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Multiscale metrics such as negative Sobolev norms are effective forquantifying the degree of mixedness of a passive scalar field advected by anincompressible flow in the absence of diffusion. In this paper we introduce amix norm that is motivated by Sobolev norm $H^{-1}$ for a general domain with ano-flux boundary. We then derive an explicit expression for the optimal flowthat maximizes the instantaneous decay rate of the mix norm under fixed energyand enstrophy constraints. Numerical simulations indicate that the mix normdecays exponentially or faster for various initial conditions and geometriesand the rate is closely related to the smallest non-zero eigenvalue of theLaplace operator. These results generalize previous findings restricted for aperiodic domain for its analytical and numerical simplicity. Additionally, weobserve that periodic boundaries tend to induce a faster decay in mix normcompared to no-flux conditions under the fixed energy constraint, while thecomparison is reversed for the fixed enstrophy constraint. In the special caseof even initial distributions, two types of boundary conditions yield the sameoptimal flow and mix norm decay.

Optimal Stirring Strategies for Passive Scalars in a Domain with a General Shape and No-Flux Boundary Condition