Extended Data Fig. 2: Profiles of ferrofluid–air interfaces measured by a 3D laser-scanning confocal microscope, the dependence of the scaling relations on the overlayer thickness, and draining ferrofluid along a curved path.

a, 3D profiles of the ferrofluid–air interface measured along the x axis at various distances from the edge of the magnet (labelled). The profiles show a gradual increase in the ferrofluid level inside the channel along the x axis, which corresponds to a gradual decrease in interfacial curvature. Measurements were performed on a 1 inch × 3 inch FLIPS sample after 2 h. Pattern 1 was used, and the channel direction was aligned to the long side of the FLIPS sample. On the right is a plot of the cross-sectional profiles of the ferrofluid–air interface. The signal is noisy in the high-curvature region near the edge, owing to the limit of the numerical aperture of the 100× long-working-distance objective used. b, 3D profiles of ferrofluid–air interface measured 1.5 cm away from the edge of the magnet over time (labelled). The FLIPS sample is the same as in a. The profiles show a gradual decrease in ferrofluid level inside the channel, which corresponds to a gradual increase in interfacial curvature. The corresponding plots of the cross-sectional profiles are presented in Fig. 1c. c, Dependence of the scaling relations on the overlayer thickness. Pattern 2 was used. Increasing the overlayer thickness decreases the prefactor but increases the power in the scaling relations. d, Effects of varying h₀, d_y and d_x on the scaling relations. The overlayer thicknesses are roughly 10–20 μm. e, Draining of ferrofluid along a curved path in a spiral pattern, demonstrating the ability of the porous-capillary flow to make turns.