Viscous fingering instability in one-end-lifted Hele-Shaw cells for producing three-dimensional hierarchical structures

Hypothesis: The focal research on viscous fingering (VF) instabilities has primarily concentrated on alleviating and suppressing this phenomenon, but the potential to enhance and leverage them for developing practical applications remains inadequately explored. VF occurs when a less viscous fluid displaces a higher viscous fluid, governed by fluid dynamics and resulting in finger-like patterns at their interface in a facile and gentle way, yet sensitive to initial and boundary conditions, such as fluid type and cell thickness, giving rise to many bifurcating structures. 

Experiments: Leveraging Hele-Shaw cells (HSCs) with changing thicknesses that induced varying boundary conditions, we systematically investigated the characteristics of VF under these conditions and developed applications based on our findings.

Findings: We introduce VF in one-end-lifted HSCs to enhance VF rather than suppression to achieve satisfactory tunability of finger size and harness it to explore application. It enables the moldless, facile, scalable, and cost-effective fabrication of diverse hierarchical structures derived from branched patterns composed of numerous dense fingers with progressively decreasing sizes. With this phenomenon, the finger width scales inversely with the capillary number raised to the power in a range between −0.53 to −1.22, depending on the varying boundary conditions. By capitalizing on the hierarchy and specific distribution ridge of the fingering patterns, the fabricated fractal structures exhibit physically nonclonable characteristics, offering promising applications in anti-counterfeiting. © 2025 The Authors.