Shape sensitivity analysis for a microchannel cooling system

We analyze the theoretical framework of a shape optimization problem for a microchannel cooling system. To this end, a cost functional based on the tracking of absorbed energy by the cooler as well as some desired flow on a subdomain of the cooling system is introduced. The flow and temperature of the coolant are modeled by a Stokes system coupled to a convection diffusion equation. We prove the well-posedness of this model on a domain transformed by the speed method. Further, we rigorously prove that the cost functional of our optimization problem is shape differentiable and calculate its shape derivative by means of a recent material derivative free adjoint approach.