Abstract
Energy stability theory is applied to a basic state of thermocapillary convection occurring in a cylindrical half-zone of finite length to determine conditions under which the flow will be stable under arbitrary perturbations. In earlier work axisymmetric disturbances have been considered while the onset of buoyancy-driven convection in a cylinder heated from below has served as a test case for the three-dimensional problem. Here, initial results will be reported for thermocapillary convection under zero gravity. A description of the numerical method used to compute the stability bounds will also be given.
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BANK, R., MITTELMANN, H. D. (1986) Continuation and multigrid for nonlinear elliptic systems. In Multigrid Methods II (ed. W. Hackbusch, U. Trottenb Lect. Notes Math. 1228, Springer.
CHARLSON, G. S., SANI, R. L. (1970) Thermoconvective instability in a bounded cylindrical fluid layer. Int. J. Heat Mass Trans. 13, 1479.
CHARLSON, G. S., SANI, R. L. (1971) On thermoconvective instability in a bound cylindrical fluid layer, Int. J. Heat Mass Trans. 14, 2157.
DAVIS, S. H. (1971) Energy stability of unsteady flows. IUTAM Symposium on Unsteady Boundary Layers, 206.
DAVIS, S. H., von KERCZEK, C. (1973) A reformulation of energy stability theory. Arch. Rat. Mech. Anal. 52, 112.
JOSEPH, D. D. (1976) Stability of Fluid Motions I, I I, Springer-Verlag, Berlin.
MITTELMANN, H. D., LAW, C. C., JANKOWSKI, D. F., NEITZEL, G. P. (1990) A large sparse and indefinite generalized eigenvalue problem from fluid mechanics. submitted to SIAM J. Sci. Stat. Comp.
MUNSON, B. R., JOSEPH, D. D. (1971) Viscous incompressible flow between concentric rotating spheres. Part 2. Hydrodynamic stability. J. Fluid Meth. 49, 305.
NEITZEL, G. P. (1984) Numerical computation of time-dependent Taylor-vortex flows in finite-length geometries. J. Fluid Mech. 141, 51.
NEITZEL, G. P., DAVIS, S. H. (1981) Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments. J. Fluid Mech. 102, 329.
PAIGE, C. C., SAUNDERS, M. A. (1975) Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617.
PREISSER, F., SCHWABE, P., SHARMANN, A. (1983) Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface. J. Fluid Mech. 126, 545.
SEN, A. K., DAVIS, S. H. (1982) Steady thermocapillary flow in two dimension slots. J. Fluid Mech. 121, 163.
SHEN, Y., NEITZEL, G. P., JANKOWSKI, D. F., MITTELMANN, H. D. (1990) Energy stability of thermocapillary convection in a model of the float-zone, crystal-growth process. J. Fluid Mech., to appear.
SINHA, S. C., CARMI, S. (1976) On the Liapunov-Movchan and the energy theories of stability. J. Appl. Math. Phys. 27, 607–612.
VELTEN, R., SCHWABE, D., SCHARMANN, A. (1990) The periodic instability of thermocapillary convection in cylindrical liquid bridges, submitted to Phys. Fluids.
XU, J.-J., DAVIS, S. H. (1983) Liquid bridges with thermocapillarity. Phys. Fluids 26, 2880.
XU, J.-J., DAVIS, S. H. (1984) Convective thermocapillary instabilities in liquid bridges. Phys. Fluids 27, 1102.
XU, J.-J., DAVIS, S. H. (1985) Instability of capillary jets with thermocapillarity. J. Fluid Mech. 161, 1.
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Mittelmann, H.D., Law, C.C., Jankowski, D.F., Neitzel, G.P. (1991). Stability of Thermocapillary Convection in Float-Zone Crystal Growth. In: Neittaanmäki, P. (eds) Numerical Methods for Free Boundary Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 99. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5715-4_3
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DOI: https://doi.org/10.1007/978-3-0348-5715-4_3
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