Theoretical Analysis of Critical Marangoni Number for Oscillatory Flows with High Prandtl Number Fluids

ORAL

Abstract

Many experiments to determine the critical Marangoni number at the onset of oscillatory flows for high Prandtl number (\textit{Pr}) fluids have showed a dependence of a characteristic length such as the diameter of the liquid column. This fact contradicts the similarity principle in fluid physics. This paper investigates the reasons why the critical Marangoni number has shown such a dependency by considering the difference in temperature distribution on the free surface that is caused by a difference in \textit{Pr}. higher \textit{Pr} fluids form a thermal boundary layer with a steep temperature gradient in the vicinity of both the hot and cold parts for Marangoni convection experiments, although the middle of the free surface between the hot and cold disks has no temperature gradients. This feature was obtained by two-dimensional simulations with container shape Marangoni experiments. By considering the partial temperature distribution on the free surface, a basic model for the understanding of Marangoni number for higher\textit{ Pr} fluids is proposed by introducing a dimensionless parameter, the effective Marangoni number. The most exact experimental results obtained from microgravity experiments were used to evaluate the effective Marangoni number which is not dependent on the characteristic length dependence at the onset of oscillatory flows. This parameter is consistent with the similarity principle in Marangoni convection behavior.

–

Authors

  • Shinichi Yoda

    Japan Aerospace Exploration Agency

  • Oscar Curet

    University of Michigan, Institute for Integrated Energy Systems, University of Victoria, Physics Department, Delhi University, Delhi 110007, India, Laboratoire de Physique des Rayonnements, Russian Academy of Sciences, Eotvos University Budapest, The University of Melbourne, Arizona State University, Nova Southeastern University, Academia Sinica, Institute of Physics, University of California at Davis, University of Illinois at Urbana-Champaign, LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100080, China, CORIA, U. Rouen, University Muenster, Carl von Ossietzky University Oldenburg, MIT, Clear Science Corporation, University of Minnesota, Dept. Mech. Eng., Tokyo University of Science., Lawrence Livermore National Laboratory, Los Alamos National Laboratory, GM Technical Center, Tohoku University, Japan Aerospace Exploration Agency, Clemson University, LPS, Ecole Normale Superieur, Paris, Ladhyx , (UMR 7646) Palaiseau, France, PMMH (UMR 7636 CNRS/ESPCI/P7/P6) Paris, France, University of Cambridge, University of Massachusetts, University of Texas at Austin, National Energy Technology Laboratory, Morgantown, WV, University of California at Santa Barbara, Dept. of Biomedical Engineering, UC Irvine, Dept. of Mechanical Engineering, Yale University, University of California, Santa Barbara, Dept. of Biomedical Engr., Medicine and Radiology, The Univ. of Iowa, Bioengineering Inst., The Univ. of Auckland, Dept. of Mech. and Industrial Engr., IIHR-Hydroscience \& Engr., The Univ. of Iowa, Syracuse University, Department of Meterology, Florida State University, Center for Studies in Physics and Biology, The Rockefeller University, ISSI, AFRL/PRTF, Institute of Industrial Science, The University of Tokyo, University of Oldenburg - Institute of Physics, Wesleyan University, California Institute of Technology, University of Science and Techology Beijing, Oakland University, Ngoya Institute of Technology, Institute of Physical Chemistry, Russia, University of Karlsruhe, Germany, University of Oldenburg, Institute of Physics, Ruhr-University Bochum, Germany, Cornell University, Ithaca, NY 14850, NCAR, Boulder, CO 80307, The University of Tokyo, Carnegie Mellon University, Seoul National University, Union College, Mechanical Engineering Department, Applied Math. Lab, Courant Institute, NYU, Department of Biology, NYU, University of Alabama, Ecole Centrale, Lyon, France, Center for Turbulence Research, Stanford, ENS-Lyon, France, Stanford University, University of Notre Dame, LMM, University Pierre et Marie Curie, France, LadHyX, Ecole Polytechnique, France, Department of Aerospace Engineering, Indian Institute of Science,Bangalore-560012, University of Utah, Inst. Inv. Materiales, Universidad Nacional Autonoma de Mexico, Division of Engineering and Applied Sciences, Harvard University, Norfolk State University, INLN, Nice, France, Max Planck Institute for Dynamics and Selforganization, Goettingen, Germany, University of California, Irvine, Northwestern University, Georgia Institute of Technology, Department of Physics, the Chinese University of Hong Kong*, Department of Physics, Hong Kong University of Science and Technology, University of Colorado, Dept. of Physics, North Carolina State University, ISR-2, Los Alamos National Laboratory, Department of Mechanical and Aerospace Engineering, Arizona State University, Department of Mechanical and Industrial Engineering, IIHR-Hydroscience and Engineering, The University of Iowa, Naval Reasearch Laboratory, University of Florida, Boston Scientific, Plymouth, MN 55442, University of Illinois at Urbana-Champaign, Urbana, IL 61801, Associate Director of Applied Research, 1 Coliseum Dr., National Center for Physical Acoustics, University of Mississippi, University, MS-38677, USA, Research Assistant, Dept. Aerospace Eng., Univ. of Seville., Dept. Mech. Eng., Univ. of Malaga, Division of Mechanics, Research Center for Applied Science, Academia Sinica, The University of Michigan, University of Southern California, Purdue University, Physics Department, Eindhoven University of Technology, Eindhoven, The Netherlands, University of Leeds, Oslo University, Trinity Christian Academy, Lexington, KY, University of Kentucky, Tokio Science University, Imperial College, London, Colorado State University, The University of Texas at Austin, LANL, Tokyo University of Agriculture and Technology, Kyushu University, Worcester Polytechnic Institute, Hokkaido University, TU-Braunschweig, Centre de Physique Mol\'eculaire Optique et Hertzienne, Universit\'e Bordeaux I, Physics Department and the James Franck Institute, the University of Chicago, Dept. of Physics, Pomona College, Claremont CA, U. S. Army, ARDEC, United States Air Force Academy, Center for Turbulence Research, Stanford University, The Ohio State University, University of Malaga (Spain), Princeton University, Princeton Plasma Physics Laboratory, Vienna University of Technology, Mechanical Engineering, University of California, Berkeley, Applied Science and Technology Group, University of California, Berkeley, Istituto Nazionale di Geofisica e Vulcanologia INGV, School of Mathematics. University of Bristol, IMEDEA (CSIC - Universitat de les Illes Balears), Columbia University, Kuban State University, Southern Research Center of Russian Academy of Sciences, Stavropolskaya St. 149, Krasnodar 350040, Russia, CNRS UMR 6595, IUSTI, Marseille, FRANCE, Department of Mechanical Engineering, Center for Nanoscale Science and Engineering, NDSU, Fargo, USA, Department of Mechanical Engineering, New Jersey Institute of Technology, Georgia Insitute of Technology, Instituto Nacional de Cardiologia, Mexico, Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK., DAMTP, University of Cambridge, Geology and Geophysics Dept and Physics Dept, Yale University, Geology and Geophysics Dept, Yale University, University of Oregon, INSEAN, Dipartimento di Meccanica e Aeronautica, Universita' di Roma La Sapienza, Seattle University, Chiba Institute of Technology, Advanced Engineering Service Co., Ltd., Dept. of Electrical and Computer Engineering, University of Colorado, Boulder, US Air Force Academy, G. W. Woodruff School of Mechanical Engineering at Georgia Institute of Technology Atlanta, GA 30032, Department of Material Engineering, Graduate School of Engineering, Nagoya Institute of Technology, National Cheng Kung University, Eindhoven University of Technology, University of Vermont, University of California, San Diego, University of Minnesota, Winner of the DFD Fluid Dynamics Prize in 2006, Russian Academy of Science, Washington University in St. Louis, Center for Aeromechanics Research, The University of Texas at Austin, Ohio State University, Univ. Politecnica de Catalunya, Niigata University, International Centre for Theoretical Physics, Iowa State University, University of Oklahoma, Benjamin Levich Insitute, City College of New York, University of Hartford, Physics of Fluids, University of Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands, Mathematics Department, Division of Applied Analysis, University of Patras, 26500 Patras, Greece, CEA / SINUMEF, University of Chicago, Harvard University, ESPCI, Physico-Chimie Th\'{e}orique, DEAS, Harvard University, G. W. Woodruff School of Mechanical Engineering at Georgia Institute of Technology Atlanta, GA, Levich Institute, City College of New York, New York, NY, Blue29, KLA-Tencor Corporation, Dept. of Mechanical Engineering, NJIT, Newark NJ, Levich Institute, City College of CUNY, New York NY, UCSD, 3S, UJF, INPG, CNRS, Grenoble, France, Applied Math Lab, Courant Institute, NYU, Department of Mathematics, Tulane University, Universidad de Catalunya, Spain, University of Marburg, University of Manchester, University of Pennsylvania, Los Alamos National Lab, VU, Imperial College London, Polytechnic University, JAMSTEC, Hokkaido Unviersity, Institute of Theoretical Geophysics, University of Cambridge, ETSI-Aeronauticos Univ. Politec. Madrid, The James Franck Institute and Department of Physics, The University of Chicago, Caltech, Leibniz-Institute for Tropospheric Research, Dept. of Physics, Michigan Technological University, Mechanical \& Aerospace Engineering, Cornell University, University of Washington, Seattle, Department of Mechanical Engineering, Osaka University, University of Wisconsin-Madison, Department of Mechanical Engineering, Northwestern University

  • Oscar Curet

    University of Michigan, Institute for Integrated Energy Systems, University of Victoria, Physics Department, Delhi University, Delhi 110007, India, Laboratoire de Physique des Rayonnements, Russian Academy of Sciences, Eotvos University Budapest, The University of Melbourne, Arizona State University, Nova Southeastern University, Academia Sinica, Institute of Physics, University of California at Davis, University of Illinois at Urbana-Champaign, LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100080, China, CORIA, U. Rouen, University Muenster, Carl von Ossietzky University Oldenburg, MIT, Clear Science Corporation, University of Minnesota, Dept. Mech. Eng., Tokyo University of Science., Lawrence Livermore National Laboratory, Los Alamos National Laboratory, GM Technical Center, Tohoku University, Japan Aerospace Exploration Agency, Clemson University, LPS, Ecole Normale Superieur, Paris, Ladhyx , (UMR 7646) Palaiseau, France, PMMH (UMR 7636 CNRS/ESPCI/P7/P6) Paris, France, University of Cambridge, University of Massachusetts, University of Texas at Austin, National Energy Technology Laboratory, Morgantown, WV, University of California at Santa Barbara, Dept. of Biomedical Engineering, UC Irvine, Dept. of Mechanical Engineering, Yale University, University of California, Santa Barbara, Dept. of Biomedical Engr., Medicine and Radiology, The Univ. of Iowa, Bioengineering Inst., The Univ. of Auckland, Dept. of Mech. and Industrial Engr., IIHR-Hydroscience \& Engr., The Univ. of Iowa, Syracuse University, Department of Meterology, Florida State University, Center for Studies in Physics and Biology, The Rockefeller University, ISSI, AFRL/PRTF, Institute of Industrial Science, The University of Tokyo, University of Oldenburg - Institute of Physics, Wesleyan University, California Institute of Technology, University of Science and Techology Beijing, Oakland University, Ngoya Institute of Technology, Institute of Physical Chemistry, Russia, University of Karlsruhe, Germany, University of Oldenburg, Institute of Physics, Ruhr-University Bochum, Germany, Cornell University, Ithaca, NY 14850, NCAR, Boulder, CO 80307, The University of Tokyo, Carnegie Mellon University, Seoul National University, Union College, Mechanical Engineering Department, Applied Math. Lab, Courant Institute, NYU, Department of Biology, NYU, University of Alabama, Ecole Centrale, Lyon, France, Center for Turbulence Research, Stanford, ENS-Lyon, France, Stanford University, University of Notre Dame, LMM, University Pierre et Marie Curie, France, LadHyX, Ecole Polytechnique, France, Department of Aerospace Engineering, Indian Institute of Science,Bangalore-560012, University of Utah, Inst. Inv. Materiales, Universidad Nacional Autonoma de Mexico, Division of Engineering and Applied Sciences, Harvard University, Norfolk State University, INLN, Nice, France, Max Planck Institute for Dynamics and Selforganization, Goettingen, Germany, University of California, Irvine, Northwestern University, Georgia Institute of Technology, Department of Physics, the Chinese University of Hong Kong*, Department of Physics, Hong Kong University of Science and Technology, University of Colorado, Dept. of Physics, North Carolina State University, ISR-2, Los Alamos National Laboratory, Department of Mechanical and Aerospace Engineering, Arizona State University, Department of Mechanical and Industrial Engineering, IIHR-Hydroscience and Engineering, The University of Iowa, Naval Reasearch Laboratory, University of Florida, Boston Scientific, Plymouth, MN 55442, University of Illinois at Urbana-Champaign, Urbana, IL 61801, Associate Director of Applied Research, 1 Coliseum Dr., National Center for Physical Acoustics, University of Mississippi, University, MS-38677, USA, Research Assistant, Dept. Aerospace Eng., Univ. of Seville., Dept. Mech. Eng., Univ. of Malaga, Division of Mechanics, Research Center for Applied Science, Academia Sinica, The University of Michigan, University of Southern California, Purdue University, Physics Department, Eindhoven University of Technology, Eindhoven, The Netherlands, University of Leeds, Oslo University, Trinity Christian Academy, Lexington, KY, University of Kentucky, Tokio Science University, Imperial College, London, Colorado State University, The University of Texas at Austin, LANL, Tokyo University of Agriculture and Technology, Kyushu University, Worcester Polytechnic Institute, Hokkaido University, TU-Braunschweig, Centre de Physique Mol\'eculaire Optique et Hertzienne, Universit\'e Bordeaux I, Physics Department and the James Franck Institute, the University of Chicago, Dept. of Physics, Pomona College, Claremont CA, U. S. Army, ARDEC, United States Air Force Academy, Center for Turbulence Research, Stanford University, The Ohio State University, University of Malaga (Spain), Princeton University, Princeton Plasma Physics Laboratory, Vienna University of Technology, Mechanical Engineering, University of California, Berkeley, Applied Science and Technology Group, University of California, Berkeley, Istituto Nazionale di Geofisica e Vulcanologia INGV, School of Mathematics. University of Bristol, IMEDEA (CSIC - Universitat de les Illes Balears), Columbia University, Kuban State University, Southern Research Center of Russian Academy of Sciences, Stavropolskaya St. 149, Krasnodar 350040, Russia, CNRS UMR 6595, IUSTI, Marseille, FRANCE, Department of Mechanical Engineering, Center for Nanoscale Science and Engineering, NDSU, Fargo, USA, Department of Mechanical Engineering, New Jersey Institute of Technology, Georgia Insitute of Technology, Instituto Nacional de Cardiologia, Mexico, Manchester Centre for Nonlinear Dynamics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK., DAMTP, University of Cambridge, Geology and Geophysics Dept and Physics Dept, Yale University, Geology and Geophysics Dept, Yale University, University of Oregon, INSEAN, Dipartimento di Meccanica e Aeronautica, Universita' di Roma La Sapienza, Seattle University, Chiba Institute of Technology, Advanced Engineering Service Co., Ltd., Dept. of Electrical and Computer Engineering, University of Colorado, Boulder, US Air Force Academy, G. W. Woodruff School of Mechanical Engineering at Georgia Institute of Technology Atlanta, GA 30032, Department of Material Engineering, Graduate School of Engineering, Nagoya Institute of Technology, National Cheng Kung University, Eindhoven University of Technology, University of Vermont, University of California, San Diego, University of Minnesota, Winner of the DFD Fluid Dynamics Prize in 2006, Russian Academy of Science, Washington University in St. Louis, Center for Aeromechanics Research, The University of Texas at Austin, Ohio State University, Univ. Politecnica de Catalunya, Niigata University, International Centre for Theoretical Physics, Iowa State University, University of Oklahoma, Benjamin Levich Insitute, City College of New York, University of Hartford, Physics of Fluids, University of Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands, Mathematics Department, Division of Applied Analysis, University of Patras, 26500 Patras, Greece, CEA / SINUMEF, University of Chicago, Harvard University, ESPCI, Physico-Chimie Th\'{e}orique, DEAS, Harvard University, G. W. Woodruff School of Mechanical Engineering at Georgia Institute of Technology Atlanta, GA, Levich Institute, City College of New York, New York, NY, Blue29, KLA-Tencor Corporation, Dept. of Mechanical Engineering, NJIT, Newark NJ, Levich Institute, City College of CUNY, New York NY, UCSD, 3S, UJF, INPG, CNRS, Grenoble, France, Applied Math Lab, Courant Institute, NYU, Department of Mathematics, Tulane University, Universidad de Catalunya, Spain, University of Marburg, University of Manchester, University of Pennsylvania, Los Alamos National Lab, VU, Imperial College London, Polytechnic University, JAMSTEC, Hokkaido Unviersity, Institute of Theoretical Geophysics, University of Cambridge, ETSI-Aeronauticos Univ. Politec. Madrid, The James Franck Institute and Department of Physics, The University of Chicago, Caltech, Leibniz-Institute for Tropospheric Research, Dept. of Physics, Michigan Technological University, Mechanical \& Aerospace Engineering, Cornell University, University of Washington, Seattle, Department of Mechanical Engineering, Osaka University, University of Wisconsin-Madison, Department of Mechanical Engineering, Northwestern University