Adjoint-based optimization of open loop control of a microfluidic acoustic flow

Drop-on-demand inkjet printing is one of the most widespread applications of microfluidics. Within the printhead, ink continuously flows through multiple microchannels. Each microchannel contains a piezo-electric actuator on the top face and a nozzle on the opposite face. When the actuator pulses, a droplet is forced through the nozzle. Acoustic oscillations then reverberate within the microchannel until they are damped by viscous and thermal dissipation. If a droplet is ejected before the reverberations from the previous droplet have been sufficiently damped, its size is affected by the reverberations, which spoils the image being printed. In this study, we design open loop control of the actuator to eliminate these reverberations. First, we derive the governing equations of the thermoviscous acoustic flow by linearising the compressible Navier-Stokes equations. Then we derive the associated adjoint problem to obtain the gradient of the objective function (the acoustic energy after a given time) with respect to the actuator deformation. Finally we formulate an optimisation problem to find the actuator waveform that minimises the reverberations within a given time.

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