Adjoint-based interfacial control of viscous drops

We develop a continuous adjoint formulation for controlling the deformation of clean, neutrally buoyant droplets in Stokes flow. The focus is on surface tension-driven flows where the interface is deformed by the local fluid velocity. We apply results from shape optimization to rigorously derive the optimality conditions for a range of interfacial problems. In the cases of interest, we make use of boundary integral methods as a natural choice for the numerical discretization of the flow variables. In the static case, our methodology is tested on a tracking-type cost functional and corresponds to classic shape optimization problems. We show agreement with black-box finite difference-based gradients and accurate minimization of the cost functionals. Finally, we demonstrate the methodology on the control of unsteady droplet deformation through external forcing.

Automated summary

A continuous adjoint formulation was developed to control droplet deformation in Stokes flow driven by surface tension. Shape optimization results were applied to derive optimal conditions for various interfacial problems, with boundary integral methods used for numerical discretization. The methodology was tested on tracking-type cost functions in static cases, showing consistency with finite difference gradients and accurate minimization of cost functionals. Finally, the methodology was demonstrated on controlling unsteady droplet deformation through external forcing.