In a strong electric field, polarizable fluids have a structural transition at a critical shear stress. When there is no flow, polarizable fluids form a body-centered tetragonal lattice which consists of closely packed chains. When the shear stress is increased from zero up to a critical value, the flow (in x-direction) has a flowing-chain (FC) structure, consisting of tilted or broken chains along the field (z-direction). At the critical shear stress, the FC structure gives way to a flowing-hexagonal-layered (FHL) structure, consisting of several two-dimensional layers which are parallel to the x-z plane. Within one layer, particles form strings in the flow direction. The effective viscosity drops dramatically at the structural change. As the shear stress reduces, the FHL structure persists even under a stress-free state if the thermal fluctuation is very weak.