## Two-fluid Marangoni-Bénard convection with a deformable interface

Citations: | 1 - 0 self |

### BibTeX

@MISC{Tavener_two-fluidmarangoni-bénard,

author = {S. J. Tavener and K. A. Cliffe},

title = {Two-fluid Marangoni-Bénard convection with a deformable interface},

year = {}

}

### OpenURL

### Abstract

Two immiscible fluid layers that are subjected to a temperature gradi-ent perpendicular to their interface, exhibit a range of behaviors that is considerably richer than for the single-fluid case. We describe a numerical technique for calculating thermally-driven flows in two fluid layers which uses a simple technique based on a Landau transformation to map the phys-ical domain into a reference domain, enabling the unknown location of the deformable interface to be determined. The coupled system of nonlinear partial differential equations, comprising mapping, continuity, momentum and energy equations and the appropriate boundary conditions, is solved using the finite-element method in two-dimensional domains. Numerical bifurcation techniques are used to investigate the multiplicity of the so-lution set. The case of heating from above is considered in some detail and the results of finite-element computations are compared with linear stability calculations performed on unbounded domains. The principal ad-vantages of the finite-element approach are the ability to determine the effect of non-90 degree contact angles (when the conducting solution no longer exists and traditional linear stability approaches fail), the ability to determine the role of finite aspect ratio domains and the relative volume fractions of the two fluids, and the capability of calculating the nonlinear development of flows beyond the critical temperature gradient.

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Citation Context ... with thermocapillary forces dominating in sufficiently thin layers. Pioneering work on the relative importance of thermocapillarity and buoyancy for single fluids was performed by Block [1], Pearson =-=[2]-=-, and Nield [3]. The range of possible behaviors when two immiscible fluid layers are confined between parallel plates and subjected to a temperature gradient perpendicular to the plates is considerab... |

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Citation Context ...tudes of the properties of the two fluids. The contact angle is not required to be close to 90 degrees, nor is the free surface curvature required to be small.TWO-FLUID BÉNARD 279 Cliffe and Tavener =-=[19]-=- used the finite-element method, coupled with numerical bifurcation techniques, to determine the quantitative effect of free-surface deformations on the bifurcation structure in two-dimensional single... |

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Citation Context ...ccur. The Galileo number was set to be 5 × 104 and the other parameters were as before. In particular, the volume ratio was fixed and equal to one. As has been reemphasized recently by Regnier et al. =-=[29]-=- and Velarde et al. [4], it is important to ensure that the Boussinesq approximation is appropriate. In order for the Boussinesq approximation to be valid for two-fluid convection, the product�i�T ∗ i... |

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Citation Context ...red layers of almost equal densities and predicted the onset of oscillatory instabilities but were unable to detect them experimentally. Their analysis was extended considerably by Renardy and Joseph =-=[10]-=- and Renardy and Renardy [11, 12]. Recently Juel et al. [13] observed experimentally both the onset of convection on heating from above and oscillatory motion at onset in two-fluid systems. Surprising... |

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Citation Context ...nsities and predicted the onset of oscillatory instabilities but were unable to detect them experimentally. Their analysis was extended considerably by Renardy and Joseph [10] and Renardy and Renardy =-=[11, 12]-=-. Recently Juel et al. [13] observed experimentally both the onset of convection on heating from above and oscillatory motion at onset in two-fluid systems. Surprisingly few two- or three-dimensional ... |

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Citation Context ...ture. Géoris et al. [14, 15] performed finite-difference computations with a nondeforming interface in a three-layer system and compared their results with experiment. Recently, Johnson and Narayanan =-=[16, 17]-=- recognized the need for stability calculations to be performed in bounded domains, due to the number of experiments performed in small-aspect ratio devices (e.g., [5, 18]) and because of the necessar... |

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Citation Context ...ture. Géoris et al. [14, 15] performed finite-difference computations with a nondeforming interface in a three-layer system and compared their results with experiment. Recently, Johnson and Narayanan =-=[16, 17]-=- recognized the need for stability calculations to be performed in bounded domains, due to the number of experiments performed in small-aspect ratio devices (e.g., [5, 18]) and because of the necessar... |

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Citation Context ...ntly, Johnson and Narayanan [16, 17] recognized the need for stability calculations to be performed in bounded domains, due to the number of experiments performed in small-aspect ratio devices (e.g., =-=[5, 18]-=-) and because of the necessarily finite domains in industrial applications. Much of the recent interest in two-fluid systems has been motivated by the liquid encapsulated vertical Bridgeman (LEVB) pro... |

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Citation Context ...he absence of gravity is at first surprising, but the role of the upper fluid when heating from above has been recognized at least back to the work of Sternling and Scriven [27] and Vidal and Acrivos =-=[28]-=-.TWO-FLUID BÉNARD 291 FIG. 4. Streamlines and isotherms when heating from above for the connected two-cell flow at Ma = 2100 and contact angle 89.89 degrees (point A, Fig. 3). Solid lines indicate po... |

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Citation Context ... date is provided by Velarde et al. [4] and by Andereck et al. [5]. The preponderance of stability analyses have been performed assuming unbounded or periodic domains. For example, Zeren and Reynolds =-=[6]-=- predicted the onset of convection on heating from above but were unable to find it experimentally. Imaishi and Fujinawa [7, 8] extended the stability analysis of Nield [3] (including both gravitation... |

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Citation Context ...med assuming unbounded or periodic domains. For example, Zeren and Reynolds [6] predicted the onset of convection on heating from above but were unable to find it experimentally. Imaishi and Fujinawa =-=[7, 8]-=- extended the stability analysis of Nield [3] (including both gravitational and surface tension effects) to two-fluid systems but assumed an undeformable interface and did not consider time-dependent ... |