Asymptotic analysis of the Navier–Stokes system in a plane domain with thin channels

Article type: Research Article

Authors: Maz’ya, V.G. | Slutskii, A.S.^;

Affiliations: Department of Mathematics, Linköping University, S‐58183 Linköping, Sweden | Institute of Problems of Mechanical Engineering, V.O., Bol’shoy pr. 61, St. Petersburg 199178, Russia

Note: [] Corresponding author.

Abstract: A flow of viscous incompressible fluid in a domain &OHgr;_ϵ depending on a small parameter ϵ is considered. The domain &OHgr;_ϵ is the union of a domain &OHgr;_0 with piecewise smooth baundary and thin channels with width of order ϵ. Every channel contains one angle point of the domain &OHgr;_0 near the channel’s inlet. We prove the existence of a solution (v_ϵ,p_ϵ) to the Navier–Stokes system such that in a neighbourhood of an angle point of the domain &OHgr;_0 the pair (v_ϵ,p_ϵ) is equal, up to a term with finite kinetic energy, to the Jeffery–Hamel solution which describes a plane viscous source (or sink) flow between the sides of the angle. In the channels the pair (v_ϵ,p_ϵ) asymptotically coincides with the Poiseuille solution. Asymptotic expressions for the kinetic energy and the Dirichlet integral of (v_ϵ,p_ϵ) are obtained.

Journal: Asymptotic Analysis, vol. 23, no. 1, pp. 59-89, 2000