Chapters (39)
Earth and Environmental Sciences (27)
Journal of Fluid Mechanics (27)
Cambridge University Press (41)
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Transition to convection in single bubble diffusive growth
Álvaro Moreno Soto, Oscar R. Enríquez, Andrea Prosperetti, Detlef Lohse, Devaraj van der Meer
Journal: Journal of Fluid Mechanics / Volume 871 / 25 July 2019
Print publication: 25 July 2019
We investigate the growth of gas bubbles in a water solution at rest with a supersaturation level that is generally associated with diffusive mass transfer. For $\text{CO}_{2}$ bubbles, it has been previously observed that, after some time of growing in a diffusive regime, a density-driven convective flow enhances the mass transfer rate into the bubble. This is due to the lower density of the gas-depleted liquid which surrounds the bubble. In this work, we report on experiments with different supersaturation values, measuring the time $t_{conv}$ it takes for convection to dominate over the diffusion-driven growth. We demonstrate that by considering buoyancy and drag forces on the depleted liquid around the bubble, we can satisfactorily predict the transition time. In fact, our analysis shows that this onset does not only depend on the supersaturation, but also on the absolute pressure, which we corroborate in experiments. Subsequently, we study how the depletion caused by the growth of successive single bubbles influences the onset of convection. Finally, we study the convection onset around diffusively growing nitrogen $\text{N}_{2}$ bubbles. As $\text{N}_{2}$ is much less soluble in water, the growth takes much longer. However, after waiting long enough and consistent with our theory, convection still occurs as for any gas–liquid combination, provided that the density of the solution sufficiently changes with the gas concentration.
Molecular dynamics study of multicomponent droplet dissolution in a sparingly miscible liquid
Shantanu Maheshwari, Martin van der Hoef, Andrea Prosperetti, Detlef Lohse
Journal: Journal of Fluid Mechanics / Volume 833 / 25 December 2017
Print publication: 25 December 2017
The dissolution of a multicomponent nanodrop in a sparingly miscible liquid is studied by molecular dynamics (MD) simulations. We studied both binary and ternary systems, in which nanodroplets are formed from one and two components, respectively. Whereas for a single-component droplet the dissolution can easily be calculated, the situation is more complicated for a multicomponent drop, as the interface concentrations of the drop constituents depend on the drop composition, which changes with time. In this study, the variation of the interface concentration with the drop composition is determined from independent 'numerical experiments', which are then used in the theoretical model for the dissolution dynamics of a multicomponent drop. The MD simulations reveal that when the interaction strengths between the drop constituents and the surrounding bulk liquid are significantly different, the concentration of the more soluble component near the drop interface may become larger than in the drop bulk. This effect is the larger the smaller the drop radius. While the present study is limited to binary and ternary systems, the same method can be easily extended to a larger number of components.
Gas depletion through single gas bubble diffusive growth and its effect on subsequent bubbles
Álvaro Moreno Soto, Andrea Prosperetti, Detlef Lohse, Devaraj van der Meer
Journal: Journal of Fluid Mechanics / Volume 831 / 25 November 2017
Print publication: 25 November 2017
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.
Dissolution and growth of a multicomponent drop in an immiscible liquid
Shigan Chu, Andrea Prosperetti
The mass flux at the surface of a drop in an immiscible host liquid is dictated by the composition of the drop surface. In a binary system, this composition is essentially constant in time and equals the solubility of the drop constituent in the host liquid. This situation has been treated in a classic study by Epstein and Plesset (J. Chem. Phys., vol. 18, 1950, pp. 1505–1509). The situation is very different for ternary and higher-order systems in which, due to the mutual interaction of the drop constituents, their concentration at the drop surface markedly differs from the respective solubilities and depends on time. This paper presents a thermodynamically consistent analysis of this situation, for both growing and dissolving drops, with and without an initial concentration of the drop constituents in the host liquid. In some cases the results, which have important implications e.g. for solvent extraction processes in the chemical and environmental remediation industries, show major deviations from the predictions of approximations in current use, including simple extensions of the Epstein–Plesset theory.
Local interfacial stability near a zero vorticity point
Yu-Hau Tseng, Andrea Prosperetti
Journal: Journal of Fluid Mechanics / Volume 776 / 10 August 2015
Published online by Cambridge University Press: 30 June 2015, pp. 5-36
Print publication: 10 August 2015
It is often observed that small drops or bubbles detach from the interface separating two co-flowing immiscible fluids. The size of these drops or bubbles can be orders of magnitude smaller than the length scales of the parent fluid mass. Examples are tip-streaming from drops or coaxial jets in microfluidics, selective withdrawal, 'skirt' formation around bubbles or drops, and others. It is argued that these phenomena are all reducible to a common instability that can occur due to a local convergence of streamlines in the neighbourhood of a zero-vorticity point or line on the interface. When surfactants are present, this converging flow tends to concentrate them in these regions weakening the effect of surface tension, which is the only mechanism opposing the instability. Several analytical and numerical calculations are presented to substantiate this interpretation of the phenomenon. In addition to some idealized cases, the results of two-dimensional simulations of co-flowing jets and a rising drop are presented.
Life and death by boundary conditions
Andrea Prosperetti
Journal: Journal of Fluid Mechanics / Volume 768 / 10 April 2015
Published online by Cambridge University Press: 27 February 2015, pp. 1-4
Picano et al. (J. Fluid Mech., vol. 764, 2015, pp. 463–487) have conducted fully resolved numerical simulations of many thousands of spherical particles in a turbulent channel flow with $\mathit{Re}=5600$ . Their results give a tantalizing demonstration of the vastness of the vistas that this line of research is about to open.
The quasi-static growth of CO2 bubbles
Oscar R. Enríquez, Chao Sun, Detlef Lohse, Andrea Prosperetti, Devaraj van der Meer
Journal: Journal of Fluid Mechanics / Volume 741 / 25 February 2014
Published online by Cambridge University Press: 04 March 2014, R1
Print publication: 25 February 2014
We study experimentally the growth of an isolated gas bubble in a slightly supersaturated water–CO2 solution at 6 atm pressure. In contrast to what was found in previous experiments at higher supersaturation, the time evolution of the bubble radius differs noticeably from existing theoretical solutions. We trace the differences back to several combined effects of the concentration boundary layer around the bubble, which we disentangle in this work. In the early phase, the interaction with the surface on which the bubble grows slows down the process. In contrast, in the final phase, before bubble detachment, the growth rate is enhanced by the onset of density-driven convection. We also show that the bubble growth is affected by prior growth and detachment events, though they are up to 15 min apart.
Highly focused supersonic microjets: numerical simulations
Ivo R. Peters, Yoshiyuki Tagawa, Nikolai Oudalov, Chao Sun, Andrea Prosperetti, Detlef Lohse, Devaraj van der Meer
Journal: Journal of Fluid Mechanics / Volume 719 / 25 March 2013
Print publication: 25 March 2013
By focusing a laser pulse inside a capillary partially filled with liquid, a vapour bubble is created that emits a pressure wave. This pressure wave travels through the liquid and creates a fast, focused axisymmetric microjet when it is reflected at the meniscus. We numerically investigate the formation of this microjet using axisymmetric boundary integral simulations, where we model the pressure wave as a pressure pulse applied on the bubble. We find a good agreement between the simulations and experimental results in terms of the time evolution of the jet and on all parameters that can be compared directly. We present a simple analytical model that accurately predicts the velocity of the jet after the pressure pulse and its maximum velocity.
Pressure-driven flow in a channel with porous walls*
QIANLONG LIU, ANDREA PROSPERETTI
Published online by Cambridge University Press: 04 May 2011, pp. 77-100
The finite-Reynolds-number three-dimensional flow in a channel bounded by one and two parallel porous walls is studied numerically. The porous medium is modelled by spheres in a simple cubic arrangement. Detailed results on the flow structure and the hydrodynamic forces and couple acting on the sphere layer bounding the porous medium are reported and their dependence on the Reynolds number illustrated. It is shown that, at finite Reynolds numbers, a lift force acts on the spheres, which may be expected to contribute to the mobilization of bottom sediments. The results for the slip velocity at the surface of the porous layers are compared with the phenomenological Beavers–Joseph model. It is found that the values of the slip coefficient for pressure-driven and shear-driven flow are somewhat different, and also depend on the Reynolds number. A modification of the relation is suggested to deal with these features. The Appendix provides an alternative derivation of this modified relation.
20 - Theory of Distributions
Book: Advanced Mathematics for Applications
Print publication: 06 January 2011, pp 568-619
Part I - Applications
Print publication: 06 January 2011, pp 51-52
Print publication: 06 January 2011, pp xvii-xviii
1 - The Classical Field Equations
Print publication: 06 January 2011, pp 3-26
It is a remarkable aspect of the "unreasonable effectiveness of mathematics in the natural sciences" (Wigner 1960) that a handful of equations are sufficient to describe mathematically a vast number of physically disparate phenomena, at least at some level of approximation. Key reasons are the isotropy and uniformity of space-time (at least locally), the attendant conservation laws, and the useful range of applicability of linear approximations to constitutive relations.
After a very much abbreviated survey of the principal properties of vector fields, we present a summary of these fundamental equations and associated boundary conditions, and then describe several physical contexts in which they arise. The initial chapters of a book on any specific discipline give a far better derivation of the governing equations for that discipline than space constraints permit here. Our purpose is, firstly, to remind the reader of the meaning accorded to the various symbols in any specific application and of the physics that they describe and, secondly, to show the similarity among different phenomena.
The final section of this chapter is a very simple-minded description of the method of eigenfunction expansion systematically used in many of the applications treated in this book. The starting point is an analogy with vectors and matrices in finite-dimensional spaces and the approach is purposely very elementary; a "real" theory is to be found in Part III of the book.
15 - Green's Functions: Ordinary Differential Equations
Green's functions permit us to express the solution of a non-homogeneous linear problem in terms of an integral operator of which they are the kernel. We have already presented in simple terms this idea in §2.4. We now give a more detailed theory with applications mainly to ordinary differential equations. The next chapter deals with Green's functions for partial differential equations.
The determination of a Green's function requires the solution of a problem similar to (although somewhat simpler than) the original one, but the effort required is balanced by several advantages. In the first place, and at the most superficial level, once the Green's function G is known, it is unnecessary to solve the problem ex novo for every new set of data: it is sufficient to allow G to act on the new data to have the solution directly. Secondly, and most importantly for our purposes, Green's function theory provides a foundation for the various eigenfunction expansion and integral transform methods used in Part I of this book. Thirdly, even if the Green's function cannot be determined explicitly, one can base on it the powerful boundary integral numerical method outlined in §16.1.3 of the next chapter. Furthermore, once an expression for the solution of a problem – even if not fully explicit – is available, it becomes possible to deduce several important features of it, including bounds existence, uniqueness and others.
Print publication: 06 January 2011, pp vii-xii
7 - Spherical Systems
In many ways the sphere is the prototypical three-dimensional body and the consideration of fields in the presence of spherical boundaries sheds light on several features of more general three-dimensional cases.
In all the examples of this chapter extensive use is made of expansions in series of Legendre polynomials, for axi-symmetric problems, or spherical harmonics, for the general three-dimensional case. After a review of the polar coordinate system, we begin with a summary of the properies of these functions which are dealt with in greater detail in Chapters 13 and 14, respectively. While the axi-symmetric situation is somewhat simpler, it is also contained as a special case in the general three-dimensional one and it is therefore expedient to treat it as a special case of the latter.
We start with the general solution of the Laplace and Poisson equations (§7.3) and apply it to several axisymmetric (§§7.4 and 7.5) and non-axisymmetric situations. In all these cases the radial part of the solution consists of powers of r. The examples in the second part of the chapter (§7.13 and §7.14) deal with the scalar Helmholtz equation, for which the radial dependence is expressed in terms of spherical Bessel functions, the fundamental properties of which are summarized in §7.12. The last four sections deal with problems involving vector fields and vector harmonics.
Part 0 - General Remarks and Basic Concepts
Print publication: 06 January 2011, pp 1-2
Part III - Some Advanced Tools
3 - Fourier Series: Applications
Part II - Essential Tools