Adjoint-based control of multi-droplet systems in Stokes flow

In this work, we present an approach to control multiple droplets by applying Lagrange

multiplier theory to two-phase Stokes flows with sharp interfaces. The focus is on the

deformation of clean, neutrally buoyant droplets with constant surface tension, where the

interface deforms with the local fluid velocity. We make use of shape optimization theory

to derive the linearized system and the corresponding optimality conditions necessary

for gradient descent methods. The forward and adjoint systems are represented through

boundary integral equations and discretized using the Nyström method and Quadrature

by Expansion. We couple the boundary integral method with a spectral representation

of the geometry (based on spherical harmonics) to obtain a highly accurate and stable

solver. We show that optimality conditions are correctly derived by comparison to black-

box finite difference approximations and minimization of select benchmark problems.

Finally, we apply the adjoint-based control to non-trivial multi-droplet systems and

evaluate the performance and robustness of the proposed methodology.