Optimal control of non-isothermal viscous fluid flow

For flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner region by controlling the heat flux along the corner boundary. The problem of matching a desired temperature along the outflow boundary is also considered. The energy equation is coupled with the mass, momentum, and constitutive equations through the assumption that viscosity depends on temperature. The latter three equations are a non-isothermal version of the three-field Stokes–Oldroyd model, formulated to have the same dependent variable set as the equations governing viscoelastic flow. The state and adjoint equations are solved using the finite element method. Previous efforts in optimal control of fluid flows assume a temperature-dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations. This assumption is not made in the current work.