We have a channel (with a bump in it) through which a fluid is moving. Adjacent to the fluid is a solid region. The fluid model we use is the incompressible Navier-Stokes system, including the energy equation for the temperature in the fluid. The solid model we use is the heat equation for the temperature in the solid. The velocity vanishes at the upper and lower walls of the channel, the heat flux vanishes at the lower wall of the fluid and the left and right ends of the solid and at the fluid outflow, and the velocity is prescribed at the fluid inflow and outflow. Continuity conditions on the temperature and heat flux are imposed at the interface between the solid and fluid.
The fluid inflow temperature and/or either the heat flux or the temperature at the top of the solid can be used for control. For the particular computations described below, the fluid inflow temperature is used for control and the heat flux at the top of the solid is set to zero.
Many different objectives have been examined. Here we only report on the matching, in an L2 sense, the temperature over a portion of the top of the solid to a given temperature. This objective functional is penalized with the L2-norm of the control. Theoretically speaking, these norms are too weak for the Dirichlet control and Dirichlet objective, but in practice they seem to work well. The two terms in the functional are multiplied by weights so that their relative importance can be changed.
The control: the inflow boundary temperature
The objective: the temperature at the top of the solid