A variational model for bubbly liquids: Reflection from a liquid‐bubbly liquid interface

1990 ◽  
Vol 88 (S1) ◽  
pp. S131-S131
Author(s):  
J. A. Hawkins ◽  
A. Bedford
Keyword(s):  
Liquid Interface ◽  
Variational Model ◽  
Bubbly Liquid ◽  
Bubbly Liquids

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.

Pressure and Shock Waves in Bubbly Liquids

2015 ◽  
Author(s):  
Shahid Mahmood ◽  
Yungpil Yoo ◽  
Ho-Young Kwak
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Sound Propagation ◽  
Relaxation Oscillation ◽  
Bubbly Flow ◽  
Gas Bubbles ◽  
Bubbly Liquid ◽  
Bubbly Liquids ◽  
Pressure Profiles

It is well known that sound propagation in liquid media is strongly affected by the presence of gas bubbles that interact with sound and in turn affect the medium. An explicit form of a wave equation in a bubbly liquid medium was obtained in this study. Using the linearized wave equation and the Keller-Miksis equation for bubble wall motion, a dispersion relation for the linear pressure wave propagation in bubbly liquids was obtained. It was found that attenuation of the waves in bubbly liquid occurs due to the viscosity and the heat transfer from/to the bubble. In particular, at the lower frequency region, the thermal diffusion has a considerable affect on the frequency-dependent attenuation coefficients. The phase velocity and the attenuation coefficient obtained from the dispersion relation are in good agreement with the observed values in all sound frequency ranges from kHz to MHz. Shock wave propagation in bubbly mixtures was also considered with the solution of the wave equation, whose particular solution represents the interaction between bubbles. The calculated pressure profiles are in close agreement with those obtained in shock tube experiments for a uniform bubbly flow. Heat exchange between the gas bubbles and the liquid and the interaction between bubbles were found to be very important factor to affect the relaxation oscillation behind the the shock front.

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.

Shock propagation in liquids containing bubbly clusters: a continuum approach

2012 ◽  
Vol 701 ◽  
pp. 304-332 ◽  
Author(s):  
H. Grandjean ◽  
N. Jacques ◽  
S. Zaleski
Keyword(s):  
Bubble Dynamics ◽  
Cluster Structure ◽  
Computation Time ◽  
Shock Propagation ◽  
Bubbly Liquid ◽  
Global Response ◽  
Bubbly Liquids ◽  
Single Bubble Dynamics ◽  
Shock Profile ◽  
Typical Length

AbstractThe present work investigates the influence of bubble clustering on the propagation of shock waves in bubbly liquids. A continuum model is developed to describe the macroscopic response of a bubbly liquid with a cluster structure, using a two-step homogenization technique. The proposed methodology allows us to simulate shock wave propagation over long distances with a small computation time and to study the effect of bubble clustering on the shock structure. It is shown that the typical length of the shock profile is related to the global response of the clusters instead of the single-bubble dynamics, as in homogeneous bubbly flows. The accuracy of the proposed modelling is assessed through comparisons with axisymmetric simulations, in which clusters are directly specified, with given positions and sizes, and with experimental data.

scholarly journals Nonlinear Maximization of the Sum-Frequency Component from Two Ultrasonic Signals in a Bubbly Liquid

Sensors ◽  
2019 ◽  
Vol 20 (1) ◽  
pp. 113
Author(s):  
María Teresa Tejedor Sastre ◽  
Christian Vanhille
Keyword(s):  
Numerical Model ◽  
Nonlinear Resonance ◽  
Frequency Component ◽  
Nonlinear Propagation ◽  
Single Frequency ◽  
Bubbly Liquid ◽  
Higher Harmonics ◽  
Sum Frequency ◽  
Bubbly Liquids ◽  
Oscillating Bubbles

Techniques based on ultrasound in nondestructive testing and medical imaging analyze the response of the source frequencies (linear theory) or the second-order frequencies such as higher harmonics, difference and sum frequencies (nonlinear theory). The low attenuation and high directivity of the difference-frequency component generated nonlinearly by parametric arrays are useful. Higher harmonics created directly from a single-frequency source and the sum-frequency component generated nonlinearly by parametric arrays are attractive because of their high spatial resolution and accuracy. The nonlinear response of bubbly liquids can be strong even at relatively low acoustic pressure amplitudes. Thus, these nonlinear frequencies can be generated easily in these media. Since the experimental study of such nonlinear waves in stable bubbly liquids is a very difficult task, in this work we use a numerical model developed previously to describe the nonlinear propagation of ultrasound interacting with nonlinearly oscillating bubbles in a liquid. This numerical model solves a differential system coupling a Rayleigh–Plesset equation and the wave equation. This paper performs an analysis of the generation of the sum-frequency component by nonlinear mixing of two signals of lower frequencies. It shows that the amplitude of this component can be maximized by taking into account the nonlinear resonance of the system. This effect is due to the softening of the medium when pressure amplitudes rise.

Transient wave propagation in bubbly liquids

1982 ◽  
Vol 119 ◽  
pp. 347-365 ◽  
Author(s):  
D. S. Drumheller ◽  
M. E. Kipp ◽  
A. Bedford
Keyword(s):  
Variational Formulation ◽  
Momentum Equation ◽  
Thermal Dissipation ◽  
Bubbly Liquid ◽  
Balance Equations ◽  
Transient Wave ◽  
One Dimensional ◽  
Bubbly Liquids ◽  
Transient Wave Propagation ◽  
Good Agreement

A theoretical and numerical investigation of the propagation of one-dimensional waves in a bubbly liquid is presented. A variational formulation of the problem is used that yields both the linear-momentum equation and the equation that describes the oscillations of the bubbles. The compressibility of the liquid is taken into account in the formulation. The thermal dissipation is treated by solving the energy-balance equations simultaneously with the mechanical equations. Solutions are obtained by a finite-difference procedure and are compared to the experimental data of Kuznetsov et al. and Noordzij & van Wijngaarden. In some cases quite good agreement is obtained, but in others substantial errors are found. It is suggested that the observed discrepancies may be due to the breakup of the bubbles in the case of very large amplitude disturbances; the fact that the formulation does not include relative motion between the liquid and the bubbles; and possible non-planarity effects in the experiments.

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