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

Using kinetic equation in the relaxation approximation (RTA), we investigate a flow generated by an infinite plate oscillating with frequency $\omega$. Geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation $0\leq \omega \tau\leq \infty$, where $\tau$ is a properly defined relaxation time. A transition from viscoelastic behavior of Newtonian fluid ($\omega\tau\to 0$) to purely elastic dynamics in the limit $\omega\tau\to \infty$ is discovered. The relation of the derived solutions to microfluidics (high-frequency micro-resonators) is demonstrated on an example of a "plane oscillator .