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 in the vertices. Non-Newtonian rheology of the flow generates nonlinear equations on the graph. A new numerical method for second order nonlinear differential equations on the graph is introduced and numerically tested.

An asymptotic ansatz for the solution of the axisymmetric problem of interaction between a thin cylindrical elastic tube and a viscous fluid filling the thin interior of the elastic tube was recently introduced and justified by G. Panasenko and R. Stavre. The thickness of the elastic medium ($\epsilon$) and that of the fluid domain (${\epsilon }_1$) are small parameters with $\epsilon <<{\epsilon }_1<<1$, while the scale of the longitudinal characteristic size is of order one. At the same time, the magnitude of the stiffness and density of the elastic tube may be large or finite parameters with respect to the viscosity and density of the fluid when the characteristic time is of order one. This ansatz can be considered as a Poiseuille-type solution for the fluid–structure interaction problem. Its substitution to the Stokes fluid–elastic wall coupled problem generates a one-dimensional model. We present a numerical experiment comparing this model with the solution of the full-dimensional fluid–structure interaction problem.

Baigtinių tūrių kintamųjų krypčių schemos hibridinės dimensijos šilumos laidumo modeliams