Adjoint-based interfacial control of axisymmetric viscous drops

We develop a continuous adjoint theory for the control of the deformation of a clean, neutrally buoyant droplet in Stokes flow. The problem reduces to the dynamics of the free drop surface, but this introduces significant complexity in the optimization process and, in particular, the formulation of necessary optimality conditions and efficient numerical handling. We make use well-known results from the field of topology optimization to derive rigorous adjoint equations and optimality conditions for this class of problems. Boundary integral methods are used to provide efficient and high-order approximations for all the quantities of interest. Finally, our methodology is then tested on axisymmetric droplets controlled by the non-dimensional Capillary number and several tracking-type
cost functionals.