Optimal control of mixing efficiency in a viscous fluid: A Navier–Stokes and advection–diffusion approach

We study an optimal control problem motivated by the physical process of mixing a soluble substance into a viscous fluid, such as dissolving sugar in coffee by stirring with a spoon. The fluid motion is modeled by the incompressible Navier–Stokes equations, while the transport of the dissolved substance is described by an advection–diffusion equation. The control variable represents the imposed stirring action, modeled here as a distributed body force acting throughout the fluid domain. The objective is to achieve fast and efficient homogenization while minimizing the energetic cost of stirring. We first establish the well-posedness of the coupled system under suitable assumptions on the data and the control. Using the direct method of the calculus of variations, we prove the existence of at least one optimal control. Subsequently, we derive first-order necessary optimality conditions through an extended Lagrangian formalism, which yields a coupled system involving the primal state (velocity, pressure, and concentration) and the adjoint states. The gradient of the cost function is explicitly computed, providing the foundation for gradient-based optimization algorithms. Finally, we implement a gradient-based numerical method and validate it on a two-dimensional test case: for a localized concentration blob stirred by a physically bounded distributed force, the optimal control reduces the terminal spatial variance of the concentration by approximately 99% relative to the uncontrolled (purely diffusive) case, and by approximately 96% relative to a constant forcing of comparable energy. The influence of the regularization parameter on the temporal structure of the optimal forcing is also discussed. The proposed framework offers a rigorous and quantitatively validated approach for optimizing mixing processes in various industrial and natural applications.