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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
I am grateful to Isaac Newton
Institute for the invitation and for the
financial support
2
Problem Geometry
z
z=H
x
3
Previous results
Linear 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).
4
Nonlinear analysis of long-wave
perturbations
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).
5
Basic assumptions
Gravity is negligible;
 Free surface is nondeformable;
 Surface tension linearly depends on both
the temperature and the concentration:
   0   T T  T0    C C  C0  ;


Soret effect plays an important role:
j  D  C  T 
The heat flux is fixed at the rigid plate;
 The Newton law of cooling governs the
heat transfer at the free surface:
kTz  q T  T   0.

6
Governing equations
v
 P 1  v    v  p   2 v,
t
T
P
 v  T   2T ,
t
C
S
 L1v  C   2C   2T ,
t
div v  0.
7
Boundary conditions
At the rigid wall:
z 0:
v  0, Tz  1, C z   ;
At the interface:
z  1:
w  0, Tz  BT  0, C z   BT  0,
u z  M 2 T  C   0.
Here v  u  wez ,
 2 is the differential operator in plane x-y
8
Dimensionless parameters

The Prandtl number

The Schmidt number

The Soret number

The Marangoni number

The Biot number

The Lewis number

P 


S
D
 C

T
 T AH 2
M 

qH
B
k
P
L
S
9
Basic state
There exist the equilibrium state corresponding to the
linear temperature and concentration distribution:
v 0  0, p0  const ,
B 1
T0  z 
,
B
C 0   z  const .
10
Equation for perturbations
ut  P 1  u    u  wu z      2u  u zz
wt  P 1  u    w  wwz    z   2w  wzz
P t  u    wz  w   2   zz
S t  L1  u    w z      2   zz     2   zz 
  u  wz  0
z 0:
z  1:
u  w  z   z  0;
w  0, z  B   0,  z   B   0,
uz  M        0
, ,  are the perturbations of the pressure, the temperature and
the concentration, respectively; here and below    2
11
Previous results
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):
Linear stability problem was studied;
 Monotonous mode was found and weakly
nonlinear analysis was performed;
 Oscillatory mode was revealed;
 The set of amplitude equations to study
2D oscillatory convective motion was
obtained.

12
Multi-scale expansion for the
analysis of long wave perturbations
Rescaled coordinates:
X   x , Y   y, Z  z
Rescaled components of the velocity:
U   u,W   2w
“Slow” times :
T   2t ,    4t
13
Multi-scale expansion for the
analysis of long wave perturbations
Small Biot number:
B   4
Expansion with respect to :
M  M0   M2 
2

0
 
2
 2

, 
,
0
       2   
0
U  U     2U   
0
2
2
 
2
 2

,
,
, W  W     2W   
0
2
14
The zeroth order solution
   F  X ,Y ,T ,  ,    G  X ,Y ,T , 
0
0

U
0
0
3
  M 0h ,
2
h  F  G,
1
1
0
  M 0 Z  3Z  2 h , W  M 0 Z 2  Z  1  2h ,
4
4
Z
X
15
The second order
The solvability conditions:
PFT  1  m0  2F  m0 2G ,
SGT   1  m0L1  2F  1  m0L1  2G
The plane wave solution:
h  A   exp ik  R  iT   c .c .,
m0
F 
A   exp ik  R  iT   c .c ., G  F  h
1  i P
16
The second order
Critical Marangoni number:
M0 1 L
m0 

48 1  
The dispersion relation:
  k 2 ,  P  
 1  L  L2   L2
1 
,
L2

   1
2
1 L  L
The solution of the second order:
 z , X ,Y ,T ,   Q  X ,Y ,T , 
 2
 2
    z , X ,Y ,T ,   R  X ,Y ,T , 

 2

 2
17
The fourth order
The solvability conditions:
m2 2
1  m0 2
m0 2

F  QT 
Q
R F
h
P
P
P
P
m0
4
1






2
m
1




L



0
2


60P
2
2
m0
m0
2
2
2 


2

P
div

h

h



h










10P 2
10P 2
48m02
312m02

div  h  F  h  
J  F ,  ,
2
35P
35P
18
m2 2
L  m0 2
G  RT   L  m0   Q 
R
h
P
P
P
m0 4 
1





2
m
1




L


0
2


60P
2
m02
m
2
2
2 
0


2

S

P
div

h

h



h










10P 2
10P 2
2
48m02
312
m
0

div h     F  L1G   h 
J G ,  ,
2
35P
35P


2

2  hY 2h X  h X 2hY .
  3 2P  m0 1     h  3 1   L  F  3LG ,
  6P h   1    F  L1G  ,
J  f , g   f X gY  fY g X .
19
PFT  1  m0  2F  m0 2G ,
SGT   1  m0L1  2F  1  m0L1  2G
20
Linear stability analysis
Oron, Nepomnyashchy, PRE, 2004
1
m2 
1 


m02k 2
 k 2  60PS V     ,


V   2  P 2  S 2  PS   3  P  S    P  2P  4S  3 
2.4
m
3
V  0 at S   P
2.2
2
2
Neutral curve m2 k  for
2
1.8
  1,   0.1;
1.6
L  0.01; S  200
1.4
0.8
1.2
1.6
k
2
21
2D regimes. Bifurcation analysis
Oron, Nepomnyashchy, PRE, 2004
Interaction of two plane waves
h  A   e
i kX T 
 B   e
i kX  T 
 c .c .
Solvability conditions:
a  2r a   K 1r a 2  K 2r b 2  a ,
b  2r b   K 1r b 2  K 2r a 2  b
Here A  ae
i A
, B  be iB
K 2r  2K1r  0, i.e. in 2D case traveling waves are selected,
standing waves are unstable
22
2D regimes. Numerical results
h

 A   e
n 

i nkX n 2k 2T
n

 c .c .
Solvability condition leads to the dynamic system for An

d
2
2
2
An  2n An  n K n An  2 j 2 A j
d
2
 
A
n

 Cnjlm A j Al Am   Cnjlm A j Al* Am*
1
 2

Cnjlm
 0 only if the resonant conditions are held:
n  j  l  m, n 2  j 2  l 2  m 2
1
 2
Cnjlm  0 only if the resonant conditions are held:
n  j  l  m, n 2  j 2  l 2  m 2
23
Numerical simulations show, that system evolve to traveling wave
An 
nl
n2

Ki 
, arg Al   2li  2lr

Kr
Kr 

2nr
index l depends on the initial conditions
Stability region for simple traveling wave An
nl
2.8
m2
Plane wave with fixed k exists
above white line and it is stable
with respect to 2D perturbations
above green line
2.4
2
1.6
1.2
0.8
1.2
1.6
k
2
24
3D-patterns. Bifurcation analysis
Interaction of two plane waves
h  A   e
i  k1 R T 
Y
k2

 B   e
i  k 2 R T 
k1
 c .c .
X
For the simplicity we set k1  k 2  k
25
Solvability conditions:
a  2r a   K 1r a 2  K 2r b 2  a ,
b  2r b   K 1r b  K 2r a  b
2
2
A  ae iA , B  be iB
Here
K 2r  2cos 2  K 1r  0
Y
The first wave is unstable with respect to any
perturbation which satisfies the condition

 

2
4
X i.e. wave vector k 2 lies inside the blue region
26
“Three-mode” solution
F  A   e
C   e
i kX T 
 B   e
i k  X Y   2T 
i kY T 
 c .c .

Y
k2
k3
k1 X
The solvability conditions gives the set of 4 ODEs for
a  A ,b  B ,c  C ,
  arg C   arg  A   arg  B 
27
Stationary solutions (a = b)
0.12
a
4
0.08

0.06
3
0.04
2
0.02
1
c
0.08
0.04
m2
0
1.5 2 2.5 3 3.5 4
m2
0
1.5 2 2.5 3 3.5 4
m2
0
1.5 2 2.5 3 3.5 4
Dashed lines correspond to the unstable solutions,
solid lines – to stable (within the framework of triplet solution)
a>c
a<c
a=0
28
Numerical results
h


 
n  m 

Anm   e


i nkX mkY  n 2 m 2 k 2T
  c .c .
The solvability condition gives the dynamic system for Anm
d
Anm  2nm Anm    Bnmljpq Alj A pq 
d
   Cnmljpqrs Alj A pq Ars 


Bnmjlpq  0 only if the resonant conditions are held:
n  j  p, m  l  q ,
n 2  m2  j 2  l 2  p2  q 2
29
Steady solution
0.12
|A 10|
0.008
|A 11|
0.006
0.08
0.004
0.04
0.002
m2
0
1.5 2 2.5 3 3.5 4
m2
0
1.5 2 2.5 3 3.5 4
Any initial condition evolves to the symmetric steady solution with
Anm  const , Anm  An ,m  An ,m
Anm  Amn , n  m is even ,
Anm  iAmn , n is odd , m is even
30
Evolution of h in T
31
Conclusions






2D oscillatory long-wave convection is studied
numerically. It is shown, that plane wave is realized
after some evolution;
The set of equations describing the 3D long-wave
oscillatory convection is obtained;
The instability of a plane wave solution with respect to
3D perturbations is demonstrated;
The simplest 3D structure (triplet) is studied;
The numerical solution of the problem shows that 3D
standing wave is realized;
The harmonics with critical wave number are the
dominant ones.
32
Thank you for the
attention!