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!