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INSTABILITIES IN A NON- HOMOGENEOUSLY HEATED
FLUID IN MARANGONI CONVECTION
A.M.Mancho Instituto de Matemáticas y Física
Fundamental, Consejo Superior de Investigaciones
Científicas, Madrid, Spain. M. C. Navarro ,
H. Herrero Departamento de Matemáticas,
Universidad de Castilla-La Mancha, Ciudad Real,
Spain. S. Hoyas Universidad Politécnica de
Madrid, Madrid, Spain.
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PHYSICAL SETUP
Domain
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RELATED THEORETICAL WORKS
We have used numerical techniques developed and
theoretically justified in these
articles Herrero, H Mancho, A.M. On pressure
boundary conditions for thermoconvective
problems. Int. J. Numer. Meth. Fluids 39 (2002),
391-402. H. Herrero, S. Hoyas, A. Donoso, A. M.
Mancho, J.M. Chacón, R.F Portugues y B. Yeste.
Chebyshev Collocation for a Convective Problem in
Primitive Variables Formulation. J. of
Scientific Computing 18 (3),315-318 (2003)
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RELATED THEORETICAL WORKS
These numerical techniques have been applied to
a similar problem with lateral constant
temperature gradient. Hoyas, S. Herrero, H.
Mancho, A. M. Bifurcation diversity in dynamic
thermocapillary liquid layers. Phys. Rev. E 66
(2002), 057301-1-057301-4. Hoyas, S. Herrero,
H. Mancho, A.M. Thermal convection in a
cylindrical annulus heated laterally. J. Phys. A
Math. Gen. 35 (2002), 4067-4083. Hoyas, S.
Mancho A.M. Herrero, H. Thermocapillar and
thermogravitatory waves in a convection problem.
Theoretical and Computational Fluid dynamics 18
(2004), 2-4, 309-321. Hoyas, S Mancho, A.M.
Herrero, H. Garnier, N. Chiffaudel, A.
Benard-Marangoni convection in a differentially
heated cylindrical cavity. Phys. of Fluids. 17,
054104-1,12 (2005).
Linear heating
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RELATED EXPERIMENTAL WORKS
These experiments describe a similar problem with
lateral constant temperature gradient. R.J.
Riley and G.P. Neitzel, Instability of
thermocapillary-buoyancy convection in shallow
layers. Part 1. Characterization of steady and
oscillatory instabilities, J. Fluid Mech. 359,
143 (1998). N. Garnier, Ondes non lineaires a
une et deux dimensions dans une mince couche de
fluide, Ph.D. thesis, Université Paris 7, France,
2000. J. Burguete, N. Mukolobwiez, F. Daviaud, N
Garnier, A. Chiffaudel, Buoyant-thermocapillary
instabilities in extended liquid layers subjected
to a horizontal temperature, Phys. of Fluids 13
(10) 2773-2787 (2001).
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FORMULATION OF THE PROBLEM
BASIC EQUATIONS
No gravity effects Pr gt Prandtl number Domain
BOUNDARY CONDITIONS for the velocity
are, M gt Marangoni number
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BOUNDARY CONDITIONS for the temperature
are, B gt Biot number b gt Gaussian
width. Heating shape S/Tugt Quotient of lateral
and vertical temperature differences Regularity
conditions at the origin Multiparametric
problem G, Pr, M, B, b, S/Tu
Control Heat related parameters
parameter
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THE BASIC STATE
Stationary and axisymmetric, Regularity
conditions for the basic state at the origin
are, We solve the basic state with a
Newton-Raphson iterative method. The equations
and boundary conditions are linearized at each
step s, around solutions at step s-1
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THE COLLOCATION METHOD
At each step unknowns are expanded in Chebyshev
polynomials Basic equations are evaluated at
collocation points Boundary Conditions are
evaluated at
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With those rules we obtain 4xLxM equations and
unknowns. Some results are
S/Tu 0.001
S/Tu 0.5
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THE LINEAR STABILITY ANALYSIS
We perturb the basic state Regularity
conditions at r0 are The perturbation
fields are expanded in Chebyshev
polynomials, A trick for m1,
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The number of unknowns and equation are for
m1, 4xLxM(L-1)xM otherwise 5xLxM
CONVERGENCE RESULTS
z-coordinate
r-coordinate
B0.05, M92Tu, G10, S1ºC, b0.8
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THE STABILITY RESULTS
G10, Pr0.4 THE INFLUENCE OF THE HEAT
PARAMETERS The shape of the heating b on the
range 0.8-10
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Biot number fixed to B0.05, b0.8 The
influence of S/Tu S/Tu 0.001-1 Thresholds
M
S/Tu
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Patterns at critical thresholds S/Tu0 S/
Tu0.01
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Patterns at critical thresholds S/Tu0.05
S/Tu0.5
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G10, B0.05 THE INFLUENCE OF THE PRANDTL
NUMBER For Pr0.1 thresholds diminish

M
S/Tu
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Pr0.4, B0.05, THE INFLUENCE OF ASPECT RATIO at
G2 thresholds are M13000 for S/Tu
0.02 Patterns have wavenumber
m1 Pr0.01, B0.05 and S/Tu -1 these
waves are possible
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COMPARISONS WITH EXPERIMENTS
G11.76, S/Tu0.05, B0.2, Pr , M 542
N. Garnier y A. Chiffaudel, Eur. Phys. J. (2001)
G 33.4, S/Tu1, B0.2, Pr , M 642
Hoyas, Mancho, Herrero, Garnier and Chiffaudel,
Physics of Fluids, (2005)
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COMPARISONS WITH EXPERIMENTS
Hoyas, Mancho, Herrero, Garnier and Chiffaudel,
Physics of Fluids, (2005)
G 52.9, S/Tu 0.6, B0.2, Pr , M 508
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CONCLUSIONS
Non-homogeneous heating develop new instabilities
on Marangoni convection. Some of them are also
present in purely buoyant convection (see MC
Navarro poster) The shape of the heating b has
been shown to be less influencial than the ratio
S/Tu S/Tu may increase considerably
instability thresholds cause spiral waves
and other oscillatory instabilities. is on
the origin of localized patterns mainly for large
values. Once S/Tu is large enough, localized
patterns are then due to combined effects of
other parameters as Pr, B and G