We report on the decay of a passive scalar in chaotic mixing protocols where the wall of the vessel is rotated, or a net drift of fluid elements near the wall is induced at each period. As a result the fluid domain is divided into a central isolated chaotic region and a peripheral regular region. Scalar patterns obtained in experiments and simulations converge to a strange eigenmode and follow an exponential decay. This contrasts with previous experiments [Gouillart, Phys. Rev. Lett. 99, 114501 (2007)] with a chaotic region spanning the whole domain, where fixed walls constrained mixing to follow a slower algebraic decay. Using a linear analysis of the flow close to the wall, as well as numerical simulations of Lagrangian trajectories, we study the influence of the rotation velocity of the wall on the size of the chaotic region, the approach to its bounding separatrix, and the decay rate of the scalar.