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

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.