## 1. Introduction

## 2. Problem Formulation

## 3. Method of Solution and Results for Positive Thermo-Diffusion Coefficient

## 4. Method of Solution and Results for Mass Fraction Dependent and Possibly Negative Thermo-Diffusion Coefficient

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Brand, H.R.; Hohenberg, P.C.; Steinberg, V. Codimension-2 bifurcations for convection in binary fluid mixtures. Phys. Rev. A
**1984**, 30, 2549–2561. [Google Scholar] [CrossRef] - Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena; John Wiley and Sons: New York, NY, USA, 1960. [Google Scholar]
- Hollinger, S.T.; Lüke, M. Influence of the Dufour effect on convection in binary gas mixtures. Phys. Rev. E
**1995**, 52, 642–657. [Google Scholar] [CrossRef] [PubMed] - Köhler, W.; Morozov, K.I. The Soret effect in liquid mixtures—A review. J. Non Equilib. Thermodyn.
**2016**, 41, 151–197. [Google Scholar] [CrossRef] - Mortimer, R.G.; Eyring, H. Elementary transition state theory of the Soret and Dufour effects. Proc. Natl. Acad. Sci. USA
**1980**, 77, 1728–1731. [Google Scholar] [CrossRef] [PubMed] - Postelnicu, A. Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Int. J. Heat Mass Transf.
**2004**, 47, 1467–1472. [Google Scholar] [CrossRef] - Weaver, J.A.; Viskanta, R. Natural convection due to horizontal temperature and concentration gradients-2. Species interdiffusion, Soret and Dufour effects. Int. J. Heat Mass Transf.
**1991**, 34, 3121–3133. [Google Scholar] [CrossRef] - Geelhoed, P.; Westerweel, J.; Kjelstrup, S.; Bedeaux, D. Thermophoresis. In Encyclopedia of Microfluidics and Nanofluidics; Li, D., Ed.; Springer: Boston, MA, USA, 2008. [Google Scholar]
- Jawad Hussam, K. Natural Convection and Soret Effect in a Multi-Layered Liquid and Porous System, Paper 1512 Digital Commons @ Ryerson. Master’s Thesis, Ryerson University, Toronto, ON, Canada, 2012. [Google Scholar]
- Lapeira, E.; Bou-Ali, M.M.; Madariaga, J.A.; Santamaria, C. Thermodiffusion coefficients of water/ethanol mixtures for low water mass fractions. Microgravity Sci. Technol.
**2016**, 28, 553–557. [Google Scholar] [CrossRef] - Madariaga, J.A.; Santamaria, C.; Bou-Ali, M.M.; Urteaga, P.; Alonso De Mezquia, D. Measurement of thermodiffusion coefficient in n-Alkane binary mixtures: Composition dependence. J. Phys. Chem.
**2010**, 114, 6937–6942. [Google Scholar] [CrossRef] [PubMed] - Yan, Y.; Blanco, P.; Saghir, M.Z.; Bou-Ali, M.M. An improved theoretical model for thermal diffusion coefficient in liquid hydrocarbon mixtures: Comparison between experimental and numerical results. J. Chem. Phys.
**2008**, 129, 194507. [Google Scholar] [CrossRef] [PubMed] - Costesèque, P.; Mojtabi, A.; Platten, J.K. Thermodiffusion phenomena. C. R. Mécanique
**2011**, 339, 275–279. [Google Scholar] [CrossRef] - Mialdun, A.; Yasnou, V.; Shevtsova, V.; Koniger, A.; Kohler, W.; Alonso de Mezquia, D.; Bou-Ali, M.M. A comprehensive study of diffusion, thermodiffusion, and Soret coefficients of water-isopropanol mixtures. J. Chem. Phys.
**2012**, 136, 244512. [Google Scholar] [CrossRef] - Chipman, J. The Soret effect. J. Am. Chem. Soc.
**1926**, 48, 2577–2589. [Google Scholar] [CrossRef] - Klein, M.; Wiegand, S. The Soret effect of mono- di- and triglycols in ethanol. Phys. Chem. Chem. Phys.
**2011**, 13, 7090–7094. [Google Scholar] [CrossRef] [PubMed] - Putnam, S.A.; Cahill, D.G.; Wong, G.C.L. Temperature dependence of thermodiffusion in aqueous suspensions of charged nanoparticles. Langmuir
**2007**, 23, 9221–9228. [Google Scholar] [CrossRef] [PubMed] - Duhr, S.; Braun, D. Why molecules move along a temperature gradient. Proc. Natl. Acad. Sci. USA
**2006**, 103, 19678–19682. [Google Scholar] [CrossRef] [PubMed] - Mojtabi, A.; Platten, J.K.; Charrier-Mojtabi, M.C. Onset of free convection in solutions with variable Soret coefficients. J. Non Equilib. Thermodyn.
**2002**, 27, 25–44. [Google Scholar] [CrossRef] - Gorban, A.N.; Sargsyan, H.P.; Wahab, H.A. Quasichemical models of multicomponent nonlinear diffusion. Math. Model. Nat. Phenom.
**2011**, 6, 184–262. [Google Scholar] [CrossRef] - Boussinesq, J. Theorie Analitique de la Chaleur; Gutheir-Villars: Paris, France, 1903; Volume 2, p. 172. [Google Scholar]
- Vadasz, P. Fluid Flow and Heat Transfer in Rotating Porous Media; Springer International Publishing AG: Cham, Switzerland, 2015. [Google Scholar] [CrossRef]

**Figure 1.**A vertical tall, fluid layer consisting of a binary mixture, heated by a constant heat flux ${q}_{o}>0$ on one side and exposed to a constant low temperature on the other side.

**Figure 2.**Graphical profile of the solution for (

**a**) the basic temperature and (

**b**) the basic mass fraction.

**Figure 4.**Qualitative description of the mass-fraction-dependent thermodiffusion coefficient, including negative values.

**Figure 5.**Graphical profile of the solution for the basic mass fraction for different Soret numbers.

**Figure 6.**Graphical profiles of the velocity solution $w$ for ${C}_{Z}=0.2$ and different Soret numbers: (

**a**) ${R}_{TC}=1$, (

**b**): ${R}_{TC}=10$, and (

**c**) ${R}_{TC}=100$.

**Figure 7.**Graphical profiles of the velocity solution $w$ for ${C}_{Z}=0.2$ and different values of ${R}_{TC}$: (

**a**) $So=0.1$, (

**b**) $So=1$, (

**c**) $So=10$, and (

**d**) $So=100$.

**Figure 8.**The variation of the location of the non-boundary zeros of the velocity $w$ with the Soret number $So$.

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