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3D Long-Wave Oscillatory Patterns in Thermocapillary Convection with Soret Effect. A. Nepomnyashchy, A. Oron Technion, Haifa, Israel, and S. Shklyaev , Technion, Haifa, Israel, Perm State University, Russia. This work is supported by the Israel Science Foundation - PowerPoint PPT Presentation
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3D Long-Wave Oscillatory 3D Long-Wave Oscillatory Patterns in Thermocapillary Patterns in Thermocapillary
Convection with Soret Convection with Soret EffectEffect
A. Nepomnyashchy, A. OronA. Nepomnyashchy, A. Oron
Technion, Haifa, Israel,Technion, Haifa, Israel,
and and S. ShklyaevS. Shklyaev,,
Technion, Haifa, Israel,Technion, Haifa, Israel,
Perm State University, RussiaPerm State University, Russia
22
This work is supported by the Israel This work is supported by the Israel Science FoundationScience Foundation
I am grateful to Isaac Newton I am grateful to Isaac Newton Institute for the invitation and for the Institute for the invitation and for the
financial supportfinancial support
33
Problem GeometryProblem Geometry
z
x
z = H
44
Previous resultsPrevious resultsLinear stability analysisLinear stability analysis
Pure liquid:• J.R.A. Pearson, JFM (1958);
• S.H. Davis, Annu. Rev. Fluid Mech. (1987).
Double-diffusive Marangoni convection:• J.L. Castillo and M.G. Velarde, JFM (1982);
• C.L. McTaggart, JFM (1983).
Linear stability problem with Soret effect:• C.F. Chen, C.C. Chen, Phys. Fluids (1994);
• J.R.L. Skarda, D.Jackmin, and F.E. McCaughan, JFM (1998).
55
Nonlinear analysis of long-wave Nonlinear analysis of long-wave perturbationsperturbations
Marangoni convection in pure liquids:• E. Knobloch, Physica D (1990);
• A.A. Golovin, A.A. Nepomnyashchy,nd L.M. Pismen, Physica D (1995);
Marangoni convection in solutions:• L. Braverman, A. Oron, J. Eng. Math. (1997);
• A. Oron and A.A. Nepomnyashchy, Phys. Rev. E (2004).
Oscillatory mode in Rayleigh-Benard convection• L.M. Pismen, Phys. Rev. A (1988).
66
Basic assumptionsBasic assumptions Gravity is negligible;Gravity is negligible; Free surface is nondeformable;Free surface is nondeformable; Surface tension linearly depends on both Surface tension linearly depends on both
the temperature and the concentration:the temperature and the concentration: 0 0 0 ;T CT T C C
D C T j
0.zkT q T T
The heat flux is fixed at the rigid plate;The heat flux is fixed at the rigid plate; The Newton law of cooling governs the The Newton law of cooling governs the heat transfer at the free surface:heat transfer at the free surface:
Soret effect plays an important role:Soret effect plays an important role:
77
Governing equationsGoverning equations
1 2
2
1 2 2
,
,
,
div 0.
P pt
TP T T
t
CS L C C T
t
vv v v
v
v
v
88
Boundary conditionsBoundary conditions
2
1: 0, 0, 0,
0.
z z
z
z w T BT C BT
M T C
u
At the rigid wall:
0 : 0, 1, ;z zz T C v
At the interface:
,zw v u eHere
is the differential operator in plane x-y 2
99
Dimensionless parametersDimensionless parameters
The Prandtl numberThe Prandtl number
The Schmidt numberThe Schmidt number
The Soret numberThe Soret number
The Marangoni numberThe Marangoni number
The Biot numberThe Biot number
The Lewis numberThe Lewis number
SD
C
T
2TAHM
P
PL
S
qHB
k
1010
Basic stateBasic state
0 0
0
0
0, ,
1,
.
p const
BT z
BC z const
v
There exist the equilibrium state corresponding to the linear temperature and concentration distribution:
1111
Equation for perturbationsEquation for perturbations
1 2
1 2
2
1 2 2
0
0 : 0;
1: 0, 0, 0,
0
t z zz
t z z zz
t z zz
t z zz zz
z
z z
z z
z
P w
w P w ww w w
P w w
S L w
w
z w
z w B B
M
u u u u u u
u
u
u
u
u
u
are the perturbations of the pressure, the temperature and the concentration, respectively; here and below 2
1212
Previous resultsPrevious results
Linear stability problem was studied;Linear stability problem was studied; Monotonous mode was found and weakly Monotonous mode was found and weakly
nonlinear analysis was performed;nonlinear analysis was performed; Oscillatory mode was revealed;Oscillatory mode was revealed; The set of amplitude equations to study The set of amplitude equations to study
2D oscillatory convective motion was 2D oscillatory convective motion was obtained.obtained.
Linear and nonlinear stability analysis of above conductive state with respect to long-wave perturbations was carried out by A.Oron and A.Nepomnyashchy (PRE, 2004):
1313
Multi-scale expansion for the Multi-scale expansion for the analysis of long wave perturbationsanalysis of long wave perturbations
, ,X x Y y Z z
2,W w U u
Rescaled coordinates:
“Slow” times :
Rescaled components of the velocity:
2 4,T t t
1414
Multi-scale expansion for the Multi-scale expansion for the analysis of long wave perturbationsanalysis of long wave perturbations
20 2
0 2 0 22 2
0 22
0 2 0 22 2
,
, ,
,
,
M M M
W W W
U U U
Expansion with respect to
Small Biot number:4B
1515
The zeroth order solutionThe zeroth order solution
0 0
00
0 0 2 20 0
, , , , , , ,
3, ,
21 1
3 2 , 1 ,4 4
F X Y T G X Y T
M h h F G
M Z Z h W M Z Z h
U
X
Z
1616
The second orderThe second order
2 20 0
1 2 1 20 0
1 ,
1 1
T
T
PF m F m G
SG m L F m L G
The solvability conditions:
The plane wave solution:
0
exp . .,
exp . .,1
h A i i T c c
mF A i i T c c G F h
i P
k R
k R
1717
The second orderThe second order
The dispersion relation:
2 22
2
2
1, ,
1
11
L L Lk P
LL L
00
148 1M L
m
Critical Marangoni number:
The solution of the second order:
2 2
2 2
, , , , , , ,
, , , , , , ,
z X Y T Q X Y T
z X Y T R X Y T
1818
The fourth orderThe fourth orderThe solvability conditions:
2 2 220 0
4 1002
2 222 20 0
2 2
2 20 0
2
1
2 160
2 div10 10
48 312div , ,
35 35
T
mm mF Q Q R F h
P P P Pm
m LP
m mP h h h
P Pm m
h F h J FP P
1919
2 2 2200
4 1002
2 222 20 0
2 2
2 210 0
2
2 160
2 div10 1048 312
div , ,35 35
T
mL mG R L m Q R h
P P Pm
m LP
m mS P h h h
P Pm m
h F L G h J GP P
0
1
3 2 1 3 1 3 ,
6 1 ,
, .X Y Y X
P m h L F LG
P h F L G
J f g f g f g
2 2 2 .Y X X Yh h h h
2020
2 20 0
1 2 1 20 0
1 ,
1 1
T
T
PF m F m G
SG m L F m L G
2121
Linear stability analysisLinear stability analysisOron, Nepomnyashchy, PRE, 2004
2 20
2 2
2 2
1,
1 60
2 3 2 4 3
m km V
k PS
V P S PS P S P P S
0.8 1.2 1.6 2k1.4
1.6
1.8
2
2.2
2.4m 2
Neutral curve for 2m k1, 0.1;
0.01; 200L S
30
2V at S P
2222
2D regimes. Bifurcation analysis2D regimes. Bifurcation analysis
. .i kX T i kX Th A e B e c c
Interaction of two plane waves
Oron, Nepomnyashchy, PRE, 2004
2 22 1 2
2 22 1 2
,r r r
r r r
a a K a K b a
b b K b K a b
,A Bi iA ae B be
Solvability conditions:
2 12 0,r rK K
Here
i.e. in 2D case traveling waves are selected, standing waves are unstable
2323
2D regimes. Numerical results2D regimes. Numerical results
2 2
. .i nkX n k T
nn
h A e c c
Solvability condition leads to the dynamic system for
nA
222 2 22
1 2 * *
2n n n n j n
njlm j l m njlm j l m
dA A n K n A j A A
d
C A A A C A A A
2 2 2 2,n j l m n j l m
1 0njlmC only if the resonant conditions are held:
2 2 2 2,n j l m n j l m
2 0njlmC only if the resonant conditions are held:
2424
n nlA Stability region for simple traveling wave
0.8 1.2 1.6 2k1.2
1.6
2
2.4
2.8m 2
Plane wave with fixed k exists above white line and it is stable with respect to 2D perturbations above green line
Numerical simulations show, that system evolve to traveling wave
22 22 , argnrnl i
n l li lrr r
KA A
n K K
index l depends on the initial conditions
2525
3D-patterns. Bifurcation analysis3D-patterns. Bifurcation analysis
21 . .i Ti Th A e B e c c k Rk R
1 2k k kFor the simplicity we set
X
Y
1k
2k
Interaction of two plane waves
2626
2 22 1 2
2 22 1 2
,r r r
r r r
a a K a K b a
b b K b K a b
,A Bi iA ae B be
Solvability conditions:
22 12cos 0r rK K
X
YThe first wave is unstable with respect to any perturbation which satisfies the condition
i.e. wave vector lies inside the blue region
2 4
2k
Here
2727
““Three-mode” solutionThree-mode” solution
X
Y
1k2k
3k
2 . .
i kX T i kY T
i k X Y T
F A e B e
C e c c
The solvability conditions gives the set of 4 ODEs for
, , ,
arg arg arg
a A b B c C
C A B
2828
Stationary solutions (Stationary solutions (aa = = bb))
a > c a < c
Dashed lines correspond to the unstable solutions, solid lines – to stable (within the framework of triplet solution)
a = 0
1 .5 2 2 .5 3 3 .5 4
m 20
0 .0 4
0 .0 8
0 .1 2a
1 .5 2 2 .5 3 3 .5 4
m 20
0 .0 2
0 .0 4
0 .0 6
0 .0 8c
1 .5 2 2 .5 3 3 .5 4
m 20
1
2
3
4
2929
Numerical resultsNumerical results
2 2 2
. .i nkX mkY n m k T
nmn m
h A e c c
The solvability condition gives the dynamic system for nmA
2nm nm nm nmljpq lj pq
nmljpqrs lj pq rs
dA A B A A
d
C A A A
2 2 2 2 2 2
, ,n j p m l q
n m j l p q
0nmjlpqB only if the resonant conditions are held:
3030
Steady solutionSteady solution
1 .5 2 2 .5 3 3 .5 4
m 20
0 .0 4
0 .0 8
0 .1 2|A 1 0 |
1 .5 2 2 .5 3 3 .5 4
m 20
0 .0 0 2
0 .0 0 4
0 .0 0 6
0 .0 0 8|A 1 1 |
Any initial condition evolves to the symmetric steady solution with
, ,,
, ,
, ,
nm nm n m n m
nm mn
nm mn
A const A A A
A A n m is even
A iA n is odd m is even
3131
Evolution of Evolution of hh in in TT
3232
ConclusionsConclusions 2D oscillatory long-wave convection is studied 2D oscillatory long-wave convection is studied
numerically. It is shown, that plane wave is realized numerically. It is shown, that plane wave is realized
after some evolution;after some evolution;
The set of equations describing the 3D long-wave The set of equations describing the 3D long-wave
oscillatory convection is obtained;oscillatory convection is obtained;
The instability of a plane wave solution with respect to The instability of a plane wave solution with respect to
3D perturbations is demonstrated;3D perturbations is demonstrated;
The simplest 3D structure (triplet) is studied;The simplest 3D structure (triplet) is studied;
The numerical solution of the problem shows that 3D The numerical solution of the problem shows that 3D
standing wave is realized;standing wave is realized;
The harmonics with critical wave number are the The harmonics with critical wave number are the
dominant ones.dominant ones.
Thank you for the Thank you for the attention!attention!