Existence of a variational solution for the stationary Boussinesq equations with thermocapillary effect and nonhomogenous boundary conditions

2015 ◽  
Vol 27 (5-6) ◽  
pp. 909-921
Author(s):  
Bernadin Ahounou ◽  
Luc Paquet
Keyword(s):  
Boundary Conditions ◽  
Boussinesq Equations ◽  
Variational Solution ◽  
Thermocapillary Effect

Optimal boundary control for the stationary Boussinesq equations with variable density

2019 ◽  
Vol 22 (05) ◽  
pp. 1950031
Author(s):  
José Luiz Boldrini ◽  
Exequiel Mallea-Zepeda ◽  
Marko Antonio Rojas-Medar
Keyword(s):  
Boundary Conditions ◽  
Optimality Conditions ◽  
Boundary Control ◽  
Boussinesq Equations ◽  
Variable Density ◽  
Control Problems ◽  
Dynamical Equations ◽  
Optimal Boundary ◽  
Optimal Boundary Control ◽  
Boundary Control Problems

Certain classes of optimal boundary control problems for the Boussinesq equations with variable density are studied. Controls for the velocity vector and temperature are applied on parts of the boundary of the domain, while Dirichlet and Navier friction boundary conditions for the velocity and Dirichlet and Robin boundary conditions for the temperature are assumed on the remaining parts of the boundary. As a first step, we prove a result on the existence of weak solution of the dynamical equations; this is done by first expressing the fluid density in terms of the stream-function. Then, the boundary optimal control problems are analyzed, and the existence of optimal solutions are proved; their corresponding characterization in terms of the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

Energy stability of the buoyancy boundary layer

1971 ◽  
Vol 47 (2) ◽  
pp. 381-403 ◽  
Author(s):  
Joseph J. Dudis ◽  
Stephen H. Davis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Boussinesq Equations ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Lagrange Equations ◽  
The Mean ◽  
Energy Theory

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the buoyancy boundary layer is the unique steady solution of the Boussinesq equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler–Lagrange equations. Analytic lower bounds to RE are obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.

Partially dissipative 2D Boussinesq equations with Navier type boundary conditions

2018 ◽  
Vol 376-377 ◽  
pp. 39-48 ◽  
Author(s):  
Weiwei Hu ◽  
Yanzhen Wang ◽  
Jiahong Wu ◽  
Bei Xiao ◽  
Jia Yuan
Keyword(s):  
Boundary Conditions ◽  
Boussinesq Equations ◽  
2D Boussinesq Equations ◽  
Type Boundary

scholarly journals Stability of Optimal Controls for the Stationary Boussinesq Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Gennady Alekseev ◽  
Dmitry Tereshko
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boussinesq Equations ◽  
Mixed Boundary Conditions ◽  
State Equation ◽  
Dirichlet Boundary Conditions ◽  
Heat Conducting ◽  
Optimal Controls ◽  
Viscous Heat ◽  
Cost Functionals

The stationary Boussinesq equations describing the heat transfer in the viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking-type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.

scholarly journals An analytical verification test for numerically simulated convective flow above a thermally heterogeneous surface

2015 ◽  
Vol 8 (6) ◽  
pp. 1809-1819 ◽  
Author(s):  
A. Shapiro ◽  
E. Fedorovich ◽  
J. A. Gibbs
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Numerical Algorithms ◽  
Boussinesq Equations ◽  
Thermal Forcing ◽  
Pressure Poisson Equation ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Buoyancy Driven Flows ◽  
Boussinesq Fluid

Abstract. An analytical solution of the Boussinesq equations for the motion of a viscous stably stratified fluid driven by a surface thermal forcing with large horizontal gradients (step changes) is obtained. This analytical solution is one of the few available for wall-bounded buoyancy-driven flows. The solution can be used to verify that computer codes for Boussinesq fluid system simulations are free of errors in formulation of wall boundary conditions and to evaluate the relative performances of competing numerical algorithms. Because the solution pertains to flows driven by a surface thermal forcing, one of its main applications may be for testing the no-slip, impermeable wall boundary conditions for the pressure Poisson equation. Examples of such tests are presented.

scholarly journals An analytical verification test for numerically simulated convective flow above a thermally heterogeneous surface

2015 ◽  
Vol 8 (3) ◽  
pp. 2847-2873
Author(s):  
A. Shapiro ◽  
E. Fedorovich ◽  
J. A. Gibbs
Keyword(s):  
Boundary Conditions ◽  
Numerical Algorithms ◽  
Boussinesq Equations ◽  
Thermal Forcing ◽  
Pressure Poisson Equation ◽  
Wall Boundary ◽  
Wall Boundary Conditions ◽  
Computer Codes ◽  
Verification Test ◽  
Boussinesq Fluid

Abstract. An analytical solution of the Boussinesq equations for the motion of a viscous stably stratified fluid driven by a surface thermal forcing with large horizontal gradients (step changes) is obtained. The solution can be used to verify that computer codes for Boussinesq fluid system simulations are free of errors in formulation of wall boundary conditions, and to evaluate the relative performances of competing numerical algorithms. Because the solution pertains to flows driven by a surface thermal forcing, one of its main applications may be for testing the no-slip, impermeable wall boundary conditions for the pressure Poisson equation. Examples of such tests are presented.

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