1. 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. INSTABILITIES IN A NON- HOMOGENEOUSLY HEATED FLUID IN MARANGONI CONVECTION

2. PHYSICAL SETUP Domain:

3. 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)

4. 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

5. 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).

6. BASIC EQUATIONS : FORMULATION OF THE PROBLEM No gravity effects Pr => Prandtl number Domain BOUNDARY CONDITIONS : for the velocity are, M => Marangoni number

7. BOUNDARY CONDITIONS : for the temperature are, B => Biot number  => Gaussian width. Heating shape S/T u => Quotient of lateral and vertical temperature differences Regularity conditions at the origin Multiparametric problem , Pr, M, B, , S/T u Control Heat related parameters parameter

8. 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 BASIC STATE

9. + + THE COLLOCATION METHOD At each step unknowns are expanded in Chebyshev polynomials: Basic equations are evaluated at collocation points Boundary Conditions are evaluated at:

10. With those rules we obtain 4xLxM equations and unknowns. Some results are: S/T u = 0.001 S/T u = 0.5

11. 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,

12. CONVERGENCE RESULTS 791113 3368.9003568.6486168.6887868.71441 3568.8894368.6456968.6873568.71319 3768.6864868.6427868.6855968.71157 r-coordinate z-coordinate B=0.05, M=92*Tu,  =10, S=1ºC,  =0.8 The number of unknowns and equation are: for m=1,4xLxM+(L-1)xM otherwise5xLxM

13.  =10, Pr=0.4 THE INFLUENCE OF THE HEAT PARAMETERS The shape of the heating  on the range {0.8-10} THE STABILITY RESULTS

14. Biot number fixed to B=0.05,  =0.8 The influence of S/Tu S/Tu ~ {0.001-1} Thresholds S/T u M

15. Patterns at critical thresholds S/T u =0 S/T u =0.01

16. Patterns at critical thresholds S/T u =0.05 S/T u =0.5

17.  =10, B=0.05 THE INFLUENCE OF THE PRANDTL NUMBER For Pr=0.1 thresholds diminish S/T u M

18. Pr=0.4, B=0.05, THE INFLUENCE OF ASPECT RATIO at  =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

19. COMPARISONS WITH EXPERIMENTS N. Garnier y A. Chiffaudel, Eur. Phys. J. (2001)  =11.76, S/Tu~0.05, B=0.2, Pr=, M= 542 Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005)  = 33.4, S/Tu~1, B=0.2, Pr=, M= 642

20. COMPARISONS WITH EXPERIMENTS  = 52.9, S/Tu~ 0.6, B=0.2, Pr=, M= 508 Hoyas, Mancho, Herrero, Garnier and Chiffaudel, Physics of Fluids, (2005)

21. 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  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 CONCLUSIONS