Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II: General case
type: Item
instance of: scholarly article
title: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. II: General case
zbMATH Open document ID: 1332.35268
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2015-09-09T00:00:00Z
review text: This paper is the second part of the authors' work (for part I, see [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 122, 125--168 (2015; Zbl 1326.35248)]). The initial-boundary value problem to the Navier-Stokes system is considered in a specific domain \(\Omega\subset \mathbb{R}^m\), \(m=2,3\). \(\Omega\) is ``a connected finite union of thin finite cylinders (in the 2D case respectively thin rectangles)''. The small positive parameter \(\varepsilon\) characterizes the thickness of cylinders. \[ \begin{aligned} &\frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla)v+\nabla p=0,\quad \mathrm{div}\,v=0,\quad x\in\Omega,\quad t>0,\\ & v=g\quad \text{on }\partial\Omega,\quad v(x,0)=0. \end{aligned} \] It is assumed that \[ g(x,0)=0,\quad \int\limits_{\partial\Omega} g\cdot n\,dS=0. \] The authors construct a family of asymptotic approximation \(u^{(j)}\) to the solution of the problem. The norms of the difference \(v-u^{(j)}\) are estimated via powers of the thickness parameter \(\varepsilon\). In the appendix, the authors present results concerning the existence, uniqueness and asymptotic behavior of the solution to the non-steady Stokes problem in unbounded domains with cylindrical outlets to infinity.
reviewed by: Il'ya Sh. Mogilevskii
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 6479797
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: thin structures
zbMATH Keywords: asymptotics
author: G. P. Panasenko
author: K. Pileckas
MaRDI profile type: Publication
full work available at URL: j.na.2015.05.018
OpenAlex ID: W818222644
cites work: Q3746966
cites work: Effective equations describing the flow of a viscous incompressible fluid through a long elastic tube
cites work: Q5550025
cites work: One-dimensional models for blood flow in arteries
cites work: Derivation of an approximate theory of shells by means of asymptotic integration of the equations of the theory of elasticity
cites work: Q5522839
cites work: Q4259427
cites work: Q5576913
cites work: Q5562267
cites work: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II. Transl. from the German by Boris Plamenevskii
cites work: Q4494670
cites work: Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure
cites work: Q4654161
cites work: Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe
cites work: Divergence equation in thin-tube structures
cites work: Flows in a tube structure: Equation on the graph
cites work: Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I: The case without boundary-layer-in-time
cites work: On the nonstationary linearized Navier-Stokes problem in domains with cylindrical outlets to infinity
cites work: Existence of solutions with the prescribed flux of the Navier-Stokes system in an infinite cylinder
DOI: 10.1016/J.NA.2015.05.018
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description: scientific article; zbMATH DE number 6479797