Stokes ’ Second Flow Problem in a High Frequency

Solving the Boltzmann BGK equation, we investigate a flow generated by an infinite plate oscillating with frequency ω. 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 behavior of Newtonian fluid (ωτ → 0) to purely elastic dynamics in the limit ωτ → ∞ is discovered. The relation of the derived solutions to microfluidics (high-frequency micro-resonators) is demonstrated on an example of a ”plane oscillator”. Introduction. During last two centuries, Newtonian fluid approximation was remarkably successful in explaining a wide variety of natural phenomena ranging from flows in pipes, channels and boundary layers to the recently discovered processes in meteorology, aerodynamics , MHD and cosmology. With advent of powerful computers and development of effective numerical methods, Newtonian hydrodynamics remains at the foundation of various design tools widely used in mechanical and civil engineering. Since technology of the past mainly dealt with large ( macroscopic) systems varying on the length/ time-scales L and T , the Newtonian fluid approximation, typically defined by the smallness of Knudsen and Weisenberg numbers Kn = λ/L ≪ 1 and Wi = τ/T ≈ λ L u c = KnMa ≪ 1, was accurate enough. The length and time-scales λ and τ ≈ λ/c, are the mean -free path and relaxation time, respectively, . With recent rapid developments in nanotechnology and bioengineering, quantitative description of high-frequency oscillating microflows, i. e. flows where Newtonian approximation breaks down, became an important and urgent task from both basic