Not yet reviewed by Pith; the record is open.

Hydrodynamic cloaking offers a promising approach for manipulating viscous flows by redirecting fluid around an obstacle without inducing external disturbances. By extending pseudo-conformal mappings into potential flow models, we introduce a new isobaric boundary condition that enables the construction of zero-index cloaks using isotropic and homogeneous media shaped into anisotropic geometries, such as elliptical shells. Compared to conventional cloaks, which suffer performance degradation under realistic viscous conditions, the zero-index design significantly reduces such losses by suppressing flow disturbances at the inner boundary. To overcome practical limitations in realizing ideal isobaric conditions, we further propose a hybrid cloak that integrates a raised fluid domain with an auxiliary flow channel above the obstacle. This architecture removes the need for viscosity tuning and, under anisotropic geometries, surpasses both conventional and zero-index cloaks in omnidirectional performance. The design is validated through simulations and experiments. Our findings offer a generalizable strategy for controlling viscous flows and open new directions for microfluidic applications including drug delivery, particle steering, and cell sorting.