Investigating the eigenfunction expansion in the analytic adjoint solution for the Blasius boundary layer via Libby-Fox perturbation theory, Borel resummation, and composite matched-asymptotic approximation.
The function g(ξ,η) is defined as a generalized Dirichlet series involving Libby-Fox eigenfunctions. While g reduces to erfc(η/(2√ξ)) in the linear Oseen limit, no closed-form expression is known for the full nonlinear case.
where φn(η) are Libby-Fox eigenfunctions satisfying a third-order ODE with the Blasius profile as coefficients, λn are eigenvalues, and an are expansion coefficients.
Shooting method with Brent's root-finding, 5000 grid points, rtol=10-12, atol=10-14 on η in [0, 12].
Scan λ in [0.5, 10.0] with 400 trial values, bisection refinement to tolerance 10-10.
Borel transform with 200-point adaptive quadrature for the Laplace-type integral.
Matched-asymptotic: inner Airy-type near wall + outer erfc, Gaussian transition function.
Explore the numerical results through interactive charts. Click legend entries to toggle series visibility.
Detailed numerical results from all analyses.
| ξ | max|g| | max|gOseen | max|g - gO | Rel. Error |
|---|---|---|---|---|
| 0.5 | 74.7366 | 1.0000 | 74.7366 | 74.7366 |
| 1.0 | 1.7152 | 1.0000 | 1.7152 | 1.7152 |
| 2.0 | 0.03937 | 1.0000 | 1.0000 | 1.0000 |
| 5.0 | 2.680e-4 | 1.0000 | 1.0000 | 1.0000 |
| 10.0 | 6.152e-6 | 1.0000 | 1.0000 | 1.0000 |
| 20.0 | 1.412e-7 | 1.0000 | 1.0000 | 1.0000 |
| 50.0 | 9.613e-10 | 1.0000 | 1.0000 | 1.0000 |
| 100.0 | 2.206e-11 | 1.0000 | 1.0000 | 1.0000 |
| ξ | gseries | gBorel | gOseen | Borel Rel. Error |
|---|---|---|---|---|
| 1.0 | 1.68823 | 1.68823 | 0.03381 | 8.39e-9 |
| 2.0 | 0.03875 | 0.03875 | 0.13342 | 5.73e-15 |
| 5.0 | 2.638e-4 | 2.638e-4 | 0.34254 | 8.87e-13 |
| 10.0 | 6.055e-6 | 6.055e-6 | 0.50212 | 8.87e-13 |
| 20.0 | 1.390e-7 | 1.390e-7 | 0.63509 | 4.47e-11 |
| 50.0 | 9.462e-10 | 9.462e-10 | 0.76406 | 1.07e-8 |
| Similarity Variable | Mean Spread | Rank |
|---|---|---|
| η/√ξ | 0.4249 | 1 (Best) |
| η/ξ1/3 | 0.4499 | 2 |
| η2/ξ | 0.5096 | 3 |
| η2/(4ξ) | 0.6628 | 4 |
| ξ | Oseen Error | Composite Error | Improvement Factor |
|---|---|---|---|
| 0.5 | 74.7366 | 74.7376 | 1.0000x |
| 1.0 | 1.7152 | 1.7153 | 1.0000x |
| 2.0 | 1.0000 | 1.0000 | 1.0000x |
| 5.0 | 1.0000 | 1.0000 | 1.0000x |
| 10.0 | 1.0000 | 1.0000 | 1.0000x |
| 20.0 | 1.0000 | 1.0000 | 1.0000x |
| 50.0 | 1.0000 | 1.0000 | 1.0000x |
| 100.0 | 1.0000 | 1.0000 | 1.0000x |
| N terms | g at ξ=1.0 | g at ξ=5.0 | g at ξ=20.0 |
|---|---|---|---|
| 1 | 1.68823 | 2.638e-4 | 1.390e-7 |
Summary of the main results and their implications for the closed-form representation of g(ξ,η).
The diffusion-type variable η/√ξ achieves the best collapse (mean spread 0.4249) among four power-law candidates, consistent with the Oseen limit structure. However, the imperfect collapse confirms that no single similarity variable captures the full nonlinear structure of g.
The Borel transform provides an exact integral representation of g, achieving machine-precision agreement with direct series evaluation. Relative errors reach as low as 5.73 x 10-15 at ξ=2.0, confirming the Borel summability of the Dirichlet series.
The computed lead eigenvalue λ0 = 5.4453 governs the algebraic decay rate of the series. The series amplitude decays from 74.74 at ξ=0.5 to 2.21 x 10-11 at ξ=100, reflecting strong ξ-λn decay.
The eigenvalue structure and non-trivial eigenfunctions suggest that a closed-form expression, if it exists, would likely involve a new special function class rather than classical elementary functions. The Borel integral representation is the most promising path.