Multiscale Analysis of Viscous Flows in Thin Tube Structures

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   The introductory chapter describes the main methods of asymptotic analysis for partial differential equations set in thin domains used further in the monograph. Namely, for simplified domains and simplified partial differential operator (Laplacian) the methods of dimension reduction and boundary layers are introduced and justified. The method of asymptotic partial decomposition for thin domains is described.

3. Chapter 2. Preliminaries

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   Chapter Preliminaries presents the essential notations and concepts from real and functional analysis. It introduces the spaces of smooth functions, Sobolev spaces, and spaces of divergence-free vector-valued functions. In particular, the results on the asymptotic behavior of solutions of the Stokes equations in domains with noncompact boundary and cylindrical outlets to infinity are presented. References to specialized literature are provided.

4. Chapter 3. Poiseuille Flows

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   Chapter Poiseuille flows is devoted to the mathematical analysis of the Poiseuille flows, solutions to the Navier–Stokes equations with no-slip boundary condition in an infinite cylinder. Existence and uniqueness of solutions is proved for stationary and non-stationary cases. The regularity of solutions is studied.

5. Chapter 4. Stokes Problem and Stokes Operator

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   Chapter Stokes Problem and Stokes Operator is devoted to the mathematical problems related to the stationary and non-stationary Stokes problems in domains with smooth boundary, domains with cylindrical outlets to infinity, thin tube structures.

6. Chapter 5. Steady-State Stokes and Navier–Stokes Equations in Tube Structures

   Grigory Panasenko, Konstantin Pileckas

   Chapter Steady-State Stokes and Navier–Stokes Equations in Tube Structures revisits the definitions of thin tube structure given in Chap. 1 and formulates the stationary problem for the Stokes and Navier–Stokes equations in thin tube structures. The boundary conditions are homogeneous no-slip on the lateral part of the boundary, and given inflow and outflow velocities. Alternatively, the pressure is given at the inflows and outflows for the Stokes equations and the Bernoulli pressure conditions for the Navier–Stokes equations. The existence and uniqueness of a solution is proved for all settings. The asymptotic expansion of the solution is constructed. The error estimates are proved for the difference of the exact solution and its asymptotic approximations. Method of asymptotic partial decomposition of the domain (MAPDD) is described and justified. Numerical experiments confirm the theoretically established error estimates.

7. Chapter 6. Nonstationary Navier-Stokes Equations in Tube Structures

   Grigory Panasenko, Konstantin Pileckas

   Chapter Nonstationary Navier–Stokes Equations in Tube Structures formulates the nonstationary initial boundary value problem for the Navier–Stokes equations in thin tube structures. The boundary conditions are homogeneous no-slip on the lateral part of the boundary and the given inflow and outflow velocities. The existence and uniqueness of a solution is proved. The asymptotic expansion of the solution is constructed. The error estimates are proved for the difference of the exact solution and its asymptotic approximations. The method of asymptotic partial decomposition of the domain (MAPDD) is described and justified. Numerical experiments confirm the theoretically established error estimates and determine the limitations of the applicability of MAPDD for high Reynolds numbers.

8. Chapter 7. Time-Periodic Case

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   Chapter Time-Periodic Case is devoted to the time-periodic setting of the Navier–Stokes problem in thin structures. The existence and uniqueness of a solution is proved. The asymptotic expansion of the solution is constructed. The error estimates are proved for the difference of the exact solution and its asymptotic approximations. Method of asymptotic partial decomposition of the domain (MAPDD) is described and justified.

9. Chapter 8. Bibliographical Remarks

   Grigory Panasenko, Konstantin Pileckas

   Abstract

   In this chapter we briefly describe the state of the art of the works on partial differential equations set in thin domains. Such domains depend on the small parameter \(\varepsilon \), which is the ratio of sizes in different directions. In particular, the thin tube structures belong to this class of domains. The studying of thin structures is motivated by the applications to modeling of industrial installations (rods, bars, frames) as well as by the biological applications (networks of blood vessels).

10. Backmatter