On an algorithm for the numerical solution of the Riemann – Hilbert problem
In the problem of viscous fluid flowing around a body, a perturbation theory for flows with small Reynolds numbers is constructed. As you know, this theory is a theory of singular perturbations, and to justify it, you need to create the appropriate mathematical apparatus. Using the theory of potentials of the linearized Navier-Stokes equations, the solution of the nonlinear problem is presented in the form of special series, re-expanding which, we can obtain asymptotic series of perturbation theory in the Stokes and Oseen regions. The question of the uniqueness of the solution of the flow problem for small Reynolds numbers is investigated in detail. The criterion of violation of uniqueness is established. A perturbation theory is developed for the solution of the flow problem when the Reynolds number is not small; it is a problem of regular perturbations. A formula is given for the force acting on the body, and thereby the Stokes formula and other formulas obtained by the method of merging asymptotic expansions are justified. Thus, in the problems under consideration, the method of coalesced asymptotic expansions was substantiated.