This is a postprint version of the following published document:
Peñas-López, P., Moreno Soto, A., Parrales, M.A., van der Meer,
D. (2017). The history effect on bubble growth and dissolution.
Part 2. Experiments and simulations of a spherical bubble
attached to a horizontal flat plate. Journal of Fluid Mechanics,
vol 820, pp 479-510.
DOI: 10.1017/jfm.2017.221
©Cambridge University Press 2017
The history effect on bubble growth and
dissolution. Part 2. Exper iments and
simulations of a spherical bubble attached
to a hori zont al flat plate
Pablo Peñas-López
1,
†, Álvaro Moreno Soto
2,
†, Miguel A. Parrales
3
,
Devaraj van der Meer
2
, Detlef Lohse
2
and Javier Rodríguez-Rodríguez
1
1
Fluid Mechanics Group, Universidad Carlos III de Madrid, Avda. de la Universidad 30,
28911 Leganés (Madrid), Spain
2
Physics of Fluids Group, Faculty of Science and Technology, University of Twente, P.O. Box 217,
7500 AE Enschede, The Netherlands
3
Departamento de Ingeniería Energética y Fluidomecánica, Escuela Técnica Superior de Ingenieros
Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain
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
(Peñas-López et al., J. Fluid Mech., vol. 800, 2016b, pp. 180–212), 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 stepwise 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, although given the supersaturation conditions studied here, the
bubble evolution is expected to be mainly diffusive.
Key words: bubble dynamics, convection, multiphase and particle-laden flows
† Email addresses for correspondence: papenasl@ing.uc3m.es, a.morenosoto@utwente.nl
1
1. Introduction
Mass transfer processes involving bubbles have gained a renewed interest over the
last few years due to their relevance in modern microfluidic applications connected
to topics such as carbon sequestration (Sun & Cubaud 2011; Volk et al. 2015). Due
to the small size of these bubbles they are spherical once they become smaller
than the channel’s size and are detached from the channel’s wall. Thus, in general
terms, the theory of Epstein & Plesset (1950) describing the diffusion-driven growth
or dissolution of an isolated, spherical particle should be applicable. However, as
discussed in Part 1 of this work (Peñas-López et al. 2016b), a number of effects not
included in the Epstein–Plesset theory, e.g. flow around the bubble, must be taken
into account to properly describe various experimental observations. Bubbles may also
interact with nearby surfaces or they may contain more than one chemical species
(Shim et al. 2014; Peñas-López, Parrales & Rodríguez-Rodríguez 2015). Another
effect that contributes to the diffusion-driven dynamics of a bubble is the so-called
history effect, discussed in Part 1 and more recently in Chu & Prosperetti (2016b).
It has been shown that any recent history of growth and/or dissolution (triggered by
past changes in ambient pressure) experienced by a particular bubble may leave, at
least for some time, a non-negligible imprint on the current state of the concentration
profile surrounding such a bubble. Consequently, the mass transfer rate is affected as
well. In Part 1, we proposed a modification to the theory of Epstein & Plesset to
take into account the history effect through a memory integral term for the case of
spherical, isolated bubbles. Moreover, we applied this modified equation to calculate
the bubble radius evolution when the bubble is subjected to some simple, yet relevant,
pressure–time histories. It is worth mentioning that history effects are common to
problems in which diffusion plays a central role, such as the viscous drag around
a body or, closer to the present mass transfer problem, the heat transfer around a
spheroid (Michaelides 2003).
The primary goal of the present paper is to quantify the relative importance of the
history effect in a canonical, yet experimentally relevant, configuration that does not
exhibit spherical symmetry, namely, that of a single spherical bubble tangent to a
horizontal flat plate that grows and dissolves in response to changes in the ambient
pressure and in the presence of gravity. In this configuration, the existence of the
history effect may become noticeable with a simple experiment: let us consider such
a spherical CO
2
bubble that dissolves when the pressure is above saturation (see
figure 1). At a given time t ≈60 s, the pressure is lowered to a new value still above
saturation (figure 1b). Despite the pressure being at all times above saturation, after
changing the pressure, the bubble is observed to grow for some time (figure 1a).
Naturally, part of this growth is due to the expansion of the gas. Thus, to observe
the effect purely due to diffusion, it is convenient to plot the ambient radius, R
0
. It
is defined as the radius one would observe if the liquid surroundings were at the
reference ambient pressure, P
0
, instead of the actual ambient pressure P
∞
(t):
R
0
(t) =R(t)
P
∞
(t)
P
0
1/3
. (1.1)
Here, R(t) is the measured bubble radius. Still, the ambient radius can be seen
to grow until approximately t ≈ 100 s, an effect purely driven by diffusion. Note
that R
0
was referred to in Part 1 of this article (Peñas-López et al. 2016b) as the
pressure-corrected radius R
corr
. However, with the purpose of maintaining the standard
nomenclature, R
0
will be used throughout this paper.
2
0.24
0.26
0.28
0.30
0.32
(a)
Simulation
t (s)
0 50 100 150
6.0
6.5
7.0
7.5
01234
0
10
20
30
40
(b)
(c)
FIGURE 1. Dissolution of a CO
2
spherical-cap bubble tangent to a flat chip immersed
in a CO
2
-water solution under pressurised conditions (see later figure 9). The bubble is
subjected to (b) a pressure jump P
∞
(t), from P
∞
(0) = P
0
= 7.4 bar to 6.5 bar. Both
pressures are above the saturation pressure, P
sat
=6.1 bar (according to simulation). Panel
(a) shows the evolution in time of the measured bubble radius R(t) (white markers)
and ambient radius R
0
(t) (dark markers). The former is compared to simulation, which
in addition was employed to depict (c) the concentration profile along the z-axis above
the bubble at three different instants in time. The employed experimental and numerical
techniques are detailed in the main text.
This phenomenon may be explained by examining the concentration of dissolved
CO
2
near the bubble (figure 1c). Indeed, although the concentration at the bubble
surface, given by Henry’s law, responds instantaneously to pressure changes, there
exists a boundary layer around the bubble where the concentration of CO
2
is higher
than the instantaneous saturation one, as a result of the dissolution stage that took
place before the pressure drop. In the example depicted in figure 1(c), it can be seen
how the concentration gradient at the bubble’s top is actually positive at t = 65 s,
which explains the growth of the ambient radius. In this figure, numerical simulations
such as the ones described in §§ 3–5 have been used to compute the concentration
field along the z axis. These simulations are validated by comparing the predicted
bubble radius with the experimental one (see figure 1a).
This simple example illustrates that, to properly describe the time evolution of the
bubble radius observed in experiments, the history effect must be taken into account.
However, a question that was left open in our previous work (Peñas-López et al.
2016b) was the relative importance of this effect in a realistic experimental condition
where other effects such as the interference with a wall and natural convection may
greatly influence the diffusion-driven bubble dynamics, as was shown by Enríquez
et al. (2014). With this idea in mind, another objective of the present work is
to propose a numerical approach able to accurately describe the evolution of a
bubble attached to a horizontal flat plate and growing/dissolving in the presence of a
gravitational field.
While this work only deals with bubbles composed of a single soluble gas, it is
important to realise that the history effect is omnipresent in multicomponent bubbles.
In Part 1, the history effect was described as ‘the acknowledgement that at any
given time the mass flux across the bubble is conditioned by the preceding time
3
history of the concentration at the bubble interface’. Thus, in dissolving/growing
multicomponent bubbles, the flow rate of a particular species across the bubble
interface will likely be different from the rest. The species composition inside the
bubble will thus change over time, which amounts to time-dependent partial pressures
and hence time-dependent interfacial concentrations. It is possible to artificially
discern the contribution of the history effect numerically, as was done by Chu &
Prosperetti (2016b) for the case of a dissolving two-gas bubble. Isolating the history
effect experimentally, on the other hand, is anticipated to be much harder.
Finally, it is worth mentioning that the history effect is naturally present in the
evaporation of multicomponent drops. Chu & Prosperetti (2016a) have recently
developed a formulation that includes a memory integral to describe the diffusion-
driven dynamics of multicomponent drops in the presence of a solvent, a phenomenon
of relevance in modern techniques of chemical analysis (Lohse 2016). In this problem,
the faster or slower dissolution of one of the components yields a time-varying
composition at the drop’s interface, which makes the inclusion of the history integral
in Fick’s law essential, even when the ambient pressure remains constant.
The paper is structured as follows: § 2 presents the experimental results that
illustrate the effect of history in the growth–dissolution of CO
2
bubbles tangent to
a flat plate. Section 3 presents the general mathematical formulation of the problem
and sheds light on the importance of the different physical effects involved in the
experiments. In § 4, a formulation based on the streamfunction–vorticity method
is described to simulate the mass transfer and flow field around the bubbles. The
simulation results are then presented and discussed in § 5. Finally, § 6 summarises
the main conclusions.
2. Experimental characterisation of the history effect
We have carried out experiments to support our theoretical and numerical analyses
by subjecting single bubbles to well-controlled, step-like pressure jumps that super- or
undersaturate the liquid alternatively. This way, we can make bubbles grow and shrink
under repeatable conditions to expose the history effect. It becomes apparent through
the differences in the responses to successive identical pressure–time histories.
2.1. Experimental set-up and procedure
Although the experimental set-up has been described in a previous work (Enríquez
et al. 2013), a brief description is included here for convenience. The facility is fed
with water that is demineralised in a purifier (MilliQ A10) and degassed by making
it flow through a filter (MiniModule, Liquicel, Membrana). This water enters into the
mixing chamber (see figure 2), that has been previously flushed with CO
2
to purge
the air from the system. There the water is stirred in the presence of CO
2
, kept at
the desired saturation pressure for approximately 45 min. Finally, the experimental
tank is pressurised with CO
2
at this same pressure and then slowly flooded with
the carbonated water, so bubbles do not appear during the filling. This preparation
procedure ensures that in the experimental tank there are no other gas species present
within the liquid or gas phases apart from CO
2
(at least in quantifiable amounts).
Placed at the centre of the experimental tank there is a silicon chip, treated to
become hydrophilic, with a black-silicon hydrophobic pit (50 µm in radius) at its
centre. The role of this pit is to force a single bubble to nucleate at a fixed location
in a repeatable way. Furthermore, in order to avoid slight temperature variations
4
