Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process

Shen, Y.; Neitzel, G. Paul; Jankowski, D. F.; Mittelmann, Hans D.

doi:10.1017/s002211209000088x

Cited by 92 publications

(23 citation statements)

References 18 publications

(23 reference statements)

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“…In this configuration a liquid drop is held between two circular cylinders by surface tension forces. In half-zones, Shen et al 5 used energy stability methods to predict the energy limit of the stability of the axisymmetric base state. The main objective of most studies relating to the half-zone has been to understand the stability characteristics of the steady basic thermocapillary flow.…”

Section: Introductionmentioning

confidence: 99%

Instabilities of thermocapillary convection in a half-zone at intermediate Prandtl numbers

2001

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The stability of thermocapillary convection inside a cylindrical liquid bridge is studied using both a direct numerical simulation of the three-dimensional problem and linear stability analysis of the axisymmetric basic state. Previously this has been studied extensively for low and high Prandtl numbers. However, the intermediate range of Prandtl numbers between approximately 0.07 and 0.8 which joins the low and high ranges is quite complicated and has not been studied to the same extent. One striking feature is that the axisymmetric base state is much more stable in this intermediate range than at high or low Prandtl numbers. We identify four different oscillatory modes in this range, which have different qualitative features. Direct numerical simulations have been carried out for representative parameter values, and show that the bifurcations are supercritical.

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Section: Introductionmentioning

confidence: 99%

Instabilities of thermocapillary convection in a half-zone at intermediate Prandtl numbers

2001

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“…Motivated by the experimental observations, several theoretical and numerical studies have been devoted to stability of the flow in a half-zone model, including energy stability theory [2,3], linear stability analysis [4,5,6,7] and 3D direct numerical simulation [8,9]. It is now well established (see Ref.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

Chen

Lizée

Roux

1997

Journal of Crystal Growth

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We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (=length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. P r < 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Re c → 1784, as P r → 0, the most dangerous mode being m = 2. While for P r ≥ 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 ≤ P r ≤ 0.8 and m = 2 for P r ≥ 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent lineartheory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a non-deformable free surface.

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“…Thus, stability analyses of thermocapillary convection in liquid bridges which consider axisymmetric perturbations solely, with or without magnetic fields (e.g. Shen et al 1990;Chen & Chin 1995), must be interpreted with the utmost caution.…”

Section: Discussionmentioning

confidence: 99%

“…Preisser, Schwabe & Scharmann 1983), numerical (e.g. Shen et al 1990;Kuhlmann & Rath 1993;Wanschura et al 1995) and analytical (e.g. Kuhlmann 1989) studies.…”

Section: Methodsmentioning

confidence: 99%

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

Prange¹,

Wanschura²,

Kuhlmann³

et al. 1999

J. Fluid Mech.

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The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.

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Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process

Cited by 92 publications

References 18 publications

Instabilities of thermocapillary convection in a half-zone at intermediate Prandtl numbers

Instabilities of thermocapillary convection in a half-zone at intermediate Prandtl numbers

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

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