Single Bubble Research Articles

Non-Spherical Cavitation Bubbles: A Review

Cavitation is a phase-change phenomenon from the liquid to the gas phase due to an increased flow velocity. As it causes severe erosion and noise, it is harmful to hydraulic machinery such as pumps, valves, and screw propellers. However, it can be utilized for water treatment, in chemical reactors, and as a mechanical surface treatment, as radicals and impacts at the point of cavitation bubble collapse can be utilized. Mechanical surface treatment using cavitation impacts is called “cavitation peening”. Cavitation peening causes less pollution because it uses water to treat the mechanical surface. In addition, cavitation peening improves on traditional methods in terms of fatigue strength and the working life of parts in the automobile, aerospace, and medical fields. As cavitation bubbles are utilized in cavitation peening, the study of cavitation bubbles has significant value in improving this new technique. To achieve this, many numerical analyses combined with field experiments have been carried out to measure the stress caused by bubble collapse and rebound, especially when collapse occurs near a solid boundary. Understanding the mechanics of bubble collapse can help to avoid unnecessary surface damage, enabling more accurate surface preparation, and improving the stability of cavitation peening. The present study introduces three cavitation bubble types: single, cloud, and vortex cavitation bubbles. In addition, the critical parameters, governing equations, and high-speed camera images of these three cavitation bubble types are introduced to support a broader understanding of the collapse mechanism and characteristics of cavitation bubbles. Then, the results of the numerical and experimental analyses of non-spherical cavitation bubbles are summarized.

An optical system for cellular mechanostimulation in 3D hydrogels

We introduce a method utilizing single laser-generated cavitation bubbles to stimulate cellular mechanotransduction in dermal fibroblasts embedded within 3D hydrogels. We demonstrate that fibroblasts embedded in either amorphous or fibrillar hydrogels engage in Ca2+ signaling following exposure to an impulsive mechanical stimulus provided by a single 250 µm diameter laser-generated cavitation bubble. We find that the spatial extent of the cellular signaling is larger for cells embedded within a fibrous collagen hydrogel as compared to those embedded within an amorphous polyvinyl alcohol polymer (SLO-PVA) hydrogel. Additionally, for fibroblasts embedded in collagen, we find an increased range of cellular mechanosensitivity for cells that are polarized relative to the radial axis as compared to the circumferential axis. By contrast, fibroblasts embedded within SLO-PVA did not display orientation-dependent mechanosensitivity. Fibroblasts embedded in hydrogels and cultured in calcium-free media did not show cavitation-induced mechanotransduction; implicating calcium signaling based on transmembrane Ca2+ transport. This study demonstrates the utility of single laser-generated cavitation bubbles to provide local non-invasive impulsive mechanical stimuli within 3D hydrogel tissue models with concurrent imaging using optical microscopy. Statement of significanceCurrently, there are limited methods for the non-invasive real-time assessment of cellular sensitivity to mechanical stimuli within 3D tissue scaffolds. We describe an original approach that utilizes a pulsed laser microbeam within a standard laser scanning microscope system to generate single cavitation bubbles to provide impulsive mechanostimulation to cells within 3D fibrillar and amorphous hydrogels. Using this technique, we measure the cellular mechanosensitivity of primary human dermal fibroblasts embedded in amorphous and fibrillar hydrogels, thereby providing a useful method to examine cellular mechanotransduction in 3D biomaterials. Moreover, the implementation of our method within a standard optical microscope makes it suitable for broad adoption by cellular mechanotransduction researchers and opens the possibility of high-throughput evaluation of biomaterials with respect to cellular mechanosignaling.

Dynamics of single cavitation bubble collapse jet under particle-wall synergy

The interaction between a particle and a cavitation bubble significantly influences the erosive effect on the wall surface of flow passage components in fluid machinery. This paper investigates the dynamics of a single bubble collapse jet under the synergetic effects of a particle and a wall, using Kelvin impulse theory and high-speed photographic experiments. A theoretical model to predict the intensity and direction of the collapse jet at arbitrary locations near the particle and the wall is constructed on the basis of the image method and Weiss's theorem. The accuracy of the model is verified by comparison with a large number of experimental results. The mechanisms underlying the relative contributions of the particle and wall to the behavior of jet intensity and direction are explored. The effects of key parameters on jet intensity and direction are also quantitatively analyzed, including the relative positions of the particle, wall, and the bubble and the dimensionless particle radius. The main conclusions are as follows: (1) the particle will cause a deflection in the direction of the collapse jet near the wall, leading to the formation of a jet attraction zone. The proposed theoretical model effectively predicts the spatial location of this zone. (2) There exists a region in which the jet is weak, and there is a jet equilibrium point with zero impulse between the particle and the wall. The position of this equilibrium point gradually approaches the wall in a nonlinear manner with increasing particle size and in a quasi-linear manner with decreasing particle–wall distance. (3) When the particle and the bubble are the same distance from the wall, the jet direction gradually changes from toward the particle to vertical to the wall in a nonlinear manner as the bubble–particle distance increases. Moreover, the effective range of the particle's influence on the jet direction decreases as the particle–wall distance decreases.

Wettability-dependent dissolution dynamics of oxygen bubbles on Ti64 substrates

In this study, the dissolution of a single oxygen bubble on a solid surface, here Titanium alloy Ti64, in ultrapure water with different oxygen undersaturation levels is investigated. For that purpose, a combination of shadowgraph technique and planar laser-induced fluorescence is used to measure simultaneously the changes in bubble geometry and in the dissolved oxygen concentration around the bubble. Two different wettabilities of the Ti64 surface are adjusted by using plasma-enhanced chemical vapor deposition. The dissolution process on the solid surface involves two distinct phases, namely bouncing of the oxygen bubble at the Ti64 surface and the subsequent dissolution of the bubble, primarily by diffusion. By investigating the features of oxygen bubbles bouncing, it was found that the boundary layer of dissolved oxygen surrounding the bubble surface is redistributed by the vortices emerging during bouncing. This establishes the initial conditions for the subsequent second dissolution phase of the oxygen bubbles on the Ti64 surfaces. In this phase, the mass transfer of O2 proceeds non-homogeneously across the bubble surface, leading to an oxygen accumulation close to the Ti64 surface. We further show that the main factor influencing the differences in the dynamics of O2 bubble dissolution is the variation in the surface area of the bubbles available for mass transfer, which is determined by the substrate wettability. As a result, dissolution proceeds faster at the hydrophilic Ti64 surface due to the smaller contact angle, which provokes a larger surface area.

Morphological Characterization of Tissue Destruction According to the Distance Between Holmium:YAG Laser Tip and Tissue Surface.

Little is known about the soft tissue destruction by holmium laser clinically used for holmium laser enucleation of the prostate (HoLEP), subject to the distance between the laser fiber tip and the tissue surface. We aimed to investigate the impact of the distance between the laser fiber tip and the phantom surface (DLP) on a soft tissue phantom (STP) in relation to the surgical modes of HoLEP. STP responses to the laser pulses produced by a commercial holmium:yttrium aluminum garnet (Holmium:YAG) laser at an output setting 2 J were observed at different values of the DLP (0, 1, 2, 3, and 4 mm) to look at (1) the single laser pulse-induced cavitation bubble and its penetration into the STP, (2) the STP destruction by a single pulse, (3) the STP destruction by 60 pulses repeated at 12 Hz, and (4) the thermal effect by the multiple pulses visualized on a thermosensitive bovine serum albumin (BSA) STP. We observed that the laser pulse produced a heated gas bubble in water centered at the laser fiber tip. The bubble shape depended on the DLP. The bubble completely penetrated into the STP at the DLP of 0 mm and the penetration decreased with the DLP. The size of the destruction of the STP by the laser pulses was shown to decrease as the DLP increased. Test with the BSA STP showed that, at the DLP of 3 mm, the destruction became insignificant while the thermal effects were still effective. We illustrated that soft tissue destruction by the Holmium:YAG laser is associated with cavitation effects. We provide for the first time experimental evidence for various surgical modes in HoLEP such as incision and hemostasis in relation to the DLP.

Dynamics of a single cavitation bubble near an oscillating boundary

Cavitation and its effects are well investigated, especially single bubble cavitation and its collapse near rigid and elastic boundaries. In our current article, we investigated novel experiments of a single cavitation bubble near an oscillatory boundary. We generated the cavitation bubble by laser focusing in water. A flat glass plate was fixed to the shaft of the magnetostriction oscillator coil. We investigated the dynamics of bubbles at two relative wall distances (ratio of the distance between the bubble center and plate surface to the maximum radius of the bubble) of the bubble from the glass plate in combination with four modes of oscillation. Each mode has specific frequency and amplitude of oscillation. The high-speed camera captured the dynamics of the bubble using the back-illumination method with a framing rate of 120Kfps and simultaneously we used an optical CMOS sensor to measure the oscillation of the glass plate. We presented a clear comparison among the bubble dynamics near stationary and oscillating plates with parameters such as oscillating modes and direction. We correlated the dynamics of the bubble with the motion of the plate. In addition, we highlighted the differences including the characteristics of bubble shape and jetting that occurred during the collapse phase. The comparison of the time histories of the bubble’s equivalent size postulated that the bubble’s collapse times vary significantly in some cases compared to the bubble’s dynamics near the stationary plate. In all cases, we noticed the shortening of the bubble’s collapsing time, i.e. accelerated collapses. In our findings, we noticed a collapse times reduction of about 4–15%. Our finding signifies the importance of introducing the oscillation of the boundaries to obtain effective energy concentration over the time during the collapse. Our study also suggests that forced oscillation of boundaries is undesirable for destructive cavitation effects. The method we suggested for the manipulation of bubble dynamics holds potential for enhancing the efficiency of applications such as lithotripsy in biomedical devices, actuation and micro pumping in microfluidic devices, and effective semiconductor surface cleaning. Not but least, obtained results can be used as benchmark in future for validating numerical methods.

Micromixer driven by bubble-induced acoustic microstreaming for multi-ink 3D bioprinting.

Recently, the 3D printing of cell-laden hydrogel structures, known as bioprinting, has received increasing attention owing to advances in tissue engineering and drug screening. However, a micromixing technology that efficiently mixes viscous bioinks under mild conditions is needed. Therefore, this study presents a novel method for achieving homogeneous mixing of multiple inks in 3D bioprinting through acoustic stimulation. This technique involves generating an acoustic microstream through bubble oscillations inside a 3D bioprinting nozzle. We determined the optimal hole design for trapping a bubble, hole arrangement, and voltage for efficient mixing, resulting in a four-fold increase in mixing efficiency compared to a single bubble arrangement. Subsequently, we propose a nozzle design for efficient mixing during bioprinting. The proposed nozzle design enabled the successful printing of line structures with a uniform mixture of different viscous bioinks, achieving a mixing efficiency of over 80% for mixing 0.5-1.0 wt% sodium alginate aqueous solutions. Additionally, acoustic stimulation had no adverse effects on cell viability, maintaining a high cell viability of 88% after extrusion. This study presents the first use of a bubble micromixer in 3D bioprinting, demonstrating gentle yet effective multi-ink mixing. We believe this approach will broaden 3D printing applications, particularly for constructing functional structures in 3D bioprinting.

Research on factors affecting bubble and particle behavior in a three-phase flotation system based on COMSOL

ABSTRACT To investigate the impact of various parameters on bubble and particle dynamics in a flotation system, COMSOL software was utilized. The “two-phase flow – Level Set” module coupled with the “fluid particle interaction” module was used to explore the behavior of single bubbles and particle groups. The bubbles were set as a discrete phase to represent a group of floating bubbles. By combining the “Single-phase flow” module with dual “fluid-particle interaction” modules, we examined the interactions between bubble populations and particles within the system. The research results show that as the bubble diameter increases, the carrying effect of the bubbles on the particles becomes stronger. When the bubble radii are 1, 1.25, 1.5, 1.75, and 2 mm respectively, the maximum movement speed of the particles in the area increases by 3.0%, 2.3%, 5.1%, and 2.7%. The average speed of the particles increases by 7.9%, 7.3%, 9.0%, and 10.0%. Increasing the mass flow rate can increase the collision probability between particles and bubbles, thus increasing the flotation rate. However, a too high mass flow rate will make the state of bubbles unstable. When the mass flow rate is less than 1.5 kg/s, the bubbles remain stable. When the mass flow rate exceeds 1.5 kg/s, the severity of bubble splitting increases. In the study area, the irregularity of the bubble clusters results in a more chaotic flow field, leading to greater speed and displacement of the particles. Increasing the number of bubbles entering at one time and their initial speed enhances the impact of the bubble clusters on the solid particles. When the number of bubbles entering at one time is 5, 10, 15, and 20 respectively, the maximum movement speed of the particles increases by 6.9%, 8.1%, and 9.0%. When the initial speed of the bubble entering is 0, 0.1, 0.2, and 0.3 m/s respectively, the maximum movement speed of the particles increases by 6.4%, 3.0%, and 8.8%.

Simulating the Rising Air Bubbles Stream through Heavy Fuel Oil Residue Bubble Column Reactor using COMSOL

Several industrial operations rely on the rise of air bubbles through heavy fuel oil residue (HFOr), including oil recovery and wastewater treatment. In order to increase efficiency and decrease expenses, it is necessary to understand the peculiarities of this process. With the help of COMSOL Multiphysics, we numerically simulate the ascent of a single air bubble via HFOr in this article. The dimensions of the container used in the simulations were (100) mm in diameter and (200) mm in height, with an air bubble diameter of 4.5 mm. For an air bubble ascending through an HFOr column, the predicted numerical outcomes are contrasted with nine experimental versions. As the bubble rose higher, the region with the greatest number of vortices was located on its side. The remaining vortex is contained inside the bubble. In addition, since drag decreases the flow, a lighter fluid usually exhibits a region of strong vortex and low pressure. Because of the oscillation effect, the results reveal that the rising velocity initially rises and then gradually falls between 0.6 and 0.65 s. The air bubbles' wavy motion is caused by the high fluid density, which becomes worse as the air velocity gets higher. The values of the drag coefficient rise sharply with decreasing air extrusion speed. It turns out that the height of the fluid has an inverse relationship with the pressure.

A New Bubble Image Model Based on the Recognition of Bubble Flow

AbstractIn this study, a new ellipse‐fitting algorithm is proposed to achieve the reconstruction of bubble shapes in bubbly flow captured by a high‐speed camera in the gas–liquid two‐phase column reactor. Bubble flow patterns and geometric parameters in the experimental images are recognized and identified successfully, represented by means of the topological parameters. Three logical steps are carried out in detail. First, the area threshold and the circularity factors are established to identify the bubbles whether belonging to a single bubble or not. The overlapping bubbles in images can be separated from single bubbles based on a watershed segmentation algorithm. Second, a single bubble image and an overlapping bubble image are combined into one image. After that, statistical analysis for the size distributions and ellipse area bubbles is performed for further analysis and discussion. The advantage of this algorithm is that it can make use of a set of major and minor axes of an ellipse to capture the ellipse parameters more effectively. Simulation results are well agreed with experimental measurements. Moreover, it can be used to detect many ellipse‐like bubbles that are dispersed in high‐speed camera images, indicating that it is a better strategy for the recognition and identification of bubbly turbulent flow accurately.

Predictions of the Effect of Non-Homogeneous Ocean Bubbles on Sound Propagation

In the ocean, bubbles rarely appear alone and are often not evenly distributed, which makes it complicated to predict the effect of ocean bubbles on sound propagation. To solve this problem, researchers have tried to use approximations such as equivalent and multiple scattering models, but these approximations are accompanied by large errors. Therefore, we propose a semi-numerical and semi-analytical calculation method for underwater sound fields containing non-homogeneous bubbles in this paper. Based on the attenuation cross section and scattering cross section of a single bubble, the non-homogeneous medium is divided into multiple layers of uniform medium. Each layer of the bubble group is regarded as a whole, which can fully reflect the influence of bubble group vibration and scattering on sound wave propagation and is conducive to faster calculation of the sound field of non-homogeneous bubbly liquids. Compared with the classic coupling model, the calculation process of this method is simpler and faster, which solves the problem of fast calculation of sound fields in bubbly liquids and simulation of distributed bubble groups containing non-homogeneous distributed bubbles.

Study on high-order-mode oscillation and migration characteristics of single bubble in the ultrasonic field

Abstract Investigating the motion characteristics of cavitation bubbles in an ultrasonic field is important for industries such as ultrasonic cleaning and ultrasonic therapy. This paper investigated the oscillation and migration characteristics of a single cavitation bubble in a 20 kHz ultrasonic field. Peristaltic pump syringe injection tests were performed at different locations in a 20 kHz ultrasonic field, and a high-speed camera was used to observe the bubble motion characteristics. The bubble was found to resonate with the ultrasonic waves at a higher order mode, changing from spherical shape to honeycomb and golf ball shapes. Also, we found that the ultrasonic waves may prevent the buoyancy of the bubble. Through revealing the motion characteristics of the bubble in the ultrasonic field, we may control the bubble motion by ultrasonic, and strengthen the bubble cavitation effect.

An improved smoothed particle hydrodynamics method for modeling multiphase flows

Multiphase flows are prevalent in both natural and engineered systems. The study of multiphase flow problems using numerical simulation is challenging due to the presence of high nonlinearities and moving interfaces. In this paper, an improved multiphase smoothed particle hydrodynamics (SPH) model is proposed for simulating multiphase flows. In the improved multiphase SPH model, an improved interface repulsive force model is proposed to reduce the interpenetration of particles at the multiphase interface and make the multiphase interface smooth and clear, and an improved kernel gradient correction is introduced to optimize the computational results. In addition, the particle shifting technology is applied to make the particle distribution uniform. Five numerical examples including the Rayleigh–Taylor instability, non-Boussinesq lock-exchange problem, square droplet deformation, single bubble rise, and circular droplet oscillation are investigated to verify the correctness and effectiveness of the improved multiphase SPH model. The results demonstrate that the improved multiphase SPH approach is effective in modeling multiphase flows.

Acoustically controlled bubble nucleation

Fundamental studies of single bubble heat transfer are crucial to improve our understanding of boiling phenomena and develop modeling and simulation tools for the design and optimization of boiling systems. However, conducting these kinds of experiments over a wide range of conditions is challenging and may involve expensive and complicated experimental techniques (e.g., lasers) to isolate and control bubble nucleation. In this work, we introduce a new technique to control nucleation with an ultrasonic beam. A low-cost and widely available piezoelectric disc driven at megahertz frequencies directed at a heated surface can increase the time-averaged pressure and suppress evaporation until the moment the transducer is turned off, allowing us to control the incipience of boiling to within 150 µs. We demonstrate this technique by measuring the physical and thermal footprint of natural and acoustically controlled bubbles with high-resolution infrared thermometry and phase detection diagnostics. We were able to raise the bubble nucleation temperature by over 30 K compared to a reference case with no acoustic control, which significantly changes bubble growth rate, microlayer formation and evaporation and departure diameter. We highlight the potential of this new low-cost and easy-to-use technique in a variety of different boiling heat transfer studies.

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.

Research of Bubble Breakup and Influencing Factors in Flotation Machine

As the core factor of flotation, bubbles have an important effect on flotation. The rotor speed, initial gas parameters and impeller structure of the flotation machine will affect the formation and movement of bubbles. It is very important to study the influence mechanism of operating parameters of flotation machine on the bubble breakup process for flotation machine design and structure optimization. This paper takes KYF-0.2m<sup>3</sup> flotation machine as the research object, establishes a single bubble analysis model, adopts the VOF (Volume of Fluid) method to analyze the influence of different initial positions of bubbles on the bubble breakup behavior, and studies the influence of impeller speed and initial position of bubbles on the bubble breakup. Result show that the breakup of bubbles mainly occurs near the stator region. With the increase of rotational speed of the impeller, the centrifugal force and the disturbance of the convection field will become greater, the time of the bubble breakup become shorter, more bubbles breakup and generate more smaller ones. With the bubble position is closer to the rotating axis of the impeller, the impact of reflow becomes stronger and the bubble breakup effect will be better, and if the bubble initial position closer to the impeller cover, the influence of impeller on the bubbles become greater and the distribution of bubbles will be more uniform.

Study of the Dynamics of a Single Bubble

The behaviour of bubbles in cavitation and boiling processes is determined by the thermodynamic parameters of the two-phase medium and the intensity of heat and mass transfer, which affect the final dynamic effects. In this review, we analyse the influences of these factors on bubble behaviour, as described in existing mathematical models. In particular, we analyse the physical processes that govern bubble behaviour, the influence of mass transfer, vapor and liquid temperature, vapour, and liquid pressure on the inertial and dynamic stages of development. In conclusion, we summarize the problems associated with modelling, the accuracy of numerical predictions, and propose directions for further research.

Bubble behavior parameters extraction and analysis during pool boiling based on deep-learning method

The nucleate pool boiling plays an important role in thermal and chemical engineering applications. Analyzing bubble dynamics at nucleation site is crucial to improve the understanding of boiling heat transfer mechanism. Quantitative extraction of bubble parameters from high-speed visualized images is a labor-intensitive and time-consuming task making it necessary for automatically detect single bubble growth and measure boiling characteristic parameters.In the present work, we proposed a deep learning based self-adaptive statistical algorithm for extraction of bubble behavior parameters quickly and automatically from numerous high-speed visualization images looking from the side view of a boiling chamber. A dataset was constructed for training and performance evaluation based on experimental data of saline solution pool boiling. The StarDist and U-Net convolutional neural network were combined in the algorithm so that more exact segmentation of the bubbles can be identified. Based on the segmentation results, a post-processing program was developed to extract the sequential variation of bubbles during consecutive cycles at nucleation sites. The dynamic characteristic parameters that affect heat transfer, such as nucleation density, bubble departure diameter, departure frequency, and wait time under different heat flux were obtained quantitatively. The comparison of automatic extraction algorithm and manual processing proves the reliability and superiority of our method. This work indicates that the proposed method has great potential to be widely applied as an efficient and universal tool for processing different types of bubble shadowgraph images.

Bubble sizes inferred from their gas composition in a temperate freshwater fish pond

ABSTRACT Rising bubbles play a fundamental role in emitting greenhouse gases from shallow waters. Their size is crucial for bubble dissolution, gas exchange with the surrounding water, and the release of gases into the atmosphere. However, little is known about bubble sizes in shallow waters. To address this, we investigated bubble diameters in a 1.2 m deep fish pond, employing 2 methods: first, we measured the bubble size distributions by optical bubble sensors; second, we used an existing single bubble dissolution model to determine diameters representative for the respective bubble size distributions at the water surface based on measured bubble oxygen contents and dissolved oxygen concentrations. Results from optical bubble sensors were relatively similar at all sites; however, subsequent analysis revealed problems, particularly in detecting small bubbles under turbid, shallow water conditions. Model-derived bubble diameters ranged from 0.5 to 10.5 mm, varied spatially within the pond, and displayed diurnal fluctuations. With increasing bubble flux, bubble diameters increased; bubbles at feeding sites were larger than in the open water area. A detailed sensitivity analysis revealed that, depending on the bubble size distribution, the uncertainty of the model increases with increasing water depth. For a typical bubble diameter of 5 mm, the simple method can provide robust estimates of representative bubble size in waters shallower than 50 m.

Single droplet or bubble and its stability: Kinetic theory and dynamical system approaches.

Steady solutions of a single droplet or bubble in the van der Waals fluid are investigated on the basis of the kinetic model equationthat has been recently proposed by Miyauchi etal. [Gas Dynamics with Applications in Industry and Life Sciences (Springer, Cham, 2023), pp. 19-39]. Under the thermal equilibrium condition and isotropic assumption with respect to the origin of the coordinates, the kinetic equationis reduced to an ordinary differential equationfor the density, which can be regarded as a low-dimensional dynamical system. The possible density distribution is studied as a flow in the low-dimensional phase space. It is clarified that a single droplet or bubble can be understood as a flow that goes into a fixed point and that the flow is qualitatively different in the unstable and metastable parameter regions. The features of the obtained density distributions in individual regions are also clarified. Finally, the stability of those solutions is studied by direct numerical experiments of the kinetic equation.