A flow of viscous incompressible fluid in a domain Ω_εdepending on a small parameter ε is considered. The domain Ω_ε is the union of a domain Ω₀ with piecewise smooth baundary and thin channels with width of order ε. Every channel contains one angle point of the domain Ω₀ 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 Ω₀ 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.