Infinite Liquid Research Articles

The diffusion stability of an externally driven cavitation bubble in micro-confinement

The diffusion stability of a single cavitation bubble in a spherical liquid cell surrounded by an infinite elastic solid is considered. The time-periodic pressure in the solid far away from the liquid cell is used as an external driving, which initiates bubble oscillations along with the gas diffusion process in the bubble-in-cell system. The work is based on the engineering approximation according to which the bubble growth/reduction is considered on average, assuming that during the period of the external driving the mass of gas in the bubble does not noticeably change. This theory predicts the existence of stably oscillating bubbles in confined liquid undergoing an external driving force. Three possible diffusion regimes are revealed: 1) total bubble dissolution, 2) partial bubble dissolution, and 3) partial bubble growth, where the last two regimes provide the diffusion stability in the bubble-in-cell system. The parametric study of the influence of the gas concentration dissolved in the liquid on the resulting stable bubble size is conducted. The obtained results are compared with the results for the case of the stable bubble oscillations in the pressure sound field in a bulk (infinite) liquid. The theoretical findings of the present study can be used for improvement of the modern applications of ultrasound technology.

Numerical simulation for pressure fluctuations caused by the growth of a single boiling bubble

The pressure fluctuations caused by the rapid volume changes during the initial growth stage for a single bubble are crucial excitations of boiling induced vibration. Therefore, it is necessary to accurately model the dynamic behavior of bubbles during this stage to obtain reasonable excitation forces, which can be used to estimate the vibration response of structures. The present study develops a dynamics model, which is verified and validated to be applicable to the growth of spherical bubbles in infinite superheated liquid and hemispherical bubbles growing on a vertical heated surface. The effects of mass transfer are considered for the vapor-liquid interface motion, and the temperature variations at the liquid side are determined by the finite difference method. The numerical data match well with the theoretical and experimental results, and the characteristics of the excitation forces caused by a single boiling bubble are obtained and analyzed. In addition, the influences of superheat, accommodation coefficient for phase change, and the temperature gradient at the liquid side on excitation forces caused by bubble growth are studied. The results show that the temperature distributions near the heated surface strongly affect the frequency domain characteristics of the excitation forces.

Thin Au/Ag/Cu-Water Nanoliquid Film Flow Over a Porous Unsteady Stretchable Cylinder

Most of the previous works on the thin nanoliquid-film (NLF) flow on a stretchable horizontal cylinder are limited to the boundary-layer flow (BLF) of infinite depth liquid. In this article, we have removed the above restrictions to examine the unsteady thin NLF flow on a porous stretchable cylinder by considering full momentum equations under the influence of thermal radiation and suction/injection. The implicit finite difference technique is utilized to obtain the numerical solutions to the problem. The present numerical solution is validated with the analytical solution obtained by Maity et al.41 for the isothermal case. It is found that the film thickness for Au/water nanoliquid is higher in comparison with Ag/water and Cu/water nanoliquids, respectively. The film thickness of nanoliquid enhances for rising magnitude of nanoparticle volume fraction (φ), porosity parameter (β), and Hartmann number (Mn). The film thickness of the nanoliquid increases or decreases as injection or suction (V) takes place through the surface of the cylinder. Furthermore, the sign of temperature gradient θR changes inside the nanoliquid depending on the values of other parameters. The percentage of changes in the film height is presented with the different parameters of the flow problem.

Shock waves at final stages of cavity collapse in non-homogeneous liquid with divergenceless flow

We show the emergence of shock waves at the final stages of the complete collapse of a spherical cavity in a liquid with a smoothly decreasing density. The well-known Rayleigh assumption of fluid homogeneity is altered, while maintaining that of divergenceless flow. The fundamental difference between both infinite liquids is that his has an infinite mass, while ours, a finite one. Given the ease of deformation of non-homogeneous media in relation to homogeneous ones, as observed in several materials, all Rayleigh results are modified, including the cavity wall speed and acceleration, total time of complete collapse, and distribution of pressure in the infinite liquid. Rather than the homogeneous Rayleigh fluid, our non-homogeneous liquid can support a finite local sound speed. As a result, we succeed to show the emergence of shock patterns at the final stages of the cavity collapse. The analytical formulation is compared with underwater implosion and explosion experiments and simulations. Possible applications as a benchmark test for hydrocodes are briefly discussed.

Modeling of vapor bubble dynamics considering the metastable fluid state

This study investigates the dynamic modeling of a spherical vapor bubble in an infinite liquid. Initially, the liquid is in a state of saturation, but then, there is a change in ambient pressure. Unlike previous studies that assumed vapor saturation, our model fully considers the metastable fluid state of subcooled vapor and superheated liquid. The theoretical model is based on the state equation of vapor and the kinetic theory of gases, allowing the visualization of the effects of superheating and subcooling on the bubble pressure and the phase transition rate. The accuracy of the numerical solutions for the bubble collapse and growth is confirmed by experimental results using water as the working fluid. Two approaches of heat transfer simplification are validated in the numerical solution, and an adhesion coefficient within the range of 0.1–1 is recommended for the calculation. Additionally, this study provides insights into how the metastable fluid state affects the bubble pressure and phase transition, especially in the early stages of bubble collapse. Furthermore, the conservation of vapor mass inside the bubble is also demonstrated.

Plume Spreading Due to Floor Conditions of a Plunging Liquid Jet Using Stereographic Backlit Imaging

Abstract Plunging liquid jets are a multiphase flow studied to understand how gas is entrained in a liquid and the resulting mixing capabilities. From existing literature, it has been hypothesized that rising bubbles play a noticeable role in the multiphase hydrodynamics of the plunging liquid jet bubble plume and that separating the rising bubbles from the incoming liquid jet can result in a significant increase in the depth of the bubble plume. This study explores the effects of separating the incoming liquid jet from the rising bubble plume through floor interactions and compression effects due to a finite tank depth. This configuration is found in many natural and industrial systems, but not within published literature. Using existing theoretical models of infinite depth plunging liquid jet systems, which align reasonably well with captured baseline data, two models are developed for when floor interactions are present, one theoretical and one empirical. The models show a correlation between plume spread and floor interaction with the incoming plunging liquid jet bubble plume. Data acquired through stereographic backlit imaging over a range of flow rates show a reasonable agreement with the proposed models.

A numerical study on bubble dynamics in hydrodynamic cavitation

This study examines the dynamic interactions of single and double bubbles in infinite liquid domains within hydrodynamic cavitation systems notably within a single-hole orifice-plate reactor. Critical parameters such as cavitation radius and implosion time are analyzed by solving the Rayleigh-Plesset and Keller-Miksis models for compressible fluids. This analysis is extended to multiple oscillations after the initial cavitation implosion has occurred. Furthermore, a numerical solution of both the Rayleigh-Plesset and Keller-Miksis models is employed to demonstrate the influence of key factors, including the polytropic index in single bubble systems and inter-bubble distances, along with secondary Bjerknes forces in double-bubble interactions. The findings highlight the significant impact of one bubble on the oscillations of neighbouring bubbles, resulting in substantial variations in the overall dynamics of the system.

Dynamics of an oscillating cavitation bubble within a narrow gap

The oscillation characteristics of a bubble in a confined space have important implications for various applications, including liquid pumping and mixing and particle conveyance in microfluidic devices. In this study, analytical solution with second-order accuracy and numerical solution are derived for the free oscillation of a single bubble in a narrow gap between parallel plates, and the applicability to dimensionless initial values of the analytical solutions is clarified. Moreover, the free-oscillation characteristics of the bubble within the gap are explored and described and are compared to those of a bubble in an infinite liquid. The primary conclusions are as follows: (1) The inherent nature of bubble oscillation in a gap is significantly influenced by the bubble equilibrium radius, and the oscillation amplitude of different orders of the analytical solution is significantly influenced by the dimensionless initial radius. (2) The difference between the natural frequency and acoustic damping constant during bubble oscillation in a gap and those in an infinite liquid decreases with increasing equilibrium radius, and the value of the difference is not less than 50%. (3) Within the gap, the bubble radius, wall velocity, and wall acceleration of a bubble in a narrow gap predicted by the bubble equation dramatically differ from those of a bubble in an infinite liquid, with the differences increasing with the dimensionless initial radius, where the values of the differences in the acceleration can be as high as the order of 104%.

Interaction of Two Rigid Spheres Oscillating in an Infinite Liquid under the Control of a Magnetic Field

The transient creeping motion of two rigid spheres oscillating in a boundless viscous fluid beneath the impact of the magnetic field is investigated. There is no slippage associated with a Stokes flow on the two rigid spherical surfaces with different sizes and radii. The solutions can be obtained using the boundary collocation scheme at low Reynolds numbers. The unsteady real and imaginary drag coefficients are estimated on the hard spherical particles. These coefficients are computed in tables for various parameters and illustrated graphically. The frequency parameter and Hartmann number also play a significant role in this study where the drag coefficients decrease or increase by 100 percent after the value of κ = 5.0 . Using available literature data, we tested the accuracy and reliability of our results.

Equilibria and stabilities of a confined floating cylinder

A two-dimensional system with a floating cylinder confined between two parallel vertical stationary plates partially immersed in an infinite liquid bath in a downward gravity field is considered. The equilibrium states of the system are investigated using the Young–Laplace equation in two dimensions. According to the symmetry of menisci at both sides of the cylinder, the equilibrium states are classified into three types: the equilibria of fully symmetric menisci, the equilibria of partially symmetric menisci and the equilibria of asymmetric menisci. The study is then extended to investigate the stabilities of the confined cylinder with bifurcation theory. Results show that there can be at most two stable regions in the bifurcation diagram. For different plates’ contact angles, there are five representative types of bifurcation behaviours for either the case of one stable region or the case of two stable regions. In comparison with the case of an unconfined cylinder, the confinement by two hydrophobic plates with a small spacing can assist the stable interfacial floatation of the confined cylinder with a large weight.

Experimental study on single bubble growth of R32 + R1234yf binary mixtures during saturated pool boiling

R32 + R1234yf mixtures are considered highly promising refrigerants. Understanding the bubble growth characteristics of mixtures is vital for understanding their boiling heat-transfer performance. In this study, pool boiling experiments were conducted to investigate the single-bubble growth characteristics of R32, R1234yf, and their binary mixtures. The bubble growth curves for each test refrigerant were obtained in the superheat range of 1.3–6.2 K. The results showed that the asymptotic bubble growth rates of the pure refrigerants were slower than those predicted using previous theoretical models, which were developed for isolated bubbles growing in an infinite superheated liquid. The bubble growth constants of the mixtures were lower than those of the pure components, whereas the growth exponents were greater than the “ideal growth exponent.” The sudden decrease in the effective wall superheat and sharp increase in mass transfer resistance slowed the initial bubble growth rate. In the late stage of growth, the microlayer that had not yet dried out provided stable heat input to the bubble, resulting in a higher asymptotic growth rate than that of pure components. A set of dimensionless correlations for the bubble growth rate of mixtures is proposed, and they predicted the data in this study within a ± 15% deviation band.

Cavitation bubbles with a tunable-surface-tension thermal lattice Boltzmann model

The effects of surface tension and initial input energy on cavitation properties based on a tunable-surface-tension large-density-ratio thermal lattice Boltzmann method pseudo-potential model are investigated. The validity and superiority of the proposed model in simulating the D2 law, Laplace law, and revised thermal two-dimensional Rayleigh–Plesset equation are demonstrated. Moreover, the lattice Boltzmann method was used to study the effects of varied surface tension on cavitation bubble properties for the first time, and the maximum surface tension-to-minimum surface tension ratio of 25 is utilized, which is highly improved compared with previous numerical simulations (<4) and makes our result more clear. The simulation results indicate that for an infinite liquid, the increase in the surface tension will improve the collapse intensity of cavitation bubbles, increasing the collapse pressure, velocity, and temperature and meanwhile reducing the bubble lifetime. For the cavitation bubbles collapsing near a neutral wall, with an increase in the surface tension, the collapse pressure, temperature, and cavitation bubble lifetime trends are the same as in the infinite liquid. However, the collapse velocity is affected by the neutral wall, and the micro-jet becomes wider and shorter. The maximum cavitation bubble radius in an infinite liquid is nearly linearly proportional to the input initial energy. An increase in the surface energy reduces the maximum radius of the cavitation bubbles, while increasing the pressure energy and thermal energy promotes the maximum radius of the cavitation bubbles. This series of simulations proves the feasibility of the proposed model to investigate the thermodynamic process of the cavitation bubbles with high density ratios, wide viscosity ratios, and various surface tensions.

Highly efficient separation of phenolic compounds from low-temperature coal tar by composite extractants with low viscosity

• Imidazolium IL-solvent composite extractants were applied for separation of phenolic compounds from oil. • The suitable composite extractants were identified by COSMO-RS calculation and viscosity determination. • [Emim][HSO 4 ]:EG (1:2) was selected due to its high phenol extraction efficiency, low oil entrainment, and low viscosity. • The separation mechanism was investigated by σ -profile, σ -potential, and FT-IR. Highly efficient separation of phenolic compounds from low-temperature coal tar (LTCT) is significant in the industry. In this work, composite extractants composed of imidazolium-based ionic liquids (ILs) and solvents were used to separate phenolic compounds from model oil and LTCT. First, the COSMO-RS model was used to screen composite extractants that combined the infinite dilution thermodynamic indexes and liquid–liquid equilibria (LLE) calculations. Meanwhile, the separation mechanism was analyzed by σ -profile and σ -potential. Then, the reliability of the COSMO-RS model was validated by LLE experiments and FT-IR. The viscosity of composite extractants with high separation performance was further determined. Among the composite extractants, 1-ethyl-3-methyl-imidazolium hydrogen sulfate ([emim][HSO 4 ])/ethylene glycol (EG) was selected as the most suitable extractant in the separation process. The m -cresol extraction efficiency and cumene entrainment were 98.1% and 8.3%, respectively, at a [emim][HSO 4 ]:EG molar ratio of 1:2, a temperature of 25 ℃, and a composite extractant: model oil mass ratio of 1:1. Moreover, the viscosity of [emim][HSO 4 ]:EG (1:2) was only 51.8 cP at 25 ℃ and decreased by 95.6% compared with pure [emim][HSO 4 ], which enhanced the mass transfer in the separation process. The [emim][HSO 4 ]:EG (1:2) could be regenerated and reused without significant reduction in separation performance. Finally, [emim][HSO 4 ]:EG (1:2) was applied to separate phenolic compounds from LTCT with a phenol extraction efficiency of more than 99.9% and a neutral oil entrainment of 8.3%, providing a promising prospect for the separation of phenolic compounds.

Natural frequency discontinuity of vertical liquid sheet flows at transcritical threshold

The natural and forced dynamic response of a gravitational plane liquid sheet (curtain) of finite length interacting with an unconfined gaseous ambient is numerically and experimentally investigated. The global eigenvalue spectrum obtained by means of a linear inviscid one-dimensional model, accounting for the coupling between the curtain motion and the ambient pressure disturbances, clearly shows an abrupt increase (jump) in the characteristic natural frequency of the flow when the supercritical ( $We>1$ ) to subcritical ( $We<1$ ) transition occurs, with the Weber number $We$ defined as the ratio between inertia and capillary forces. On the other hand, the numerical simulation of the forced sheet response does not show any discontinuity between supercritical and subcritical conditions, as recently found by Torsey et al. (J. Fluid Mech., vol. 910, 2021, pp. 1–14) in the case of an infinite liquid sheet subjected to imposed ambient pressure disturbances not coupled with the curtain motion. It is argued that the forced liquid sheet behaviour varies continuously in shape and amplitude between the two regimes, not depending on the specific liquid–gas interaction model considered, whilst the natural frequency of the finite flow system does undergo a discontinuity, which can be theoretically predicted by the model of sheet–ambient interaction employed here. As a major result, the experimental evidence of the natural frequency jump is for the first time provided as well.

The Rayleigh prolongation factor at small bubble to wall stand-off distances

The Rayleigh collapse time is the time it would take to shrink an empty spherical bubble in an infinite liquid domain to zero size, which is a function of ambient pressure and initial bubble radius. If a solid boundary is located in the vicinity of the shrinking or collapsing bubble, then liquid flow is hindered, such that the collapse time is prolonged. This can be quantified with the Rayleigh prolongation factor $k$ . Here, we provide $k$ for intermediate to smallest bubble to wall stand-off distances. It is measured with single laser-induced cavitation bubbles in water close to a solid boundary. Maximum bubble radii are determined from microscopic high-speed imaging at one million frames per second. Collapse times are measured acoustically via the acoustic transients emitted during bubble seeding and collapse. The experimental findings are compared, with good agreement, to numerical simulations based on a volume of fluid method. As a result, a polynomial fit of $k$ versus stand-off distance is given for the near-wall bubble collapse in water. Then the influence of the viscosity on $k$ is studied numerically in the near-wall regime.

Nuclear matter as a liquid phase of spontaneously broken semiclassical SU(2)L×SU(2)R chiral perturbation theory: Static chiral nucleon liquids

The standard model of particle physics (SM), augmented with neutrino mixing, is either the complete theory of interactions of known particles at energies naturally accessible on earth, or very nearly so, with a Lagrangian symmetric under the global $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{R}$ symmetry of two-massless-quark QCD, spontaneously broken to $\mathrm{SU}{(2)}_{L+R}$. Using naive dimensional operator power counting that enables perturbation and truncation in inverse powers of ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}\ensuremath{\approx}1\phantom{\rule{4.pt}{0ex}}\text{GeV}$, we show that, to $\mathcal{O}({\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB})$ and $\mathcal{O}({\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}^{0})$, SU(2) chiral perturbation theory [SU(2)$\ensuremath{\chi}\mathrm{PT}$] of protons, neutrons, and pions admits a liquid phase, with energy required to increase or decrease the nucleon density. We further show that in the semiclassical approximation---i.e., quantum nucleons and classical pions---``pionless SU(2)$\ensuremath{\chi}$PT'' emerges in that chiral liquid: soft static infrared Nambu-Goldstone-boson pions decouple from ``static chiral nucleon liquids'' (Static$\ensuremath{\chi}$NLs). This vastly simplifies the derivation of saturated nuclear matter (the infinite liquid phase) and of finite microscopic liquid drops (ground-state heavy nuclides). Static$\ensuremath{\chi}$NLs are made entirely of nucleons. They have even parity, total spin zero, even proton number $Z$, and even neutron number $N$. The nucleons are arranged so local expectation values for spin and momentum vanish. We derive the Static$\ensuremath{\chi}$NL effective Lagrangian from semiclassical SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$symmetries to order ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}$ and ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}^{0}$, including all relativistic four-nucleon operators that survive Fierz rearrangement in the nonrelativistic limit and SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$fermion exchange operators and isovector exchange operators which are important when $Z\ensuremath{\ne}N$. Mean-field Static$\ensuremath{\chi}$NL nontopological solitons are true solutions of SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$semiclassical symmetries; e.g., they obey all conserved vector current (CVC) and partially conserved axial current (PCAC) conservation laws. They have zero internal and external pressure. The nuclear liquid-drop model and Bethe--von Weizs\"acker semiempirical mass formula emerge---with correct nuclear density and saturation and asymmetry energies---in an explicit Thomas-Fermi construction.

Linear stability of electrocapillary convection in an infinite liquid layer

Electrocapillary convection is the flow driven by the interfacial shear stress induced by a surface-charge density and electric fields. It has become an important topic in electrohydrodynamics recently. The present work examine the convective instability for electrocapillary liquid layers by the modal and non-modal approaches. The modal analysis finds two kinds of modes, namely purely hydrodynamic and electrohydrodynamic modes, while both of them are stable. In the non-modal analysis, the amplifications of initial and external disturbances are measured by the transient growth and the response function, respectively. Both the perturbation velocity and surface-charge density can be amplified significantly.

The singular hydrodynamic interactions between two spheres in Stokes flow

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centers, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs), which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by the use of a bipolar coordinate system, we derive the stream function for the flow as the Reynolds number (Re) tends to zero and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behavior of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon the known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with the existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix R in a number of situations in which positivity is destroyed for multipole/perturbative methods.

Energy characteristics of a radiating thin layer of bubbles in a liquid

The radiation of a thin bubble layer in an infinite liquid excited by a single wave of large amplitude is studied. It is shown that, after excitation, the layer emits a strong nonlinear wave in both directions. The radiation of the layer becomes close to sinusoidal with one line in the spectrum at large times. It is shown that in the nonlinear phase of the sound wave radiation, the time variation of the bubble energy in the layer is characterized by a stepwise dependence.

Hydrostatics and stability of a floating elliptical cylinder with surface tension effects for different eccentricities and Bond numbers

A model dealing with the conditions of equilibrium and initial stability of floating bodies in two dimensions with an arbitrary cross-section has been proposed recently (J. Fluid Mech. 847:489-519, 2018). The method is applied to investigate the hydrostatics and stability of the configuration of an elliptical cylinder floating on an infinite liquid bath for a range of two dimensionless parameters: the eccentricity of the elliptical cross-section e and the Bond number Bo. It is shown that the prolate cylinder with a greater eccentricity e can bear a greater pressing force and a greater moment and therefore is more stable. For a great Bond number, the greater the eccentricity e of the cylinder, the greater the inclination angle corresponding to maximum moment. For a small Bond number, the opposite is true. However, the inclination angle of the cylinder corresponding to the maximum moment does not obey this rule. The maximum inclination angle occurs at an intermediate Bond number.