  This work develops an efficient and accurate optimization algorithm to study
the optimal mixing problem driven by boundary control of unsteady Stokes flows,
based on the theoretical foundation laid by Hu and Wu in a series of work. The
scalar being mixed is purely advected by the flow and the control is a force
exerted tangentially on the domain boundary through the Navier slip conditions.
The control design has potential applications in many industrial processes such
as rotating wall driven mixing, micromixers with acoustic waves, and artificial
cilia mixing.
  The numerical algorithms have high complexity, high accuracy demand, and high
computing expense, due to the multiscale nature of the mixing problem and the
optimization requirements. A crucial problem is the computation of the
G$\^a$teaux derivative of the cost functional. To this end, a hybrid approach
based on variational formula and finite difference is built with high accuracy
and efficiency to treat various types of control input functions. We have
experimented with various optimization schemes including the steepest descent
algorithm, the conjugate gradient method and two line search options
(backtracking and exact line search). We are able to identify and implement the
best combinations.
  The numerical simulations show that the mixing efficacy is limited when only
one single type of control is applied, but can be enhanced when more diverse
control types and more time segmentation are utilized. The mix-norm in the
optimal mixings decays exponentially. The numerical study in this work
demonstrates that boundary control alone could be an effective strategy for
mixing in incompressible flows.
