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We study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

We demonstrate monolithic aluminum gallium arsenide (AlGaAs) optical anoantennas. Using a selective oxidation technique, we fabricate such epitaxial semiconductor nanoparticles on an aluminum oxide substrate. Second harmonic generation from an AlGaAs nanocylinder of height h=400 nm and varying radius pumped with femtosecond pulses delivered at 1554-nm wavelength has been measured, revealing a peak conversion efficiency exceeding 10-5 for nanocylinders with an otpimized geometry.

We prove unconditional long-time stability for a particular velocity-vorticity discretization of the 2D Navier-Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity-pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit-explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method's vorticity and velocity are both long-time stable in the $L^2$ and $H^1$ norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method.

Vibrating nano- and micromechanical resonators have been the subject of research aiming at ultrasensitive mass sensors for mass spectrometry, chemical analysis and biomedical diagnosis. Unfortunately, their merits diminish dramatically in liquids due to dissipative mechanisms like viscosity and acoustic losses. A push towards faster and lighter miniaturized nanodevices would enable improved performances, provided dissipation was controlled and novel techniques were available to efficiently drive and read-out their minute displacement. Here we report on a nano-optomechanical approach to this problem using miniature semiconductor disks. These devices combine mechanical motion at high frequency above the GHz, ultra-low mass of a few picograms, and moderate dissipation in liquids. We show that high-sensitivity optical measurements allow to direct resolve their thermally driven Brownian vibrations, even in the most dissipative liquids. Thanks to this novel technique, we experimentally, numerically and analytically investigate the interaction of these resonators with arbitrary liquids. Nano-optomechanical disks emerge as probes of rheological information of unprecedented sensitivity and speed, opening applications in sensing and fundamental science.

GaAs disk resonators (typical disk size 5 \mum * 200 nm in our work) are good candidates for boosting optomechanical coupling thanks to their ability to confine both optical and mechanical energy in a sub-micron interaction volume. We present results of optomechanical characterization of GaAs disks by near-field optical coupling from a tapered silica nano-waveguide. Whispering gallery modes with optical Q factor up to a few 10^5 are observed. Critical coupling, optical resonance doublet splitting and mode identification are discussed. We eventually show an optomechanical phenomenon of optical force attraction of the silica taper to the disk. This phenomenon shows that mechanical and optical degrees of freedom naturally couple at the micro-nanoscale.

We report on miniature GaAs disk optomechanical resonators vibrating in air in the radiofrequency range. The flexural modes of the disks are studied by scanning electron microscopy and optical interferometry, and correctly modeled with the elasticity theory for annular plates. The mechanical damping is systematically measured, and confronted with original analytical models for air damping. Formulas are derived that correctly reproduce both the mechanical modes and the damping behavior, and can serve as design tools for optomechanical applications in fluidic environment.