Self-similarity of the nonlinear adjoint Blasius solution with an alternative similarity variable

Determine whether the adjoint x-momentum solution y(x,y) for the Prandtl boundary layer over a flat plate with integrated friction drag objective admits any self-similar representation with respect to a similarity variable different from η or ξ, by establishing or refuting the existence of such a similarity profile that satisfies the adjoint equation (16) and boundary conditions (17).

Background

The authors show the Oseen-linearized problem admits a self-similar adjoint solution with the same profile as the primal. For the full nonlinear Blasius adjoint solution, they demonstrate that the proposed similarity form in the literature does not satisfy the adjoint equation, and numerical evaluations do not indicate similarity in η or ξ. They explicitly leave open whether a different similarity variable might yield a self-similar formulation.