Towards the design of an optimal mixer

We define as an optimal mixer a mixing device able to deliver a uniformly optimal mixing performance over a wide range of operating and initial conditions. We consider the conceptual problem of designing an optimal mixer starting from a well-known reference mixer, the sine flow. We characterize the mixing performance of the reference mixer, and show that it performs poorly and erratically over a wide range of operating conditions and is quite sensitive to the geometry of the initial concentration field. We define as a target performance the best mixing performance the reference mixer is able to achieve. In steps we modify the design of the reference mixer. First, we optimize the time sequence of the switching protocols and show that the mixing performance of the time-optimized mixer, although substantially improved with respect to the reference mixer, is still far from achieving the target performance and being insensitive to the geometry of the initial concentration field. The analysis of the performance of the time-optimized mixer brings to light the deficiency of the actuating system used, which delivers always the same amount of shear at the same locations. We modify the actuating system by allowing the stirring velocity fields to shift along their coordinate axes. A new mixer, the space-optimized mixer, is created by equipping the reference mixer with the new actuating system and optimizing the shift of the stirring velocity field at each iteration. The space-optimized mixer is able to deliver the target performance over the upper two-thirds of the operating range. In the lower one-third, the performance of the space-optimized mixer deteriorates because of the use of a periodic protocol. A optimal mixer is finally obtained using the actuating system of the space-optimized mixer and coupling the time and shift optimizations. The resulting optimal mixer is able to deliver a uniform target performance, insensitive to the geometry of the initial conditions, over the entire operating range.