A PHASE-PLANE DESCRIPTION OF NONLINEAR TRAVELING WAVES IN BUBBLY LIQUIDS

1999 ◽  
Vol 07 (02) ◽  
pp. 71-82
Author(s):  
A. NADIM ◽  
D. GOLDMAN ◽  
J. J. CARTMELL ◽  
P. E. BARBONE
Keyword(s):  
Equation Of State ◽  
Fixed Points ◽  
Traveling Wave ◽  
Phase Plane ◽  
Volume Fraction ◽  
Low Frequency ◽  
Traveling Wave Solutions ◽  
Bubbly Liquid ◽  
Phase Plane Analysis ◽  
Bubbly Liquids

One-dimensional traveling wave solutions to the fully nonlinear continuity and Euler equations in a bubbly liquid are considered. The elimination of velocity from the two equations leaves a single nonlinear algebraic relation between the pressure and density profiles in the mixture. On assuming the bubbles to have identical size and taking the volume fraction of bubbles in the medium to be small, an equation of state which relates the mixture pressure to the density and its first two material time-derivatives is derived. When this equation of state is linearized and combined with the laws of conservation of mass and momentum, a nonlinear, second-order, ordinary differential equation is obtained for the density as a function of the single traveling wave coordinate. A phase-plane analysis of this equation reveals the existence of two fixed points, one of which is a saddle and the other a node. A single trajectory connects the two fixed points and corresponds to a traveling shock wave solution when the Mach number of the wave, defined as the ratio of traveling wave speed to the low-frequency speed of sound in the bubbly liquid, exceeds unity. The analysis provides a qualitative explanation of the oscillations behind shocks seen in experiments on bubbly liquids.

Wave propagation in bubbly liquids at finite volume fraction

1985 ◽  
Vol 160 ◽  
pp. 1-14 ◽  
Author(s):  
Russel E. Caflisch ◽  
Michael J. Miksis ◽  
George C. Papanicolaou ◽  
Lu Ting
Keyword(s):  
Wave Propagation ◽  
Finite Volume ◽  
Small Volume ◽  
Volume Fraction ◽  
Low Frequency ◽  
Multiple Scale ◽  
Bubbly Liquid ◽  
Bubbly Liquids ◽  
Frequency Regime ◽  
Small Volume Fraction

We derive effective equations for wave propagation in a bubbly liquid in a linearized low-frequency regime by a multiple-scale method. The effective equations are valid for finite volume fraction. For periodic bubble configurations, effective equations uniformly valid for small volume fraction are obtained. We compare the results to the ones obtained in a previous paper (Caflisch et al. 1985) for a nonlinear theory at small volume fraction.

scholarly journals Vibration of a stiffened pipe filled with a bubbly liquid: analysis of resonance frequencies in function of bubble fraction

2021 ◽  
Vol 263 (5) ◽  
pp. 1008-1018
Author(s):  
Sanae Serbout ◽  
Laurent Maxit ◽  
Frédéric Michel
Keyword(s):  
Cylindrical Shell ◽  
Volume Fraction ◽  
Low Frequency ◽  
Acoustic Properties ◽  
Bubbly Liquid ◽  
Low Frequencies ◽  
Stiffened Cylindrical Shell ◽  
Resonance Frequencies ◽  
Gas Volume Fraction ◽  
Gas Volume

The characterization of the presence of bubbles in industrial fluid circuits may be extremely important for many safety issuses. It is well known that the acoustic properties of liquids can be drastically modified by a small amount of gaz content in the liquid. At sufficiently low frequencies, the speed of sound depends primarily on the gas volume fraction. The variation of the gas fraction may then induce some variations in the vibroacoustic behavior of the pipe transporting the liquid. Analysis of the pipe vibrations can then help in the monitoring of the bubble presence. In such a context, the aim of this study is to show how the the presence of bubbles in the liquid could affect the resonance frequencies of the pipe. A numerical vibroacoustical model has been developed to predict the vibroacoustical behavior of a stiffened cylindrical shell filled with a bubbly liquid exhibiting low frequency resonances. The model, experimentally verified with a well-characterized bubbly liquid, is then used to analyse the frequency shifts of the shell resonances in function of the bubble. Keywords : pipe, heavy fluid, numerical modelling, circumferential admittance approach, cylindrical shell, resonance frequency, void fraction

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

A Novel Fault-location Method for HVDC Transmission Lines Based on Concentric Relaxation Principle and Wavelet Packet

2020 ◽  
Vol 13 (5) ◽  
pp. 705-716
Author(s):  
Congshan Li ◽  
Ping He ◽  
Feng Wang ◽  
Cunxiang Yang ◽  
Yukun Tao ◽  
...  
Keyword(s):  
Transmission Lines ◽  
Traveling Wave ◽  
Fault Location ◽  
Wave Attenuation ◽  
Wavelet Packet ◽  
Low Frequency ◽  
Evaluation Process ◽  
Location Method ◽  
Hvdc Transmission ◽  
Instantaneous Energy

Background: A novel fault location method of HVDC transmission line based on a concentric relaxation principle is proposed in this paper. Methods: Due to the different position of fault, the instantaneous energy measured from rectifier and inverter are different, and the ratio k between them is the relationship to the fault location d. Through the analysis of amplitude-frequency characteristics, we found that the wave attenuation characteristic of low frequency in the traveling wave is stable, and the amplitude of energy is larger, so we get the instantaneous energy ratio by using the low-frequency data. By using the method of wavelet packet decomposition, the voltage traveling wave signal was decomposed. Results: Finally, calculate the value k. By using the data fitting, the relative function of k and d can be got, that is the fault location function. Conclusion: After an exhaustive evaluation process considering different fault locations, fault resistances, and noise on the unipolar DC transmission system, four-machine two-area AC/DC parallel system, and an actual complex grid, the method presented here showed a very accurate and robust behavior.

Traveling wave solutions of the one-dimensional Boussinesq paradigm equation

2013 ◽  
Author(s):  
V. M. Vassilev ◽  
P. A. Djondjorov ◽  
M. Ts. Hadzhilazova ◽  
I. M. Mladenov
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
One Dimensional ◽  
Wave Solutions ◽  
The One
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