ISSN 0001-4346, Mathematical Notes, 2016, Vol. 99, No. 5, pp. 636–642. © Pleiades Publishing, Ltd., 2016.Existence of the Stationary Solution of a Rayleigh-Type Equation *D. I. Borisov 1,2,3** and R. K. Gaydukov 4***1 Institute of Mathematics with Computer Center, Ufa Scientif i c Center,Russian Academy of Sciences, Ufa, Russia2 Akhmulla Bashkir State Pedagogical University, Ufa, Russia3 University of Hr ádec Kr álov é, Hr ádec Kr álov é, Czech Republic4 National Research University Higher School of Economics, Moscow, RussiaReceived March 23, 2016Abstract—A f l uid f l ow along a semi-inf i nite plate with small periodic irregularities on the surfaceis considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thinboundary layer(“lowerdeck”)and a classicalPrandtl boundarylayer(“upper deck”). The aim ofthispaperistoprovethe existenceand uniquenessofthe stationary solutionofaRayleigh-typeequation,which describesoscillations of the vertical velocity component in the classical boundary layer.DOI: 10.1134/S0001434616050023Keywords: double-deck structure, boundary-layer theory, f l uid mechanics, Navier–Stokesequations, Rayleigh-type equation, eigenvalue problem.1. INTRODUCTIONIn this paper, we continue our study of the Rayleigh-type equation (1.8), which was started in [1];see also [2]. It is known that the Rayleigh equation (see [3]) plays an important role in f l uid mechanicsproblems; see [4]. In this paper, this equation is considered on a semi-inf i nite cylinder (see (1.8), (1.9)),and it describes oscillations of the vertical velocity component in the classical Prandtl boundary layer(in the “upper deck” of a boundary layer with a double-deck boundary layer structure, see region II inFig. 2) in the problem of an incompressible viscosity f l uid f l ow along a semi-inf i nite f l at plate with smallperiodic perturbations on the surface (see Fig. 1) for large Reynolds number Re; for more details, seebelow.In [1], it was proved that the stationary solution of the Rayleigh-type equation (1.8) exists and isunique for all x > δ and δ > M, where x is the distance from the edge of the plate and M is a constant;see (2.1). The aim of this paper is to prove that a stationary solution of the Rayleigh-type equation (1.8)exists and is unique for all x > δ and δ ∈ (0,M] (i.e., at the edge of the plate).As will be shown in Section 2, the proof of the existence of the solution in this case is reduced toproving that the discrete spectrum of a Schr ödinger-type operator on the half-space with a potential inthe form of a well of a small depth (see Fig. 3) is empty, and the last statement is proved (see Lemma 1).We note that the results of this paper (see Theorem 2) together with the results in [1] prove that thestationary solution of the Rayleigh-type equation (2.2) exists and is unique for all x > δ and δ > 0 (i.e.,in the entire region under study) and this fact actually means the existence of the double-deck structure(because all equations describing this structure are solvable).In this section, we present the main results from [1, 2], which we need for further discussion.We assume that the plate surface is described by the relationy s = ε 4/3 μ ? x,x/ε ? , (1.1)∗ The article was submitted by the authors for the English version of the journal.** E-mail: borisovdi@yandex.ru*** E-mail: roma1990@gmail.com636
