Steady state non-Newtonian flow in a thin tube structure: equation on the graph

The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of an asymptotic analysis of the stationary non-Newtonian fluid motion in a thin tube structure with no-slip boundary condition on the lateral boundary.