objectives of the present paper. First, we propose a general theory of scattering on resonance and capture into resonance in volume-preserving systems. In our study we rely heavily on the close analogy between volume-preserving and Hamiltonian systems, for which the theory of scattering on resonance and capture into resonance was developed in [@Neishtadt:1999]. Second, we would like to demonstrate that the resonance phenomena can be used in real settings to create flows with coexisting mixing and non-mixing (KAM) domains with the respective size and mixing rates easily and independently manageable. In the process, we want to emphasize that the presence of resonances is not a sufficient condition for chaotic advection and to illustrate some of the necessary ingredients for the resonance phenomena to create the chaotic advection.

The structure of the paper is as follows. In Sections 2 to 4 we discuss a general theory of passages through resonances in near-integrable volume-preserving 3-D systems. In Sect. 2 we introduce basic equations and study the unperturbed and the averaged systems. In Sect. 3 we consider the structure of a resonance and discuss the breakdown of the method of averaging. Section 4 contains a detailed description of the general properties of the 