Corrections to Kirchhoff's law for the flow of viscous fluid in thin bifurcating channels and pipes

Abstract

We consider the flow of a viscous Newtonian fluid in a bifurcation of thin pipes with a diameter-to-length ratio of order O(ε). The model is based on the stationary Navier–Stokes equations with pressure conditions on the outflow boundaries. Existence and local uniqueness is established under the assumption of small data represented by a Reynolds number Re_ε of order O(ε). We construct an asymptotic expansion in powers of ε and Re_ε for the solution consisting of Stokes flow in the junction part of the bifurcation and Poiseuille flow in the pipes. We introduce a correction to Kirchhoff's law of the balancing fluxes in O(ε) which allows to establish error estimates for the gradient of velocity. These estimates result from the analysis of the decay properties of the flow in the layer near the bifurcation.