# Soret Effect On The Boundary Layer Regime In A Binary Mixture

## Transcript Of Soret Effect On The Boundary Layer Regime In A Binary Mixture

SORET EFFECT ON THE BOUNDARY LAYER REGIME IN A BINARY MIXTURE CONFINED IN AN INCLINED POROUS ENCLOSURE

A. RTIBI, M. HASNAOUI and A. AMAHMID

Faculty of Sciences Semlalia, Physics Department LMFE, BP 2390, Marrakech, Morocco [email protected]

Abstract

Characteristics of flow and heat and mass transfer induced in an inclined porous layer subject to constant flux of heat combined with Soret effect are studied analytically. An approximate solution is derived for boundary layer regime in an inclined cavity and its validity is examined when the separation parameter varies.

Introduction

In initially homogeneous systems, the Soret phenomenon (also called thermal diffusion) corresponds to the appearance of separation components inside a mixture when the latter is submitted to a thermal gradient. Through the decades, the Soret effect has received a growing interest due to its implication in many engineering applications including the migration of moisture in fibrous insulation, the contaminant transport in saturated soil, the underground disposal of nuclear wastes and drying processes. A literature review showed that theoretical and experimental efforts have been devoted to understand these phenomena and also to the measurement of the Soret coefficient [1], which is usually unknown and depends on various flow mixture properties. Different techniques used by researchers to measure the Soret coefficient are described in this paper. The author concluded that there is no universal technique that works for measuring the Soret coefficient of any binary mixture; each technique has its own limitation.

When multicomponent fluid flows are involved, crossdiffusion (Soret effect) is more often the main driver of various convective phenomena that occur within a thermal stratified media. Subcritical and oscillatory flows, boundary layer regime and hysteresis effects are some examples of these phenomena which are omnipresent in nature and in many engineering applications. In this study, the literature review is restricted to double diffusive works evoking boundary layer flows which can develop within vertical porous enclosures either in the presence or in the absence of Soret effect. Earlier, Trevisan and Bejan [2] derived an analytical solution in the case of a vertical porous cavity submitted to Neumann boundary conditions on vertical walls. The solution proposed is valid only for Le =1 since they assumed a horizontally uniform temperature and solute concentration outside the boundary layer. Later on, it was learned that, in the boundary layer regime, a horizontal temperature and solute concentration stratification in the horizontal direction is possible even for Le ≠ 1. Alavyoon [3] and Mamou et al. [4] developed analytical solutions for the boundary layer regime valid for Le ≠1. After that, for

opposing buoyancy forces, Amahmid et al. [5] investigated a particular situation where the boundary layer scale exponent is different from that derived in [3-4]. It was found that, for large Rayleigh number, the boundary layer regime could be observed for the velocity and density profiles even though neither the temperature nor the concentration profiles exhibited such behaviours. More recently, Er-Raki et al. [6] focused their study on the effect of Soret diffusion on the boundary layer flow regime in a vertical tall porous cavity subject to horizontal heat and mass fluxes. It was demonstrated analytically that, depending on the sign of the buoyancy ratio, the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied.

On the basis of the literature review, it appears that no study in the boundary layer regime was devoted to examine the combined effect of thermodiffusion and the remaining governing parameters in inclined cavities. The conditions of the validity of the boundary layer approximation are examined in this study.

Mathematical formulation and numerical solution

Porous medium

θ= 45°

Fig.1.-Schematical diagram of the physical model and coordinate system

The flow configuration under investigation, sketched in

Fig. 1, is a 2-D Darcy porous cavity of length L′ and height

H′, inclined with an angle 45° with respect to the horizontal

surface and saturated with a binary mixture. The long walls

of the cavity are subject to a uniform flux of heat, q′, while

the short walls are considered adiabatic. All the boundaries

of the porous enclosure are impermeable to mass transfer.

Using the vorticity-stream function formulation, the

dimensionless governing equations are given by:

η ∂ζ ∂t

+ζ =

2 2

RT

⎜⎜⎛ ∂∂x

−

∂∂y ⎟⎟⎞(T

+ϕS )

(1)

⎝

⎠

∂T + u ∂T + v ∂T = ∇2T

(2)

∂t ∂x ∂y

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9ième Congrès de Mécanique, FS Semlalia, Marrakech

523

ε ∂S + u ∂S + v ∂S = 1 [∇2S − ∇2T]

(3)

∂t ∂x ∂y Le

ζ = −∇ 2ψ

(4)

The associated hydrodynamic, thermal and solutal

boundary conditions are as follows:

y = ± Ar : ψ=0,

∂T ∂S = = −1

(5)

2

∂y ∂y

x = ± 1 : ψ = 0 , ∂ T = ∂S = 0 (6)

2

∂x ∂x

The problem is governed by four dimensionless parameters,

namely the thermal Darcy-Rayleigh number RT , the Soret

parameter ϕ , the Lewis number Le and the cavity aspect

ratio Ar , defined as:

R = gβT ∆T′H′3 , ϕ = βS∆S′ , Le = α and A = L′

T

αν

βT∆T′ D r H ′

where

∆T′ =

q′H′

and

∆S′ =

s′ (1− s′ )D ∆T ′

0 0T .

λ

D

The numerical solution of the full governing equations

is based on a central finite-difference technique. The

iterative procedure is performed using the Alternate

Direction Implicit method (A.D.I) for Eqs. (1)-(3). The

stream function field is determined from Eq. (4) using the

point successive-over-relaxation method.

Analytical Solution

For enclosures with sufficiently large aspect ratio ( Ar >> 1 ), the present problem can be significantly simplified and analytically solved using the parallel flow approximation. Taking into account Eqs. (5) and (6) in the integration of Eqs. (2) to (3), Eq. (1) yields the expression: ∂2Ψ − Ω2Ψ(y) = G ∂y 2

where Ω2 = RT sinθ[CT + ϕ(CT + LeCS )] and

G = 22 R T [(C T + ϕ C s ) + (1 + ϕ ) ]/ Ω 2 .

The analytical solution, predicted by the parallel flow approximation, depends on the sign of Ω2 . Note that only the case corresponding to Ω2 > 0 may lead to boundary layer profiles. Hence the case of negative Ω2 will not be considered here. The analytical solution obtained for Ω2 > 0 is given by: Ψ ( y ) = − B Ω cosh( Ω y ) + G

u ( y ) = − B Ω 2 sinh( Ω y )

(7)

T ( x , y ) = C T x − C T B sinh( Ω y ) + (C T G − 1) y

S ( x, y) = Cs x − (CT + Cs Le )B sinh( Ωy) + ((CT + Cs Le )G − 1) y

where B =

G

. The expressions of CT and

Ω cosh( Ω / 2)

CS are given by :

CT = α1B2CT + B(1− GCT )α 2

⎪⎫

⎬ (8)

CS = CT + Le[α1B2 (LeCs + CT ) + Bα 2 (1− G(CT + LeCs ))]⎪⎭

where α 1 = Ω2 [sinh( Ω ) − Ω ]

and α = Ω cosh( Ω ) − 2 sinh(

2

2

Ω). 2

For given values of RT, Le and ϕ , the thermal and solutal

gradients CT and CS can be obtained by solving

numerically Eq. (8).

The Nusselt and Sherwood numbers are given by:

Nu = 1/(1 − Bα 2CT )

⎫

Sh = 1/(1 −

Bα 2 (CT

+

⎬ LeCS ))⎭

(9)

Note that Sh represents the inverse of the solute concentration difference induced across the layer by the thermodiffusion phenomenon.

Boundary layer regime

In the boundary layer regime (large Ω) the analytical solution can be considerably simplified to yield

( ) ( ) ⎡

Ω ⎜⎛ y − 1 ⎟⎞ ⎤

y

Ω⎜⎛ y − 1 ⎟⎞

ψ y = G ⎢1 − e ⎝ 2 ⎠ ⎥ , U y = − GΩe ⎝ 2 ⎠

⎢⎣

⎥⎦

y

( ) y C G Ω⎜⎛ y −1 ⎟⎞

T (x, y) = CT x + CT G − 1 y + y ΩT e ⎝ 2 ⎠

S ( x, y) = Cs x − ((CT + LeCs )G − 1)y

y (C + LeC )G Ω⎜⎛ y − 1 ⎟⎞

+

T

s

e ⎝ 2⎠

y

Ω

[ ][ ] Nu =

1 + α4G 2

1 + G 2 (1 − 4 / Ω ) / Ω

Sh=

1+α4G2 1+ Le2G2α4

1+ 1 Le2G2α + G2α + Le2G4α α / Ω− (1+ Le)G2α 2

Ω

5

4

45

3

( ) α G

(1+ Le)Gα

CT

=

3

1+α

G2

,

Cs

= 1+

1 + Le2

G 2α

3

+ Le2G 4α 2

4

4

4

Where α 3 = 1 − 2 / Ω , α 4 = 1 − 3 / Ω , α 5 = 1 − 4 / Ω

Results and Discussion

In the following section, the effect of the Rayleigh number RT on the flow intensity Ψc and heat and mass transfer characteristics represented by Nu and Sh is discussed. Figs. 2a-b illustrate the results obtained for these quantities as functions of RT for Le=10 and different values of φ (Ar≥12). The numerical solution of the full governing equations, depicted by dots, is seen to be in good agreement with the analytical solution. At very small RT, the Nusselt and Sherwood numbers are close to unity and Ψc is close to 0, indicating that the heat and mass transfer is mainly dominated by the diffusion regime. Note that the Nusselt number and the flow intensity are always increasing with RT. However the Sherwood number exhibits a somewhat complex behavior. In fact, its variations are characterized by a decrease with RT in an intermediate range of this parameter (Fig. 2c). As a result the curve of Sh presents

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9ième Congrès de Mécanique, FS Semlalia, Marrakech

524

relative maximum and minimum before it tends towards the boundary layer curve. Note also that for a given thermal Rayleigh number, the flow intensity decreases with φ, while the tendency is inverted in the case of Nusselt and Sherwood numbers.

18

12

Sh

6

Parallel flow Numerical Boudary layer

(a)

φ = 1. φ = 0.5 φ = -0.5

0

1

10

Ra 100

1000

12

8

Nu

Analytical Numerical Boundary layer

φ = 0.5 φ = 1.

(b)

4

φ = -0.5

1

10

Ra

100

1000

12

8

Ψc

4

Analytical Numerical Boundary layer

(c)

φ = 0.5 φ = -0.5

valid for Nu and Ψc (i.e for the flow and heat transfer) even when RT is of order 20, while in the case of Sh (mass transfer) values of RT higher that 100 are required.

Conclusion

Soret convection in an inclined Darcy porous layer, filled with a binary fluid, subject to constant fluxes of heat is studied analytically and numerically. An analytical solution is derived by assuming that the flow is parallel in the core region of the enclosure. The conditions of the validity of the boundary layer approximation are seen to be dependent on the separation parameter and are more severe in the case of mass transfer.

References

[1] J.K. Platten, The Soret effect: a review of recent experimental results, J. Appl. Mech. 73 (2006) 5–15. [2] O.V. Trevisan, A. Bejan, Mass and heat transfer by natural convection in a vertical slot filled with porous medium, Int. J. Heat Mass Transfer 29 (1986) 403–415. [3] F. Alavyoon, On natural convection in vertical porous enclosures due to prescribed fluxes of heat and mass at the vertical boundaries, Int. J. Heat Mass Transfer 36 (1993) 2479–2498. [4] M. Mamou, P., Vasseur, E., Bilgen, and D. Gobin, Double-diffusive convection in an inclined slot filled with porous medium, European J. of Mechanics B/Fluids 14 (1995) 629-652.

[5] A. Amahmid, M. Hasnaoui, M. Mamou, P. Vasseur, Boundary layer flows in a vertical porous enclosure induced by opposing buoyancy forces, Int. J. Heat Mass Transfer 42 (1999) 3599-3608. [6] M. Er-Raki, M. Hasnaoui, A. Amahmid and M. Mamou, Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes, Int. J. Heat and Mass Transfer, Part A, vol. 49, Issues 17-18, pp. 3111-3120, August 2006.

φ = 1.

0

1

10

100

1000

Ra

Fig. 2: Effect of RT on: (a) Nu, (b) Sh, (c) Ψc for Le= 10

Figs.2a-b show that the boundary layer curves predict well

the results of numerical and parallel flow solutions when

the Rayleigh number is large enough except for the case of φ=-0.5 for which the boundary layer regime is not completely achieved even at RT=103. In fact the value of RT beyond which the boundary layer regime is achieved depends on the parameters φ and Le. Furthermore, negative values of φ lead to opposing thermal and solutal buoyancy

forces which retard the development of the boundary layer

regime. Finally it can be deduced from Figs. 2a-b that, for φ= 1 and 0.5, the boundary layer expressions derived are

___________________________________________________________________________________________________

9ième Congrès de Mécanique, FS Semlalia, Marrakech

525

A. RTIBI, M. HASNAOUI and A. AMAHMID

Faculty of Sciences Semlalia, Physics Department LMFE, BP 2390, Marrakech, Morocco [email protected]

Abstract

Characteristics of flow and heat and mass transfer induced in an inclined porous layer subject to constant flux of heat combined with Soret effect are studied analytically. An approximate solution is derived for boundary layer regime in an inclined cavity and its validity is examined when the separation parameter varies.

Introduction

In initially homogeneous systems, the Soret phenomenon (also called thermal diffusion) corresponds to the appearance of separation components inside a mixture when the latter is submitted to a thermal gradient. Through the decades, the Soret effect has received a growing interest due to its implication in many engineering applications including the migration of moisture in fibrous insulation, the contaminant transport in saturated soil, the underground disposal of nuclear wastes and drying processes. A literature review showed that theoretical and experimental efforts have been devoted to understand these phenomena and also to the measurement of the Soret coefficient [1], which is usually unknown and depends on various flow mixture properties. Different techniques used by researchers to measure the Soret coefficient are described in this paper. The author concluded that there is no universal technique that works for measuring the Soret coefficient of any binary mixture; each technique has its own limitation.

When multicomponent fluid flows are involved, crossdiffusion (Soret effect) is more often the main driver of various convective phenomena that occur within a thermal stratified media. Subcritical and oscillatory flows, boundary layer regime and hysteresis effects are some examples of these phenomena which are omnipresent in nature and in many engineering applications. In this study, the literature review is restricted to double diffusive works evoking boundary layer flows which can develop within vertical porous enclosures either in the presence or in the absence of Soret effect. Earlier, Trevisan and Bejan [2] derived an analytical solution in the case of a vertical porous cavity submitted to Neumann boundary conditions on vertical walls. The solution proposed is valid only for Le =1 since they assumed a horizontally uniform temperature and solute concentration outside the boundary layer. Later on, it was learned that, in the boundary layer regime, a horizontal temperature and solute concentration stratification in the horizontal direction is possible even for Le ≠ 1. Alavyoon [3] and Mamou et al. [4] developed analytical solutions for the boundary layer regime valid for Le ≠1. After that, for

opposing buoyancy forces, Amahmid et al. [5] investigated a particular situation where the boundary layer scale exponent is different from that derived in [3-4]. It was found that, for large Rayleigh number, the boundary layer regime could be observed for the velocity and density profiles even though neither the temperature nor the concentration profiles exhibited such behaviours. More recently, Er-Raki et al. [6] focused their study on the effect of Soret diffusion on the boundary layer flow regime in a vertical tall porous cavity subject to horizontal heat and mass fluxes. It was demonstrated analytically that, depending on the sign of the buoyancy ratio, the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied.

On the basis of the literature review, it appears that no study in the boundary layer regime was devoted to examine the combined effect of thermodiffusion and the remaining governing parameters in inclined cavities. The conditions of the validity of the boundary layer approximation are examined in this study.

Mathematical formulation and numerical solution

Porous medium

θ= 45°

Fig.1.-Schematical diagram of the physical model and coordinate system

The flow configuration under investigation, sketched in

Fig. 1, is a 2-D Darcy porous cavity of length L′ and height

H′, inclined with an angle 45° with respect to the horizontal

surface and saturated with a binary mixture. The long walls

of the cavity are subject to a uniform flux of heat, q′, while

the short walls are considered adiabatic. All the boundaries

of the porous enclosure are impermeable to mass transfer.

Using the vorticity-stream function formulation, the

dimensionless governing equations are given by:

η ∂ζ ∂t

+ζ =

2 2

RT

⎜⎜⎛ ∂∂x

−

∂∂y ⎟⎟⎞(T

+ϕS )

(1)

⎝

⎠

∂T + u ∂T + v ∂T = ∇2T

(2)

∂t ∂x ∂y

___________________________________________________________________________________________________

9ième Congrès de Mécanique, FS Semlalia, Marrakech

523

ε ∂S + u ∂S + v ∂S = 1 [∇2S − ∇2T]

(3)

∂t ∂x ∂y Le

ζ = −∇ 2ψ

(4)

The associated hydrodynamic, thermal and solutal

boundary conditions are as follows:

y = ± Ar : ψ=0,

∂T ∂S = = −1

(5)

2

∂y ∂y

x = ± 1 : ψ = 0 , ∂ T = ∂S = 0 (6)

2

∂x ∂x

The problem is governed by four dimensionless parameters,

namely the thermal Darcy-Rayleigh number RT , the Soret

parameter ϕ , the Lewis number Le and the cavity aspect

ratio Ar , defined as:

R = gβT ∆T′H′3 , ϕ = βS∆S′ , Le = α and A = L′

T

αν

βT∆T′ D r H ′

where

∆T′ =

q′H′

and

∆S′ =

s′ (1− s′ )D ∆T ′

0 0T .

λ

D

The numerical solution of the full governing equations

is based on a central finite-difference technique. The

iterative procedure is performed using the Alternate

Direction Implicit method (A.D.I) for Eqs. (1)-(3). The

stream function field is determined from Eq. (4) using the

point successive-over-relaxation method.

Analytical Solution

For enclosures with sufficiently large aspect ratio ( Ar >> 1 ), the present problem can be significantly simplified and analytically solved using the parallel flow approximation. Taking into account Eqs. (5) and (6) in the integration of Eqs. (2) to (3), Eq. (1) yields the expression: ∂2Ψ − Ω2Ψ(y) = G ∂y 2

where Ω2 = RT sinθ[CT + ϕ(CT + LeCS )] and

G = 22 R T [(C T + ϕ C s ) + (1 + ϕ ) ]/ Ω 2 .

The analytical solution, predicted by the parallel flow approximation, depends on the sign of Ω2 . Note that only the case corresponding to Ω2 > 0 may lead to boundary layer profiles. Hence the case of negative Ω2 will not be considered here. The analytical solution obtained for Ω2 > 0 is given by: Ψ ( y ) = − B Ω cosh( Ω y ) + G

u ( y ) = − B Ω 2 sinh( Ω y )

(7)

T ( x , y ) = C T x − C T B sinh( Ω y ) + (C T G − 1) y

S ( x, y) = Cs x − (CT + Cs Le )B sinh( Ωy) + ((CT + Cs Le )G − 1) y

where B =

G

. The expressions of CT and

Ω cosh( Ω / 2)

CS are given by :

CT = α1B2CT + B(1− GCT )α 2

⎪⎫

⎬ (8)

CS = CT + Le[α1B2 (LeCs + CT ) + Bα 2 (1− G(CT + LeCs ))]⎪⎭

where α 1 = Ω2 [sinh( Ω ) − Ω ]

and α = Ω cosh( Ω ) − 2 sinh(

2

2

Ω). 2

For given values of RT, Le and ϕ , the thermal and solutal

gradients CT and CS can be obtained by solving

numerically Eq. (8).

The Nusselt and Sherwood numbers are given by:

Nu = 1/(1 − Bα 2CT )

⎫

Sh = 1/(1 −

Bα 2 (CT

+

⎬ LeCS ))⎭

(9)

Note that Sh represents the inverse of the solute concentration difference induced across the layer by the thermodiffusion phenomenon.

Boundary layer regime

In the boundary layer regime (large Ω) the analytical solution can be considerably simplified to yield

( ) ( ) ⎡

Ω ⎜⎛ y − 1 ⎟⎞ ⎤

y

Ω⎜⎛ y − 1 ⎟⎞

ψ y = G ⎢1 − e ⎝ 2 ⎠ ⎥ , U y = − GΩe ⎝ 2 ⎠

⎢⎣

⎥⎦

y

( ) y C G Ω⎜⎛ y −1 ⎟⎞

T (x, y) = CT x + CT G − 1 y + y ΩT e ⎝ 2 ⎠

S ( x, y) = Cs x − ((CT + LeCs )G − 1)y

y (C + LeC )G Ω⎜⎛ y − 1 ⎟⎞

+

T

s

e ⎝ 2⎠

y

Ω

[ ][ ] Nu =

1 + α4G 2

1 + G 2 (1 − 4 / Ω ) / Ω

Sh=

1+α4G2 1+ Le2G2α4

1+ 1 Le2G2α + G2α + Le2G4α α / Ω− (1+ Le)G2α 2

Ω

5

4

45

3

( ) α G

(1+ Le)Gα

CT

=

3

1+α

G2

,

Cs

= 1+

1 + Le2

G 2α

3

+ Le2G 4α 2

4

4

4

Where α 3 = 1 − 2 / Ω , α 4 = 1 − 3 / Ω , α 5 = 1 − 4 / Ω

Results and Discussion

In the following section, the effect of the Rayleigh number RT on the flow intensity Ψc and heat and mass transfer characteristics represented by Nu and Sh is discussed. Figs. 2a-b illustrate the results obtained for these quantities as functions of RT for Le=10 and different values of φ (Ar≥12). The numerical solution of the full governing equations, depicted by dots, is seen to be in good agreement with the analytical solution. At very small RT, the Nusselt and Sherwood numbers are close to unity and Ψc is close to 0, indicating that the heat and mass transfer is mainly dominated by the diffusion regime. Note that the Nusselt number and the flow intensity are always increasing with RT. However the Sherwood number exhibits a somewhat complex behavior. In fact, its variations are characterized by a decrease with RT in an intermediate range of this parameter (Fig. 2c). As a result the curve of Sh presents

___________________________________________________________________________________________________

9ième Congrès de Mécanique, FS Semlalia, Marrakech

524

relative maximum and minimum before it tends towards the boundary layer curve. Note also that for a given thermal Rayleigh number, the flow intensity decreases with φ, while the tendency is inverted in the case of Nusselt and Sherwood numbers.

18

12

Sh

6

Parallel flow Numerical Boudary layer

(a)

φ = 1. φ = 0.5 φ = -0.5

0

1

10

Ra 100

1000

12

8

Nu

Analytical Numerical Boundary layer

φ = 0.5 φ = 1.

(b)

4

φ = -0.5

1

10

Ra

100

1000

12

8

Ψc

4

Analytical Numerical Boundary layer

(c)

φ = 0.5 φ = -0.5

valid for Nu and Ψc (i.e for the flow and heat transfer) even when RT is of order 20, while in the case of Sh (mass transfer) values of RT higher that 100 are required.

Conclusion

Soret convection in an inclined Darcy porous layer, filled with a binary fluid, subject to constant fluxes of heat is studied analytically and numerically. An analytical solution is derived by assuming that the flow is parallel in the core region of the enclosure. The conditions of the validity of the boundary layer approximation are seen to be dependent on the separation parameter and are more severe in the case of mass transfer.

References

[1] J.K. Platten, The Soret effect: a review of recent experimental results, J. Appl. Mech. 73 (2006) 5–15. [2] O.V. Trevisan, A. Bejan, Mass and heat transfer by natural convection in a vertical slot filled with porous medium, Int. J. Heat Mass Transfer 29 (1986) 403–415. [3] F. Alavyoon, On natural convection in vertical porous enclosures due to prescribed fluxes of heat and mass at the vertical boundaries, Int. J. Heat Mass Transfer 36 (1993) 2479–2498. [4] M. Mamou, P., Vasseur, E., Bilgen, and D. Gobin, Double-diffusive convection in an inclined slot filled with porous medium, European J. of Mechanics B/Fluids 14 (1995) 629-652.

[5] A. Amahmid, M. Hasnaoui, M. Mamou, P. Vasseur, Boundary layer flows in a vertical porous enclosure induced by opposing buoyancy forces, Int. J. Heat Mass Transfer 42 (1999) 3599-3608. [6] M. Er-Raki, M. Hasnaoui, A. Amahmid and M. Mamou, Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes, Int. J. Heat and Mass Transfer, Part A, vol. 49, Issues 17-18, pp. 3111-3120, August 2006.

φ = 1.

0

1

10

100

1000

Ra

Fig. 2: Effect of RT on: (a) Nu, (b) Sh, (c) Ψc for Le= 10

Figs.2a-b show that the boundary layer curves predict well

the results of numerical and parallel flow solutions when

the Rayleigh number is large enough except for the case of φ=-0.5 for which the boundary layer regime is not completely achieved even at RT=103. In fact the value of RT beyond which the boundary layer regime is achieved depends on the parameters φ and Le. Furthermore, negative values of φ lead to opposing thermal and solutal buoyancy

forces which retard the development of the boundary layer

regime. Finally it can be deduced from Figs. 2a-b that, for φ= 1 and 0.5, the boundary layer expressions derived are

___________________________________________________________________________________________________

9ième Congrès de Mécanique, FS Semlalia, Marrakech

525