Existence of boundary flex control for the systems governed by Boussinesq equation with the press boundary condition and mixed boundary condition

TL;DR

This paper investigates boundary flex control for a non-stationary Boussinesq system with complex boundary conditions, proving the existence of weak solutions and optimal controls in a viscous incompressible fluid model.

Contribution

It establishes the existence of weak solutions and optimal controls for a Boussinesq system with non-standard boundary conditions involving temperature-dependent viscosity and heat conductivity.

Findings

Existence of weak solutions for the state equation.

Existence of optimal boundary control.

Handling of mixed and dynamical pressure boundary conditions.

Abstract

In this paper, the boundary flex control problem of non stationary equation governing the coupled mass and heat flow of a viscous incompressible fluid in a generalized Boussinesq approximation by assuming that viscosity and heat conductivity are dependent on temperature has been studied. The boundary condition for velocity of fluid is non -standard boundary condition: specifically the case where dynamical pressure is given on some part of the boundary and the boundary condition for temperature of fluid is mixed boundary condition has been considered. First, we have proved the existence of existence of the weak solution for state equation.Then the optimal condition has been proved the existence of optimal control.