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

TL;DR

This paper investigates the applicability of transformation optics principles to hydrodynamics, revealing that certain flow equations are invariant under transformations, enabling the design of structured hydrodynamic metamaterials with spatially varying geometries.

Contribution

It establishes the conditions under which transformation theory applies to fluid flows and proposes a multilayered structure design for hydrodynamic metamaterials based on Hele-Shaw flow.

Findings

Hele-Shaw flow retains form under transformations

Stokes and Navier-Stokes equations do not remain invariant

Design of multilayered structures with spatially varying depth

Abstract

The method of transformation optics has been a powerful tool to manipulate physical fields if governing equations are formally invariant under coordinate transformations. However, regulation of hydrodynamics is still far from satisfactory due to the lack of rigorous arguments on the validation of transformation theory for various categories of fluids. In this paper, we systematically investigate the applicability of transformation optics to fluid mechanics. We find that the Stokes equation and the Navier-Stokes equations, respectively describing the Stokes flow and general flow, will alter their forms under curvilinear transformations. On the contrary, the Hele-Shaw flow characterized with shallow geometries rigidly retain the form of its governing equation under arbitrary transformations. Based on the derived transformation rules, we propose the design of multilayered structures with spatially varying cell depth, instead of engineering the rank-2 shear viscosity tensor, to realize the required anisotropy of transformation Hele-Shaw hydrodynamic metamaterials. The theoretical certify and fabrication method revealed in this work may pave an avenue for precisely controlling flow distribution with the concept of artificial structure design.