# Influence of Second Viscosity on Pressure Pulsation

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## Abstract

**:**

## 1. Introduction

## 2. Model Derivation

#### 2.1. Continuity Equation

#### 2.2. Momentum Equation

#### 2.3. Transfer Matrix

#### 2.4. Transfer Matrix of the Pipe

#### 2.5. Transfer Matrix of the Entire System

## 3. Experiment

#### Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Definition | Symbol | Defintion |

a | speed of sound in the fluid (m s${}^{-1}$) | ${s}^{\ast}$ | parameter of the Laplace transform (m${}^{-1}$) |

${a}_{c}$ | complex sound speed (m s${}^{-1}$) | S | pipe cross-section (m${}^{2}$) |

${a}_{0}$ | speed of sound in the pipe (m s${}^{-1}$) | ${S}_{p}$ | pipe wall cross-section (m${}^{2}$) |

b | damping (Pa s) | ${s}_{s}$ | eigenvalue of the system (s${}^{-1}$) |

D | pipe diameter (m) | x | space coordinate (m) |

E | Young’s modulus (Pa) | t | time (s) |

${E}_{c}$ | complex Young’s modulus (Pa) | ${t}_{w}$ | pipe wall thickness (m) |

f | frequency (Hz) | u | displacement (m) |

F | force (N) | $\mathbf{u}$ | state vector |

${f}_{w}$ | width of the peak (Hz) | v,$\mathbf{v}$ | velocity (m s${}^{-1}$) |

$\mathbf{I}$ | identity matrix (-) | V | volume (m${}^{3}$) |

m | mass (kg) | W | memory function (-) |

$\mathbf{n}$ | normal vector (m) | $\epsilon $ | deformation (-) |

p | pressure (Pa) | $\zeta $ | damping ratio (Pa) |

P | area of pipe wall (m${}^{2}$) | $\eta $ | shear (dynamic) viscosity (Pa s) |

${\mathbf{P}}_{\mathbf{f}}$ | transfer matrix of fluid | ${\eta}_{2}$ | second viscosity (Pa s) |

${\mathbf{P}}_{\mathbf{m}}$ | transfer matrix of mass | $\mathbf{\Pi}$ | stress tensor (Pa) |

${\mathbf{P}}_{\mathbf{p}}$ | transfer matrix of pipe | $\rho $ | density (kg m${}^{-3}$) |

Q | flow rate (m${}^{3}$ s${}^{-1})$ | $\sigma $ | stress (Pa) |

s | parameter of the Laplace transform (s${}^{-1}$) | $\tau $ | time in convolution integral (s) |

bar | variable after the Laplace transform |

## References

- Daily, J.W.; Hankey, W.L.; Olive, R.W.; Jordaan, J.M. Resistance coefficients for accelerated and decelerated flows through smooth tubes and orifices. Trans. ASME
**1956**, 78, 1071–1077. [Google Scholar] - Caulk, D.A.; Naghdi, P.M. Axisymmetric motion of a viscous fluid inside a slender surface of revolution. J. Appl. Mech.
**1987**, 54, 190–196. [Google Scholar] [CrossRef] - Robertson, A.M.; Sequeira, A. A Director Theory Approach for Modelling Blood Flow in the Arterial System: An Alternative to Classical 1D Models. Math. Mod. Meth. Appl. Sci.
**2005**, 15, 871–906. [Google Scholar] [CrossRef] - Carapau, F.; Correia, P. Numerical simulations of a third-grade fluid flow on a tube through a contraction. Eur. J. Mech. B-Fluid
**2017**, 65, 45–53. [Google Scholar] [CrossRef] - Brunone, B.; Golia, U.M.; Greco, M. Effects of two-dimensionality on pipe transients modeling. J. Hydr. Eng.
**1995**, 121, 906–912. [Google Scholar] [CrossRef] - Abreu, J.M.; Almeida, A.B. Wall shear stress and flow behavior under transient flow in a pipe. In Proceedings of the 9th Conference on Pressure Surge, Chester, UK, 24–26 March 2004; pp. 457–476. [Google Scholar]
- Pezzinga, G. Evaluation of unsteady flow resistances by quasi-2D or 1D models. J. Hydr. Eng.
**2000**, 126, 778–785. [Google Scholar] [CrossRef] - Stokes, G.G. On the theories of internal friction of fluid in motion. Trans. Camb. Philos. Soc.
**1845**, 8, 287–305. [Google Scholar] - Tisza, L. Supersonic absorption and Stokes’ viscosity relation. Phys. Rev.
**1942**, 61. [Google Scholar] [CrossRef] - Meier, K.; Laesecke, A.; Kabelac, S. Transpor coefficients of the Lennard-Johes model fluid. III. Bulk viscosity. J. Chem. Phys.
**2005**, 122, 14513. [Google Scholar] [CrossRef] [PubMed] - Fanourgakis, G.S.; Medina, J.S.; Prosmiti, F. Determining the bulk viscosity of rigid water models. J. Phys. Chem. A
**2012**, 116, 2564–2570. [Google Scholar] [CrossRef] [PubMed] - Guo, G.J.; Zhang, Y.G. Equilibrium molecular dynamics calculation of the bulk viscosity of liquid water. Mol. Phys.
**2001**, 99, 283–289. [Google Scholar] [CrossRef] - Karim, S.M. Second viscosity coefficient of liquid. J. Acoust. Soc. Am.
**1953**, 25, 997–1002. [Google Scholar] [CrossRef] - Foldyna, J.; Sitek, L.; Habán, V. Acoustic wave propagation in high-pressure system. Ultrasonics
**2006**, 44, 1457–1460. [Google Scholar] [CrossRef] [PubMed] - Foldyna, J.; Habán, V.; Pohylý, F.; Sitek, L. Transmition of acoustic waves. In Proceedings of the International Congress On Ultrasonics, Vienna, Austria, 9–12 April 2007. [Google Scholar]
- Dukhin, A.S.; Goetz, P.J. Bulk viscosity and compressibility measurement using acoustic spectroscopy. J. Chem. Phys.
**2009**, 130, 124519. [Google Scholar] [CrossRef] [PubMed] - Holmes, M.J.; Parker, N.G.; Povey, M.J.W. Temperature dependence of bulk viscosity in water using acoustic spectroscopy. J. Phys. Conf. Ser.
**2011**, 269, 012011. [Google Scholar] [CrossRef] [Green Version] - He, X.; Wei, H.; Shi, J.; Liu, J.; Li, S.; Chen, W.; Mo, X. Experimental measurement of bulk viscosity of water based on stimulated Brillouin scattering. Opt. Commun.
**2012**, 285, 4120–4124. [Google Scholar] [CrossRef] - Streeter, V.L.; Wylie, E.B. Fluid Mechanics; Clark, B.J., Maisel, J.W., Eds.; McGraw-Hill, Inc.: New York, NY, USA, 1975. [Google Scholar]
- Keramat, A.; Tijsseling, A.S.; Hou, Q.; Ahmadi, A. Fluid-structure interaction with pipe-wall viscoelasticity during water hammer. J. Fluids Struct.
**2012**, 28, 434–455. [Google Scholar] [CrossRef] [Green Version] - Weinerowska-Bords, K. Viscoelastic model of waterhammer in single pipeline—Problems and questions. Arch. Hydro-Eng. Environ. Mech.
**2006**, 53, 331–351. [Google Scholar] - Zielke, W. Frequency-dependent friction in transient pipe flow. J. Basic Eng.
**1968**, 90, 109–115. [Google Scholar] [CrossRef] - Zielke, W. Frequency Dependent Friction in Transient Pipe Flow; The University of Michigan: Ann Arbor, MI, USA, 1966. [Google Scholar]

**Figure 7.**Model sensitivity on the second viscosity. The spot marks the value corresponding to the measurement (pipe length 1 m).

**Figure 8.**Model sensitivity on the shear viscosity. The spot marks the value corresponding to the measurement (pipe length 1 m), when the second viscosity equals zero.

Length (m) | Diameter (mm) | Wall | Eigenvalue (s${}^{-1}$) | Speed of | Damping | Young’s | Second |
---|---|---|---|---|---|---|---|

Thickness | Sound | Ratio | Modulus | Viscosity | |||

(mm) | (m s${}^{-1}$) | (MPa) | (MPa) | (kPa s) | |||

0.8 | 32.0 | 6.5 | −22.5 + $5411.9i$ | 1369.0 | 702.4 | 227.0 | 2.30 |

1.0 | 32.0 | 6.5 | −37.2 + $4436.5i$ | 1383.6 | 704.0 | 227.0 | 6.98 |

1.0 | 50.0 | 3.0 | −34.2 + $4820.8i$ | 1462.8 | 1841.8 | 225.6 | 6.44 |

2.0 | 32.0 | 6.5 | −25.2 + $2263.6i$ | 1430.0 | 702.4 | 227.0 | 18.81 |

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**MDPI and ACS Style**

Himr, D.; Habán, V.; Fialová, S.
Influence of Second Viscosity on Pressure Pulsation. *Appl. Sci.* **2019**, *9*, 5444.
https://doi.org/10.3390/app9245444

**AMA Style**

Himr D, Habán V, Fialová S.
Influence of Second Viscosity on Pressure Pulsation. *Applied Sciences*. 2019; 9(24):5444.
https://doi.org/10.3390/app9245444

**Chicago/Turabian Style**

Himr, Daniel, Vladimír Habán, and Simona Fialová.
2019. "Influence of Second Viscosity on Pressure Pulsation" *Applied Sciences* 9, no. 24: 5444.
https://doi.org/10.3390/app9245444