## 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.

### Citations

36 |
On convection cells induced by surface tension
- Pearson
- 1958
(Show Context)
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... |

20 |
Eigenvalues of the discretized Navier-Stokes equations with application to the detection of Hopf bifurcations
- Cliffe, Garratt, et al.
- 1993
(Show Context)
Citation Context ...n efficient numerical technique for finding these “most dangerous eigenvalues” when a mixed finite-element method is used to discretize the Navier–Stokes equations has been developed by Cliffe et al. =-=[20]-=- and has been applied, for example, to determine the linear stability of flow past a sphere in a pipe [21]. This technique makes extensive use of the special block structure of the generalized eigenva... |

18 |
The computation of symmetry-breaking bifurcation points
- Werner, Spence
- 1984
(Show Context)
Citation Context ... − û ∂h ∂� + � ˆv = 0, (14) where h, u, and v are components of the solution, ˆh, û, and ˆv are components of the perturbation, and � is the aspect ratio. When using the Werner–Spence extended system =-=[24]-=- to locate symmetry-breaking points, Eq. (14) is used to determine the h-component of the null eigenvector ˆh at each node along the free surface. For all symmetric solutions, regardless of the value ... |

16 |
Surface tension and buoyancy effects in cellular convection
- Nield
- 1964
(Show Context)
Citation Context ...illary 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 considerably richer. For ... |

12 |
Surface tension as the cause of Benard cells and surface deformation in a liquid lm
- Block
- 1956
(Show Context)
Citation Context ... fluid layer, 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 ... |

11 |
The calculation of Hopf points by a direct method
- Griewank, Reddien
- 1983
(Show Context)
Citation Context ...two different fluids are coupled via a deformable interface are currently under investigation. While it would be possible to construct an extended system to locate Hopf bifurcation points (see, e.g., =-=[22]-=-), without a good estimate for the critical parameter values and eigenvector, discovery of a Hopf bifurcation point in this manner would be merely serendipitous. We concentrate instead on regions of p... |

9 |
Eigenvalue problems with the symmetry of a group and bifurcations
- Werner
- 1990
(Show Context)
Citation Context ...even-cell flows is associated with the breaking of a subspace in which the velocity field is zero. (For a discussion of subspace breaking and methods for computing subspace-breaking bifurcations, see =-=[25, 26]-=-.) An extended system similar to the Werner–Spence system is constructed, and by appropriate application of boundary conditions, the conducting solution is required to lie within a subspace in which t... |

6 |
Surface tension-driven convection patterns in two liquid layers
- Juel, Burgess, et al.
- 2000
(Show Context)
Citation Context ... 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 computations have been repo... |

6 | Marangoni-Bénard convection with a deformable free surface
- Cliffe, Tavener
- 1998
(Show Context)
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... |

5 |
Interfacial turbulence: hydrodynamic instability and the Marangoni e ect. A.I.Ch.E
- Sternling, Scriven
- 1959
(Show Context)
Citation Context ...the density ratio even in the 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. ... |

4 | Stability of the O(2)-symmetric flow past a sphere in a pipe
- Tavener
- 1994
(Show Context)
Citation Context ... method is used to discretize the Navier–Stokes equations has been developed by Cliffe et al. [20] and has been applied, for example, to determine the linear stability of flow past a sphere in a pipe =-=[21]-=-. This technique makes extensive use of the special block structure of the generalized eigenvalue problem. Efficient ways to utilize the more complicated block structure that arises when two different... |

3 |
Oscillatory instability in a Bénard problem of two fluids
- Renardy, Joseph
- 1985
(Show Context)
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... |

3 | Implementation of extended systems using symbolic algebra
- Cliffe, Tavener
(Show Context)
Citation Context ... and parameters are required. All subroutines to evaluate the weak form and its derivatives were written using a preprocessor, ENTCODE, based on Mathematica. A discussion of this technique appears in =-=[23]-=-. The nonlinear system of equations resulting from the finite-element discretization may be written as f (a, b) = 0, f :R N × R p ↦→ R N . (12) Let S be a finite-dimensional representation of the refl... |

3 |
Regular systems for bifurcation points with underlying symmetries
- Werner
- 1984
(Show Context)
Citation Context ...even-cell flows is associated with the breaking of a subspace in which the velocity field is zero. (For a discussion of subspace breaking and methods for computing subspace-breaking bifurcations, see =-=[25, 26]-=-.) An extended system similar to the Werner–Spence system is constructed, and by appropriate application of boundary conditions, the conducting solution is required to lie within a subspace in which t... |

2 |
Onset of Oscillatory Interfacial Instability and Wave Motions in Bénard Layers
- Velarde, Nepomnyaschy, et al.
- 2000
(Show Context)
Citation Context ...lls). Unlike the single fluid case, time-dependent flows may develop directly from the conducting solution. A comprehensive review of the considerable literature to 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 o... |

2 |
Perturbation analysis of steady and oscillatory onset in a Bénard problem with two similar liquids
- Renardy, Renardy
- 1985
(Show Context)
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 ... |

2 |
Investigation of thermocapillary convection in a three-liquid-layer system
- Géoris, Lebon, et al.
- 1999
(Show Context)
Citation Context ...t of convection on heating from above and oscillatory motion at onset in two-fluid systems. Surprisingly few two- or three-dimensional computations have been reported in the literature. 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... |

2 |
Marangoni Convection in Multiple Bounded Fluid Layers and its Application to
- Johnson, Narayanan
- 1998
(Show Context)
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... |

2 |
Nature of the neutral state in surface tension driven convection. Phys. Fluids 9
- Vidal, Acrivos
- 1966
(Show Context)
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... |

2 |
Thermocapillary convection in a multilayer, Phys. Fluids A
- Géoris, Simanovskii, et al.
- 1993
(Show Context)
Citation Context ...t divergence constraints are required. A volume constraint is therefore not applied to the eigenvector. The linearized kinematic condition along the interface is −u ∂ ˆh ∂� + �v − û ∂h ∂� + � ˆv = 0, =-=(14)-=- where h, u, and v are components of the solution, ˆh, û, and ˆv are components of the perturbation, and � is the aspect ratio. When using the Werner–Spence extended system [24] to locate symmetry-bre... |

2 |
Linear and nonlinear Rayleigh-Bénard-Marangoni instability with surface deformations, Phys
- Regnier, Dauby, et al.
- 2000
(Show Context)
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... |

1 |
Renardy, Instabilities in two layer Rayleigh-Bénard convection: Overview and outlook
- Andereck, Colovas, et al.
- 1998
(Show Context)
Citation Context ...id case, time-dependent flows may develop directly from the conducting solution. A comprehensive review of the considerable literature to 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 we... |

1 | in two-fluid horizontal layers - Zeren, Reynolds - 1972 |

1 |
Thermal instability in two-fluid layers
- Imaishi, Fujinawa
- 1974
(Show Context)
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 ... |

1 |
A theoretical and experimental study of double-layer convection
- Rasenat, Busse, et al.
- 1989
(Show Context)
Citation Context ...sis 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 disturbances. Rasanat et al. =-=[9]-=- considered 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... |

1 |
Bifurcating solutions at the onset of convection in the Bénard problem of two fluids
- Renardy, Renardy
- 1988
(Show Context)
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 ... |

1 | Thermocapillary convection in a multilayer,” Phys - Wertgeim - 1993 |

1 |
A tutorial on the Rayleigh-Marangoni-Bénard problem with multiple layers and side-wall effects, Chaos 9
- Johnson, Narayanan
- 1999
(Show Context)
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... |

1 |
What is responsible for thermal coupling in layered convection
- Nataf, Moreno, et al.
- 1988
(Show Context)
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... |

1 |
Thermal instabilities in two-fluid horizontal layers
- Zeren, Reynolds
- 1972
(Show Context)
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... |

1 |
Theoretical study of the stability of two-fluid layers
- Imaishi, Fujinawa
- 1974
(Show Context)
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 ... |