so does the resistance to fluid motion, which causes a decrease in velocity. The behaviour of the non-uniform heat source on temperature distributions is seen in Figure 22. The figure makes it clear that improving the non-uniform heat source settings improves the temperature profiles. This validates the heat source's overall physical behaviour, which shows that positive values for A1 and B1 function as heat generators. The thickness of the thermal boundary layer generally increases with an increase in the heat source. Figure 23 illustrates how different values of the DuFour number Du affect the heat profile. It is evident that the temperature profile is significantly impacted by Du. It has been discovered that an increase in the Du causes the temperature of the boundary layer as a whole to rise and then gradually decrease from the surface to the free flow value. The concentration distribution for the DuFour parameter is displayed in Figure 24, which indicates that concentration grows steadily as Du increases. Figure 25 shows the influence of the chemical reaction parameter (Kr) on the temperature field. A rise in the thermal boundary layer is observed with an increase in the chemical reaction parameter. The thermal boundary layer's change with the chemical reaction parameter (Kr) is depicted in Figure 26. It is found that as the chemical reaction parameter increases, the thickness of the thermal boundary layer decreases.

Table 2 provides skin friction values for M, Gr, Gc, R, S, λ, and K. We see that as we increase the values of Gr and R, skin friction also increases, but in other circumstances it decreases.

Table 3 shows the local share wood values for M, Gr, Gc, Nb, Nt, Du, Kr, and Le. We can observe that as we increase the values of Gr, Gc Nb, and Le, the share wood number grows, but in other cases it falls.

Table 4 displays the local Nusselt numbers (heat transfer) for M, Gr, Gc, Nb, Nt, Pr, R, Du, Kr, and A1&B1. We can see that as we increase the parameters Pr, Nt, and Du, the nusselt number increases whereas in other cases it decreases.

The effects of thermal diffusion (Soret) and diffusion-thermo (DuFour) on MHD mixed convective flow of a viscous nanofluid across a stretching sheet in the presence of a non-uniform heat source/sink and a chemical reaction embedded in a porous media under a magnetic field were investigated. The results are refined using a numerical technique known as the Keller box technique. The following points summarise the study's main findings.

1) Velocity profiles grow with stretching sheet parameter values and reverse trend is observed for permeability parameter.

2) Rising suction/injection parameters (S) and magnetic field parameters (M) cause a drop in the velocity profile.

3) As thermal radiation increases, the temperature profile also rises.

4) Temperature profiles decrease as the Prandtl number (Pr) value increases.

5) The Brownian motion parameter (Nb) value is growing, as is the temperature, while the concentration profiles are following the opposite pattern.

6) The thermophoresis parameter (Nt) value is growing, while temperature is falling, and concentration profiles are in reverse.

7) The value of the Lewis parameter (Le) is increasing The concentration is falling.

8) Thermal Grashof number (Gr) is increasing, velocity is increasing, and temperature and concentration are on the reverse trend.

9) The concentration, temperature, and velocity values drop as the Concentration Grashof number (Gc) increases.

10) Temperature was found to rise as the space-dependent (A1) and temperature-dependent (B1) parameters for heat source/sink increased.

11) Increasing Du values leads to a decrease in temperature, while concentration profiles show the opposite.

12) Chemical reaction parameter Kr values increase as temperature rises, while the concentration profile decreases.

Prashanth: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

Srinivasa Rao: Supervision, Visualization, Writing – review & editing

The authors declare no conflicts of interest.