Electro-Osmotic Flow within Multilayer Microfluidic Structures and an Algebraic Framework for Hydrodynamic Cloaking and Shielding

Multilayer structures are ubiquitous in constructing microfluidic devices for manipulating fluids to realize various frontier applications, including energy harvesting and invisibility cloaking. In this study, we develop a mathematical framework for analyzing electro-osmotic flow (EOF) within multilayer microfluidic structures. These structures are formed by Hele-Shaw configurations, with cross-sectional shapes that are concentric disks or confocal ellipses, and each layer is filled with a fluid material characterized by a zeta potential. The number of layers can vary, and the zeta potentials in each layer may differ. By dynamically adjusting the zeta potential, the multilayer EOF structure can effectively manipulate fluids without requiring metamaterials in microfluidic devices. Considering the impingement of an arbitrary nonuniform incident field on the multilayer 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 multipolar expansion. Utilizing Fourier series and spectral theory, we derive a closed-form solution for the coupled system in the control region with multilayer concentric disks or confocal ellipses. This framework provides a convenient algebraic approach for studying hydrodynamic cloaking and shielding in multilayer structures, laying the groundwork for various microfluidic applications. © 2024 Society for Industrial and Applied Mathematics.