An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows

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Abstract

The primary objective of this short work is the identification of alternate routes for the determination of exact and numerical solutions of the Navier-Stokes equations in the specific case of surface-tension driven thermal convection. We aim to introduce a theoretical approach in which the typical kinematic boundary conditions required at the free surface by this kind of flows can be replaced by a homogeneous Neumann condition. More precisely, the novelty of the present framework lies in the adoption of a class of ‘continuous’ distribution functions by which no discontinuities or abrupt variations are introduced in the model. The rationale for such a line of inquiry can be found 1) in the potential to overcome the typical bottlenecks created by the need to account for a shear stress balance at the free surface in the context of analytic models for viscoelastic and other non-Newtonian fluids and/or 2) in the express intention to support existing numerical (commercial or open-source) tools where the possibility to impose non-homogeneous Neumann boundary conditions is not an option. Both analytic solutions and (two-dimensional and three-dimensional) numerical “experiments” (concerned with the application of the proposed strategy to thermocapillary and Marangoni-Bénard flows) are presented. The implications of the proposed approach in terms of the well-known existence and uniqueness problem for the Navier-Stokes equations are also discussed to a certain extent, indicating possible directions of future research and extension.
LanguageEnglish
Pages407–418
Number of pages12
JournalInternational Journal of Heat and Mass Transfer
Volume114
Early online date26 Jun 2017
DOIs
Publication statusPublished - 30 Nov 2017

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Navier-Stokes equation
Navier Stokes equations
surface tension driven convection
derivation
Boundary conditions
boundary conditions
uniqueness
free convection
shear stress
Distribution functions
Surface tension
Shear stress
discontinuity
Kinematics
kinematics
distribution functions
routes
Fluids
fluids
Experiments

Keywords

  • Navier Stokes equations
  • thermal convection
  • surface-tension
  • kinematic boundary conditions
  • Marangoni flow
  • analytic solutions
  • two-dimensional and three-dimensional numerical results

Cite this

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title = "An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows",
abstract = "The primary objective of this short work is the identification of alternate routes for the determination of exact and numerical solutions of the Navier-Stokes equations in the specific case of surface-tension driven thermal convection. We aim to introduce a theoretical approach in which the typical kinematic boundary conditions required at the free surface by this kind of flows can be replaced by a homogeneous Neumann condition. More precisely, the novelty of the present framework lies in the adoption of a class of ‘continuous’ distribution functions by which no discontinuities or abrupt variations are introduced in the model. The rationale for such a line of inquiry can be found 1) in the potential to overcome the typical bottlenecks created by the need to account for a shear stress balance at the free surface in the context of analytic models for viscoelastic and other non-Newtonian fluids and/or 2) in the express intention to support existing numerical (commercial or open-source) tools where the possibility to impose non-homogeneous Neumann boundary conditions is not an option. Both analytic solutions and (two-dimensional and three-dimensional) numerical “experiments” (concerned with the application of the proposed strategy to thermocapillary and Marangoni-B{\'e}nard flows) are presented. The implications of the proposed approach in terms of the well-known existence and uniqueness problem for the Navier-Stokes equations are also discussed to a certain extent, indicating possible directions of future research and extension.",
keywords = "Navier Stokes equations, thermal convection, surface-tension, kinematic boundary conditions, Marangoni flow, analytic solutions, two-dimensional and three-dimensional numerical results",
author = "Marcello Lappa",
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language = "English",
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AB - The primary objective of this short work is the identification of alternate routes for the determination of exact and numerical solutions of the Navier-Stokes equations in the specific case of surface-tension driven thermal convection. We aim to introduce a theoretical approach in which the typical kinematic boundary conditions required at the free surface by this kind of flows can be replaced by a homogeneous Neumann condition. More precisely, the novelty of the present framework lies in the adoption of a class of ‘continuous’ distribution functions by which no discontinuities or abrupt variations are introduced in the model. The rationale for such a line of inquiry can be found 1) in the potential to overcome the typical bottlenecks created by the need to account for a shear stress balance at the free surface in the context of analytic models for viscoelastic and other non-Newtonian fluids and/or 2) in the express intention to support existing numerical (commercial or open-source) tools where the possibility to impose non-homogeneous Neumann boundary conditions is not an option. Both analytic solutions and (two-dimensional and three-dimensional) numerical “experiments” (concerned with the application of the proposed strategy to thermocapillary and Marangoni-Bénard flows) are presented. The implications of the proposed approach in terms of the well-known existence and uniqueness problem for the Navier-Stokes equations are also discussed to a certain extent, indicating possible directions of future research and extension.

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