Steady Streaming flow induced by two oscillating parallel infinite plates

Singular perturbation method was used to investigate the steady streaming flow pattern of a viscous fluid between two infinite parallel plates. The upper and the lower plates were allowed to oscillate with two different frequencies (ω₁ and ω₂),phase shift (φ₁andφ₂ )between the vertical and the lateral velocities of the two plates, and amplitudes of oscillations ( ε₁a and ε₂a) . Where a is the width of the microchannel. These assumptions can create two completely different frequency parameters (|M₁|² and |M₂|² ) for the upper and the lower plates. The frequency parameter |M|² =ωa²/ν is the ratio channel width to the viscous length and it is assumed to be large, |M|²>>1 . When these two distinct steady streaming flows induced by the upper and the lower oscillating plates, interact with each other they create different steady streaming flow pattern. The resulting steady streaming can be used to optimize mixing or pumping of the sandwiched fluid in the microchannel.