Feedback control of flow vorticity at low Reynolds numbers

Abstract

Abstract. Our aim is to explore strategies of feedback control to design and stabilize novel dynamic flow patterns in model systems of complex fluids. To introduce the control strategies, we investigate the simple Newtonian fluid at low Reynolds number in a circular geometry. Then, the fluid vorticity satisfies a diffusion equation. We determine the mean vorticity in the sensing area and use two control strategies to feed it back into the system by controlling the angular velocity of the circular boundary. Hysteretic feedback control generates self-regulated stable oscillations in time, the frequency of which can be adjusted over several orders of magnitude by tuning the relevant feedback parameters. Time-delayed feedback control initiates unstable vorticity modes for sufficiently large feedback strength. For increasing delay time, we first observe oscillations with beats and then regular trains of narrow pulses. Close to the transition line between the resting fluid and the unstable modes, these patterns are relatively stable over long times.

Dissipative systems with nonlocal delayed feedback control

“…While their amplitudes depend on the specific realization of the sigmoid as hyperbolic tangent, algebraic sigmoid, and ramp function, their frequencies are similar. Both, the stability-instability transition and the appearance of limit cycles match to the findings in [8]. This suggests that our model captures the essential features of a spatially extended dissipative systems when subjected to nonlocal delayed feedback.…”

“…We close with two comments. First, we note the similarities between our dominant eigenvalues and eigenvalues described in [8] for the specific case of delayed feedback control applied to vortex diffusion in a circular geometry. The characteristic function for that system is derived by solving the spatiotemporal problem explicitly.…”

“…We mimic this behavior here by implementing a bounded control force. Bounded delayed feedback was previously studied, e.g.to suppress overshooting due to overcompensating control forces [40], or for hydrodynamic vortex diffusion in a circular geometry to stabilize unstable modes [8].…”

“…From mixing two liquids [1], sorting colloids [2,3], controlling reaction rates [4] and heat transport in microfluidic devices [5], to fluid optics [1] and spiral patterns in liquid crystals [6], soft matter systems often display their most useful or interesting properties under external stresses. Several control and driving schemes have already been applied to these systems, including optimal control [7], hysteresis control [8], and time-delayed feedback [8][9][10]. These methods sense the characteristic response of a material and adapt their control to it.…”

“…Earlier theoretical studies [8,10,20] and experiments [9,21] applied delayed feedback to the spatiotemporal dynamics of specific systems. One study showed that a system with two delay times can be mapped to the complex Ginzburg-Landau equation [13].…”