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Transformation hydrodynamic metamaterials: Rigorous arguments on form invariance and structural design with spatial
Physical Review. E
|June 17, 2023
Summary
Transformation optics can manipulate physical fields, but not generally with Navier-Stokes equations. Lubrication approximation, however, allows form invariance for Hele-Shaw flows, enabling hydrodynamic metamaterial design.
Area of Science:
- Fluid dynamics
- Metamaterials
- Transformation optics
Background:
- Transformation optics manipulates physical fields using form-invariant governing equations.
- Recent efforts explore applying transformation optics to hydrodynamic metamaterials governed by Navier-Stokes equations.
- Rigorous analysis of transformation optics applicability to Navier-Stokes equations is lacking.
Purpose of the Study:
- To establish a criterion for form invariance in transformation optics.
- To determine the applicability of transformation optics to Navier-Stokes and related fluid equations.
- To propose designs for hydrodynamic metamaterials based on form-invariant fluid models.
Main Methods:
- Derived a criterion for form invariance involving metric and affine connections.
- Analyzed the Navier-Stokes and Stokes equations for form invariance.
- Investigated creeping flows under lubrication approximation (Hele-Shaw model) for form invariance.
- Proposed multilayered structures with spatially varying cell depth.
Main Results:
- Navier-Stokes and Stokes equations lack form invariance due to affine connections in viscous terms.
- Creeping flows under lubrication approximation (Hele-Shaw model) exhibit form invariance.
- Spatially varying cell depth can mimic anisotropic shear viscosity for Hele-Shaw flows.
- Demonstrated the crucial role of lubrication approximation in achieving form invariance.
Conclusions:
- Transformation optics is not generally applicable to Navier-Stokes equations.
- Lubrication approximation is key for form invariance in hydrodynamic metamaterial design.
- Proposed designs offer a feasible approach for experimental fabrication of hydrodynamic metamaterials.
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