Asymptotic expansions of the eigenvalues and norms for perturbations about the Falkner--Skan solution

Abstract

We derive large-mode asymptotic expansions for the eigenvalues and eigenfunction norms of the Chen-Libby perturbation problem about Falkner-Skan boundary-layer profiles. The analysis extends Brown's matched-asymptotic construction for the Blasius problem to Falkner-Skan solutions with positive wall shear -- both favorable gradients and the upper branch of the mild adverse range. Although the outer and middle layers retain the same structure as in the Blasius case, the wall layer changes qualitatively when the pressure-gradient parameter $\beta$ is nonzero: the Falkner-Skan wall expansion singularly perturbs the inner Bessel problem at relative order $\Lambda^{-1/3}$, where $\Lambda$ is the large eigenvalue parameter. This produces a new wall-induced contribution $s^{-1/3}$ (where $s=n-1$ and $n$ is the large eigenvalue index) to the large-mode eigenvalue expansion, ahead of Brown's $s^{-1/2}$ correction. The new term vanishes in the Blasius limit, where Brown's ordering is recovered. The matched eigenfunction also yields asymptotic estimates for the Chen-Libby norms. The eigenvalue and norm formulae are compared with direct numerical shooting calculations for $\beta=1/2$, showing the expected asymptotic convergence and confirming the role of the new wall correction.