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 $\varphi$ 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 ${\beta }_{\mathrm{s}}={\ell }^3∕\left(8{\pi }^3{a}^3\right)$. 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