Boundary behavior of D0 in the Falkner–Skan adjoint expansion

Prove that the function D0(η) defined by D0(η) = Σk (2k + β − 1) Dk(η) satisfies the boundary conditions D0(0) = 1 and limη→∞ D0(η) = 0 for the Falkner–Skan boundary layer, thereby establishing from the series representation (equation (119)) that the analytic adjoint solution obeys the required adjoint boundary conditions.

Background

In extending the analysis to Falkner–Skan flows, the adjoint solution is represented as a series over adjoint eigenfunctions. Unlike the Blasius case where D0 = 1, the sum defining D0(η) is not constant and must satisfy a third-order adjoint ODE with specific boundary conditions. The authors state they were unable to prove directly from the series representation that these boundary conditions hold, highlighting a need for a rigorous justification.

References

The reason (which also explains the behavior of the solution that will be illustrated below by explicit evaluation of eq. (118)) is that, unlike the Blasius case, the function Do(n)=>- 00 k=1 2k + B-1. Dk (n) (119) is not constant but rather has to obey the equation -Do,nnn + FBDk.nn + (2-2 +2B) FB. Dk.n +(2+23-22k) FB.m.Dk =0 with boundary conditions Do(0) =1 and Do(o)=0, neither of which have we been able to prove directly from eq. (119).

— Libby-Fox perturbations and the analytic adjoint solution for laminar viscous flow along a flat plate  (2601.16718 - Lozano et al., 23 Jan 2026) in Section 8