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Spectral analysis of a viscoelastic tube conveying fluid with generalised boundary conditions

Xiao Xuan Feng, Mahyar Mahinzaeim, Gen Qi Xu

Abstract

We study the spectral problem associated with the equation governing the small transverse motions of a viscoelastic tube of finite length conveying an ideal fluid. The boundary conditions considered are of general form, accounting for a combination of elasticity and viscous damping acting on both the slopes and the displacements of the ends of the tube. These include many standard boundary conditions as special cases such as the clamped, free, hinged, and guided conditions. We derive explicit asymptotic formulae for the eigenvalues for the case of generalised boundary conditions and specialise these results to the clamped case and the case in which damping acts on the slopes but not on the displacements. In particular, the dependence of the eigenvalues on the parameters of the problem is investigated and it is found that all eigenvalues are located in certain sectorial sets in the complex plane.

Spectral analysis of a viscoelastic tube conveying fluid with generalised boundary conditions

Abstract

We study the spectral problem associated with the equation governing the small transverse motions of a viscoelastic tube of finite length conveying an ideal fluid. The boundary conditions considered are of general form, accounting for a combination of elasticity and viscous damping acting on both the slopes and the displacements of the ends of the tube. These include many standard boundary conditions as special cases such as the clamped, free, hinged, and guided conditions. We derive explicit asymptotic formulae for the eigenvalues for the case of generalised boundary conditions and specialise these results to the clamped case and the case in which damping acts on the slopes but not on the displacements. In particular, the dependence of the eigenvalues on the parameters of the problem is investigated and it is found that all eigenvalues are located in certain sectorial sets in the complex plane.
Paper Structure (6 sections, 6 theorems, 90 equations)

This paper contains 6 sections, 6 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Abstract problem formulation and spectrum
  3. Asymptotics of eigenvalues
  4. Eigenvalues for clamped boundary conditions
  5. Asymptotics of eigenvalues for $k_{04}=k_{14}= 0$
  6. Further remarks and open problems

Key Result

Proposition 2.1

Let $A_0$ be defined by 208xy, 208xyz. The following assertions hold:

Theorems & Definitions (14)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • ...and 4 more