Fig. 2: Transmigration of an active drop across a constriction.

a–f If the size of the constriction is comparable with that of the droplet (λ ≃ 0.8, h = 72, R_p = 49, D = 90), the latter moves unidirectionally (the green arrow indicates the direction of motion) undergoing weak shape deformations, such as a slight longitudinal stretching (b) and mild compression (c and d). Out of the constriction, the circular shape is restored (e and f). The splay distortion (highlighted by large red arrows) of the polarization remains essentially unaltered. g–n If λ ≃ 0.5 (h = 46, \({R}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime} }=62\), D = 90), the incipient unidirectional motion slows down as the droplet approaches the constriction, where large portions of the fluid interface stick to (g and h) because of adhesion forces, larger at the entry and weaker at the exit (here γ_L = 3 × 10⁻², γ_R = 7.5 × 10⁻³). Afterwards, the droplet undergoes an intense folding (i) followed by a significant elongation (j–l) and decompression (m). Within the constriction, the polarization aligns essentially perpendicularly to the direction of motion, exhibiting a temporary bend deformation (highlighted with blue arrows), initially at the front (j and k) and then shifted towards the back (m). At the exit of the pore, the droplet detaches from the pillars and proceeds along the direction imposed by the splay deformation (n). o and v If the size of the constriction is very narrow (λ ≃ 0.2, h = 20, \({R}_{{{{{{{{\rm{p}}}}}}}}}^{{\prime\prime} }=75\), D = 90), the droplet initially adheres to the pillars and then shifts downwards to protrude its leading edge within the pore (o–q). Afterwards, it dramatically stretches along the direction of motion acquiring an initial ampule-like shape (r), subsequently replaced by an intermediate hourglass structure (s). Finally, the droplet retracts its rear, pushes its front forward (t), and leaves the pore (u and v), a process facilitated by the lower adhesion forces at the exit (here γ_L = 2.5 × 10⁻², γ_R = 10⁻²). Within the pore, the polar field shows long-lasting bend distortions, either along with splay deformations (r–t) or alone and spread to the whole drop (s).