Hele-Shaw Viscous Fingering
Saffman-Taylor instability: low-viscosity fluid invades high-viscosity medium
Fluid Parameters
Viscosity ratio M=μ₂/μ₁
10.0
Surface tension σ
0.10
Injection rate V
1.0
Noise ε
0.30
Reset Simulation
Growth Rate
σ(k) = —
k_c (neutral) = —
k_max (fastest) = —
Theory
Saffman-Taylor (1958)
: Linear stability gives growth rate:
σ(k) = Vk·(M-1)/(M+1) − σk³/[μ(M+1)]
Instability onset: M > 1 (less viscous invades more viscous). Critical wavenumber:
k_c = √[V(M-1)/σ]
Fastest growing mode k_max = k_c/√3. Tip-splitting leads to fractal fingers.