@汥瑀瑯步渠 thermodiffusion equations: lie symmetry and...

Symmetries of thermodi�usion equations

THERMODIFFUSION EQUATIONS: LIE SYMMETRY

AND EXACT SOLUTIONS

Irina Stepanova

Institute of Computational Modelling, Krasnoyarsk, Russia

Work is supported by RFBR Grant �14-01-31038 and Grant of Scienceand Technology Support Fund of Krasnoyarsk Territory

Irina Stepanova Symmetries of thermodi�usion equations


Thermodi�usion

Thermal di�usion (Soret) e�ect

Thermal di�usion (or the Soret e�ect) is a molecular transport of masscaused by the thermal gradient in a �uid mixture.

The di�usive �ux in a binary mixture is

J = −ρ(D∇C + C(1− C)DT∇T

).

In a closed system at the stationary stateand mechanical equilibrium, J = 0, so

∇C = −C(1− C)DTD∇T.

∇T�

∇C�



Thermodi�usion

Transport properties

density ρ

viscosity ν

thermal conductivity κ

thermal di�usivity χ

heat capacity cp

di�usion coe�cient D

thermal di�usion coe�cient DT

These are essential to design ofchemical processes and equipmentinvolving �uid �ow, heat and masstransfer, chemical reactions.

It is not possible to determine thesecoe�cients by theoretical considerationonly. The principle of this study hasbeen the use of mathematical modelscomplemented with some empiricalparameters.

These empirical parameters aredetermined by comparison betweenmeasurements in specially designedexperiments and the results ofmathematical models that describe theprocess.



Thermodi�usion

Experimental data (density and thermal conductivity)

Density and thermal conductivity of sea water for di�erent salinity.

M.H. Sharqawy, J.H. Lienhard V, S.M.Zubair Thermophysical properties of seawater: a

review of existing correlation and data// Desalination and Water Treatment. 16, (2010),

354�380.



Thermodi�usion

Experimental data(di�usion and thermodi�usion coe�cients)

Di�usion and thermal di�usion of NaCl ions in water for salinity 0.0285.

D.R.Caldwell Thermal and Fickian di�usion chloride in a solution of oceanic

concentration// Deep-Sea Research. 20, (1973), 1029�1039.



Lie Ovsyannikov method

The Lie-Ovsyannikov method

Partial di�erential equations with a paramemter

The group classi�cation problem:To �nd all forms of parameters and

corresponding transformations of variables

Group of transformations Algebras of generators

The basic algebra L0: A setof generators admitted by

PDE at arbitrary parameters

Speci�cations of parametersand extension of Lie algebra L0

Olver P., Applications of Lie groups to di�erential equations// Springer�Verlag, NewYork, 1993.

Ovsyannikov L.V., Group analysis of di�erential equations// Academic Press, NewYork, 1982.




Lev Vasilyevich Ovsyannikov (1919-2014)

Academician L.V. Ovsyannikov is anoutstanding Russian scientist, who madea great contribution to the developmentof mechanics and applied mathematics.

His works have generated the beginningof new research �elds, activelydeveloping in Russia and over the world.

The results of Ovsyannikov in gasdynamics, the theory of �uid motionwith free boundaries in the �eld ofmathematical study models ofcontinuum mechanics have becomeclassical.

L.V. Ovsyannikov is one of the foundersof the group analysis of di�erentialequations. All his paper features a clearstatement of the problem, elegant andrigorous mathematical apparatus.




Application of symmetry analysis to convection equations

Stepanova I. V. Symmetry analysis of nonlinear heat and mass transfer

equations under Soret e�ect. Commun Nonlinear Sci Numer Simulat,2015. V. 20.

Stepanova I. V. Group classi�cation for equation of thermodi�usion in

binary mixture. Commun Nonlinear Sci Numer Simulat, 2013. V. 18.

Ryzhkov I. I. Symmetry analysis of equations for convection in binary

mixture. Journal of Siberian Federal University. Mathematics andPhysics, 2008. V. 1(4).

Andreev V.K., Stepanova I. V. Symmetry of thermodi�usion equations

under nonlinear dependence of buoyancy force on temperature and

concentration. Computational Technologies Journal, 2010. V. 15(4)(in Russian).

Goncharova O.N. Group classi�cation of the free convection equations.

Continuum dynamics: collection of papers, Novosibirsk 1987. V. 79(in Russian).



Group classi�cation problem

Heat and mass transfer equations under Soret e�ect

∂T

∂t= χ

(∂2T

∂(x1)2+

∂2T

∂(x2)2+

∂2T

∂(x3)2

)+

+∂χ

∂T

(( ∂T∂x1

)2+( ∂T∂x2

)2+( ∂T∂x3

)2)+ (1)

+∂χ

∂C

(∂T

∂x1

∂C

∂x1+

∂T

∂x2

∂C

∂x2+

∂T

∂x3

∂C

∂x3

),

∂C

∂t= D

(∂2C

∂(x1)2+

∂2C

∂(x2)2+

∂2C

∂(x3)2

)+∂D

∂T

(∂T

∂x1

∂C

∂x1+

∂T

∂x2

∂C

∂x2+

∂T

∂x3

∂C

∂x3

)+

+∂D

∂C

(( ∂C∂x1

)2+( ∂C∂x2

)2+( ∂C∂x3

)2)+DT

(∂2T

∂(x1)2+

∂2T

∂(x2)2+

∂2T

∂(x3)2

)+ (2)

+∂DT

∂T

(( ∂T∂x1

)2+( ∂T∂x2

)2+( ∂T∂x3

)2)+∂DT

∂C

(∂T

∂x1

∂C

∂x1+

∂T

∂x2

∂C

∂x2+

∂T

∂x3

∂C

∂x3

),




References

L.V.Ovsyannikov, Group properties of nonlinear thermal conductivityequations, Doklady AS USSR, 125 3 (1959) 492-495.

∂tu = ∂x(θ(u)∂xu)

N.M. Ivanova and C. Sophocleous, On the group classi�cation ofvariable-coe�cient nonlinear di�usion-convection equations,J. Comput. Appl. Math. 197 (2006) 322-344.

f(x)ut = (g(x)D(u)ux)x +K(u)ux

I.B. Kovalenko and A.G. Kushner, The non-linear di�usion andthermal conductivity equation: group classi�cation and exact solutions,Regular and Chaotic Dyn. 8 (2003) 167�189.

ut = (uαux)x + F (u)




The basic group and additional generators

L0 = 〈∂t, ∂xi , 2t∂t +

3∑i=1

xi∂xi , xj∂xi − x

i∂xj 〉, i, j = 1, ..., 3, i 6= j.

generator transformation

∂t time transition

∂xi, i = 1, . . . , 3 space transition

xj∂xi− xi∂

xj, i, j = 1, ..., 3, i 6= j rotation

2t∂t +3∑i=1

xi∂xi

scale transformation

Additional generators

Z = (m+ n)

(t∂t +

3∑i=1

xi∂xi

),

T 1 = T∂T , T3 = ∂T , C

1 = C∂C , C2 = T∂C , C

3 = ∂C .




Results of group classi�cation at χ = χ(T,C)

χ, D DT generators w

e(m+n)C/T fi e(m+n)C/T (C/T (f1 − f2) + f3) Z+C2 T

e(m+n)T fi e(m+n)T (kT (f1 − f2) + f3) Z+T3+kC2+qC3 C−kT2/2− qT

(kT+C)m+nfi (kT+C)m+n(k(f2−f1)+(kT+C)f3) Z+kC2+C1 T

e(m+n)T fi e(m+n+l)T (k(f2 − f1)e−lT + f3) Z+T3+lkC2+lC1 (C+kT+k/l)e−lT

Tm+nfi Tm+n−1(kT (f1 − f2) + f3) Z+T1+kC2+qC3 C−kT−q lnT

Tm+nfi Tm+n(k(f2−f1)/(h−1)+Th−1f3) Z+T1+kC2+hC1 (C+kT/(h−1))T−h

Tm+nfi Tm+n(k(f1 − f2) lnT + f3) Z+T1+kC2+C1 C/T−k lnT




Results of group classi�cation at χ = χ∗ = const

D DT generators w

f λ2 lnT (χ∗−f)+F T1+λ2C2+C1 C/T−λ2 lnT

f λ0T (χ∗−f)+F T3+λ0C2+λ1C

3 C−λ0T2/2−λ1T

f C(χ∗ − f)/T + F C2, C1 T

f λ1(χ∗−f)/(1−λ3)+F T1+λ1C2+λ3C

1+λ1λ2λ3C3/(λ3−1) (C−λ1(λ2+T )/(1−λ3))T

−λ3

f λ0(χ∗−f)+F/T T1+λ0C2−λ1C

3 C−λ0T+λ1 lnT

f λ1(χ∗−f)+e−λ2TF T3+λ1λ2C2−λ2C

1−λ1C3 (C+λ1T )e−λ2T



Exact solutions

Exact solutions

Choose admissible generators

〈∂t, ∂x1 , ∂x3〉

Choose the transport coe�cients

κ = κ(T,C), ρ = ρ(T,C), D = D(T,C), DT = DT (T,C)

Choose the geometry of the problem

z

0

L



Exact solutions

Reduced system and boundary conditions

Governing equations:

d

dz

(κ(T,C)

dT

dz

)= 0, (3)

d

dz

(ρ(T,C)

(D(T,C)

dC

dz+ C(1− C)DT (T,C)

dT

dz

))= 0 (4)

Conservation of mass:

S′∫ L′

0

ρ(T,C)C dz = ρ0C0SL, S′∫ L′

0

ρ(T,C) dz = ρ0SL, (5)

Boundary conditions:

T (0) = T1, T (L′) = T2, (6)

ρ(T,C)

(D(T,C)

dC

dz+ C(1− C)DT (T,C)

dT

dz

)= 0 at z = 0, L′. (7)



Exact solutions

General solution

Let us integrate governing equations once using boundary condition (5):

κ(T,C)dT

dz= κ0,

dC

dz+ C(1− C)ST (T,C)

dT

dz= 0,

where ST (T,C) = DT /D is the Soret coe�cient.To solve the problem, we assume that z = z(T ), C = C(T ):

dz

dT=κ(T,C)

κ0, z(T1) = 0, z(T2) = L, (8)

dC

dT= −C(1− C)ST (T,C), C(T1) = C1, (9)

where C1 is found from mass conservation conditions (5).The solution of problem (8) is given by

z(T ) =1

κ0

∫ T

T1

κ(τ, C(τ)

)dτ, κ0 =

1

L′

∫ T2

T1

κ(T,C(T )

)dT. (10)



Exact solutions

General solution

Consider the problem for concentration C:

dC

dT= −C(1− C)ST (T,C), C(T1) = C1. (11)

The mass conservation conditions (5) lead to∫ T2

T1

ρCκ dT

(∫ T2

T1

ρκ dT

)−1

= C0,V ′

V= ρ0

∫ T2

T1

κ dT

(∫ T2

T1

ρκ dT

)−1

, (12)

where C = C(T ), ρ = ρ(T,C(T )), κ = κ(T,C(T )), V ′ = S′L′, V = SL.

The general method of solving the problem:

Solve problem (11) analytically or numerically to �nd C(T ). Determineconstant C1 from �rst equation in (12).

Find the dependence of space coordinate on temperature z(T ) from

z(T ) =1

κ0

∫ T

T1

κ(τ, C(τ)

)dτ. (13)

Given the functions z = z(T ) and C = C(T ), determine thetemperature and concentration pro�les T (z) and C(z).



Exact solutions

Examples of solution construction

Case 1. Constant ρ, κ and S′T = C0(1− C0)ST0.The pro�les of temperature and concentration are linear:

T (z) = T1 +T2 − T1

Lz, (14)

C(z) = C0 − C0(1− C0)ST0(T2 − T1)

(z

L− 1

2

).

Consider ethanol�water mixture at T0 = 30oC, C0 = 0.14.The Soret coe�cient is ST0 = 5.11 · 10−3K−1.



Exact solutions


Case 2. Constant ρ, κ and S′T = C0(1− C0)ST (T ).The temperature pro�le T (z) is linear (see (10)). The general solution for C is

C = C0 − C0(1− C0)

[ T∫T1

ST (τ) dτ −1

T2 − T1

T2∫T1

T∫T1

ST (τ) dτ dT

].

For sodium chloride in water near the oceanic concentration C0 = 0.0285

ST = (−1.232 + 0.1128T − 0.00087T 2) · 10−3 K−1.

This formula is valid in the range 0�45 oC. We take T0 = 25oC.



Exact solutions


Case 3. Constant ρ, κ and S′T = CST (T ).For small C, one has C(1− C) ∼ C.The temperature pro�le T (z) is linear (see (14)). The general solution for C:

C = C1 exp

(−

T∫T1

ST (τ) dτ

).

After determining C1 from the condition of mass conservation(�rst equation in (12)), we �nd

C =

C0 (T2 − T1) exp

(−

T∫T1

ST (τ) dτ

)T2∫T1

exp

(−

T∫T1

ST (τ) dτ

)dT

.



Exact solutions


Consider aqueous suspension of polystyrene spheres(M. Braibanti, D. Vigolo, R. Piazza, Phys. Rev. Lett. 2008. V. 100, 108303)

The Soret coe�cient in aqueous colloids is written as

ST = S∞T

[1− exp

(T ∗ − TTc

)].

For PS spheres with the radius R = 123 nm

T ∗ = 12.20 oC, Tc = 25.63 K, S∞T = 2.016 K−1.

We take T0 = 20oC and initial concentration of PS spheres C0 = 0.0035.



Summary

Summary

The equations describing thermodi�usion without convection in binarymixture with Soret e�ect under buoyancy force action are considered.

The Lie symmetries are found. Group classi�cation with respect to thephysical parameters is performed.

Di�erent exact solutions of the governing equations are constructedand analyzed.

These results are useful for analytical and numerical modeling of heatand mass transfer in binary mixtures with Soret e�ect.


@汥瑀瑯步渠 thermodiffusion equations: lie symmetry and...

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