# Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials

^{1}

^{2}

^{3}

^{*}

*Fractal Fract’s*Editorial Board Members)

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Mathematical Preliminaries

**Definition**

**1.**

#### 2.2. Motion Equation of Viscoelastic Pipeline Conveying Fluid

## 3. Numerical Study

#### 3.1. Shifted Legendre Polynomials

#### 3.2. Function Approximation

#### 3.3. Differential Operator Matrix

#### 3.4. Numerical Example

## 4. Numerical Results and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Peng, G.; Xiong, Y.M.; Liu, L.M.; Gao, Y.; Wang, M.H.; Zhang, Z. 3D nonlinear dynamics of inclined pipe conveying fluid supported at both ends. J. Sound Vib.
**2019**, 449, 405–426. [Google Scholar] [CrossRef] - Chang, G.H.; Sadeghi, Y.M. Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation. J. Sound Vib.
**2014**, 333, 4265–4280. [Google Scholar] [CrossRef] - He, F.; Dai, H.L.; Huang, Z.H.; Wang, L. Nonlinear dynamics of a fluid-conveying pipe under the combinedaction of cross-flow and top-end excitations. Appl. Ocean Res.
**2017**, 62, 199–209. [Google Scholar] [CrossRef] - Rahmati, M.; Mirdamadi, H.R.; Goli, S. Divergence instability of pipes conveying fluid with uncertain flow velocity. Physica A
**2018**, 491, 650–665. [Google Scholar] [CrossRef] - Zhang, Y.F.; Yao, M.H.; Zhang, W.; Wen, B.C. Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance. Aerosp. Sci. Technol.
**2017**, 68, 441–453. [Google Scholar] [CrossRef] - Zhang, Y.L.; Chen, L.Q. External and internal resonances of the pipe conveying fluid in the supercritical regime. J. Sound Vib.
**2013**, 332, 2318–2337. [Google Scholar] [CrossRef] - Wang, L.; Liu, Z.Y.; Abdelkefi, A.; Wang, Y.K.; Dai, H.L. Nonlinear dynamics of cantilevered pipes conveying fluid: Towards a further understanding of the effect of loose constraints. Int. J. Non-Linear Mech.
**2017**, 95, 19–29. [Google Scholar] [CrossRef] - Tang, Y.; Zhen, Y.X.; Fang, B. Nonlinear vibration analysis of a fractional dynamic model for the viscoelastic pipe conveying fluid. Appl. Math. Model.
**2018**, 56, 123–136. [Google Scholar] [CrossRef] - Mitsotakis, D.; Dutykh, D.; Li, Q.; Peach, E. On some model equations for pulsatile flow in viscoelastic vessels. Wave Motion
**2019**, 90, 139–151. [Google Scholar] [CrossRef] - Yano, D.; Ishikawa, S.; Tanaka, K.; Kijimoto, S. Vibration analysis of viscoelastic damping material attached to a cylindrical pipe by added mass and added damping. J. Sound Vib.
**2019**, 454, 14–31. [Google Scholar] [CrossRef] - Husain, S.A.; Anderssen, R.S. Modelling the relaxation modulus of linear viscoelasticity using Kohlrausch functions. J. Non-Newton. Fluid Mech.
**2005**, 125, 159–170. [Google Scholar] [CrossRef] - Yan, W.; Ying, J.; Chen, W.Q. The behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces in cylindrical bending. Compos. Struct.
**2007**, 78, 551–559. [Google Scholar] [CrossRef] - Machiraju, C.; Phan, A.V.; Pearsall, A.W.; Madanagopal, S. Viscoelastic studies of human subscapularis tendon: Relaxation test and a Wiechert model. Comput. Methods Programs Biomed.
**2006**, 83, 29–33. [Google Scholar] [CrossRef] [PubMed] - Peng, Y.; Zhao, J.Z.; Li, Y.M. A wellbore creep model based on the fractional viscoelastic constitutive equation. Pet. Explor. Dev.
**2017**, 44, 1038–1044. [Google Scholar] [CrossRef] - Long, J.M.; Xiao, R.; Chen, W. Fractional viscoelastic models with non-singular kernels. Mech. Mater.
**2018**, 127, 55–64. [Google Scholar] [CrossRef] - Mokhtari, M.; Permoon, M.R.; Haddadpour, H. Aeroelastic analysis of sandwich cylinder with fractional viscoelastic core described by Zener model. J. Fluids Struct.
**2019**, 85, 1–16. [Google Scholar] [CrossRef] - Yu, Y.; Perdikaris, P.; Karniadakis, G.E. Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms. J. Comput. Phys.
**2016**, 323, 219–242. [Google Scholar] [CrossRef] - Meng, R.F.; Yin, D.; Drapaca, C.S. A variable order fractional constitutive model of the viscoelastic behavior of polymers. Int. J. Non-Linear Mech.
**2019**, 113, 171–177. [Google Scholar] [CrossRef] - Meng, R.F.; Yin, D.; Drapaca, C.S. Variable-order fractional description of compression deformation of amorphous glassy polymers. Comput. Mech.
**2019**, 64, 163–171. [Google Scholar] [CrossRef] - Sene, N.; Fall, A.N. Homotopy perturbation ρ-Laplace transform method and its application to the fractional diffusion equation and the fractional diffusion–reaction equation. Fractal Fract.
**2019**, 3, 14. [Google Scholar] [CrossRef] - Baleanu, D.; Jassim, H.K.; Qurashi, M.A. Solving Helmholtz equation with local fractional derivative operators. Fractal Fract.
**2019**, 3, 43. [Google Scholar] [CrossRef] - Malmir, I. A new fractional integration operational matrix of Chebyshev wavelets in fractional delay systems. Fractal Fract.
**2019**, 3, 46. [Google Scholar] [CrossRef] - Chen, Y.M.; Liu, L.Q.; Li, B.F.; Sun, Y.N. Numerical solution for the variable order linear cable equation with Bernstein polynomials. Appl. Math. Comput.
**2014**, 238, 329–341. [Google Scholar] [CrossRef] - Chen, Y.M.; Liu, L.Q.; Li, X.; Sun, Y.N. Numerical solution for the variable order time fractional diffusion equation with Bernstein polynomials. Comput. Model. Eng. Sci.
**2014**, 97, 81–100. [Google Scholar] - Chen, Y.M.; Wei, Y.Q.; Liu, D.Y.; Yu, H. Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets. Appl. Math. Lett.
**2015**, 46, 83–88. [Google Scholar] [CrossRef] - Chen, Y.M.; Liu, L.Q.; Liu, D.Y.; Boutat, D. Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials. Ain Shams Eng. J.
**2018**, 9, 1235–1241. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**The numerical example results when n values 4 for (

**a**) absolute error ${e}_{w}(x,t)$, (

**b**) ${({\displaystyle \frac{\partial w(x,t)}{\partial x}})}^{2}$, and (

**c**) comparison of exact solution and numerical solution.

**Figure 3.**Dynamic response on the viscoelastic pipeline conveying fluid when n values 4 within 50 s for (

**a**) three-dimensional displacement map and (

**b**) contour displacement map.

**Figure 4.**Dynamic response with the fluid velocity varying when n values 4 and x values 1 m within 50 s for (

**a**) displacement response of $w(x,t)$ and (

**b**) displacement oscillation of $w(x,t)$.

**Figure 5.**Dynamic response with the force excitation varying when n values 4 and x values 1 m within 50 s for (

**a**) excitation amplitude ${f}_{0}$ and (

**b**) excitation frequency $\omega $.

**Figure 6.**Oscillation response with the variable fractional order varying when n values 4 within 50 s for (

**a**) basic order ${\alpha}_{0}$ and (

**b**) order varying rate ${\alpha}_{k}$.

Physical Quantity | Symbol | Value | Dimension |
---|---|---|---|

External radius | D | $0.4$ | $\mathrm{m}$ |

Internal radius | d | $0.32$ | $\mathrm{m}$ |

Length | H | 2 | $\mathrm{m}$ |

Area moment of inertia | I | $0.0119$ | ${\mathrm{m}}^{4}$ |

Constitutive model parameter | E | 600 | $\mathrm{MPa}$ |

Constitutive model parameter | $\theta $ | $0.1$ | 1 |

Density of viscoelastic pipeline | ${\rho}_{p}$ | $1.2\times {10}^{3}$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Density of internal fluid | ${\rho}_{f}$ | $1.05\times {10}^{3}$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

Mass per unit length of viscoelastic pipeline | ${m}_{p}$ | $217.2$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$ |

Mass per unit length of internal fluid | ${m}_{f}$ | $168.84$ | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$ |

Tensional force | ${T}_{0}$ | 500 | $\mathrm{N}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Chen, Y.
Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials. *Fractal Fract.* **2019**, *3*, 52.
https://doi.org/10.3390/fractalfract3040052

**AMA Style**

Wang Y, Chen Y.
Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials. *Fractal and Fractional*. 2019; 3(4):52.
https://doi.org/10.3390/fractalfract3040052

**Chicago/Turabian Style**

Wang, Yuanhui, and Yiming Chen.
2019. "Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials" *Fractal and Fractional* 3, no. 4: 52.
https://doi.org/10.3390/fractalfract3040052