Eur. Phys. J. E (2015) 38: 22
https://doi.org/10.1140/epje/i2015-15022-7

Regular Article

Feedback control of flow vorticity at low Reynolds numbers

¹ Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623, Berlin, Germany
² Institut für Mathematik 1, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

^* e-mail: maria.zeitz@tu-berlin.de
^** e-mail: gurevich@math.fu-berlin.de
^*** e-mail: holger.stark@tu-berlin.de

Received: 5 December 2014
Revised: 27 February 2015
Accepted: 5 March 2015
Published online: 30 March 2015

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.