### Abstract

Original language | English |
---|---|

Title of host publication | Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics |

Editors | Alexander Gelfgat |

Place of Publication | Cham. |

Publisher | Springer |

Chapter | 12 |

Pages | 419-458 |

Number of pages | 40 |

Volume | 50 |

ISBN (Print) | 9783319914930 |

DOIs | |

Publication status | E-pub ahead of print - 7 Jul 2018 |

### Publication series

Name | Springer Mathematical Series |
---|

### Fingerprint

### Keywords

- two-dimensional flow
- thermogravitational flows
- oscillatory buoyancy
- shallow cavities

### Cite this

*Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics*(Vol. 50, pp. 419-458). (Springer Mathematical Series). Cham.: Springer. https://doi.org/10.1007/978-3-319-91494-7_12

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*Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics.*vol. 50, Springer Mathematical Series, Springer, Cham., pp. 419-458. https://doi.org/10.1007/978-3-319-91494-7_12

**A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section.** / Lappa, Marcello.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section

AU - Lappa, Marcello

N1 - Part of the Computational Methods in Applied Sciences book series

PY - 2018/7/7

Y1 - 2018/7/7

N2 - It is often assumed that two-dimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing two-dimensional and three-dimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to 1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and 2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backward-facing or forward-facing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned categories of materials, namely molten silicon (Pr<1) and silicone oil (Pr>1). The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of two-dimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.

AB - It is often assumed that two-dimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing two-dimensional and three-dimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to 1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and 2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backward-facing or forward-facing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned categories of materials, namely molten silicon (Pr<1) and silicone oil (Pr>1). The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of two-dimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.

KW - two-dimensional flow

KW - thermogravitational flows

KW - oscillatory buoyancy

KW - shallow cavities

UR - http://www.springer.com/gb/

U2 - 10.1007/978-3-319-91494-7_12

DO - 10.1007/978-3-319-91494-7_12

M3 - Chapter

SN - 9783319914930

VL - 50

T3 - Springer Mathematical Series

SP - 419

EP - 458

BT - Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics

A2 - Gelfgat, Alexander

PB - Springer

CY - Cham.

ER -