Rigorous justification of the Kirchhoff law for junction of thin pipes filled with viscous fluid

Abstract

We study the junction of m pipes that are either thin or long (i.e., they have small ratio between the cross‐section and the length, denoted by ε). Pipes are filled with incompressible Newtonian fluid and the values of the pressure p_i at end of each pipe are prescribed. By rigorous asymptotic analysis, as ε→0, we justify the analog of the Kirchhoff law for computing the junction pressure. In interior of each pipe the effective flow is the Poiseuille flow governed by the pressure drop between the end of the pipe and the junction point. The pressure at the junction point is equal to a weighted mean value of the prescribed p_i‐s (Kirchhoff law). In the vicinity of the junction there is an interior layer, with thickness εlog (1/ε). To get a better approximation and to control the velocity gradient in vicinity of the junction, first order asymptotic approximation has to be corrected by solving an appropriate Leray problem. We prove the asymptotic error estimate for the approximation.