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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
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�
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Lie Ovsyannikov method
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Lie Ovsyannikov method
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).
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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
),
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Group classi�cation problem
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)
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Group classi�cation problem
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 .
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Group classi�cation problem
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
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Group classi�cation problem
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
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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)
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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)
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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).
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Exact solutions
Examples of solution construction
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Exact solutions
Examples of solution construction
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
.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
Exact solutions
Examples of solution construction
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.
Irina Stepanova Symmetries of thermodi�usion equations
Symmetries of thermodi�usion equations
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.
Irina Stepanova Symmetries of thermodi�usion equations