3D LongWave Oscillatory Patterns in Thermocapillary Convection with Soret Effect

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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

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This work is supported by the Israel Science
Foundation I am grateful to Isaac Newton
Institute for the invitation and for the
financial support
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Problem Geometry
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Previous resultsLinear 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).

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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).

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Basic assumptions

• Gravity is negligible
• Free surface is nondeformable
• Surface tension linearly depends on both the
  temperature and the concentration

• Soret effect plays an important role

• The heat flux is fixed at the rigid plate
• The Newton law of cooling governs the heat
  transfer at the free surface

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Governing equations
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Boundary conditions
At the rigid wall
At the interface
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Dimensionless parameters

• The Prandtl number
• The Schmidt number
• The Soret number
• The Marangoni number
• The Biot number
• The Lewis number

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Basic state
There exist the equilibrium state corresponding
to the linear temperature and concentration
distribution
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Equation for perturbations
P, Q, S are the perturbations of the pressure,
the temperature and the concentration,
respectively here and below
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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.

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Multi-scale expansion for the analysis of long
wave perturbations
Rescaled coordinates
Rescaled components of the velocity
Slow times
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Multi-scale expansion for the analysis of long
wave perturbations
Small Biot number
Expansion with respect to e
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The zeroth order solution
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The second order
The solvability conditions
The plane wave solution
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The second order
Critical Marangoni number
The dispersion relation
The solution of the second order
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The fourth order
The solvability conditions
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Linear stability analysis
Oron, Nepomnyashchy, PRE, 2004
Neutral curve for
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2D regimes. Bifurcation analysis
Oron, Nepomnyashchy, PRE, 2004
Interaction of two plane waves
Solvability conditions
Here
i.e. in 2D case traveling waves are selected,
standing waves are unstable
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2D regimes. Numerical results
only if the resonant conditions are held
only if the resonant conditions are held
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Numerical simulations show, that system evolve to
traveling wave
index l depends on the initial conditions
Plane wave with fixed k exists above white line
and it is stable with respect to 2D perturbations
above green line
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3D-patterns. Bifurcation analysis
Interaction of two plane waves
For the simplicity we set
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Solvability conditions
Here
The first wave is unstable with respect to any
perturbation which satisfies the condition i.e.
wave vector lies inside the blue region
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Three-mode solution
The solvability conditions gives the set of 4
ODEs for
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Stationary solutions (a b)
Dashed lines correspond to the unstable
solutions, solid lines to stable (within the
framework of triplet solution)
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Numerical results
The solvability condition gives the dynamic
system for
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Steady solution
Any initial condition evolves to the symmetric
steady solution with
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Evolution of h in T
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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.

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Thank you for the attention!