Keywords: liquid bridges, crystals, finite elements, nonlinear analysis.

Flow and thermal effects concerned with liquid crystal growth are studied nonlinearly for low Prandtl numbers, in an axisymmetric steady configuration with endwalls present. The reason for this is that numerous major properties in liquid crystal behaviour are both thoroughly nonlinear and of nonparallel form. The concern here is in full direct simulations as well as nonlinear theoretical analyses.

Full solutions are obtained by finite element simulation to examine the influences of aspect ratio, Marangoni number and Rayleigh number. The method is described in some detail, based on the works of Bänsch & Höhn [1], Davis & Smith [2], Bänsch [3], Tenhaeff [4] and others, and the results which cover a quite wide range of parameter values are then discussed. In particular a `lemonhead' phenomenon is found in which the velocity profiles acquire a very localised bell-shaped form as jet like flow starts to emerge, when the relative size of the Marangoni number increases above a finite critical value which is identified.

Wide-domain and narrow-domain analyses are then presented. The analysis for wide domains hinges on the main flow and thermal activity being concentrated in a zone near the outer wall. The analysis for narrow domains, on the other hand, is based on recirculatory slender flow modelling which covers the entire domain apart from end zones where nonlinear inviscid mechanics is dominant.

Comparisons are presented, showing favourable agreement with the full solutions and explaining the impact of the end walls on the motion. The agreement applies in practice for aspect ratios above about 2 as far as the wide-domain context is concerned and below about 0.3 in the case of narrow domains. Sensitive scalings for the relative effects of the Marangoni and Rayleigh numbers emerge. One such example concerns the scaling of the Rayleigh number with the sixth power of the aspect ratio for the context of a narrow domain. The lemonhead phenomenon is also accounted for as a substantial change in the convective flow structure. This involves study of the solution features for an ordinary differential system followed by the implications for the structure of the global solution.

1

E. Bänsch, B. Höhn, "Numerical simulation of a silicon floating zone witha free capillary surface", Rep.98-08, Zentrum für Technomathematik, University of Bremen, Germany, 1998.

2

D.A.R. Davis, F.T. Smith, "Flow and thermal convection in full-zone liquid bridges of wide-ranging aspect ratio", Report no. 720, WIAS Berlin, ISSN 0946-8633, 2002.

3

E. Bänsch, "Finite element discretization of the Navier-Stokes equations with a free capillary surface", Numer. Math. 88, 203-235, 2001. doi:10.1007/PL00005443

4

M. Tenhaeff, "Berechnung inkompressibler rotationsymmetrischer Strömungen elektrisch leitender Flüssigkeiten unter dem Einfluss von rotierenden Magnetfeldern" Diploma disseration, University of Freiburg, Germany, 1997.

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