Controlling a Thermal Fluid: Theoretical and Computational Issues
Monday, March 28, 2016 - 2:25pm - 3:25pm
We first discuss the problem of designing a feedback law which locally stabilizes a two dimensional thermal fluid modeled by the Boussinesq equations. The investigation of stability for a fluid flow in the natural convection problem is important in the theory of hydrodynamical stability. The challenge of stabilization of the Boussinesq equations arises from the stabilization of the Navier-Stokes equations and its coupling with the convection-diffusion equation for temperature. In our current work, we are interested in stabilizing a possible unstable steady state solution to the Boussinesq equations on a bounded and connected domain. We show that a finite number of controls acting on a part of the boundary through Neumann/Robin boundary conditions is sufficient to stabilize the full nonlinear equations in the neighborhood of this steady state solution. Dirichlet boundary conditions are imposed on the rest of the boundary. Moreover, we prove that a stabilizing feedback control law can be obtained based on the partial estimation of the system state by solving an extended Kalman filter problem for the linearized Boussinesq equations. In particular, a reduced order model is derived to construct a finite dimensional estimator. Numerical results are provided to illustrate the idea. In the end, we discuss the problem of control design for the Boussinesq equations with zero diffusivity and its application to optimal mixing, mass and energy transport during processing.