The question of what fluid flow maximizes mixing rate, slows it down, or even steers a quantity of interest toward a desired target distribution draws great attention from a broad range of scientists and engineers in the area of complex dynamical systems. Our methodology is to place these problems within a flexible computational framework, and to develop a solution strategy based on optimal control tools. Theoretically, we investigate the well-posedness, regularity of the solution for various control designs. Computationally, we propose a novel model order reduction method to reduce computational costs. Numerical analysis and experiments demonstrate the effectiveness of our reduced order models.