Stabilities of surface-tension-driven convection in an open cylinder are investigated numerically. The cylinder is heated laterally through its sidewall and is cooled at free surface by radiation. A seeding crystal at constant temperature is in contact with the free surface. Axisymmetric base flow is solved using the high-order finite difference method. Three-dimensional perturbation is applied to the obtained base flow to determine the critical Marangoni numbers at which the axisymmetry is broken. The eigenvalue matrix equation is solved using linear fractional transformation with banded matrix structure taken into account. Critical Marangoni-Reynolds numbers are obtained at various boundary conditions.
1.
Hurle
, D. T. J.
, 1966, “Temperature Oscillations in Molten Metals and Their Relationship to Growth Striae in Melt-Grown Crystal
,” Philos. Mag.
0031-8086, 13
, pp. 305
–310
.2.
Pearson
, J. R. A.
, 1958, “On Convection Cells Induced by Surface Tension
,” J. Fluid Mech.
0022-1120, 4
, pp. 489
–500
.3.
Levich
, H. G.
, and Krylov
, V. S.
, 1969, “Surface Tension-Driven Phenomena
,” Annu. Rev. Fluid Mech.
0066-4189, 1
, pp. 293
–316
.4.
Davis
, S. H.
, 1987, “Thermocapillary Instabilities
,” Annu. Rev. Fluid Mech.
0066-4189, 19
, pp. 403
–435
.5.
Schatz
, M. F.
, and Neitzel
, G. P.
, 2001, “Experiments on Thermocapillary Instabilities
,” Annu. Rev. Fluid Mech.
0066-4189, 33
, pp. 93
–127
.6.
Peltier
, L. J.
, and Biringen
, S.
, 1993, “Time-Dependent Thermocapillary Convection in a Rectangular Cavity: Numerical Results for a Moderate Prandtl Number Fluid
,” J. Fluid Mech.
0022-1120, 257
, pp. 339
–357
.7.
Sab
, V.
, Kuhlmann
, H. C.
, and Rath
, H. J.
, 1996, “Investigation of Three-Dimensional Thermocapillary Convection in a Cubic Container by Multi-grid Method
,” Int. J. Heat Mass Transfer
0017-9310, 39
, pp. 603
–613
.8.
Xu
, J.
, and Zebib
, A.
, 1998, “Oscillatory Two- and Three-Dimensional Thermocapillary Convection
,” J. Fluid Mech.
0022-1120, 364
, pp. 187
–209
.9.
Vrentas
, J. S.
, Narayanan
, R.
, and Agrawal
, S. S.
, 1981, “Free Surface Convection in Bounded Cylindrical Geometry
,” Int. J. Heat Mass Transfer
0017-9310, 24
(9
), pp. 1513
–1529
.10.
Wagner
, C.
, Friedrich
, R.
, and Narayanan
, R.
, 1994, “Comments on the Numerical Investigation of Rayleigh and Marangoni Convection in a Vertical Circular Cylinder
,” Phys. Fluids
1070-6631, 6
(4
), pp. 1425
–1433
.11.
Dauby
, P. C.
, Lebon
, G.
, and Bouhy
, E.
, 1997, “Linear Bénard-Marangoni Instability in Rigid Circular Containers
,” Phys. Rev. E
1063-651X, 56
(1
), pp. 520
–530
.12.
Schwabe
, D.
, Möller
, U.
, Schneider
, J.
, and Scharmann
, A.
, 1992, “Instabilities of Shallow Dynamic Thermocapillary Liquid Layer
,” Phys. Fluids A
0899-8213, 4
, pp. 2368
–2381
.13.
Ezersky
, A. B.
, Garcimartín
, A.
, Burguete
, J.
, Mancini
, H. L.
, and Pérez-García
, C.
, 1993, “Hydrothermal Waves in Marangoni Convection in a Cylindrical Container
,” Phys. Rev. E
1063-651X, 47
(2
), pp. 1126
–1131
.14.
Favre
, E.
, Blumendeld
, L.
, and Daviaud
, F.
, 1997, “Instabilities of a Liquid Layer Locally Heated on its Free Surface
,” Phys. Fluids
1070-6631, 9
(5
), pp. 1473
–1475
.15.
Garnier
, N.
, and Chiffaudel
, A.
, 2001, “Two Dimensional Hydrothermal Waves in an Extended Cylindrical Vessel
,” Eur. Phys. J. B
1434-6028, 19
, pp. 87
–95
.16.
Hoyas
, S.
, Herrero
, H.
, and Mancho
, A. M.
, 2002, “Thermal Convection in a Cylindrical Annulus Heated Laterally
,” J. Phys. A
0305-4470, 35
, pp. 4067
–4083
.17.
Sim
, B.-C.
, Zebib
, A.
, and Schwabe
, D.
, 2003, “Oscillatory Thermocapillary Convection in Open Cylindrical Annuli. Part 2. Simulations
,” J. Fluid Mech.
0022-1120, 491
, pp. 259
–274
.18.
Jing
, C. J.
, Imaishi
, N.
, Yasuhiro
, S.
, and Miyazawa
, Y.
, 1999, “Three-Dimensional Numerical Simulation of Spoke Pattern in Oxide Melt
,” J. Cryst. Growth
0022-0248, 200
, pp. 204
–212
.19.
Ding
, Y.
, and Kawahara
, M.
, 1998, “Linear Stability of Incompressible Fluid Flow Using a Mixed Finite Element Method
,” J. Comput. Phys.
0021-9991, 139
(2
), pp. 243
–273
.20.
Ai
, X.
, 2004, “The Instability Analysis and Direct Numerical Simulation of Turbulent Flows in Electromagnetically Levitated Droplets
,” Ph.D. thesis, Washington State University, Pullman, WA.21.
Le
, H.
, and Moin
, P.
, 1991, “An Improvement of Fractional Step Methods for the Incompressible Navier-Stokes Equations
,” Int. J. Numer. Methods Fluids
0271-2091, 92
, pp. 369
–379
.22.
Kim
, J.
, and Moin
, P.
, 1985, “Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations
,” J. Comput. Phys.
0021-9991, 59
, pp. 308
–323
.23.
Chénier
, E.
, Delcarte
, C.
, Kasperski
, G.
, and Labrosse
, G.
, 2002, “Sensitive of the Liquid Bridge Hydrodynamics to Local Capillary Contributions
,” Phys. Fluids
0031-9171, 14
(9
), pp. 3109
–3117
.24.
Chénier
, E.
, Delcarte
, C.
, and Labrosse
, G.
, 1999, “Stability of the Axisymmetric Buoyant-Capillary Flows in a Laterally Heated Liquid Bridge
,” Phys. Fluids
0031-9171, 11
(3
), pp. 527
–541
.25.
Johnson
, D.
, and Narayanan
, R.
, 1999, “A Tutorial on the Rayleigh-Marangoni-Bénard Problem with Multiple Layers and Side Wall Effects
,” Chaos
1054-1500, 9
(1
), pp. 124
–140
.26.
Rubinov
, A.
, Erenburg
, V.
, Gelfgat
, A. Yu.
, Kit
, E.
, Bar-Yoseph
, P. Z.
, and Solan
, A.
, 2004, “Three-Dimensional Instabilities of Natural Convection Flow in a Vertical Cylinder With Partially Heated Sidewall
,” ASME J. Heat Transfer
0022-1481, 126
, pp. 586
–599
.Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
