VISCOUS FINGERING — Laplacian Growth (Hele-Shaw / Saffman-Taylor)

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Saffman-Taylor instability (1958): When a less viscous fluid (e.g., air) invades a more viscous one (e.g., oil) in a Hele-Shaw cell, the interface is unstable — small bumps grow into fingers. Growth ∝ ∇²φ = 0 (Laplace equation) → fractal dimension D_f ≈ 1.71 (DLA universality class). Surface tension σ regularizes the fingertip singularity — larger σ → wider, fewer fingers.