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The method of transformation optics has been a powerful tool to manipulatephysical fields if governing equations are formally invariant under coordinatetransformations. However, regulation of hydrodynamics is still far fromsatisfactory due to the lack of rigorous arguments on the validation oftransformation theory for various categories of fluids. In this paper, wesystematically investigate the applicability of transformation optics to fluidmechanics. We find that the Stokes equation and the Navier-Stokes equations,respectively describing the Stokes flow and general flow, will alter theirforms under curvilinear transformations. On the contrary, the Hele-Shaw flowcharacterized with shallow geometries rigidly retain the form of its governingequation under arbitrary transformations. Based on the derived transformationrules, we propose the design of multilayered structures with spatially varyingcell depth, instead of engineering the rank-2 shear viscosity tensor, torealize the required anisotropy of transformation Hele-Shaw hydrodynamicmetamaterials. The theoretical certify and fabrication method revealed in thiswork may pave an avenue for precisely controlling flow distribution with theconcept of artificial structure design.

Transformation hydrodynamic metamaterials: Rigorous arguments on form invariance and structured design with spatial variance