Method of Small Perturbations, Linearized Theory

The method of small perturbations was first used to solve the problem of the flow around a thin body in a uniform stream of a compressible fluid. If the thickness ratio δ of the thin body is much smaller than unity, we may develop the velocity potential of the irrotational flow as a power series of the thickness ratio ¹ . For two-dimensional flows without a stagnation point, this power series expansion is valid. On the other hand, for twodimensional flows which exhibit a stagnation point, terms of δⁿ log δ appear in this expansion of the velocity potential. Since this kind of singularity does not appear explicitly in the expansion of series of stream function, it is more convenient to solve this latter problem using the stream function instead of the velocity potential.