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Flow control is the manipulation of fluid flow into a desired behaviour with the purpose to either enhance mixing or suppress turbulence. However, because all flow systems are essentially distributed parameter systems, which are often described as partial differential equations, systematic analysis of the effectiveness and control design of the flow control is often difficult to perform. Oscillatory flows have the potential to enhance mixing by providing high-amplitude shear oscillations. For example, oscillating electro-osmotic flow caused by an externally induced electric field can be applied to membrane systems to reduce polarisation or fouling. Therefore, an approach that can model and analyse the relationship between oscillatory flows and the resulting mixing enhancement is needed. This paper proposes an approach that systematically analyses the effects of external inputs, which vary in both spatial and temporal directions, on the mixing enhancement in channel flows. A two-dimensional (2D) frequency response is obtained from the reduced-order model of the Navier-Stokes equations. The 2D frequency response describes the system behaviour at different frequencies and wavenumber. The analysis results are useful for system and control design, e.g. in identifying the locations to install actuators and sensors, which is often very difficult for distributed parameter systems.

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