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This study investigates instabilities in a fluid heated non-homogeneously in Marangoni convection. Numerical techniques and theoretical justifications are used to analyze the fluid dynamics. The influence of various parameters like heat-related parameters, Prandtl number, and aspect ratio are examined.
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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.
PHYSICAL SETUP Domain:
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 Computing18 (3),315-318 (2003)
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
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).
FORMULATION OF THE PROBLEM BASIC EQUATIONS: No gravity effects Pr => Prandtl number Domain BOUNDARY CONDITIONS: for the velocity are, M => Marangoni number
BOUNDARY CONDITIONS: for the temperature are, B => Biot number b => Gaussian width. Heating shape S/Tu=> Quotientof lateral and vertical temperature differences Regularity conditions at the origin Multiparametric problem G, Pr, M, B, b, S/Tu Control Heat related parameters parameter
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
+ + THE COLLOCATION METHOD At each step unknowns are expanded inChebyshev polynomials: Basic equations are evaluated at collocation points Boundary Conditions are evaluated at:
With those rules we obtain 4xLxM equations and unknowns. Some results are: S/Tu =0.001 S/Tu =0.5
THE LINEAR STABILITY ANALYSIS We perturb the basic state : Regularity conditions at r=0 are: The perturbation fields are expanded in Chebyshev polynomials, A trick for m=1,
The number of unknowns and equation are: for m=1, 4xLxM+(L-1)xM otherwise 5xLxM CONVERGENCE RESULTS z-coordinate r-coordinate B=0.05, M=92*Tu, G=10, S=1ºC, b=0.8
THE STABILITY RESULTS G=10, Pr=0.4 THE INFLUENCE OF THE HEAT PARAMETERS The shape of the heating bon the range {0.8-10}
Biot number fixed to B=0.05, b=0.8 The influence of S/Tu S/Tu ~ {0.001-1} Thresholds M S/Tu
Patterns at critical thresholds S/Tu=0 S/Tu=0.01
Patterns at critical thresholds S/Tu=0.05 S/Tu=0.5
G=10, B=0.05 THE INFLUENCE OF THE PRANDTL NUMBER For Pr=0.1 thresholds diminish M S/Tu
Pr=0.4, B=0.05, THE INFLUENCE OF ASPECT RATIO at G=2 thresholds are M~13000 for S/Tu =0.02 Patterns have wavenumber m=1 Pr=0.01, B=0.05 and S/Tu =-1 these waves are possible
COMPARISONS WITH EXPERIMENTS G=11.76, S/Tu~0.05, B=0.2, Pr= , M= 542 N. Garnier y A. Chiffaudel, Eur. Phys. J. (2001) G= 33.4, S/Tu~1, B=0.2, Pr= , M= 642 Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005)
COMPARISONS WITH EXPERIMENTS Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005) G= 52.9, S/Tu~ 0.6, B=0.2, Pr= , M= 508
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
