Abstract

We develop a general mathematical framework for studying electro-osmotic flow (EOF) within multi-layer microfluidic structures. Considering the impingement of an arbitrary non-uniform incident field on the multi-layer structure, we first establish the representation formula of the solution of the coupled system using the layer potential techniques, and then the concept of contracted hydrodynamic Generalized Polarization Tensors (GPTs) is defined by multi-polar expansion. Based on Fourier series and spectral theory, we derive the closed-form solution of the coupled system for the control region with the cross-sectional shape being multi-layer concentric disks or multi-layer confocal ellipses. This enables us to establish a handy algebraic framework for studying hydrodynamic cloaking and shielding associated with multi-layer structures, which forms the fundamental basis for many microfluidic applications.