Orr-Sommerfeld Equation: Viscous Flow Stability

Orr-Sommerfeld equation governs the stability of viscous parallel flows. Linearizing Navier-Stokes about U(y) and writing perturbations as φ(y)·exp(i(αx−ωt)), the streamfunction amplitude φ satisfies:

(U−c)(φ'' − α²φ) − U''φ = (1/iαRe)(φ'''' − 2α²φ'' + α⁴φ)

This is an eigenvalue problem for c = ω/α (complex wave speed). Instability requires Im(c) > 0. For plane Poiseuille flow, the critical Reynolds number is Re_c ≈ 5772 (Heisenberg/Lin 1955) at α ≈ 1.02. The paradox: viscosity both damps oscillations AND destabilizes via Tollmien-Schlichting waves. Couette flow is linearly stable for all Re (nonlinear transition at Re~360).