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.
Original languageEnglish
Pages (from-to)407–418
Number of pages12
JournalInternational Journal of Heat and Mass Transfer
Volume114
Early online date26 Jun 2017
DOIs
Publication statusPublished - 30 Nov 2017

Keywords

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

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