Fig. 4

Generated flow fields and fluid mixing using the quarter and staggered patterns. a Generated flow field as a function of the external field direction \(\phi\) for \(\alpha =10.0\) and \(\beta =0.05\). Streamline colours represent the vortex strength. b Tracer particle positions after 50 cycles of the external field actuation. c Phase diagram from the simulation in \({N}_{x}={N}_{y}=4\) array. Vertical dotted line shows the theoretical prediction of critical value \({\alpha }_{\mathrm{c}}=1/\pi\), while the horizontal dotted line shows the value \({\beta }_{\mathrm{s}}={\ell }^{3}/(8{\pi }^{3}{a}^{3})\). d–f Rotational patterns under three different conditions; red denotes counter-clockwise rotation, while blue denotes clockwise rotation. Bottom row shows generated flow field from the rotational patterns