The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ϵ in the case of the Reynolds number of order ϵ. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(ϵ ^J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.

The steady-state Navier-Stokes equations in thin structures lead to some elliptic second order equation for the macroscopic pressure on a graph. At the nodes of the graph the pressure satisfies Kirchoff-type junction conditions. In the non-steady case the problem for the macroscopic pressure on the graph becomes nonlocal in time. In the paper we study the existence and uniqueness of a solution to such one-dimensional model on the graph for a pipe-wise network. We also prove the exponential decay of the solution with respect to the time variable in the case when the data decay exponentially with respect to time.

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter . Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameter is obtained.

The mathematical tools for the up-scaling from micro to macro in periodic media are considered: the homogenization techniques, the multiscale modelling, etc. The main applications of these tools are the mechanics of composite materials, flows in porous media, and discrete models for nanostructures.

In this article the Stokes equations are considered in a domain simulating a capillary bed system. The capillaries are supposed to be thin, parallel and periodic. An asymptotic approximation is constructed. The macroscopic pressure satisfies a Robin interface condition whose coefficients are calculated numerically through a finite element approximation of a boundary layer problem, which is inspired to a domain decomposition technique.