Stokes' second flow problem in a high-frequency limit: application to nanomechanical resonators

Abstract

Solving the Boltzmann–BGK equation, we investigate a flow generated by an infinite plate oscillating with frequency ω. The geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation 0 ≤ ωτ ≤ ∞, where τ is a properly defined relaxation time. A transition from viscoelastic behaviour of a Newtonian fluid (ωτ → 0) to purely elastic dynamics in the limit ωτ → ∞ is discovered. The relation of the derived solutions to nanofluidics is demonstrated on a solvable example of a ‘plane oscillator’. The results from the derived formulae compare well with experimental data on various nanoresonators operating in a wide range of both frequency and pressure variation.