Closed-Form Expressions for Libby-Fox Eigenvalues and Norms

Computational investigation of the eigenvalue problem governing perturbations to the Blasius boundary layer. An open problem in fluid dynamics.

12 Eigenvalues Computed

1.908 Asymptotic Slope α

< 0.06 Residual (k ≥ 6)

12 digits Sum-Rule Precision

Problem Overview

The Blasius boundary layer arises from the self-similar reduction of the steady incompressible boundary-layer equations on a flat plate. Perturbations about the Blasius profile lead to the eigenvalue problem:

φ''_k + (1/2) f φ'_k - A_k f' φ_k = 0, φ_k(0) = 0, φ_k(η → ∞) → 0

where A_k are the Libby-Fox eigenvalues and f(η) is the Blasius profile satisfying f''' + (1/2)ff'' = 0. No closed-form expressions are known beyond Brown's asymptotic approximation for large k.

Interactive Results

Eigenvalue Spectrum vs. Brown's Asymptotic Fit

Brown's Asymptotic Residuals

Eigenvalue Spacings: A(k+1) - A(k)

Spacing Ratio: Delta(k) / alpha

Normalization Constants C(k) log scale

Weighted Norms log scale

Sum-Rule Convergence

Numerical Data

Eigenvalues and Normalization Constants

Eigenvalue Spacings and Brown's Fit

Sum-Rule Partial Sums

Key Findings

Brown's asymptotic formula: A(k) ~ 1.9084k - 2.6642 fits with residuals below 0.06 for k >= 6.
Super-exponential norm growth: Norms grow as approximately 10^(1.8k), from 0.333 (k=0) to 4.9e19 (k=11).
Rapid sum-rule convergence: Partial sums stabilize to 12-digit precision by K=7, dominated by the first two terms.
Principal obstacle: The transcendental nature of the Blasius profile f(eta) prevents standard special-function identification.
Promising direction: The change of variable xi = f'(eta) maps the problem to [0,1), potentially connecting to the Heun class of ODEs.