Analysis and Computations of Optimal Control Problems for Boussinesq Equations

Abstract

The main purpose of engineering applications for fluid with natural and mixed convection is to control or enhance the flow motion and the heat transfer. In this paper, we use mathematical tools based on optimal control theory to show the possibility of systematically controlling natural and mixed convection flows. We consider different control mechanisms such as distributed, Dirichlet, and Neumann boundary controls. We introduce mathematical tools such as functional spaces and their norms together with bilinear and trilinear forms that are used to express the weak formulation of the partial differential equations. For each of the three different control mechanisms, we aim to study the optimal control problem from a mathematical and numerical point of view. To do so, we present the weak form of the boundary value problem in order to assure the existence of solutions. We state the optimization problem using the method of Lagrange multipliers. In this paper, we show and compare the numerical results obtained by considering these different control mechanisms with different objectives.

Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation

“…Suppose that when ũ is equal to ũ1 and ũ2 ∈ V in the Equation ( 10), the Equations ( 11) and ( 13) have, respectively, solutions (u (11), we obtain:…”

“…Up to now, there have been few studies on the steady Boussinesq equation (Problem 1), and some numerical methods have been provided (see [1][2][3][4][5][6][7][8][9][10][11]). However, in these numerical methods, the analytical solution for the steady Boussinesq equation (Problem 1) is directly assumed to exist.…”

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

“…The analysis of the results obtained in [3,4] made it possible to identify interesting regularities related to the interaction of hydrodynamic and thermal fields in binary and/or heat-conducting media and, in particular, to establish the most effective mechanisms for controlling thermohydrodynamic processes in viscous liquids. The close problems of boundary or distributed control for the HT equations in the Boussinesq approximation have also been investigated in the works [5][6][7][8][9][10].…”

Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

“…Their results have been extended to a flow model with nonlocal boundary conditions, where the fluid slips along an impermeable solid boundary of the flow domain [28], as well as to an optimal flow control problem [29] and a model for a rigid viscoplastic media of the Bingham kind with threshold slippage [30]. It also should be mentioned at this point that there exists extensive literature devoted to studying optimization and control problems for equations governing heat and mass transfer in a fluid with constant viscosity [31][32][33][34][35][36][37][38]. Finally, we mention the works [39][40][41], in which the unique solvability of the evolutionary Boussinesq system with constant viscosity and energy dissipation is proved under suitable smallness assumptions on data.…”