Low-viscosity fluid displacing high-viscosity fluid in a Hele-Shaw cell — fingers grow via Laplacian instability.
Hele-Shaw Cell Simulation
Parameters
Darcy: u = -(k/η)∇p
∇²p = 0 (Laplace in bulk)
Unstable if k > 1/3 (Saffman-Taylor)
Growth rate σ_k = k(A·V−k²B)
A (Atwood) = —
Step: 0
Max finger length: —
Most unstable mode: —
The Saffman-Taylor instability occurs when a less viscous fluid (η₁) pushes a more viscous one (η₂) in a Hele-Shaw cell. The pressure field satisfies Laplace's equation; surface tension stabilizes short wavelengths but longer fingers grow without bound (in the absence of surface tension).