The history effect in bubble growth and dissolution. Part 1. Theory
TL;DR: In this article, the history effect of past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble is discussed.
Abstract: The term ‘history effect’ refers to the contribution of any past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble. The history effect arises from the (non-instantaneous) development of the dissolved gas concentration boundary layer in the liquid in response to changes in the concentration at the bubble interface caused, for instance, by variations of the ambient pressure in time. Essentially, the history effect amounts to the acknowledgement that at any given time the mass flux across the bubble is conditioned by the preceding time history of the concentration at the bubble boundary. Considering the canonical problem of an isolated spherical bubble at rest, we show that the contribution of the history effect in the current interfacial concentration gradient is fully contained within a memory integral of the interface concentration. Retaining this integral term, we formulate a governing differential equation for the bubble dynamics, analogous to the well-known Epstein–Plesset solution. Our equation does not make use of the quasi-static radius approximation. An analytical solution is presented for the case of multiple step-like jumps in pressure. The nature and relevance of the history effect is then assessed through illustrative examples. Finally, we investigate the role of the history effect in rectified diffusion for a bubble that pulsates under harmonic pressure forcing in the non-inertial, isothermal regime.
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05 Apr 2018
TL;DR: In this paper, asymptotic analysis is used to characterize collective shielding effects in bubble lattices, the resulting bubble lifetime and dissolution dynamics, which is driven by the diffusion of gas within the surrounding liquid.
Abstract: Microscopic bubble dissolution is driven by the diffusion of gas within the surrounding liquid. Asymptotic analysis is used to characterize collective shielding effects in bubble lattices, the resulting bubble lifetime and dissolution dynamics.
37 citations
TL;DR: The current results show that the dynamical evolution of bubbles is influenced by comprehensive effects combining chemical catalysis and physical mass transfer, and the size of the bubbles at the moment of detachment is determined by the balance between buoyancy and surface tension and by the detailed geometry at the bubble’s contact line.
Abstract: Whereas bubble growth out of gas-oversatured solutions has been quite well understood, including the formation and stability of surface nanobubbles, this is not the case for bubbles forming on catalytic surfaces due to catalytic reactions, though it has important implications for gas evolution reactions and self-propulsion of micro/nanomotors fueled by bubble release. In this work we have filled this gap by experimentally and theoretically examining the growth and detachment dynamics of oxygen bubbles from hydrogen peroxide decomposition catalyzed by gold. We measured the bubble radius R(t) as a function of time by confocal microscopy and find R(t) ∝ t1/2. This diffusive growth behavior demonstrates that the bubbles grow from an oxygen-oversaturated environment. For several consecutive bubbles detaching from the same position in a short period of time, a well-repeated growing behavior is obtained from which we conclude the absence of noticeable depletion effect of oxygen from previous bubbles or increasing oversaturation from the gas production. In contrast, for two bubbles far apart either in space or in time, substantial discrepancies in their growth rates are observed, which we attribute to the variation in the local gas oversaturation. The current results show that the dynamical evolution of bubbles is influenced by comprehensive effects combining chemical catalysis and physical mass transfer. Finally, we find that the size of the bubbles at the moment of detachment is determined by the balance between buoyancy and surface tension and by the detailed geometry at the bubble's contact line.
34 citations
TL;DR: In this paper, the authors study the successive quasi-static growth of many bubbles from the same nucleation site described in this paper and show that the radius-versus-time curves of subsequent bubbles differ from each other due to this phenomenon.
Abstract: When a gas bubble grows by diffusion in a gas-liquid solution, it affects the distribution of gas in its surroundings. If the density of the solution is sensitive to the local amount of dissolved gas, there is the potential for the onset of natural convection, which will affect the bubble growth rate. The experimental study of the successive quasi-static growth of many bubbles from the same nucleation site described in this paper illustrates some consequences of this effect. The enhanced growth due to convection causes a local depletion of dissolved gas in the neighbourhood of each bubble beyond that due to pure diffusion. The quantitative data of sequential bubble growth provided in the paper show that the radius-versus-time curves of subsequent bubbles differ from each other due to this phenomenon. A simplified model accounting for the local depletion is able to collapse the experimental curves and to predict the progressively increasing bubble detachment times.
24 citations
TL;DR: In this article, the formation of gas bubbles at gas cavities located in walls bounding the flow has been studied in many technical applications, but it is usually hard to observe.
Abstract: The formation of gas bubbles at gas cavities located in walls bounding the flow occurs in many technical applications, but is usually hard to observe. Even though, the presence of a fluid flow undoubtedly affects the formation of bubbles, there are very few studies that take this fact into account. In the present paper new experimental results on bubble formation (diffusion-driven nucleation) from surface nuclei in a shear flow are presented. The observed gas-filled cavities are micrometre-sized blind holes etched in silicon substrates. We measure the frequency of bubble generation (nucleation rate), the size of the detaching bubbles and analyse the growth of the surface nuclei. The experimental findings support an extended understanding of bubble formation as a self-excited cyclic process and can serve as validation data for analytical and numerical models.
23 citations
Cites background from "The history effect in bubble growth..."
...In recent works the so called history effect is considered (Peñas-López et al (2016, 2017))....
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TL;DR: In this paper, the history effect of the local concentration boundary layer around the bubble has been studied for a spherical bubble attached to a horizontal flat plate and in the presence of gravity.
Abstract: The accurate description of the growth or dissolution dynamics of a soluble gas bubble in a super- or undersaturated solution requires taking into account a number of physical effects that contribute to the instantaneous mass transfer rate. One of these effects is the so-called history effect. It refers to the contribution of the local concentration boundary layer around the bubble that has developed from past mass transfer events between the bubble and liquid surroundings. In Part 1 of this work \citep{penas2016}, a theoretical treatment of this effect was given for a spherical, isolated bubble. Here, Part 2 provides an experimental and numerical study of the history effect regarding a spherical bubble attached to a horizontal flat plate and in the presence of gravity. The simulation technique developed in this paper is based on a streamfunction--vorticity formulation that may be applied to other flows where bubbles or drops exchange mass in the presence of a gravity field. Using this numerical tool, simulations are performed for the same conditions used in the experiments, in which the bubble is exposed to subsequent growth and dissolution stages, using step-wise variations in the ambient pressure. Besides proving the relevance of the history effect, the simulations highlight the importance that boundary-induced advection has to accurately describe bubble growth and shrinkage, i.e. the bubble radius evolution. In addition, natural convection has a significant influence that shows up in the velocity field even at short times, though, given the supersaturation conditions studied here, the bubble evolution is expected to be mainly diffusive.
21 citations
References
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TL;DR: In this article, approximate solutions for the rate of solution by diffusion of a gas bubble in an undersaturated liquid-gas solution are presented, with the neglect of the translational motion of the bubble.
Abstract: With the neglect of the translational motion of the bubble, approximate solutions may be found for the rate of solution by diffusion of a gas bubble in an undersaturated liquid‐gas solution; approximate solutions are also presented for the rate of growth of a bubble in an oversaturated liquid‐gas solution. The effect of surface tension on the diffusion process is also considered.
1,343 citations
TL;DR: In this paper, the equations governing spherically symmetric phase growth in an infinite medium are first formulated for the general case and then simplified to describe growth controlled by the transport of heat and matter.
832 citations
"The history effect in bubble growth..." refers background or methods in this paper
...…approximation (Plesset & Zwick 1954), perturbation techniques (Duda & Vrentas 1969), infinite series (Tao 1978), integral methods (Rosner & Epstein 1972) and self-similar solutions for bubble growth starting from zero initial size (Birkhoff, Margulies & Horning 1958; Scriven 1959), to cite a few....
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...Perhaps the most widely used ones are the Epstein–Plesset solutions, valid for growth and dissolution, based on the quasi-stationary approximation (Epstein & Plesset 1950), and Scriven’s exact solution for growth that accounts for the advection term in the diffusion equation (Scriven 1959)....
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...To prove this, we can take the well-known asymptotic solutions of Plesset & Zwick (1954), Birkhoff et al. (1958) or Scriven (1959) for thermal diffusion growth – under assumptions (i) and (ii) – driven by a temperature difference 1T between the bubble boundary and the bulk fluid....
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...While growth is well determined by existing solutions (Epstein & Plesset 1950; Scriven 1959), the dissolution experienced by this bubble is greatly affected by the low-concentration boundary layer left by the preceding growth stage....
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...Equivalently, for mass-diffusion-controlled growth driven by a (molar) concentration difference 1C between the bubble boundary and the bulk fluid, Epstein & Plesset (1950) and Scriven (1959) among others obtained R(t)∼ Jam √ Dmt, with Jam = Mg1C ρg ....
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TL;DR: In this article, a solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius, since the radius at which it becomes valid is near the lower limit of experimental observation.
Abstract: The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius‐time behavior is made with experimental observations in superheated water, and very good agreement is found.
771 citations
"The history effect in bubble growth..." refers methods in this paper
...To prove this, we can take the well-known asymptotic solutions of Plesset & Zwick (1954), Birkhoff et al. (1958) or Scriven (1959) for thermal diffusion growth – under assumptions (i) and (ii) – driven by a temperature difference 1T between the bubble boundary and the bulk fluid....
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...The methods employed are based on the quasi-static approximation (Epstein & Plesset 1950), thin boundary layer approximation (Plesset & Zwick 1954), perturbation techniques (Duda & Vrentas 1969), infinite series (Tao 1978), integral methods (Rosner & Epstein 1972) and self-similar solutions for…...
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01 Aug 1953
TL;DR: In this paper, a solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius, since the radius at which it becomes valid is near the lower limit of experimental observation.
Abstract: The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius‐time behavior is made with experimental observations in superheated water, and very good agreement is found.
729 citations
TL;DR: In this paper, a linearized theory of the forced radial oscillations of a gas bubble in a liquid is presented, with particular attention devoted to the thermal effects of the bubble.
Abstract: A linearized theory of the forced radial oscillations of a gas bubble in a liquid is presented. Particular attention is devoted to the thermal effects. It is shown that both the effective polytropic exponent and the thermal damping constant are strongly dependent on the driving frequency. This dependence is illustrated with the aid of graphs and numerical tables which are applicable to any noncondensing gas–liquid combination. The particular case of an air bubble in water is also considered in detail.
368 citations
"The history effect in bubble growth..." refers background in this paper
...Following the work of Prosperetti (1977) (see figure 1 of that paper), the assumption of isothermal oscillations (polytropic exponent equal to unity) may be safely assumed, provided the thermal Péclet number based on the oscillation frequency is smaller than one: Ωth ≡ 2πfcR2c/Dth 1, with Dth =…...
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