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

Research output: Chapter in Book/Report/Conference proceedingChapter

2 Citations (Scopus)

Abstract

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.
LanguageEnglish
Title of host publicationComputational Modeling of Bifurcations and Instabilities in Fluid Mechanics
EditorsAlexander Gelfgat
Place of PublicationCham.
PublisherSpringer
Chapter12
Pages419-458
Number of pages40
Volume50
ISBN (Print)9783319914930
DOIs
Publication statusE-pub ahead of print - 7 Jul 2018

Publication series

NameSpringer Mathematical Series

Fingerprint

Buoyancy
Liquids
Containers
Molten materials
Fluids
Crystal growth
Topography
Salts
Semiconductor materials
Silicon
Oxides
Convection
Gases

Keywords

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

Cite this

Lappa, M. (2018). A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section. In A. Gelfgat (Ed.), 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
Lappa, Marcello. / A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section. Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics. editor / Alexander Gelfgat. Vol. 50 Cham. : Springer, 2018. pp. 419-458 (Springer Mathematical Series).
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abstract = "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.",
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Lappa, M 2018, A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section. in A Gelfgat (ed.), 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.

Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics. ed. / Alexander Gelfgat. Vol. 50 Cham. : Springer, 2018. p. 419-458 (Springer Mathematical Series).

Research output: Chapter in Book/Report/Conference proceedingChapter

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

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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.

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KW - two-dimensional flow

KW - thermogravitational flows

KW - oscillatory buoyancy

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Lappa M. A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section. In Gelfgat A, editor, Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics. Vol. 50. Cham.: Springer. 2018. p. 419-458. (Springer Mathematical Series). https://doi.org/10.1007/978-3-319-91494-7_12