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

2 This work is supported by the Israel Science Foundation I am grateful to Isaac Newton Institute for the invitation and for the financial support

3 Problem Geometry z x z = H

4 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).

5 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).

6 Basic assumptions  Gravity is negligible;  Free surface is nondeformable;  Surface tension linearly depends on both the temperature and the concentration:  The heat flux is fixed at the rigid plate;  The Newton law of cooling governs the heat transfer at the free surface:  Soret effect plays an important role:

7 Governing equations

8 Boundary conditions At the rigid wall: At the interface: Here is the differential operator in plane x-y

9 Dimensionless parameters  The Prandtl number  The Schmidt number  The Soret number  The Marangoni number  The Biot number  The Lewis number

10 Basic state There exist the equilibrium state corresponding to the linear temperature and concentration distribution:

11 Equation for perturbations  are the perturbations of the pressure, the temperature and the concentration, respectively; here and below

12 Previous results  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. 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):

13 Multi-scale expansion for the analysis of long wave perturbations Rescaled coordinates: “Slow” times : Rescaled components of the velocity:

14 Multi-scale expansion for the analysis of long wave perturbations Expansion with respect to  Small Biot number:

15 The zeroth order solution X Z

16 The second order The solvability conditions: The plane wave solution:

17 The second order The dispersion relation: Critical Marangoni number: The solution of the second order:

18 The fourth order The solvability conditions:

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21 Linear stability analysis Oron, Nepomnyashchy, PRE, 2004 Neutral curve for

22 2D regimes. Bifurcation analysis Interaction of two plane waves Oron, Nepomnyashchy, PRE, 2004 Solvability conditions: Here i.e. in 2D case traveling waves are selected, standing waves are unstable

23 2D regimes. Numerical results Solvability condition leads to the dynamic system for only if the resonant conditions are held:

24 Stability region for simple traveling wave 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 index l depends on the initial conditions

25 3D-patterns. Bifurcation analysis For the simplicity we set X Y Interaction of two plane waves

26 Solvability conditions: X Y The first wave is unstable with respect to any perturbation which satisfies the condition i.e. wave vector lies inside the blue region Here

27 “Three-mode” solution X Y The solvability conditions gives the set of 4 ODEs for

28 Stationary solutions ( a = b ) a > ca < c Dashed lines correspond to the unstable solutions, solid lines – to stable (within the framework of triplet solution) a = 0

29 Numerical results The solvability condition gives the dynamic system for only if the resonant conditions are held:

30 Steady solution Any initial condition evolves to the symmetric steady solution with

31 Evolution of h in T

32 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.

Thank you for the attention!