Long-wave oscillatory Marangoni instability in a horizontal liquid layer |
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| Institution: | 1. School of Mathematical Sciences, Peking University, Beijing 100871, China;2. BIC-ESAT, College of Engineering, Peking University, Beijing 100871, China;3. State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China;4. Institute of Ocean Research, Peking University, Beijing 100871, China;5. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource & Hydropower, Sichuan University, Chengdu, 610065, China;6. Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Tsinghua University, Beijing 100084, China;7. Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Swabi, KPK, Pakistan;1. Department of Mathematics, M.S. Ramaiah Institute of Technology, Bengaluru 560 054, India;2. Department of Mathematics, Bengaluru Central University, Central College Campus, Bengaluru 560 001, India |
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| Abstract: | The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values. |
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