Abstract
The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem.
Highlights
The present paper is motivated by the problem of computer modeling of the blood vessel network
That is why we suggest the hybrid dimension models, combining the one-dimensional reduction in the regular zones with threedimensional zooms in small zones of singular behavior. This approach was addressed to the non-stationary initial boundary value problems for the Navier–Stokes equations in thin tube structures [17]
In the first part of the paper we study the existence and uniqueness of the time periodic Navier–Stokes equations for both settings and derive the a priori estimates
Summary
The present paper is motivated by the problem of computer modeling of the blood vessel network. That is why we consider here the periodic in time problem for the Navier–Stokes equation, prove the existence and uniqueness theorems for this setting, and construct the asymptotic expansion of a solution with respect to the small parameter. The constructed asymptotic expansion is used for the construction of a special numerical method combining one-dimensional description with three-dimensional zooms, the method of asymptotic partial decomposition of the domain (MAPDD) This method reduces the full geometry setting to the computations in neighborhoods of bifurcation zone of diameter of order ε| ln ε| as in [4] but in the time periodic setting. Let g be the divergence-free time periodic extension of the boundary function g
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