Saffman-Taylor Viscous Fingering

DLA-based Hele-Shaw Cell Simulation · Laplacian Growth

Parameters

Laplacian Growth

∇²P = 0 (pressure in fluid)
v_n = −(μ₁−μ₂)/μ₁μ₂ ∇P·n̂
P_interface = σκ (Young-Laplace)
Growing fingers...

Theory

When a less viscous fluid (μ₁) displaces a more viscous one (μ₂) in a Hele-Shaw cell, pressure satisfies Laplace's equation. The flat interface is Saffman-Taylor unstable — perturbations grow by diffusion-limited aggregation (DLA). Surface tension regularizes the instability, selecting a single finger of width λ/2 at high velocity. The fractal dimension of DLA clusters: D_f ≈ 1.71.