## Onset of surface-tension-driven Bénard convection (1995)

Citations: | 12 - 0 self |

### BibTeX

@MISC{Vanhook95onsetof,

author = {J. Vanhook and Michael F. Schatz and J. B. Swift and W. D. Mc Cormick and Harry and L. Swinney},

title = {Onset of surface-tension-driven Bénard convection},

year = {1995}

}

### OpenURL

### Abstract

below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (< 0.017 cm for 0.102 cm 2 s −1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (> 0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the longwavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient.

### Citations

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Citation Context ...transfer coefficient, the Biot number, which for a flat interface is H(q) =q kg k coth q dg d (2.1)50 S. VanHook, M. Schatz, J. Swift, W. McCormick and H. Swinney (see, for example, equation 8.91 of =-=Cross & Hohenberg 1993-=- or the appendix of Hadji et al. 1991). As q → 0, H → kgd/kdg (for the remainder of the paper, H will mean H(q → 0)). The earliest linear stability analyses assumed a non-deformable interface (Pearson... |

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Citation Context ...en repeated (Koschmieder & Biggerstaff 1986) and the most recent experiment by Schatz et al. (1995) found onset of the instability in close agreement with the prediction of linear stability analysis (=-=Pearson 1958-=-; Nield 1964). In addition, Schatz et al. found that the bifurcation was subcritical with 3.2% hysteresis, larger than predicted by nonlinear theory (Scanlon & Segel 1967, 2.3%; Cloot & Lebon 1984, 0.... |

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Citation Context ...oschmieder & Biggerstaff 1986) and the most recent experiment by Schatz et al. (1995) found onset of the instability in close agreement with the prediction of linear stability analysis (Pearson 1958; =-=Nield 1964-=-). In addition, Schatz et al. found that the bifurcation was subcritical with 3.2% hysteresis, larger than predicted by nonlinear theory (Scanlon & Segel 1967, 2.3%; Cloot & Lebon 1984, 0.2%). A secon... |

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Citation Context ...0 100 2 3 G 0 1 2 3 4 0 1 2 3 4 q q Figure 2. Marginal stability curves for ν = 0.10 cm 2 s −1 silicone oil in terrestrial gravity according to linear stability theory for d/dg ≪ 1 (see, for example, =-=Smith 1966-=-). (a) Ford= 0.03 cm, the primary instability appears at M = 80 with q =1.99. (b) Ford= 0.02 cm, the minimum at q =0 becomes the global minimum and the primary instability occurs at q = 0 and M =2G/3 ... |

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Citation Context ... cool regions of the interface to pull liquid from warm regions of the interface. This thermocapillary mechanism can drive fluid flow and lead to instabilities in non-isothermal free-surface systems (=-=Davis 1987-=-). In surface-tension-driven Bénard convection (henceforth called ‘Marangoni convection’), a liquid layer is heated from below and cooled from above (see figure 1). In thin liquid layers or in microgr... |

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Citation Context ...linear stability theory successfully predicts the onset of the q =1.99 Pearson/Bénard instability, it is less successful with the long-wavelength instability. The only experiment to test this theory (=-=VanHook et al. 1995-=-) found that the onset in the experiments occurs 35% earlier than predicted by the one-layer linear theory. A one-layer model nonlinear evolution equation for the long-wavelength instability was deriv... |

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Citation Context ...m above (see figure 1). In thin liquid layers or in microgravity, the buoyancy mechanism of Rayleigh–Bénard convection becomes negligible, leaving thermocapillarity as the dominant driving mechanism (=-=Davis & Homsy 1980-=-). Bénard (1900) performed some of the earliest studies of convection in liquid layers almost a century ago. Heating thin layers (≈ 1 mm deep) of spermaceti from below, he found a transition from cond... |

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Citation Context ... and λ is the wavelength of the cells. Block (1956) and Pearson (1958) first recognized that surface tension, not buoyancy, was the source of the instability. Bénard’s experiments have been repeated (=-=Koschmieder & Biggerstaff 1986-=-) and the most recent experiment by Schatz et al. (1995) found onset of the instability in close agreement with the prediction of linear stability analysis (Pearson 1958; Nield 1964). In addition, Sch... |

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Citation Context ...ffect of thermocapillarity; the second, gravity; and the third, surface tension. The equation of Davis (1983, 1987) is recovered for F = 0; the one-layer model equation (Kopbosynov & Pukhnachev 1986; =-=Oron & Rosenau 1992-=-; Deissler & Oron54 S. VanHook, M. Schatz, J. Swift, W. McCormick and H. Swinney 0 0 c q q 2 max =Bε (a) ε�0 –0.5 0 1 2 q (b) ε�0 –0.5 0 1 2 q Figure 4. Growth rates γq as a function of q for (a) ɛ n... |

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Citation Context ... formed a large-scale drained region (‘dry spot’). The experiments found onset of the instability to occur 35% below the prediction of linear stability analysis of a one-layer model (Takashima 1981a; =-=Goussis & Kelly 1990-=-; Pérez-García & Carneiro 1991; Wilson 1993). Another long-wavelength instability, unrelated to the one described in the previous paragraph, was predicted by Pearson (1958) and further discussed by Si... |

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Citation Context ...drained region (‘dry spot’). The experiments found onset of the instability to occur 35% below the prediction of linear stability analysis of a one-layer model (Takashima 1981a; Goussis & Kelly 1990; =-=Pérez-García & Carneiro 1991-=-; Wilson 1993). Another long-wavelength instability, unrelated to the one described in the previous paragraph, was predicted by Pearson (1958) and further discussed by Sivashinsky (1982), Garcia-Ybarr... |

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Citation Context ...le and eventually formed a large-scale drained region (‘dry spot’). The experiments found onset of the instability to occur 35% below the prediction of linear stability analysis of a one-layer model (=-=Takashima 1981-=-a; Goussis & Kelly 1990; Pérez-García & Carneiro 1991; Wilson 1993). Another long-wavelength instability, unrelated to the one described in the previous paragraph, was predicted by Pearson (1958) and ... |

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Citation Context ...re cell. Once the dry spot forms, fluid flow consists of steady-state convection concentrated at the edge of the dry spot. The dry spot is not completely dry since a very thin adsorbed layer remains (=-=Burelbach et al. 1990-=-). The dry spot appears at different locations in different experiments, although it usually appears at the same location for multiple runs in a given experiment. When it appears near the centre of th... |

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Citation Context ...analysis (Pearson 1958; Nield 1964). In addition, Schatz et al. found that the bifurcation was subcritical with 3.2% hysteresis, larger than predicted by nonlinear theory (Scanlon & Segel 1967, 2.3%; =-=Cloot & Lebon 1984-=-, 0.2%). A second and completely different instability due to surface tension also exists in this same system and should become primary in very thin or very viscous liquid layers or in microgravity. T... |

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Citation Context ...ul in studying the predicted transition between dry spots and high spots at F =1/2. We use an optical system (see figure 18) that serves as both an interferometer and a shadowgraph (see, for example, =-=Rasenat et al. 1989-=-). When the deformation is small, we use the optical system as an interferometer to give an indication of the deformation of the interface. The mirror–window fringes are much68 S. VanHook, M. Schatz,... |

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Citation Context ...diction of linear stability analysis (Pearson 1958; Nield 1964). In addition, Schatz et al. found that the bifurcation was subcritical with 3.2% hysteresis, larger than predicted by nonlinear theory (=-=Scanlon & Segel 1967-=-, 2.3%; Cloot & Lebon 1984, 0.2%). A second and completely different instability due to surface tension also exists in this same system and should become primary in very thin or very viscous liquid la... |

1 | Nonlinear surface waves driven by the Marangoni instability in a heat transfer system - Funada - 1987 |

1 | A nonlinear evolution equation for Bénard-Marangoni convection with deformable boundary. Phys - Garcia-Ybarra, Castillo, et al. - 1987 |

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Citation Context ...e the most likely culprit and we have begun a theoretical study of the effect of the sidewalls upon the formation of the longwavelength instability. Marangoni convection experiments aboard Apollo 14 (=-=Grodzka & Bannister 1972-=-) and Apollo 17 (Grodzka & Bannister 1975) yielded Bénard convection cells, and not the long-wavelength instability, even though G<1. Several factors explain why the long-wavelength instability was no... |

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Citation Context ...un a theoretical study of the effect of the sidewalls upon the formation of the longwavelength instability. Marangoni convection experiments aboard Apollo 14 (Grodzka & Bannister 1972) and Apollo 17 (=-=Grodzka & Bannister 1975-=-) yielded Bénard convection cells, and not the long-wavelength instability, even though G<1. Several factors explain why the long-wavelength instability was not observed in the Apollo experiments. Fas... |

1 |
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(Show Context)
Citation Context ...ch for a flat interface is H(q) =q kg k coth q dg d (2.1)50 S. VanHook, M. Schatz, J. Swift, W. McCormick and H. Swinney (see, for example, equation 8.91 of Cross & Hohenberg 1993 or the appendix of =-=Hadji et al. 1991-=-). As q → 0, H → kgd/kdg (for the remainder of the paper, H will mean H(q → 0)). The earliest linear stability analyses assumed a non-deformable interface (Pearson 1958; Nield 1964) and thus never dis... |

1 | Long-wave instabilities of heated falling films - Joo, Davis, et al. - 1991 |

1 |
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(Show Context)
Citation Context ...curly brackets describes the effect of thermocapillarity; the second, gravity; and the third, surface tension. The equation of Davis (1983, 1987) is recovered for F = 0; the one-layer model equation (=-=Kopbosynov & Pukhnachev 1986-=-; Oron & Rosenau 1992; Deissler & Oron54 S. VanHook, M. Schatz, J. Swift, W. McCormick and H. Swinney 0 0 c q q 2 max =Bε (a) ε�0 –0.5 0 1 2 q (b) ε�0 –0.5 0 1 2 q Figure 4. Growth rates γq as a func... |

1 |
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(Show Context)
Citation Context ...linear stability analysis based on this equation agrees with (2.2). Numerical simulations of the equation find the formation of a large-scale dry spot (Joo, Davis & Bankoff 1991; VanHook et al. 1995; =-=Krishnamoorthy & Ramaswamy 1995-=-), in qualitative accord with what is seen in experiments (VanHook et al. 1995). 3. Two-layer model A thermocapillary model with two fluid layers was first considered by Smith (1966). He found a negli... |

1 | Long-wave thermocapillary convection in layers with deformable interfaces - Nepomnyashchii, Simanovskii - 1990 |

1 | Rayleigh-Benard-Marangoni instability: New experimental results - VanHook, Schatz, et al. - 1979 |

1 |
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Citation Context ...or most experiments in this paper we use a ν =0.102 cm2 s−1 (at 50 ◦C) silicone oil that has been distilled once to remove low-vapour-pressure components, which can condense on the cool, upper plate (=-=Schatz & Howden 1995-=-). Use of a much larger-viscosity liquid increases the difficulty of the experiments by increasing the relaxation time of the liquid. The predicted viscosity independence of onset is tested in experim... |

1 | Surface-tension-driven convection with boundary slab of finite conductivity - Takashima - 1970 |