# Note on the globalized Soret and Dufour formulations in the buoyancy-driven instability in binary mixtures. What do they mean? How are they useful?

Daniel Henry

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Daniel Henry. Note on the globalized Soret and Dufour formulations in the buoyancy-driven instability in binary mixtures. What do they mean? How are they useful?. 2022. hal-03738582

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# Note on the globalized Soret and Dufour formulations in the buoyancy-driven instability in binary mixtures. What do they mean? How are they useful?

D. Henry

Laboratoire de Mécanique des Fluides et d'Acoustique

CNRS / Université de Lyon

Ecole Centrale de Lyon / Université Lyon 1 / INSA de Lyon

ECL, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France

July 26 2022

## Abstract

Globalized formulations have been proposed since at least 40 years for taking into account Soret and Dufour effects in buoyancy-driven instabilities in binary fluids. The purpose of this note is to explain what is the physical meaning of these formulations and how useful they can be.

## 1 Introduction

For the study of buoyancy driven instabilities in binary fluids, we generally use the Boussinesq approximation giving a linear relation between the density and both temperature and mass fraction, together with the phenomenological equations relating the heat and mass fluxes to the thermal and solutal gradients. In the more general case, the cross terms corresponding to the Soret and Dufour effects are involved in these equations and they will induce cross influences between temperature and mass fraction perturbations in the stability analyses. These cross influences may seem complex. However, as shown for example by Gutkowicz-Krusin et al. [1], compact globalized formulations can be derived, which reduce the number of parameters governing the instability results. These globalized formulations, which can be very useful, are not always well known. Moreover, the physical meaning they involve has not been clearly put into light. In this note, we will try to explain the physical meaning of these globalized compact formulations and to show how they can be useful.

## 2 Governing equations

We consider a binary Newtonian fluid mixture confined in a rigid and impervious cavity (the orthogonal coordinates are $\bar{x}$, $\bar{y}$, and $\bar{z}$, $\bar{z}$ being in the vertical direction). The bottom and top boundaries are kept at different constant temperatures (prescribed temperature case) or
are submitted to a constant vertical heat-flux (prescribed heat-flux case), which in any case determines a vertical temperature gradient. We use the Boussinesq approximation: the physical properties of the fluid are assumed to be constant, except for the density $\bar{\rho}$ which is assumed to be a linear function of the temperature $\bar{T}$ and of the mass-fraction $\bar{X}$ (of the denser component) in the buoyancy term:

$$ \bar{\rho} = \bar{\rho}_0[1 - \alpha(\bar{T} - \bar{T}_0) + \beta(\bar{X} - \bar{X}_0)], \quad (1) $$

where $\alpha$ and $\beta$ are the thermal and solutal expansion coefficients, respectively, and an overbar denotes a dimensional quantity. We have to use the phenomenological equations relating the heat and mass fluxes ($\bar{J}_Q$ and $\bar{J}_X$) to the thermal and solutal gradients (Onsager laws). For the heat flux, two expressions will be considered, depending on the Dufour model considered. The first expression $\bar{J}_{Q_1}$ (Dufour model $D_1$) is used by Gutkowicz-Krusin et al. [1], Knobloch [2], and Hardin et al. [3]. The second expression $\bar{J}_{Q_2}$ (Dufour model $D_2$) is used by Lee et al. [4], Hort et al. [5], and Hollinger and Lücke [6], which follow the indications and notations of Landau and Lifshitz [7]. Compared to the first expression, the second expression includes an extra term, which is the diagonal contribution of the Dufour effect to the heat balance.

$$ \bar{J}_X = -\bar{\rho} (D k_T / \bar{T}_0) \nabla \bar{T} - \bar{\rho} D \nabla \bar{X}, \quad (2) $$

$$ \bar{J}_{Q_1} = -\lambda \nabla \bar{T} - \bar{\rho} D k_T (\partial \mu_c / \partial \bar{X}) \nabla \bar{X}, \quad \text{Dufour model } D_1 \quad (3) $$

$$ \begin{aligned} \bar{J}_{Q_2} &= -\lambda \nabla \bar{T} + k_T (\partial \mu_c / \partial \bar{X}) \bar{J}_X, \quad \text{Dufour model } D_2 \\ &= -(\lambda + \bar{\rho} (D k_T / \bar{T}_0) k_T (\partial \mu_c / \partial \bar{X})) \nabla \bar{T} - \bar{\rho} D k_T (\partial \mu_c / \partial \bar{X}) \nabla \bar{X}, \quad (4) \end{aligned} $$

where $\lambda$ is the thermal conductivity, $D$ is the mass diffusivity, $\mu_c$ is the chemical potential, and $k_T$ is the thermal diffusion ratio. $k_T$ enters the expression of both Soret and Dufour effects. $D k_T / \bar{T}_0$ is often replaced by $D' X_1 X_2$, where $D'$ is the Soret diffusion coefficient and $X_1$ and $X_2$ are the volume averaged mass fraction of the components, as in [3].

In the situation considered in this study (binary fluid heated in an impervious box), the cross effect of main importance is the contribution of the temperature gradient to the flux of matter (Soret effect) as it creates the separation of the constituents. The coefficient $k_T$ must have opposite signs for each component of the binary mixture. If $X$ and $k_T$ refer to the denser component, as we have chosen here, $k_T > 0$ means that the denser component will preferentially migrate to the colder region. The influence of the Dufour effect, negligible in binary liquid mixtures, may become important in binary gas mixtures.

The equations that govern the fluid motion in the studied situation describe the conservation of mass, momentum, energy, and mass of the denser component. The introduction of the density variation (1) and of the phenomenological equations (2)-(4) leads to the governing systems to be solved with appropriate boundary conditions.

In such heated binary mixture, there is a trivial steady state solution which is a state of rest characterized by vertical and constant temperature and mass-fraction gradients. This Soret equilibrium corresponds to zero mass flux $\bar{J}_X$ and is given by $\frac{\partial \bar{T}_b}{\partial \bar{z}} = -\frac{\Delta \bar{T}}{H} \bar{z}$ and $\frac{\partial \bar{X}_b}{\partial \bar{z}} = \frac{k_T}{T_0} \frac{\Delta \bar{T}}{H}$, where $\Delta \bar{T}$ is the temperature difference between the bottom and the top of the cavity at this rest state and $H$ is the height of the cavity. This leads to a mass fraction difference between bottom and top boundaries, $\Delta \bar{X} = -k_T \Delta \bar{T} / \bar{T}_0$, which corresponds to the diffusive Soret separation. In fact, such an equilibrium can be stable or unstable depending on the values of the characteristic parameters of the problem.

To study the stability of this trivial solution, a linear stability analysis can be performed. The linearized equations governing the infinitesimal perturbations (superposed on this basic state trivial solution) are

$$ \nabla \cdot \mathbf{V}_p = 0, \quad (5) $$
$$ \frac{\partial \mathbf{V}_p}{\partial \bar{t}} = -\nabla P_p + (\alpha T_p - \beta X_p)g\mathbf{e}_z + \nu \nabla^2 \mathbf{V}_p, \qquad (6) $$

$$ \frac{\partial T_p}{\partial \bar{t}} + (\mathbf{V}_p)_z \frac{\partial \bar{T}_b}{\partial \bar{z}} = \kappa \nabla^2 T_p + \gamma_1 \nabla^2 X_p, \quad \text{model } D_1 \qquad (7) $$

$$ \frac{\partial T_p}{\partial \bar{t}} + (\mathbf{V}_p)_z \frac{\partial \bar{T}_b}{\partial \bar{z}} = (\kappa + \gamma_1 \gamma_2 / D) \nabla^2 T_p + \gamma_1 \nabla^2 X_p, \quad \text{model } D_2 \qquad (8) $$

$$ \frac{\partial X_p}{\partial \bar{t}} + (\mathbf{V}_p)_z \frac{\partial \bar{X}_b}{\partial \bar{z}} = \gamma_2 \nabla^2 T_p + D \nabla^2 X_p, \qquad (9) $$

where $\mathbf{V}_p$, $P_p$, $T_p$, and $X_p$ are perturbations for velocity, pressure, temperature, and mass-fraction, respectively, and $\mathbf{e}_z$ is the unit vector in this $z$ direction. In these equations, following Gutkowicz-Krusin et al. [1], we have used the following notations:

$$ \gamma_1 = \frac{D k_T (\partial \mu_c / \partial \bar{X})}{C_p}, $$

and

$$ \gamma_2 = \frac{D k_T}{\bar{T}_0}, $$

these values coming from (2)-(4) and estimated at mean temperature and mass-fraction values. In addition, $\nu$ is the kinematic viscosity, $\kappa$ is the thermal diffusivity, and $C_p$ is the calorific capacity. The system (5)-(9) can be made dimensionless with the use of reference values: $H$ is used for length, $\nu/H$ for velocity, $H^2/\nu$ for time, $\Delta\bar{T} = (\bar{T}_B - \bar{T}_T)$ (prescribed temperature case), $\Delta\bar{T} = \bar{q}H/(\lambda(1 - D_f S))$ (prescribed heat-flux case, model $D_1$) or $\Delta\bar{T} = \bar{q}H/\lambda$ (prescribed heat-flux case, model $D_2$) for temperature, and $\Delta\bar{X} = -\alpha \Delta\bar{T}/\beta$ for mass-fraction. In the prescribed temperature case, $\Delta\bar{T}$ is directly defined from $\bar{T}_B$ and $\bar{T}_T$, the imposed temperature values at the bottom and top boundaries, respectively. In the prescribed heat-flux case, it is defined through the applied vertical heat-flux $\bar{q}$ ($D_f$ and $S$ which appear with the model $D_1$ will be defined hereafter).

The system governing the dimensionless perturbation variables $\mathbf{V}$, $P$, $T$, and $X$ is then

$$ \nabla \cdot \mathbf{V} = 0, \qquad (10) $$

$$ \frac{\partial \mathbf{V}}{\partial t} = -\nabla P + Gr(T + X)\mathbf{e}_z + \nabla^2 \mathbf{V}, \qquad (11) $$

$$ Pr \frac{\partial T}{\partial t} = Pr V_z + \nabla^2 T - D_f \nabla^2 X, \quad \text{model } D_1 \qquad (12) $$

$$ Pr \frac{\partial T}{\partial t} = Pr V_z + (1 + D_f S) \nabla^2 T - D_f \nabla^2 X, \quad \text{model } D_2 \qquad (13) $$

$$ Sc \frac{\partial X}{\partial t} = Sc S V_z - S \nabla^2 T + \nabla^2 X. \qquad (14) $$

The dimensionless parameters appearing in these equations are the Grashof number,

$$ Gr = \frac{\alpha g \Delta \bar{T} H^3}{\nu^2}, $$

($Gr > 0$ when heating is from the bottom, and $Gr < 0$ when heating is from the top), the Prandtl number,

$$ Pr = \frac{\nu}{\kappa}, $$
the Schmidt number,

$$ Sc = \frac{\nu}{D}, $$

the separation or Soret parameter,

$$ S = \frac{\beta \gamma_2}{\alpha D}, $$

and the Dufour parameter,

$$ D_f = \frac{\alpha \gamma_1}{\beta \kappa}. $$

The Soret parameter gives the ratio between the density change due to the Soret separation and that due to the applied temperature difference.

The associated boundary conditions for the perturbations are

* $T = 0$ or $(J_Q)_n = 0$ at $z = 0$ and $z = 1$ (bottom and top walls) corresponding to the prescribed temperature case or the prescribed heat-flux case $((J_Q)_n)$ is the dimensionless perturbation heat-flux along the normal to the walls; $(J_Q)_n = -\partial T/\partial n + D_f \partial X/\partial n$ with the model $D_1$ and $(J_Q)_n = -(1 + D_f S) \partial T/\partial n + D_f \partial X/\partial n$ with the model $D_2$);

* $V = 0$ and $(J_X)_n = S \partial T/\partial n - \partial X/\partial n$ on all the boundaries corresponding to no mass-flux through the rigid and impervious walls $((J_X)_n)$ is the dimensionless perturbation mass-flux along the normal to the walls);

* $T = 0$ or $(J_Q)_n = 0$ on the lateral boundaries corresponding to conducting or adiabatic (zero heat flux) conditions for the temperature at the lateral walls (these cases are referred in the following as conducting and adiabatic cases, respectively).

The results obtained from the solution of system (10)-(14) with the associated boundary conditions correspond to threshold values below which the trivial solution is stable and beyond which it becomes unstable due to the development of a perturbation. These threshold values correspond to critical temperature differences expressed through critical values of the Grashof number ($Gr_c$) or rather of the Rayleigh number ($Ra_c = Gr_c Pr$) as a function of the other parameters. In the following, we will only consider steady thresholds (development of steady perturbations). In this case, it is well known that for a mono-component fluid the results are given through a critical value of the Rayleigh number ($Ra_c$) only depending on the geometry of the cavity and on the temperature boundary conditions. For example, for the extended fluid layer, $Ra_c \approx 1708$ for the prescribed temperature case and $Ra_c \approx 720$ for the prescribed heat-flux case (Nield (1967)), and for a tall cylinder (height over radius equal to 6) in the prescribed temperature case, $Ra_c = 3.02 \times 10^5$ for the conducting case and $Ra_c = 1.01 \times 10^5$ for the adiabatic case (Hardin et al. (1990)). In the next sections, we will first consider the case of a mixture with Soret effect alone (without Dufour effect), and then the case of a mixture with Soret and Dufour effects.

## 3 Soret formulation without Dufour effect

In the case where the Dufour effect is neglected, the governing equations correspond to the system (10)-(14) with $D_f = 0$. We want to derive a formulation of the equations with the smallest number of parameters at steady state and to understand its physical meaning.

Consider first equation (14). If the convective transport term is zero, equation (14) at steady
state becomes $-S \nabla^2 T + \nabla^2 X = 0$ which, with the associated boundary condition $(J_X)_n = S \partial T / \partial n - \partial X / \partial n = 0$, gives the solution $X = S T$. This means that at steady state and without convective transport, the Soret effect leads to perturbations of mass-fraction connected to those of temperature by the parameter $S$. This suggests to write $X$ as

$$X = S T + X_c, \qquad (15)$$

the new variable $X_c$ being the perturbation of the mass fraction due to the convective transport term, called self-perturbation of the mass fraction. This change of variable then expresses that at steady state, the mass fraction perturbation $X$ can be decomposed in a part coming from the temperature perturbation, through the Soret effect, and in another part connected to the self-perturbation of the mass fraction. With this change of variable the zero mass-flux boundary condition becomes $\partial X_c / \partial n = 0$ and the steady state system to solve is

$$\nabla \cdot \mathbf{V} = 0, \qquad (16)$$

$$-\nabla P + Gr((1+S)T + X_c)e_z + \nabla^2 \mathbf{V} = 0, \qquad (17)$$

$$Pr V_z + \nabla^2 T = 0, \qquad (18)$$

$$Sc S V_z + \nabla^2 X_c = 0. \qquad (19)$$

By taking $T = Pr T'$ and $X_c = Sc S X'_c$, we will define new perturbations $T'$ and $X'_c$ which will satisfy identical equations at steady states and will then be of the same order of magnitude. The governing system is then

$$\nabla \cdot \mathbf{V} = 0, \qquad (20)$$

$$-\nabla P + Gr(Pr(1+S)T' + Sc S X'_c)e_z + \nabla^2 \mathbf{V} = 0, \qquad (21)$$

$$V_z + \nabla^2 T' = 0, \qquad (22)$$

$$V_z + \nabla^2 X'_c = 0, \qquad (23)$$

with the associated boundary conditions:

* $T' = 0$ or $\partial T' / \partial n = 0$ at $z = 0$ and $z = 1$;

* $\mathbf{V} = 0$ and $\partial X'_c / \partial n = 0$ on all the boundaries;

* $T' = 0$ or $\partial T' / \partial n = 0$ on the lateral boundaries.

Equation (21) indicates that the buoyancy forces which generate the motion are proportional to the perturbations of temperature $T'$ with the factor $Gr Pr (1+S)$ and to the perturbations of mass-fraction $X'_c$ with the factor $Gr Sc S$. It must be pointed out that the term containing the perturbations of temperature, $T'$, corresponds to the direct perturbations of temperature, but also to its repercussion on the mass-fraction field by the Soret effect, whereas the term containing the perturbations of mass-fraction, $X'_c$, only corresponds to the self-perturbations of the mass-fraction due to convective transport. $T'$ and $X'_c$ being now perturbations of the same order of magnitude, the relative importance of the two terms will directly depend on the factors $Gr Pr (1+S)$ and $Gr Sc S$ and will be correctly represented by the parameter $\Psi$:

$$\Psi = \frac{Sc}{Pr} \frac{S}{1+S},$$

a globalized Soret parameter which was first proposed by Schechter et al. [8] ($\Psi$ is denoted as $H$ in their study), and then used by Gutkowicz-Krusin et al. [1], for example.
From the above discussion, we see that if $\Psi \to 0$, the main influence comes from the perturbation of the temperature, whereas if $\Psi \to \infty$, the main influence comes from the self-perturbations of the mass fraction. Equation (21) can then be written as a function of this parameter $\Psi$ and of a Rayleigh number $\tilde{Ra}$:

$$ \tilde{Ra} = Ra(1 + S) = GrPr(1 + S), $$

under the form:

$$ -\nabla P + \tilde{Ra}(T' + \Psi X_c')e_z + \nabla^2 V = 0. \quad (24) $$

Alternatively (but less usual for Soret situations), it could also be written as a function of this Rayleigh number $\tilde{Ra}$ and of a solutal Rayleigh number defined as $Ra_s = \tilde{Ra}\Psi = GrScS$. Taking apart the dependence on the geometry of the cavity and on the temperature boundary conditions, two parameters are then enough to describe the steady problem, and the steady thresholds can then be given by a curve expressing the critical Rayleigh number $\tilde{Ra}_c$ as a function of the value of $\Psi$ (or as a function of the value of $Ra_s$).

## 4 First Soret-Dufour formulation (development for the Dufour model $D_1$)

We now consider both Soret and Dufour effects. Starting with equations (10)-(14), we can apply the change of variable used for $X$ in the Soret formulation without Dufour effect, $X = ST+X_c$, and similar changes for $T$,

$$ T = D_f X + T_c, \quad \text{model } D_1 \quad (25) $$

$$ T = -D_f S T + D_f X + T_c. \quad \text{model } D_2 \quad (26) $$

Now both $X_c$ and $T_c$ are self-perturbations of $X$ and $T$, respectively, due to convective transport. $D_f X$ is the perturbation of $T$ coming from the perturbation of $X$ through the Dufour effect. With the model $D_2$, the Dufour effect also gives a perturbation on temperature coming from $T$ itself through the contribution $-D_f S T$. With this new change of variable for temperature, the zero heat-flux boundary condition becomes $\partial T_c / \partial n = 0$. We now consider the case with the Dufour model $D_1$, whereas the case with the Dufour model $D_2$ is given in Annexe A.

From the relations (15) and (25), we get

$$ X = \frac{X_c + S T_c}{1 - D_f S} \quad \text{and} \quad T = \frac{T_c + D_f X_c}{1 - D_f S}, $$

which are used together with $T_c = Pr T'_c$ and $X_c = Sc S X'_c$, as in the Soret formulation without Dufour effect, to get a new system of equations:

$$ \nabla \cdot \mathbf{V} = 0, \quad (27) $$

$$ -\nabla P + \left[ \left( \frac{GrPr(1+S)}{1-D_f S} \right) T'_c + \left( \frac{GrScS(1+D_f)}{1-D_f S} \right) X'_c \right] e_z + \nabla^2 V = 0, \quad (28) $$

$$ V_z + \nabla^2 T'_c = 0, \quad (29) $$

$$ V_z + \nabla^2 X'_c = 0, \quad (30) $$

with the associated boundary conditions:

$$ \bullet \quad T = 0 \ (T'_c = -Sc D_f S X'_c / Pr) \text{ or } \partial T'_c / \partial n = 0 \text{ at } z = 0 \text{ and } z = 1; $$
* $V = 0$ and $\partial X'_c / \partial n = 0$ on all the boundaries;

* $T = 0$ ($T'_c = -Sc D_f S X'_c / Pr$) or $\partial T'_c / \partial n = 0$ on the lateral boundaries.

Equation (28) indicates that here the buoyancy forces which generate the motion are proportional to the perturbations of temperature $T'_c$ with the factor $Gr Pr (1+S)/(1-D_f S)$ and to the perturbations of mass-fraction $X'_c$ with the factor $Gr Sc S (1+D_f)/(1-D_f S)$. $T'_c$ and $X'_c$ being still perturbations of the same order of magnitude, the relative importance of the two terms will directly depend on the factors $Gr Pr (1+S)/(1-D_f S)$ and $Gr Sc S (1+D_f)/(1-D_f S)$, and will be correctly represented by the parameter $\Psi'_{D_1}$:

$$ \Psi'_{D_1} = \frac{Sc}{Pr} \frac{S}{1+S} (1+D_f) = \Psi (1+D_f), $$

a globalized Soret parameter accounting for Dufour effect. Equation (28) can then be written as a function of this parameter $\Psi'_{D_1}$ and of a Rayleigh number $\tilde{Ra}'_{D_1}$:

$$ \tilde{Ra}'_{D_1} = Ra \frac{1+S}{1-D_f S} = \frac{\tilde{Ra}}{1-D_f S}, $$

under the form:

$$ -\nabla P + \tilde{Ra}'_{D_1} (T'_c + \Psi'_{D_1} X'_c) e_z + \nabla^2 V = 0. \quad (31) $$

Alternatively, it could be also written as a function of this Rayleigh number $\tilde{Ra}'_{D_1}$ and of a solutal Rayleigh number defined as $(Ra'_{D_1})_s = Gr Sc S (1+D_f)/(1-D_f S)$.

In this case including Dufour effect, it is also possible to physically explain how the two terms of the buoyancy force, denoted $B_{T'_c}$ and $B_{X'_c}$ in the following are obtained. The first term corresponds to all the contributions of the self-perturbations of temperature. The direct contribution is proportional to $Gr Pr$, but, by Soret effect, it will create a perturbation on mass-fraction proportional to $Gr Pr S$, and then, by Dufour effect, a perturbation on temperature proportional to $Gr Pr S D_f$, and so on, alternatively on mass-fraction and temperature by Soret and Dufour effects. For this first term, it then gives

$$ B_{T'_c} = T'_c Gr Pr (1 + S + S D_f + S D_f S + S D_f S D_f + S D_f S D_f S + ...) \quad (32) $$

The second term corresponds to all the contributions of the self-perturbations of mass-fraction. The direct contribution is proportional to $Gr Sc S$, but, by Dufour effect, it will create a perturbation on temperature proportional to $Gr Sc S D_f$, and then, by Soret effect, a perturbation on mass-fraction proportional to $Gr Sc S D_f S$, and so on, alternatively on temperature and mass-fraction by Dufour and Soret effects. For this second term, it then gives

$$ B_{X'_c} = X'_c Gr Sc S (1 + D_f + D_f S + D_f S D_f + D_f S D_f S + D_f S D_f S D_f + ...) \quad (33) $$

These two expressions can be written differently as

$$ B_{T'_c} = T'_c Gr Pr (1 + S) (1 + K + K^2 + K^3 + ...) \quad (34) $$

$$ B_{X'_c} = X'_c Gr Sc S (1 + D_f) (1 + K + K^2 + K^3 + ...) \quad (35) $$

with $K = D_f S$. In fact from Onsager's reciprocal relations and second law of thermodynamics, we have

$$ 0 \le \frac{\gamma_1 \gamma_2}{\kappa D} = D_f S = K \le 1. \quad (36) $$
So the geometrical series found in (34) and (35) is convergent and has a well defined limit for an infinite expansion, which is $1/(1 - D_f S)$. We then obtain the two final expressions for the buoyancy forces:

$$B_{T_c'} = T_c' \frac{GrPr(1+S)}{(1 - D_f S)}, \qquad (37)$$

and

$$B_{X_c'} = X_c' \frac{GrScS(1+D_f)}{(1 - D_f S)}, \qquad (38)$$

which are the expressions we found during the derivation of the equations. We see that, compared with the pure Soret case, the intervention of the Dufour effect leads to successive contributions (as we have a cross-coupling between the equations), which introduce a factor $1/(1 - D_f S)$ for the contributions of the self-perturbations of temperature and a factor $(1 + D_f)/(1 - D_f S)$ for the contributions of the self-perturbations of mass-fraction. At steady state, the governing equations are then similar to those obtained for the Soret formulation without Dufour effect, but with the parameters $(\tilde{R}a, \Psi)$ replaced by $(\tilde{R}a'_{D_1}, \Psi'_{D_1})$, and the variable $T'$ changed to $T_c'$.

In fact, the exact equivalence between the two problems can be done only for certain boundary conditions. Concerning velocity and mass-fraction, there is no difference in the boundary conditions. For the temperature, it depends on the type of boundary conditions. For prescribed temperature values at the top and bottom walls, and conducting lateral walls, the two problems are not similar, as the boundary condition is $T = 0$ ($T_c' = -ScD_fS X_c'/Pr$), whereas the equivalence with the problem with only Soret effect would lead to $T_c' = 0$. Finally, the exact equivalence between the two problems can only be obtained for prescribed heat-flux at the top and bottom walls and adiabatic lateral walls, as the boundary condition, $\partial T_c'/\partial n = 0$, is similar to that used in the problem with only Soret effect. In this specific case, the results obtained in the formulation $(\tilde{R}a, \Psi)$ without Dufour effect can be exactly applied with Dufour effect in the formulation $(\tilde{R}a'_{D_1}, \Psi'_{D_1})$.

# 5 Second Soret-Dufour formulation
(development for the Dufour model $D_1$)

The previous Soret formulation with Dufour effect has allowed us to understand in which way the coupling by Soret and Dufour effects affects the transition, but the correspondence with the problem without Dufour effect was obtained only for certain boundary conditions. In order to get a more general correspondence, in particular for the more usual case of prescribed temperatures at the top and bottom walls, we need a slightly different formulation. We now break the symmetry between temperature and mass-fraction and choose variables similar to that used in the Soret formulation without Dufour effect, namely $T$ and $X_c$ such that $X = ST + X_c$. $T$ contains all the perturbations of the temperature (which also affect $X$ through the Soret effect), those created by the motion (self-perturbations of temperature) and all the others caused by successive repercussions by Dufour and Soret effect of the self-perturbations of temperature and mass-fraction, and $X_c$ only corresponds to the self-perturbations of mass-fraction caused by the motion. The system of equations is identical to (16)-(19), except for equation (18), which is changed due to the introduction of the Dufour effect and becomes

$$PrV_z + \nabla^2 T (1 - D_f S) - D_f \nabla^2 X_c = 0, \quad \text{model } D_1 \qquad (39)$$

$$PrV_z + \nabla^2 T - D_f \nabla^2 X_c = 0. \quad \text{model } D_2 \qquad (40)$$
We now consider the case with the Dufour model $D_1$, whereas the case with the Dufour model $D_2$ is given in Annexe B.

The equation (39) can also be combined with (19) to obtain

$$ (Pr + Sc D_f S) V_z + \nabla^2 T(1 - D_f S) = 0. \quad (41) $$

By taking $X_c = Sc S X'_c$ and $T = (Pr + Sc D_f S) T'/(1 - D_f S)$ to have perturbations $X'_c$ and $T'$ of the same order of magnitude, we get a system similar to (20)-(23) with the Navier-Stokes equation (21) changed to

$$ -\nabla P + \left( Gr Pr(1+S) \frac{(1+\frac{Sc}{Pr}D_f S)}{(1-D_f S)} T' + Gr Sc S X'_c \right) e_z + \nabla^2 V = 0. \quad (42) $$

Then, we have two buoyancy forces, one generated by all the perturbations of temperature and their repercussion on mass-fraction by Soret effect (called total force connected to temperature fluctuations), which is

$$ Gr Pr(1+S) \frac{(1+\frac{Sc}{Pr}D_f S)}{(1-D_f S)} T', \quad (43) $$

and the other only corresponding to the self-perturbations of the mass-fraction, which is

$$ Gr Sc S X'_c. \quad (44) $$

As $T'$ and $X'_c$ are of the same order of magnitude, the ratio between these two forces can be given by

$$ \Psi_{D_1} = \frac{Sc}{Pr} \frac{S}{(1+S)} \frac{(1 - D_f S)}{(1 + \frac{Sc}{Pr} D_f S)} = \Psi F_{D_1}, \quad (45) $$

with

$$ F_{D_1} = \frac{(1 - D_f S)}{(1 + \frac{Sc}{Pr} D_f S)} \quad (46) $$

and the force connected to $T'$ is described by

$$ \tilde{Ra}_{D_1} = \frac{\tilde{Ra}}{F_{D_1}}. \quad (47) $$

These parameters $\Psi_{D_1}$ and $\tilde{Ra}_{D_1}$ are those found by Gutkowicz-Krusin et al. [1] and denoted as $\psi$ and $\tilde{R}$, respectively, in their paper. $\Psi_{D_1}$, denoted as $\psi$, is also used in Hardin et al. [3]. With these parameters, equation (42) becomes

$$ -\nabla P + \tilde{Ra}_{D_1} (T' + \Psi_{D_1} X'_c) e_z + \nabla^2 V = 0. \quad (48) $$

In this second formulation, we can also retrieve the buoyancy forces with the previous considerations:

* the second force only corresponds to $Gr Sc S$,

* the first force corresponds to all the other forces, i.e., those coming from the self-perturbations of $T$ (see equation (37)), which correspond to

$$ \frac{Gr Pr(1+S)}{(1 - D_f S)}, $$
and those coming from the self-perturbations of X (see equation (38)), except the initial perturbation, which correspond to

$$ \frac{Gr Sc S (1 + D_f)}{(1 - D_f S)} - Gr Sc S = \frac{Gr Sc D_f S (1 + S)}{(1 - D_f S)}. $$

This first force then corresponds to

$$ Gr Pr(1 + S) \frac{(1 + \frac{Sc}{Pr} D_f S)}{(1 - D_f S)}, $$

which is what has been found in the derivation of the equations.

With this second formulation, with the model $D_1$ as with the model $D_2$ (see Annexe B), we obtain a Navier-Stokes equation which can be written as

$$ -\nabla P + \tilde{R}a_D (T' + \Psi_D X_c') e_z + \nabla^2 V = 0, \quad (49) $$

with the parameters denoted as $\tilde{R}a_D$ and $\Psi_D$ in a general way. Alternatively, this equation could be also written as a function of the Rayleigh number $\tilde{R}a_D$ and of a solutal Rayleigh number defined as $(Ra_D)_s = GrScS = Ra_s$, i.e. equivalent to the solutal Rayleigh number without Dufour effect.

We then still obtain identical equations at steady state with the case without Dufour effect. Concerning the boundary conditions, we still have a correspondence for the velocity and the mass-fraction. For the temperature, the correspondence now occurs for the case with prescribed values at the boundaries, which corresponds to $(T = 0) \sim (T' = 0)$ for the top and bottom walls (prescribed temperature case) as well as for the lateral walls (conducting case). But it can be shown that it is also valid for adiabatic conditions, conditions occuring at the lateral walls (adiabatic case) or at the top and bottom walls (prescribed heat-flux case). Indeed, the condition is a condition of zero heat-flux, which corresponds to

$$ A \frac{\partial T}{\partial n} + B \frac{\partial X}{\partial n} = 0, $$

the expressions of A and B depending on the Dufour model considered ($D_1$ or $D_2$). But, from the condition of zero mass-flux, we also have

$$ S \frac{\partial T}{\partial n} - \frac{\partial X}{\partial n} = 0. $$

These two equations lead to $\partial T/\partial n = 0$ and $\partial X/\partial n = 0$. So, the adiabatic condition can be expressed through $\partial T'/\partial n = 0$, which corresponds well to the adiabatic condition in the case without Dufour effect.

So, in any case, the results obtained in the formulation $(\tilde{R}a, \Psi)$ without Dufour effect can be exactly applied with Dufour effect in the formulations $(\tilde{R}a_D, \Psi_D)$. In the following, we will denote the different thermal boundary conditions by indexes, T or F for the cases with prescribed temperature or prescribed heat-flux at the top and bottom boundaries, c or a for the cases with conducting or adiabatic conditions at the lateral boundaries.

# 6 Discussion on the stability thresholds with only Soret effect

## 6.1 Simple analytic solutions

From the system (20),(22),(23),(24), we see that the critical thresholds can be expressed as simple curves $\tilde{R}a = f(\Psi)$ or $\tilde{R}a = g(Ra_s)$ only depending on the geometry of the system and on
the thermal boundary conditions. In fact, simple analytic solutions for the critical thresholds can be found in particular cases. In the following, we will not write the incompressibility condition, and concerning the boundary conditions, we will only precise those for temperature and mass-fraction, as those for velocity are unchanged.

* In the pure thermal case ($S = 0$, so that $\Psi = 0$ and $\tilde{Ra} = Ra$), the system becomes

$$-\nabla P + RaT' e_z + \nabla^2 V = 0, \qquad (50)$$

$$V_z + \nabla^2 T' = 0. \qquad (51)$$

For a given geometrical configuration, there are well defined positive values for the critical Rayleigh number $Ra$ (indicating, as it is well known, that the instabilities are obtained when heating is from the bottom) for the different thermal boundary conditions (note that the indication for criticality is omitted). These values will be denoted in a general way as $R$, and more specifically with the indexes corresponding to the boundary conditions, i.e., $R_{Tc}$, $R_{Ta}$, $R_{Fc}$, or $R_{Fa}$. For the pure thermal case, the critical threshold will then be expressed as

$$Ra = R. \qquad (52)$$

* Considering a mixture with Soret effect, but such that $\Psi \sim 0$ (i.e., $Sc/Pr$ or $\beta$ very small), the system becomes

$$-\nabla P + \tilde{Ra}T'e_z + \nabla^2 V = 0, \qquad (53)$$

$$V_z + \nabla^2 T' = 0. \qquad (54)$$

The perturbations $X'_c$ have no influence on the thresholds, which are connected to perturbations of temperature, either directly or through the Soret effect. The system is identical to that obtained in the pure thermal case but with $Ra$ replaced by $\tilde{Ra}$ and has the same boundary conditions. In a general way, the critical thresholds will then be expressed as

$$\tilde{Ra} = Ra(1 + S) = R. \qquad (55)$$

* In the asymptotic case $\Psi \to \infty$ (or $Ra_s \gg \tilde{Ra}$), the system becomes

$$-\nabla P + \tilde{Ra}\Psi X'_c e_z + \nabla^2 V = 0, \qquad (56)$$

$$V_z + \nabla^2 X'_c = 0. \qquad (57)$$

Here, the perturbations $T'$ have no influence on the thresholds, as the instability arises from the perturbations of mass-fraction, $X'_c$. The system is still identical to that obtained in the pure thermal case but with $Ra$ replaced by $\tilde{Ra}\Psi = Ra_s$ and the variable $T'$ replaced by $X'_c$. The boundary conditions are those for $X'_c$, i.e. $\partial X'_c/\partial n = 0$, and then correspond to the prescribed heat-flux and adiabatic case of the pure thermal situation. The critical thresholds will then be expressed as

$$\tilde{Ra}\Psi = Ra_s = R_{Fa}. \qquad (58)$$

or

$$Ra S \frac{Sc}{Pr} = R_{Fa}. \qquad (59)$$
* An interesting case to consider is the case where $T'$ and $X'_c$ have similar boundary conditions, i.e., the prescribed heat-flux and adiabatic case (called reference case in the following). In such situation, from the general system (20),(22),(23),(24), we can combine equations (22) and (23) to get a new system:

$$-\nabla P + \tilde{R}a(T' + \Psi X'_c) e_z + \nabla^2 V = 0, \qquad (60)$$

$$V_z(1 + \Psi) + \nabla^2(T' + \Psi X'_c) = 0, \qquad (61)$$

or

$$-\nabla P + \tilde{R}a(1 + \Psi) \left( \frac{T' + \Psi X'_c}{1 + \Psi} \right) e_z + \nabla^2 V = 0, \qquad (62)$$

$$V_z + \nabla^2 \left( \frac{T' + \Psi X'_c}{1 + \Psi} \right) = 0. \qquad (63)$$

The system is still identical to that obtained in the pure thermal case but with $Ra$ replaced by $\tilde{R}a(1 + \Psi) = \tilde{R}a + Ra_s$ and the variable $T'$ replaced by $(T' + \Psi X'_c)/(1 + \Psi)$. The boundary conditions for this new variable are those of the prescribed heat-flux and adiabatic case, so that the critical thresholds can be expressed as

$$\tilde{R}a(1 + \Psi) = \tilde{R}a + Ra_s = R_{Fa}, \qquad (64)$$

or

$$Ra \left( 1 + S \left( 1 + \frac{Sc}{Pr} \right) \right) = R_{Fa}. \qquad (65)$$

## 6.2 Analysis of the stability thresholds

We have seen that, in the pure Soret case, the stability problem only depends on two parameters which are representative of the perturbation buoyancy forces acting in the system. The first parameter (generally used as critical parameter) is $\tilde{R}a$ which expresses the total force connected to temperature perturbations. It also corresponds to the perturbation density gradient. The other parameter is $Ra_s$ which expresses the force connected to self-perturbations of mass-fraction, or, alternatively, $\Psi$ which expresses the ratio between these two forces. All the figures illustrating the comments will concern the reference case (prescribed heat-flux and adiabatic case) for which analytical laws have been obtained.

### 6.2.1 ($\tilde{R}a, Ra_s$) representation

First, we have to note that the total force connected to temperature perturbations is destabilizing for $\tilde{R}a > 0$ and stabilizing for $\tilde{R}a < 0$, and similarly the force due to self-perturbations of mass-fraction is destabilizing for $Ra_s > 0$ and stabilizing for $Ra_s < 0$. In this representation, we obtain a single curve which separates the unstable domain (above the curve) and the stable domain (below the curve), as increases of $\tilde{R}a$ and $Ra_s$ are both destabilizing. When $Ra_s = 0$, the force is only connected to temperature perturbations with a threshold $\tilde{R}a = R$, and when $\tilde{R}a = 0$, the force is only connected to self-perturbations of mass-fraction with a threshold $Ra_s = R_{Fa}$. In the reference case ($R = R_{Fa}$; see figure 1), the curve corresponds to a straight line with a slope equal to -1,

$$\tilde{R}a = R_{Fa} - Ra_s.$$

This simplification of the curve comes from the fact that, in this case, the boundary conditions are the same for the temperature perturbations and the self-perturbations of mass-fraction.
<figure><img src="image_2.png" /><figcaption>Figure 1: Critical thresholds in the situation with Soret effect expressed through $\tilde{Ra}/R_{Fa}$ as a function of $Ra_s/R_{Fa}$.</figcaption></figure>

These conditions of Neumann type would correspond to a perturbation with a zero wave-number for the extended layer (with $R_{Fa} = 720$, see Nield (1967)) or to a single roll in confined cavities, and these flow structures are valid for the whole parameter range. On the contrary, in the prescribed temperature and conducting case, the wave-number is finite for $Ra_s = 0$ ($k = 3.12$, corresponding to $\tilde{Ra} = 1708$) due to the boundary conditions on temperature of Dirichlet type (multi-rolls in the confined cavity) and this wave-number evolves to zero for $\tilde{Ra} = 0$ (corresponding to $Ra_s = 720$) and remains zero for the negative values of $\tilde{Ra}$ which correspond to larger $Ra_s$, as the instability for these values of $\tilde{Ra}$ is connected to the destabilizing self-perturbations of mass-fraction with boundary conditions of Neumann type. Compared to the reference case, the boundary conditions for temperature in the other cases correspond to more constrained temperature perturbations, leading to stronger $\tilde{Ra}$ (less destabilizing temperature perturbations) for $\tilde{Ra} > 0$ and to larger $|\tilde{Ra}|$ (less stabilizing temperature perturbations) for $\tilde{Ra} < 0$, the value $Ra_s = R_{Fa}$ for $\tilde{Ra} = 0$ being unchanged. The curves corresponding to the other cases are then above for $\tilde{Ra} > 0$ and below for $\tilde{Ra} < 0$, and they cross at the point $(Ra_s = R_{Fa}, \tilde{Ra} = 0)$.

### 6.2.2 ($\tilde{Ra}, \Psi$) representation

It is interesting to consider the results in such a representation giving a critical threshold (depending on the temperature difference) as a function of a parameter only depending on the physical properties of the fluid, which are approximately constant for a given experiment. In this representation, $\Psi > 0$ means that the force due to self-perturbations of mass-fraction acts in the same way as the total force connected to temperature perturbations, and $\Psi < 0$ means that they act in the opposite way. We have then the influence of the forces in the different quadrants of the ($\tilde{Ra}, \Psi$) representation:

* for $\tilde{Ra} < 0$ and $\Psi > 0$, both forces are stabilizing,

* for $\tilde{Ra} > 0$ and $\Psi > 0$, both forces are destabilizing,
<figure><img src="image_3.png" /><figcaption>Figure 2: Critical thresholds in the situation with Soret effect expressed through $\tilde{Ra}/R_{Fa}$ as a function of $\Psi$ (solid lines). Asymptotic law for large $|\Psi|$ (dashed lines).</figcaption></figure>

* for $\tilde{Ra} < 0$ and $\Psi < 0$, the total force connected to temperature perturbations is stabilizing and the force due to self-perturbations of mass-fraction is destabilizing,

* for $\tilde{Ra} > 0$ and $\Psi < 0$, the total force connected to temperature perturbations is destabilizing and the force due to self-perturbations of mass-fraction is stabilizing.

With this representation, as shown in figure 2 with the reference case where the thresholds are given by

$$ \tilde{Ra} = R_{Fa}/(1 + \Psi), $$

the stability results are given by two curves, one for $\tilde{Ra} > 0$ and the other for $\tilde{Ra} < 0$ and $\Psi < 0$, the fourth quadrant ($\tilde{Ra} < 0$ and $\Psi > 0$) being unconditionnaly stable.

In the quadrant ($\tilde{Ra} > 0$ and $\Psi > 0$), both forces are destabilizing. The threshold expressed by $\tilde{Ra}$ decreases when $\Psi$ increases (corresponding to an increasing influence of the force due to self-perturbations of mass-fraction), going from the value $R$ (or $R_{Fa}$ in figure 2) at $\Psi = 0$ (no influence of this force) to zero for $\Psi \to \infty$ (preponderant influence of this force). This stability curve continues in the quadrant ($\tilde{Ra} > 0$ and $\Psi < 0$), with a strong increase of the thresholds when $|\Psi|$ increases due to the stabilizing contribution of the force due to self-perturbations of mass-fraction. For the reference case shown in figure 2, an asymptote is found for $\Psi = -1$.

In the quadrant ($\tilde{Ra} < 0$ and $\Psi < 0$), we still have $\tilde{Ra} = 0$ for $|\Psi| \to \infty$ (preponderant influence of the force due to self-perturbations of mass-fraction). When $|\Psi|$ decreases, the stabilizing influence of the total force connected to temperature perturbations increases more and more, leading to an increase of the thresholds expressed by $|\tilde{Ra}|$. Here also, for the reference case shown in figure 2, an asymptote is found for $\Psi = -1$.

In any cases, the variations of $\tilde{Ra}$ found for $\Psi \to \infty$ correspond to an asymptotic variation already mentionned, $\tilde{Ra}\Psi = Ra_s = R_{Fa}$, which is found to be quite accurate for $\Psi > 10$.

Compared to the reference case of figure 2, the other cases (as already mentionned in the previous section) will correspond to more stabilized situations for $\tilde{Ra} > 0$ and to less stabilized situations for $\tilde{Ra} < 0$, the asymptotic domain being similar. The stability curves in these cases
<table><caption>Table 1: Correspondence between values of S and values of Ψ.</caption><tr><td>S</td><td>0.1</td><td>-1.1</td><td>-0.9</td><td>-0.1</td></tr><tr><td>Ψ/(Sc/P<sub>r</sub>)</td><td>0.091</td><td>11</td><td>-9</td><td>-0.111</td></tr></table>

will then be above those given in figure 2 and the asymptotes with $|\tilde{R}a| \to \infty$ will correspond to values of Ψ greater than -1.

### 6.2.3 (Ra, S) representation for different Sc/Pr ratios

Since the experimentally meaningful parameters are rather Ra and S (directly related to the temperature difference and the Soret coefficient, respectively), it is also interesting to consider the curves expressing Ra as a function of S for different values of the ratio Sc/Pr. In the reference case, these curves are obtained from (65) as

$$Ra = \frac{R_{Fa}}{\left(1 + S\left(1 + \frac{Sc}{Pr}\right)\right)}, \qquad (66)$$

where Ra is an hyperbolic function of S, with coefficients of S depending on Sc/Pr (see figure 3). We notice that $|Ra|$ is infinite for $S = -1/(1 + Sc/Pr)$.

It is first useful to see the correspondence between the values of Ψ and the values of S through the definition of Ψ: $Ψ = ScS/(Pr(1+S))$. The domain with $Ψ > 0$ and $\tilde{R}a > 0$ corresponds to the variation of S in the positive domain between 0 and $\infty$ with $Ra > 0$, and in the negative domain between $-\infty$ and -1 with $Ra < 0$. The domain with $Ψ < 0$ and $\tilde{R}a < 0$ corresponds to $-1 < S < 0$ with $Ra < 0$. The value $S \to \infty$ corresponds to $Ψ = Sc/Pr$, which induces an asymptotic behaviour for $S \to \infty$ corresponding to $RaS = cst(RaS = R_{Fa}/(1 + Sc/Pr)$ in the reference case). Some other correspondences, which will be useful in the discussion, are given in table 1.

* If Sc/Pr is small, Ψ remains close to zero in a large domain of S: for all positive S (with $Ra > 0$) because for $S \to \infty$, Ψ tends to Sc/Pr which is small, and likewise for large absolute values of S negative (with $Ra < 0$), with the only exception around $S = -1$. In this large domain, as seen previously (55), we have

$$Ra = R/(1 + S) \qquad (67)$$

(see figure 3 obtained with $R = R_{Fa}$), which is what is expected for the instability in binary fluids with Soret effect if the results are only based on the total density of the basic state. This relation is then only valid if the self-perturbations of mass-fraction are negligible. Near $S = -1$ (but with $S < -1$), this relation gives $Ra \to -\infty$. In fact, in this domain with $Ra < 0$, we can use the asymptotic relation (59) which, for $S = -1$, gives

$$Ra = -R_{Fa}/(Sc/Pr),$$

and indicates that for small Sc/Pr, the instability threshold is really very large. For $S > -1$, the system is very stable, because on one hand it must be more stable than for $S = -1$, and on the other hand the instability can only be generated by the self-perturbations of mass-fraction which are negligible.
Summarizing, for small *Sc/Pr*, there are instabilities in the domain of destabilizing global density gradient ($\tilde{Ra} > 0$, that is $Ra > 0$ with $S > -1$ and $Ra < 0$ with $S < -1$) expressed by the relation (67). If the global density gradient is stabilizing ($\tilde{Ra} < 0$, that is $Ra < 0$ with $S > -1$), there is no instability in the limit of small *Sc/Pr*.

<figure><img src="image_4.png" /><figcaption>Figure 3: Critical thresholds in the situation with Soret effect expressed through $Ra/R_{Fa}$ as a function of $S$ for three $Sc/Pr$ ratios, 0 (solid lines), 1 (dashed lines) and 100 (dotted lines). In case $Sc/Pr = 1$, the asymptotic relation (59) is shown around $S = -1$.</figcaption></figure>

* If $Sc/Pr \sim 1$ (as for many gases), in the domain $0 < S < +\infty$ (corresponding to $0 < \Psi < 1$), $\tilde{Ra}$ will slightly decrease with increasing $S$, indicating the intervention on the instability of the self-perturbations of mass-fraction together with the total perturbations connected to temperature (see figure 3 obtained in the reference case). $Ra$ will then decrease from $R$ to 0, but with a quicker variation than (67). For $-\infty < S < -1$ (corresponding to $1 < \Psi < \infty$), as $S$ increases from $S = -\infty$, $\tilde{Ra}$ continuously decreases from the value of $\tilde{Ra}$ obtained for $\Psi = -1$. This corresponds, for $Ra$ negative, to an increase of $|Ra|$ between 0 and the asymptotic value for $S = -1$ given by (59), $|Ra| = R_{Fa}/(Sc/Pr) \approx R_{Fa}$; this increase will be slower than (67) because of the intervention of the destabilizing self-perturbations of mass-fraction. For $-1 < S < 0$ (corresponding to $-\infty < \Psi < 0$), the value of $|Ra|$ still increases from $R_{Fa}$ to reach infinity for a certain value of $S$ in this interval. It can be seen from table 1 that $-1.1 < S < -0.9$ corresponds to $11 < \Psi < +\infty$ and $-\infty < \Psi < -9$, which constitutes the whole domain of asymptotic variations of $\Psi$, where the self-perturbations of mass-fraction are preponderant (recall section 6.2.2). This means that for fluids with $Sc/Pr \sim 1$ such as gases, the asymptotic domain for $\Psi$ is very small and practically only corresponds to the point $S = -1$. This is confirmed by figure 3 where we see that the asymptotic curve (59) giving here $Ra = R_{Fa}/S$ is only valid in a small neighbourhood of $S = -1$. In fact, this point is rather a singular point, as the preponderance of the self-perturbations of mass-fraction comes from the fact that the perturbations of temperature have no influence because they create, by Soret effect, perturbations of mass-fraction which are exactly opposite, so that the total perturbations connected to temperature are zero.
* If $Sc/Pr$ is large (as for some liquids, e.g. molten salts with $Sc/Pr \approx 100$ or liquid metals), in the domain $0 < S < +\infty$ (corresponding to $0 < \Psi < 100$), $\tilde{Ra}$ decreases strongly, reaching quite small values belonging to the asymptotic domain ($\Psi > 10$). In this case, the self-perturbations of mass-fraction will not only play a role in the instability, but will become preponderant for large $S$. $Ra$ decreases more quickly than for $Sc/Pr \sim 1$ from $R$ to $0$ (figure 3). For $-\infty < S < -1$ (corresponding to $100 < \Psi < \infty$, the diminution of $\tilde{Ra}$ occurs in a domain of small values. This will correspond to a slow increase of $|Ra|$ between $0$ and the small asymptotic value for $S = -1$ given by (59), $|Ra| = R_{Fa}/(Sc/Pr) \approx R_{Fa}/100$, but all this variation will occur in the asymptotic domain. For $-1 < S < 0$ (corresponding to $-\infty < \Psi < 0$), the value of $|Ra|$ will still increase from $R_{Fa}/100$, first quite slowly (asymptotic domain for $\Psi < -10$), then strongly to reach infinity for a certain value of $S$ in this interval (figure 3). In this case (see table 1), the asymptotic domain ($9 < \Psi < +\infty$ and $-\infty < \Psi < -11$) corresponds to a large domain of $S$, $0.1 < S < +\infty$ and $-\infty < S < -0.1$. This means that, in this entire domain, the influence of the total perturbations connected to temperature will be negligible compared to the influence of the self-perturbations of mass-fraction. Practically, temperature will not be perturbed, and only perturbations of mass-fraction will be responsible for the onset of motions. The asymptotic relation (59) giving here $Ra = R_{Fa}/(100S)$ indicates that two situations with reverse heating and opposite $S$ values present an instability for an identical critical Rayleigh number $|Ra|$. As the asymptotic domain corresponds to the values $S > 0.1$ ($Ra > 0$) and $S < -0.1$ ($Ra < 0$), we will have identical threshold values for opposite $S$ values in the whole domain $|S| > 0.1$. This phenomenon agrees well with the fact that only mass-fraction, which is in the two cases identically stratified, participates to the onset of the motion. Numerical simulations of convective situations in the asymptotic domain with opposite $S$ values are presented by Henry and Roux [9]: for $Sc = 60$ and $Pr = 0.6$, and in slightly supercritical conditions ($Gr = 3$ in situations (with $S = 0.5$ and bottom heating or $S = -0.5$ and top heating) for which the critical Grashof number is 2.66), we observe a similarity in the convective situations characterized by almost identical flows and deformations of the mass-fraction fields, and by no deformation of the temperature field.

* Compared to the reference case given in figure 3, and following the observations obtained in the $(\tilde{Ra}, \Psi)$ formulation, the other cases with different boundary conditions will correspond to more stabilized situations for $Ra > 0$, and for $Ra < 0$ with $S < -1$ and to less stabilized situations for $Ra < 0$ with $S > -1$, with a crossing of the curves for $S = -1$. In fact, for the different boundary conditions, the curves will be similar in the asymptotic domain, a large domain of $S$ for $Sc/Pr = 100$ ($0.1 < S < +\infty$ and $-\infty < S < -0.1$), but the single point $S = -1$ for $Sc/Pr = 1$. For $Sc/Pr$ small, only more stabilized situations will be observed. Concerning the asymptotes for $|Ra| \to \infty$, they will be obtained for values of $S$ greater than $-1/(1 + Sc/Pr)$.

# 7 Discussion on the stability thresholds with Soret and Dufour effects

## 7.1 Simple analytic solutions

We have shown in previous sections that the results obtained with pure Soret effect can be used in the case with Dufour effect by changing the governing parameters from $(\tilde{Ra}, \Psi)$ to $(\tilde{Ra}_D, \Psi_D)$
(second Soret formulation with Dufour effect). In particular, this can be done for the general analytic expression found in the reference case. In the case with Dufour effect, equation (64) then becomes

$$ \tilde{R}a_D (1 + \Psi_D) = \tilde{R}a_D + Ra_s = R_{Fa}, $$

i.e.,

$$ \tilde{R}a = (R_{Fa} - Ra_s) F_D, \qquad (68) $$

or

$$ \tilde{R}a = \frac{R_{Fa} F_D}{(1 + \Psi F_D)}, \qquad (69) $$

or still

$$ Ra = \frac{R_{Fa} (1 - D_f S)}{(1 + S) + \frac{Sc}{Pr} S (1 + D_f)} \quad (\text{model } D_1), \qquad (70) $$

$$ Ra = \frac{R_{Fa}}{(1 + S) + \frac{Sc}{Pr} S (1 + D_f + D_f S)} \quad (\text{model } D_2). \qquad (71) $$

The two last expressions (70) and (71) can also easily be found from the first Soret formulation with Dufour effect, as both formulations are valid for the reference case.

## 7.2 Analysis of the stability thresholds

We have seen that in the Soret situation with Dufour effect, the stability problem still only depends on two parameters which are representative of the perturbation buoyancy forces acting in the system. We will consider the second Soret-Dufour formulation which gives an equivalence with the pure Soret case. The first parameter (generally used as critical parameter) is then $\tilde{R}a_D$ which expresses the total force connected to temperature perturbations. The other parameter is $Ra_s$ which expresses the force connected to self-perturbations of mass-fraction, or, alternatively, $\Psi_D$ which expresses the ratio between these two forces. In this section also, all the figures illustrating the comments will concern the reference case (prescribed heat-flux and adiabatic case) for which analytical laws have been obtained.

### 7.2.1 ($\tilde{R}a_D, Ra_s$) and ($\tilde{R}a_D, \Psi_D$) formulations

Using the equivalence with the problem with pure Soret effect, these two formulations give similar results to those obtained in the ($\tilde{R}a, Ra_s$) and ($\tilde{R}a, \Psi$) formulations, respectively. In particular, for the reference case, the results will be those given in figures 1 and 2, respectively, and similar comments on the influence of the two buoyancy forces involved can be made. In the ($\tilde{R}a_D, \Psi_D$) formulation, the asymptotic variation $\tilde{R}a_D \Psi_D \sim \tilde{R}a \Psi = R_{Fa}$ is now obtained for $|\Psi_D| > 10$.

To precise the influence of the Dufour effect on the instability, it is interesting to consider a less general formulation as the ($\tilde{R}a, Ra_s$) formulation for which the influence of Dufour effect is concentrated in the parameter $F_D$. We will see later how $F_D$ varies with the parameters involved in its definition, but it is first interesting to note that, with both $D_1$ and $D_2$ models, the values of $F_D$ are between 1 and 0, the value 1 corresponding to no Dufour effect and the decrease of $F_D$ being associated with more Dufour effect.
<figure><img src="image_5.png" /><figcaption>Figure 4: Critical thresholds in the situation with Soret and Dufour effects expressed through $\tilde{Ra}/R_{Fa}$ as a function of $Ra_s/R_{Fa}$ for four values of $F_D$, 1 (solid lines), 0.8 (dashed lines), 0.6 (dotted lines) and 0.4 (dashed-and-dotted lines).</figcaption></figure>

### 7.2.2 ($\tilde{Ra}, Ra_s$) formulation for different values of $F_D$

In the ($\tilde{Ra}, Ra_s$) formulation and for the reference case, the influence of the Dufour effect on the stability thresholds through the parameter $F_D$ is given by the relation (68) and the results are

<figure><img src="image_6.png" /><figcaption>Figure 5: Critical thresholds in the situation with Soret and Dufour effects expressed through $\tilde{Ra}/R_{Fa}$ as a function of $\Psi$ for four values of $F_D$, 1 (solid lines), 0.8 (dashed lines), 0.6 (dotted lines) and 0.4 (dashed-and-dotted lines).</figcaption></figure>
shown in figure 4. We see that the straight line with slope -1 obtained without Dufour effect is changed to a straight line with a less steep slope (decrease of the slope proportional to $F_D$) with a rotation around the point at $\tilde{Ra} = 0$ which is not changed, as the only force at this point is the force connected to self-perturbations of mass-fraction, which is insensitive to Dufour effect. This change of slope indicates that when the Dufour effect is increased (through a decrease of $F_D$), the system is more unstable for $\tilde{Ra} > 0$ and less unstable for $\tilde{Ra} < 0$. This is directly connected to the fact that the total force connected to temperature perturbations has been increased with Dufour effect (through the successive repercussions of perturbations involved), going from $\tilde{Ra}$ to $\tilde{Ra}/F_D$, thus being more destabilizing for $\tilde{Ra} > 0$ and more stabilizing for $\tilde{Ra} < 0$, the force connected to self-perturbations of mass-fraction being unchanged.

### 7.2.3 ($\tilde{Ra}, \Psi$) formulation for different values of $F_D$

In the more usual ($\tilde{Ra}, \Psi$) formulation and for the reference case, the influence of the Dufour effect on the stability thresholds through the parameter $F_D$ is given by the relation (69) and the results are shown in figure 5. We can still see the destabilizing effect induced by the Dufour effect for $\tilde{Ra} > 0$ in the upper quadrants and the stabilizing effect for $\tilde{Ra} < 0$ in the lower left quadrant. The asymptotic variation $\tilde{Ra}\Psi = R_{Fa}$ is valid in a smaller domain of $\Psi$ as $F_D$ is decreased.

<figure><img src="image_7.png" /><figcaption>Figure 6: Relationship between $F_D$ and $D_f S$ for both Dufour models, $D_1$ (black curves, equation (46)) and $D_2$ (red curves, equation (82)), and three $Sc/Pr$ ratios, 0 (solid lines), 1 (dashed lines) and 100 (dotted lines).</figcaption></figure>
### 7.2.4 Variation of $F_D$

We want to characterize the variation of $F_D$ when Dufour effect is increased through the parameter $D_f$. In fact, as shown from (46) (Dufour model $D_1$) and (82) (Dufour model $D_2$), $F_D$ really depends on the product $D_f S$ and on $Sc/Pr$. From (36), we know that $0 \le D_f S \le 1$, which induces that $0 \le F_D \le 1$. The variation of $F_D$ as a function of $D_f S$ is given in figure 6 for different $Sc/Pr$ ratios and both Dufour models. For the model $D_1$ (black curves, equation (46)), we see that $F_D$ decreases continuously from 1 to 0 when $D_f S$ is increased from 0 to 1, but this decrease changes with $Sc/Pr$. It is linear for $Sc/Pr \sim 0$, but occurs more and more for small values of $D_f S$ when $Sc/Pr$ is increased. For the model $D_2$ (red curves, equation (82)), the change of $F_D$ with $D_f S$ is less marked as the curves do not go to 0 for $D_f S = 1$. For $Sc/Pr \sim 0$, $F_D$ is constant and equal to 1, which means that the Dufour effect has no influence in this case. For $Sc/Pr = 1$, the decrease of $F_D$ occurs down to only 0.5, here also very differently from what occurs for the $D_1$ model. Finally, for $Sc/Pr = 100$, the decrease of $F_D$ occurs down to about 0.01 and the curve is relatively close to that obtained with the $D_1$ model. Returning to the direct influence of the Dufour parameter $D_f$, as $S$ and $D_f$ have a same sign, an increase of $|D_f|$ for a given $S$ will always correspond to an increase of $D_f S$ and so to a decrease of $F_D$. Then a decrease of $F_D$ really means more Dufour effect, this effect being positive or negative, but with the same sign as the Soret effect.

<figure><img src="image_8.png" /><figcaption>Figure 7: Critical thresholds in the situation with Soret and Dufour effects expressed through $Ra/R_{Fa}$ as a function of $S$ for the Dufour model $D_1$ (equation (70)) and four values of $D_f S$, 0 (solid lines), 0.2 (dashed lines), 0.4 (dotted lines) and 0.6 (dashed-and-dotted lines) ($Sc/Pr = 0$).</figcaption></figure>

### 7.2.5 ($Ra, S$) formulation for different values of $D_f S$

For this formulation, the results depend on the choice of the Dufour model, as we have a different expression of $F_D$ for each model. In any case, $D_f S$ is between 0 and 1, so that values in this range must be taken. We will generally choose four values, 0, 0.2, 0.4 and 0.6. We first discuss the results for the Dufour model $D_1$ and will then give a comparison with the Dufour model
$D_2$. For this model $D_1$, the stability thresholds in the reference case are expressed through the relation (70).

• Case with $Sc/Pr \sim 0$

For $Sc/Pr \sim 0$, equation (70) is reduced to

$$Ra = \frac{R_{Fa}(1 - D_f S)}{(1 + S)} \qquad (72)$$

The curve obtained without Dufour effect (see the case $Sc/Pr = 0$ in figure 3) is modulated by the factor $(1 - D_f S)$ (positive, but less than 1), which decreases as the Dufour effect is increased. This leads to the decrease of the absolute amplitude of the thresholds, and so, in any case, to a destabilisation. This is illustrated in figure 7 for four values of $D_f S$. For all these cases, the curves have an asymptote at $S = -1$. The physical explanation of this general destabilisation can be found in the comments given for $Sc/Pr = 1$.

<figure><img src="image_9.png" /><figcaption>Figure 8: Critical thresholds in the situation with Soret and Dufour effects expressed through $Ra/R_{Fa}$ as a function of $S$ for the Dufour model $D_1$ (equation (70)) and four values of $D_f S$, 0 (solid lines), 0.2 (dashed lines), 0.4 (dotted lines) and 0.6 (dashed-and-dotted lines) ($Sc/Pr = 1$).</figcaption></figure>

• Case with $Sc/Pr = 1$

For $Sc/Pr = 1$, the results in the reference case are shown in figure 8. They show the destabilization effect due to Dufour effect (expressed by the parameter $D_f S$) obtained for $Ra > 0$, and for $Ra < 0$ with $S < -1$, and the stabilization effect obtained for $Ra < 0$ with $S > -1$. We can find some simple justifications of these results in the different quadrants of the $(Ra, S)$ formulation.

When the system is heated from below ($Ra > 0$) with $S > 0$, temperature and mass-fraction are destabilizing, and the Dufour coefficient $D_f$ being positive (same sign as $S$), it will transfer heat to the less rich region, i.e., the heated bottom. As a first tendency, the Dufour effect then tends to increase the destabilizing effect of temperature and, as a consequence by Soret effect, the destabilizing effect of mass-fraction, giving a general effect of destabilization.

When the system is heated from above ($Ra < 0$) with $S < 0$, temperature is stabilizing and mass-fraction destabilizing. The Dufour coefficient $D_f$ being negative, it will transfer heat
<figure><img src="image_10.png" /><figcaption>Figure 9: Critical thresholds in the situation with Soret and Dufour effects expressed through $Ra/R_{Fa}$ as a function of $S$ for the Dufour model $D_1$ (equation (70)) and four values of $D_f S$, 0 (solid lines), 0.2 (dashed lines), 0.4 (dotted lines) and 0.6 (dashed-and-dotted lines) ($Sc/Pr = 100$).</figcaption></figure>

to the richer region, i.e., the heated top. As a first tendency, the Dufour effect then tends to increase the stabilizing effect of temperature, but, as a consequence by Soret effect, to increase the destabilizing effect of mass-fraction. In fact when $S < -1$ ($|S| > 1$), the increase of the destabilizing effect of mass-fraction will be preponderant and will destabilize the system, whereas when $-1 < S < 0$ ($|S| < 1$), the increase of the stabilizing effect of temperature will be preponderant and will stabilize the system.

In fact, this description corresponds to the consideration of the first terms in (32) and (33) (two terms for (32) and three terms for (33)). A more accurate description must include all the repercussions caused by Dufour effect interacting with Soret effect, as we have done before. These other terms (deduced from the previous ones by multiplying successively by $D_f S$ which is positive and less than 1) will give contributions in the same direction, but with a decreasing importance, and they will not modify the qualitative description given previously.

### Case with $Sc/Pr = 100$

For $Sc/Pr = 100$, the results in the reference case are shown in figure 9. Note first that the comments about the changes induced by the Dufour effect in the different quadrants, given in the previous subsection for $Sc/Pr = 1$, are still valid for $Sc/Pr = 100$. For large values of $Sc/Pr$, the influence of $D_f S$ for large $|S|$, in the asymptotic domain, is very small. The influence of $D_f S$ is clearly larger for small values of $S$, around $|S = 0|$. It should be noted, however, that in the vicinity of $|S = 0|$, rather small values of $D_f S$ are expected (except if $D_f$ was infinite, which is totally unexpected), so that the influence of $D_f S$ will still remain small.

### Comparison between $D_1$ and $D_2$ Dufour models

We will rather focus on the moderate values of $Sc/Pr$ for this comparison. From what we saw before in section 7.2.4, the influence of $D_f S$ is expected to be smaller for model $D_2$ (larger values of $F_D$).

For $Sc/Pr = 0$, an influence of $D_f S$ was observed with the model $D_1$ (figure 7). In contrast, there is no influence at all given by the model $D_2$ ($F_D = 1$).

For $Sc/Pr = 1$, the comparison between the two Dufour models is given in figure 10. For
the model $D_2$, we observe the same type of influence of $D_f S$ on the stability curves as for the model $D_1$. This influence, however, is quite strongly reduced, indicating that the two Dufour models give clearly different influences of the Dufour effect and are really not equivalent for these stability purposes.

<figure><img src="image_11.png" /><figcaption>Figure 10: Critical thresholds in the situation with Soret and Dufour effects expressed through $Ra/R_{Fa}$ as a function of $S$ for both Dufour models, $D_1$ (black curves, equation (70)) and $D_2$ (red curves, equation (71)), and four values of $D_f S$, 0 (solid lines), 0.2 (dashed lines), 0.4 (dotted lines) and 0.6 (dashed-and-dotted lines) ($Sc/Pr = 1$).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11: Critical thresholds in the situation with Soret and Dufour effects expressed through $Ra/R_{Fa}$ as a function of $S$ for the Dufour model $D_1$ (equation (70)) and three values of $|D_f|$, 0 (solid lines), 0.1 (dashed lines) and 0.2 (dotted lines) ($Sc/Pr = 1$).</figcaption></figure>
### 7.2.6 ($Ra, S$) formulation for different values of $D_f$

We finally give the influence of the Dufour effect on the stability thresholds directly through the influence of the $D_f$ parameter in the reference case (figure 11). We take the example of the Dufour model $D_1$ and choose two values of $|D_f|$, 0.1 and 0.2. The values of $D_f$ cannot be chosen by chance as we must keep $0 \le D_f S \le 1$. $D_f$ must then have the same sign as $S$ and be positive for $S > 0$ and negative for $S < 0$. This is what we have chosen in the figure. Moreover, with the two values of $|D_f|$ chosen, the range of $S$ values presented in the figure allows to keep $D_f S$ within its theoretical range. With this representation, we have still the crossing of the stability curves for $S = -1$, but we have also the absence of influence of the Dufour effect for $S = 0$, a behaviour which was hidden in figure 8 where the stability curves were plotted for fixed values of $D_f S$, independently of the value of $S$. But the increase of the effect of the Dufour effect when $S$ is increased is now due to the fact that $D_f S$, which is the governing parameter, increases with $S$.

Finally, each representation has its interest, knowing that each of them will give the same stability threshold for given values of the dimensional physical parameters. Besides, the two Dufour models will still give different results. In any case, the influence of the Dufour effect on the stability thresholds is shown to remain weak, much weaker than the influence of the Soret effect in such Rayleigh-Bénard situations.

## 8 Concluding remarks

In this note, we have considered the Rayleigh-Bénard instability in binary fluids, accounting for both the Soret and Dufour effects. We have derived step by step the globalized formulations allowing to reduce the number of parameters governing the instability equations. We have shown that the expression of these parameters can be justified by considering the contributions of elementary temperature and mass-fraction perturbations, complemented by their successive repercussions through Soret and Dufour effects. A better insight in the instability phenomenon in binary mixtures is thus obtained. Note that two different models have been proposed in the literature for the expression of the Dufour effect in the phenomenological laws. Both models can lead to a globalized formulation, but the expressions of the obtained parameters are different.

For a given solid geometry (with no-slip boundary conditions), the instability thresholds in the pure thermal case are given by a critical value of the Rayleigh number, only depending on the boundary conditions associated with the temperature. The different Soret and Soret-Dufour globalized formulations allow to make connections between the critical values of the globalized parameters in these more complex situations and those of the pure thermal case. In particular, the case with prescribed heat flux at the top and bottom boundaries and adiabatic conditions at the lateral walls is particularly interesting, as the instability thresholds in the Soret and Soret-Dufour situations can be obtained analytically with expressions only depending on the critical Rayleigh number of the associated pure thermal case, by which they can be normalized.

In this reference case, the effect of the Soret and Dufour parameters on the normalized instability thresholds can then be easily expressed through different general curves, allowing to better see their contributions. A single curve is obtained for the globalized Soret formulation when expressing the Rayleigh number $\tilde{Ra}$ as a function of the globalized Soret parameter $\Psi$, whereas different curves depending on the ratio $Sc/Pr$ are obtained when using the usual Rayleigh number $Ra$ as a function of the usual Soret parameter $S$. For the globalized Soret-Dufour formulation, the same curve as in the Soret case is obtained when now expressing the Rayleigh number $\tilde{Ra}_D$ as a function of the globalized parameter $\Psi_D$. When expressing $\tilde{Ra}$ as a function of $\Psi$, the curves now depend on a kind of Dufour parameter $F_D$. And when expressing
$Ra$ as a function of $S$, the curves depend on the Dufour parameter $D_f$ or on the $D_f S$ product and on the $Sc/Pr$ ratio. Each of these representations of the stability thresholds has its interest to understand the influence of the Soret and Dufour effects, knowing that they all correspond to the same threshold for given values of the dimensional physical parameters, only expressed in different manners.

The two Dufour models give different results for the influence of the Dufour effect on the instability thresholds. More precisely, the model $D_2$ gives a smaller influence than the model $D_1$. In any case, this influence of the Dufour effect on the stability thresholds is shown to remain weak, much weaker than the influence of the Soret effect in such Rayleigh-Bénard situations.

Accounting for the small influence of the Dufour effect on the instability, it is rather the more simple globalized Soret formulation which has a practical interest. Nevertheless, the globalized Soret-Dufour formulation can be useful for people interested by the tricky behaviours related to the Dufour effect. And it was also interesting to put into light the intricate interplay between Soret and Dufour effects, which occurs in these Soret-Dufour situations.

A CC-BY public copyright licence has been applied by the author to the present document.

## References

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[3] G. R. Hardin, R. L. Sani, D. Henry, and B. Roux. Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part I: General theory and stationary stability results. *Int. J. Numer. Meth. Fluids*, 10:79–117, 1990.

[4] G. W. T. Lee, P. Lucas, and A. Tyler. Onset of Rayleigh-Bénard convection in binary liquid mixtures of ³He in ⁴He. *J. Fluid Mech.*, 135:235–259, 1983.

[5] W. Hort, S. J. Linz, and M. Lücke. Onset of convection in binary gas mixtures: Role of the Dufour effect. *Phys. Rev. A*, 45:3737–3748, 1992.

[6] St. Hollinger and M. Lücke. Influence of the Dufour effect on convection in binary gas mixtures. *Phys. Rev. E*, 52:642–657, 1995.

[7] L. D. Landau and E. M. Lifshitz. *Fluid mechanics*, volume 6 of *Course of Theoretical Physics (2nd edition)*. Butterworth-Heinemann, Oxford, 1987.

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[9] D. Henry and B. Roux. Three-dimensional numerical study of convection in a cylindrical diffusion cell: inclination effect. *Phys. Fluids*, 30:1656–1666, 1987.
Annexe A

First Soret-Dufour formulation with the Dufour model $D_2$

From the relations (15) and (26), we get

$$X = (1 + D_f S) X_c + S T_c \quad \text{and} \quad T = T_c + D_f X_c,$$

which are used together with $T_c = PrT'_c$ and $X_c = ScS X'_c$ to get a new system of equations. This system and the boundary conditions are similar to the case with the model $D_1$, except that equation (28) is replaced by

$$-\nabla P + [GrPr(1+S)T'_c + GrScS(1+D_f+D_fS)X'_c]e_z + \nabla^2 V = 0. \quad (73)$$

Equation (73) indicates that, $T'_c$ and $X'_c$ being still perturbations of the same order of magnitude, the ratio of the buoyancy forces $B_{X'_c}$ and $B_{T'_c}$ is represented by the parameter $\Psi'_{D_2}$:

$$\Psi'_{D_2} = \frac{Sc}{Pr} \frac{S}{1+S} (1 + D_f + D_f S) = \Psi (1 + D_f + D_f S),$$

another globalized Soret parameter accounting for Dufour effect. Equation (73) can then be written as a function of this parameter $\Psi'_{D_2}$ and of a Rayleigh number $\tilde{Ra}_{D_2}$:

$$\tilde{Ra}_{D_2} = Ra(1+S) = \tilde{Ra},$$

under the form

$$-\nabla P + \tilde{Ra}'_{D_2} (T'_c + \Psi'_{D_2} X'_c) e_z + \nabla^2 V = 0. \quad (74)$$

Alternatively, we can use a solutal Rayleigh number defined as $(Ra'_{D_2})_s = GrScS(1+D_f+D_fS)$.

In this case corresponding to the model $D_2$, it is also possible to physically explain how the two terms of the buoyancy force are obtained. A specificity of this case is that the Dufour effect acts in two ways: from a perturbation on mass-fraction, it induces a perturbation on temperature with the factor $D_f$, but it also creates a perturbation on temperature from a temperature perturbation with the factor $-D_f S$. For $B_{T'_c}$, which corresponds to all the contributions of the self-perturbations of temperature, the direct contribution is proportional to $GrPr$, but, by Soret effect, it will create a perturbation on mass-fraction proportional to $GrPrS$, and then, by Dufour effect, it will also create perturbations on temperature coming first from $GrPr$ with the factor $-D_f S$ and then from $GrPrS$ with the factor $D_f$. These last temperature perturbations, which can be written as $-D_f S GrPr + D_f GrPrS$, vanish and cannot give further contributions. We then have

$$B_{T'_c} = GrPr(1+S). \quad (75)$$

For $B_{X'_c}$, which corresponds to all the contributions of the self-perturbations of mass-fraction, the direct contribution is proportional to $GrScS$, but by Dufour effect it will create a perturbation on temperature proportional to $GrScS D_f$, and then by Soret effect a perturbation on mass-fraction proportional to $GrScS D_f S$. Finally by Dufour effect, we will have two new contributions which, as for $B_{T'_c}$, will vanish. We then have

$$B_{X'_c} = GrScS(1+D_f+D_fS). \quad (76)$$

In this case corresponding to the model $D_2$, we see that the specific form of the phenomenological equation (4), coming from the dependence of the Dufour term on the mass flux $\bar{J}_X$, does not
allow multiple contributions as for the model $D_1$. In this case too, the governing equations at steady state are similar to those obtained for the Soret formulation without Dufour effect, but with the parameters ($\tilde{R}a, \Psi$) replaced by ($\tilde{R}a'_{D_2}, \Psi'_{D_2}$), and the variable $T'$ changed to $T'_c$, and the exact equivalence between the two problems can only be obtained for prescribed heat-flux at the top and bottom walls and adiabatic lateral walls

Annexe B

Second Soret-Dufour formulation with the Dufour model $D_2$

Equation (40) can also be combined with (19) to obtain

$$ (Pr + Sc D_f S) V_z + \nabla^2 T = 0. \qquad (77) $$

By taking $X_c = Sc S X'_c$ and $T = (Pr + Sc D_f S) T'$ to have perturbations $X'_c$ and $T'$ of the same order of magnitude, we get a system similar to (20)-(23) with the Navier-Stokes equation (21) changed to

$$ -\nabla P + \left( Gr Pr (1+S) \left(1 + \frac{Sc}{Pr} D_f S\right) T' + Gr Sc S X'_c \right) e_z + \nabla^2 V = 0. \qquad (78) $$

Then, we have two buoyancy forces, one generated by all the perturbations of temperature and their repercussion on mass-fraction by Soret effect (called total force connected to temperature fluctuations), which is

$$ Gr Pr (1 + S) \left(1 + \frac{Sc}{Pr} D_f S\right) T', \qquad (79) $$

and the other only corresponding to the self-perturbations of mass-fraction, which is

$$ Gr Sc S X'_c. \qquad (80) $$

As $T'$ and $X'_c$ are of the same order of magnitude, the ratio between these two forces can be given by

$$ \Psi_{D_2} = \frac{Sc}{Pr} \frac{S}{(1+S)} \frac{1}{(1 + \frac{Sc}{Pr} D_f S)} = \Psi F_{D_2}, \qquad (81) $$

with

$$ F_{D_2} = \frac{1}{(1 + \frac{Sc}{Pr} D_f S)}, \qquad (82) $$

and the force connected to $T'$ is described by

$$ \tilde{R}a_{D_2} = \tilde{R}a \left(1 + \frac{Sc}{Pr} D_f S\right) = \frac{\tilde{R}a}{F_{D_2}}. \qquad (83) $$

The parameter $\Psi_{D_2}$ is denoted as $H$ in Lee et al. [4] and as $p$ in Hollinger and Lücke [6]. Lee et al. [4] denote $\tilde{R}a_{D_2}$ and the solutal Rayleigh number as $R_2$ and $R_1$, respectively. Hollinger and Lücke [6] use another Rayleigh number which is the sum of $\tilde{R}a_{D_2}$ and the solutal Rayleigh number.

With these parameters, equation (78) becomes

$$ -\nabla P + \tilde{R}a_{D_2} (T' + \Psi_{D_2} X'_c) e_z + \nabla^2 V = 0. \qquad (84) $$

In this second formulation, we can also retrieve the buoyancy forces with the previous considerations:
* the second force only corresponds to $GrScS$,

* the first force corresponds to all the other forces, i.e., those coming from the self-perturbations of $T$, which correspond to

$$GrPr(1+S),$$

and those coming from the self-perturbations of $X$, except the initial perturbation, which correspond to

$$GrPr(1+S)\left(\frac{Sc}{Pr}D_fS\right) - GrScS = GrScD_fS(1+S).$$

This first force then corresponds to

$$GrPr(1+S)\left(1+\frac{Sc}{Pr}D_fS\right),$$

which is what has been found in the derivation of the equations.