Consequences of Soret–Dufour Effects, Thermal Radiation, and Binary Chemical Reaction on Darcy Forchheimer Flow of Nanoﬂuids

: This research article aims to investigate the consequences of binary chemical reaction, thermal radiation, and Soret–Dufour effects on a steady incompressible Darcy–Forchheimer ﬂow of nanoﬂuids. Stretching surface is assumed to drive the ﬂuid along positive horizontal direction. Brownian motion, and the Thermophoresis are accounted in particular. The governing highly nonlinear system of problems which are advanced version of Navier–Stokes equations are transformed into ordinary differential equations (ODEs) using appropriately adjusted transformations invoking symmetric property of the independent variables. The numerical approach using RK45 in connection with shooting technique is adopted to solve the ﬁnal equations. Graphical approach is used to interpret the results and the values of important physical quantities are given in tabular data form. Velocity ﬁeld, temperature distribution and concentration distribution are graphically analyzed for variation in respective ﬂuid parameters. Furthermore, density graphs and stream lines are sketched for the present model. The outputs indicate a rise of temperature ﬁeld in connection with thermal radiation parameter. A clear decline is noticed in velocity ﬁeld for elevated values of Forchheimer number and porosity factor. The Dufour effect anticipates a rising factor for temperature distribution and the same is noticed for concentration distribution in lieu of Soret effect. Thermal radiation and binary chemical reaction has strong impact on heat transport mechanism. The results for physical quantities such as skin friction, heat and mass ﬂux rates are given in tabular data form in last section of this study.


Introduction
Coined by Choi [1], the advent of nanofluids drastically improved the heat transport efficiency of various fluids especially water and glycol. The formulation is purely based on the dispersion of nanometric materials such as metals, polymers, non-metals, etc. in the given base fluid such as water and other typical fluids. For efficient and improved thermal management situation that arose on the extensive and quick displacement of heat in various machinery, state of the art devices, etc. The need for nanofluids is typically based on the logic that classic fluids have deficient performing capacity due to low thermophysical properties whereas, the technological development has greatly advanced the machinery and other equipment. Such equipment and machinery become heated while working with the typical base fluids; however, the invention of nanofluids especially those manufactured with most suited metallic ingredients have drastically controlled the heating situation of machinery. The suspension of nanoparticles that are highly capable of thermal conductivity are responsible for the enhancement in thermo-physical properties of the base fluid. The volume to surface ratio of these nanoparticles is core factor in this improvement. Some relevant studied are discussed here. For example, in the last two years numerous articles are reported targeting the enhanced heat transport mechanism on the basis of nanofluids. Aslfattahi et al. [2] reported some interesting findings in the Mxene-based emerging category of silicone oil type nanofluids, from the prospect of photo-voltaic collector, where they used Transient Hot Bridge 500 to measure the thermal conductivity of the silicone oil type Mxene nanofluid. For viscosity measurement they used Rheometer at different temperatures such as 25:25:125 degrees centigrade. Abbas et al. [3] reported an article to analyze the potential evaluation in the automotive radiators. In this study, they have confirmed the assistance of nanofluids offered to automotive radiators for enhancement in thermal management accepting the challenge of sustainability issue. Bakthavatchalam et al. [4] presented a review article on nanofluids and ionanofluids to see the comparative analysis of heat transfer improvements. A detailed comparative chart is given in this study supporting the enhancement and improvement in thermophysical attributes of the base fluids. Ghahremanian et al. [5] reported an investigation to analyze the nanofluid flow via nano-channel to see the consequences and impact of nanoparticles on condensation. They used Green-Kubo formula in their study to calculate the results. In another study, Cacua et al. [6] reported some interesting findings on stability issues of nanofluids in thermosyphon performance. The results were supported by Two-Phase-Closed mechanism. More interesting articles can be seen in  and cross references.
Flow through Darcy-medium has wide spread applications and significant worth in pharmaceutical, concoction, and modern ecological frameworks. Procedures such as geothermal exchange formats in the context of thermal management, oil supplication in various procedures, waster transfer procedures in atomic heads, water purification and water developments and numerous others. Classic Darcy model was composed of non-Darcian permeable medium which was limited version of this medium when it comes to the above mentioned applications and procedures. It was totally a restricted version for lower porosity and velocity field. For higher velocities Forchheimer updated this model accounting the squared velocity term in momentum equations. A detailed description of this model is given in Darcy [30] and Forchheimer [31]. Later on, Muskat [32] termed this medium as Darcy-Forchheimer medium (the porous medium). Relatively some works have been regarded very promising in the context of Darcy medium. For instance, Muhammad et al. [33] analyzed Darcy-Forchheimer model-based nanofluid flow bounded by an exponentially stretching flat surface in curved shape. They incorporated Cattaneo-Christov double diffusion model in their study to the enhancement in heat and mass transport via relaxation time parameters. Ambreen et al. [34] reported analytic study on hydro-thermal characteristics of nanofluid using Darcy-Forchheimer medium together with Brinkman model. The coupled multiphase Eulerian model resulted in enhancement of performance subject to low porosity factor. Ullah et al. [35] disclosed the features of Arrhenius energy in nanofluid flow under the umbrella of Darcy-Forchheimer model-based three-dimensional rotating flow subject to radiative heat transfer. They adopted NDSolve procedure for the final solutions. Huda et al. [36] produced the results for implementation of Cattaneo-Christov model in ethylene glycol-based nanofluid flow subject to Darcy-Forchheimer medium. Carbon nanotubes were considered in this theoretical study of nanofluid flow. Furthermore, Sajid et al. [37] analyzed the convenience of a Darcy-Forchheimer flow modeled by Maxwell using nonlinear type of thermal radiation via stretching surface.
The combined study of heat and mass flux phenomena involves the flow driven by the difference of densities caused by the gradients of temperature and concentration distributions at the same time together with material composition. Soret relates with the mass flux phenomena caused by the thermal diffusion, while heat flux (energy flux) developed by the solute difference is typically known as the Dufour effect. The mixtures between gases with lighter and medium molecular weights are dealt with by the Soret effect. There are many practical applications linked with this phenomena, such as chemical and geophysical engineering. Researchers showed immense interest in these two aspects and they were as a result involved in various studies. For instance, Jiang et al. [38] reported interesting results in simultaneously heat and mass transport mechanisms via species inter-diffusion process under the direct influence of Dufour and Soret effect. Natural convection was examined via evaporation and condensation phenomena. Liu et al. [39] incorporated multiple-relaxation phenomena via lattice Boltzmann theory to implement the dual-diffusive natural convective flow. Dufour and Soret impacts are involved. They revealed that double-diffusive natural convection can be done easily under the direct influence of Soret and Dufour effects. Sardar et al. [40] analyzed mixed convection phenomena in Carreau nanofluid flow bounded by wedge with the effects of Soret and Dufour. The outcomes indicated that there is a considerable decrease in heat flux for enhanced values of Brownian diffusion parameter.
A typical chemical reaction is always dependent on a decent quantity of activation energy no matter it is linear or binary, to start off. The Arrhenius equations is therefore, necessary for a model involving chemical reaction to calculate the amount of this activation energy. This equations describes the variation of temperature within the system due to the chemical reaction phenomena. Several industrial applications are linked with fluid flow analysis based on chemical reaction due to which researchers have adopted this factor frequently in their models. For instance, Hamid et al. [41] reported some good results on the impact of such chemical reaction using the equation of activation energy in the MHD Williamson nanofluid flow. The heat flux de-escalates for larger chemical reaction parameter. Dhlamini et al. [42] analyzed the impact of activation energy as well as the binary chemical reaction on mixed convection yielded by the use of convective boundary. The Biot number increases the concentration distribution of chemical species. Khan et al. [43] reported dual solutions for the impact of activation energy on the flow MHD titanium alloy (Ti 6 Al 4 V)-based nanofluid with a cross flow structure. A chemical reaction (binary) term and thermal radiation was involved in this study. Important results have been reported by Dulal Pal et al. [44] regarding the impact of chemical reaction and Surot and Dufour factors in fluid flow analysis.
Numerous research articles are reported in the context of nanofluids flow bounded by stretching surfaces under the umbrella of Buongiorno's model. Different methodologies were adopted to tackle the heat and transport problems as reported in the literature mentioned above. However, there are still gap in the studies on nanofluids in many aspects. Here in this study, we incorporated the Darcy-Forchheimer relation together with Soret-Dufour effects and binary chemical reaction subject to linear stretching with uniform impact of MHD to fill the gap in research mentioned in preceding section. Thermal radiation is another important factor involved in this study. Furthermore, Brownian motion, and thermophoresis are considered in particular. The modeled problem is solved numerically using numerical RK45-Scheme with the help of Shooting technique. Up to the understanding and knowledge of authors, this formulation has not been reported in the literature. The results are highly recommended for industrial and engineering applications of nanofluids as well as in chemical engineering. The paper is organized as follows: Modeling of the problem with a literature review of the topic is followed by the numerical solution of the problem. The detailed discussion followed by a comparison with previously published literature is concluded with salient findings noted in last section.

Mathematical Modeling
We consider a steady incompressible Darcy-Forchheimer model-based nanofluid flow maintained over a linearly stretching sheet under the umbrella of Buongiorno's model to tackle the Brownian motion and thermophoresis phenomena. The major contribution comes from Soret-Dufour effects and binary chemical reaction which are the novel factors in this study. The linear stretching property of the sheet with thermal radiation impact are the driving factors for the fluid. The two dimensions are taken in Cartesian coordinates with x-axis alongside the flow and y-axis normal the surface of flow. The fluid flow is taken along positive x-direction while the velocity along surface normal to the x-direction is assumed to be zero. Typical velocity components are assumed as u, v for horizontal and vertical directions, respectively. The governing problems (see for example [21,34,41,44,45]) are given below, ∂u ∂x subject to Define, Using (7) in (1)- (6), we have Here, M 1 is magnetic number such that M 2 1 = is Thermophoretic force factor and Sc = ν D B is Schmidt number, is the radiation derived from q r using the Taylors expansion by neglecting higher order terms.
kT ∞ is used for activation energy with K 1 = (C w − C ∞ ) as the chemical reaction parameter.

Methodology
The numerical approach using RK45 in connection with shooting technique is adopted to solve the final governing system of Ordinary differential equations. Scheme is given as follows: Subject to Results are plotted in graphs.

Results and Discussion
We consider a viscous incompressible nanofluid fluid flow under the umbrella of Darcy-Forchheimer model using a linearly stretchable sheet. The major contribution comes from Soret-Dufour effects and binary chemical reaction which are the novel factors in this study. The linear stretching property of the sheet with thermal radiation impact, Thermophoresis and Brownian diffusion all together are the driving factors for the fluid. From Figures 1-3 we notice the impact of Porosity factor, magnetic parameter and Forchheimer (inertia factor) number on the fluid momentum. Velocity profile receives much reduction for elevated values of these three parameters within the domain of suitable convergent values for the given parameters. Clearly the rising effect of porosity and inertial boost up the friction offered to the fluid flow and thereafter, the result is opposite. In case of magnetic parameter, the sudden bumps in the way of fluid motion offered by the in-field magnetic impact are causing disturbance for smooth movement of the fluid. Thus, the profile receives much reduction in all the three cases. In Figures 4 and 5 the impact of Brownian diffusion and Thermophoresis is analyzed for their elevated values versus the thermal distribution of the fluid flow analysis. In both cases, a rise in thermal state is noticed. The in-predictive motion appearing within the nanoparticles due to the Brownian diffusion speeds up with stronger thermophoretic force. In addition, consequent results is the enhancement of thermal state of the fluid. The radiation parameter is also an important factor in this study and relevant information about the behavior of thermal profile for variation in radiation parameter is given in Figure 6. One can see a prominent rising trend in thermal state of the fluid due to larger values of radiation factor especially when radiation impact approaches 1 (infinity). This trend is justified because the radiation parameter provides convenient and easier convection ground to the thermal distribution. The thermal state receives enhancement for larger values of Dufour number due to the heat generated by the temporary concentration gradient which is the constituent element of Dufour effect shown in Figure 7. Figures 8 and 9 are the influential displays of variation noted in concentration of the nanoparticles for elevated values of Brownian diffusion parameter and thermophoresis. An equally opposite trend is noticed in this case. The rising concentration distribution represents the variation against Brownian diffusion whereas, the reduction is linked to the thermophoresis. The consequent results of Schmidt number on concentration distribution is given in Figure 10. The inverse proportion of kinematic viscosity with Brownian diffusion results a rise in concentration distribution. Concentration distribution receives prominent rise for augmented values of Soret factor given in Figure 11. The impact of Binary chemical reaction is given in Figure 12. A reduction near the surface is noticed in higher values of chemical reaction parameter. The non-dimensional number δ 1 impacts on the concentration profile as a declining factor as shown in Figure 13. Three dimensional display of variation in Nusselt number is given in Figures 14 and 15 for Brownian diffusion and Dufour number, respectively. Figures 16 and 17 are the sketch of stream density at fixed values of porosity factor and magnetic impact parameter. A slight variation is noticed which disappears towards infinity. Table 1 gives a validation of present results for velocity field with previously published literature which is found in agreement. The numerical results of skin-friction and Nusselt number are given in Tables 2 and 3, respectively. As per the data given in Tables 2 and 3

Conclusions
In this research article, we investigated the consequences of Soret-Dufour effects, thermal radiation, and binary chemical reaction on steady incompressible Darcy-Forchheimer flow of nanofluids under the umbrella of Buongiorno's model. Brownian motion and thermophoresis are accounted in particular. The salient features of this research are listed below: • Velocity profile receives much reduction for elevated values of porosity factor and Forchheimer number due to enhancement in friction offered to the fluid flow.
• Thermal state is seen rising for elevated Brownian diffusion and thermophoresis.
• The radiation parameter is responsible for a rising trend in thermal state of the fluid for easy and convenient heat convection.
• Concentration distribution receives prominent rise for augmented values of Soret factor.
• A reduction near the surface in concentration distribution is noticed in higher values of chemical reaction parameter.
• Thermal state receives enhancement for augmented numerical values of Dufour number.
• Skin friction is augmented for all the three parameters that are used in momentum equation i.e., Magnetic impact, Forchheimer number and Porosity.

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The skin-friction is positively affected by the three parameters involved in momentum equation.
• Dufor and Soret numbers appear as a source of enhancement in Nusselt number.
• Chemical reaction results in reduction of the heat flux.
Author Contributions: G.R. formulated the problem, derived the equations, generated the results, wrote the analysis and discussion, and concluded the paper. A.S. generated the results and validated the model. D.B. and M.S. helped in revision and provided funding. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Acknowledgments:
The authors are highly obliged and thankful to unanimous reviewers for their valuable comments and suggestions on the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.