Asymptotical form of Possio integral equation in theoretical aeroelasticity

Article type: Research Article

Authors: Shubov, Marianna A.

Affiliations: Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA. Tel.: +1 603 862 2731; E-mail: [email protected]

Abstract: The paper is the second in a series of works devoted to the solvability of the Possio singular integral equation. This equation relates the pressure distribution over a typical section of a slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (downwash). In spite of the importance of the Possio equation, the question of the existence of its solution has not been settled yet. In the first paper “Reduction of the boundary-value problem to Possio integral equation in theoretical aeroelasticity”, we reformulated the initial boundary-value problem, involving a partial differential equation for the velocity potential and highly nonstandard boundary conditions, in the form of a singular integral equation, the Possio equation. In our derivation, we have used the ideas and techniques totally different from the ones used by C. Possio in his original work. In the present paper, we show that the integral equation can be split up into two parts in such a way that the first part is not small with respect to a complex parameter entering the equation, while the second part is asymptotically small and tends to zero as the above parameter tends to infinity. In this paper, we justify such a splitting and in the next paper, we will prove solvability of the nonvanishing part of the Possio equation and then prove that asymptotically small terms cannot destroy the aforementioned unique solvability.

Keywords: compressible flow, velocity potential, Mikhlin multipliers, Hilbert integral transform

DOI: 10.3233/ASY-2009-0941

Journal: Asymptotic Analysis, vol. 64, no. 3-4, pp. 213-238, 2009