The influence of surface deformations on the Rayleigh–Bénard–Marangoni instability of a uniform layer of a non-Boussinesq fluid heated from below is investigated. In particular, the stability of the conductive state of a horizontal fluid layer with a deformable surface, a flat isothermal rigid lower boundary, and a convective heat transfer condition at the upper free surface is considered. The fluid is assumed to be isothermally incompressible. In contrast to the Boussinesq approximation, density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. Two different types of temperature dependence of the density are considered: linear and exponential. The longwave instability is studied analytically, and instability to perturbations with finite wavenumber is examined numerically. It is found that there is a decrease in stability of the system with respect to the onset of longwave Marangoni convection. This result could not be obtained within the framework of the conventional Boussinesq approximation. It is also shown that at Ma = 0 the critical Rayleigh number increases with Ga (the ratio of gravity to viscous forces or Galileo number). At some value of Ga, the Rayleigh–Bénard instability vanishes. This stabilization occurs for each of the density equations of state. At small values of Ga and when deformation of the free surface is important, it is shown that there are significant differences in stability behavior as compared to results obtained using the Boussinesq approximation.
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February 2018
Research Article|
February 26 2018
Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface Available to Purchase
D. V. Lyubimov;
D. V. Lyubimov
1
Theoretical Physics Department, Perm State University
, Perm 614099, Russia
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T. P. Lyubimova;
T. P. Lyubimova
a)
1
Theoretical Physics Department, Perm State University
, Perm 614099, Russia
2
Institute of Continuous Media Mechanics UB RAS
, Perm 614013, Russia
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N. I. Lobov;
N. I. Lobov
1
Theoretical Physics Department, Perm State University
, Perm 614099, Russia
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J. I. D. Alexander
J. I. D. Alexander
3
School of Engineering, University of Alabama at Birmingham
, Birmingham, Alabama 35294, USA
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D. V. Lyubimov
1
T. P. Lyubimova
1,2,a)
N. I. Lobov
1
J. I. D. Alexander
3
1
Theoretical Physics Department, Perm State University
, Perm 614099, Russia
2
Institute of Continuous Media Mechanics UB RAS
, Perm 614013, Russia
3
School of Engineering, University of Alabama at Birmingham
, Birmingham, Alabama 35294, USA
Physics of Fluids 30, 024103 (2018)
Article history
Received:
September 30 2017
Accepted:
February 01 2018
Citation
D. V. Lyubimov, T. P. Lyubimova, N. I. Lobov, J. I. D. Alexander; Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface. Physics of Fluids 1 February 2018; 30 (2): 024103. https://doi.org/10.1063/1.5007117
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