An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions

Skalák, Zdenĕk; Kučera, Petr

doi:10.1023/a:1022224328555

Cited by 21 publications

(15 citation statements)

References 14 publications

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“…[9,21,54,56,67]). When one deals with the mixed boundary conditions of velocity and the component of strain ε(u)n, for variational formulation of the problem the bilinear form…”

Section: Introductionmentioning

confidence: 99%

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

Kim

Cao

2015

Nonlinear Analysis: Theory, Methods & Applications

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“…[9,21,54,56,67]). When one deals with the mixed boundary conditions of velocity and the component of strain ε(u)n, for variational formulation of the problem the bilinear form…”

Section: Introductionmentioning

confidence: 99%

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

Kim

Cao

2015

Nonlinear Analysis: Theory, Methods & Applications

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“…Most of these theoretical studies have been directed to the case in which the driving force is a volume force (as implicit in the problem statement per se). For the case of buoyancy flow, as an example, the interested reader may consider [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79]. Surprisingly, however, no results specific for the case of Marangoni flow have been obtained or presented so far.…”

Section: Discussionmentioning

confidence: 99%

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

Lappa

2017

International Journal of Heat and Mass Transfer

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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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“…In [14,15], the problem on exterior domains with the homogeneous Dirichlet boundary condition for velocity and nonhomogeneous Dirichlet boundary condition for temperature was studied. In [16], problem (1) was studied under the mixture of the nonhomogeneous Dirichlet boundary condition and the stress boundary condition for fluid and the mixture of nonhomogeneous Dirichlet, Neumann, and Robin boundary conditions for temperature. ey proved the existence of a unique local-in-time solution under a compatibility condition at the initial instance (see (27) and (31) of [16]).…”

Section: Introductionmentioning

confidence: 99%

The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

Kim

2020

International Journal of Differential Equations

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In this paper, we are concerned with the nonsteady Boussinesq system under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, static (or total) pressure, rotation, and stress (or total stress) together, and the boundary conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity, and shape of the boundary surface, we get variational formulation. The formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. For the case of boundary conditions including the static pressure and stress, we prove that if the data of the problem are small enough and compatibility conditions at the initial instance are satisfied, then there exists a unique solution on the given interval. For the case of boundary conditions including the total pressure and total stress, we prove the existence of a solution without restriction on the data and parameters of the problem.

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An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions

Cited by 21 publications

References 14 publications

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

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

The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

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