Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 2.7.1 - Simulation of interfacial dynamics in the presence of surfactant using the Lattice Boltzmann approach
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 Material Information
Title: 2.7.1 - Simulation of interfacial dynamics in the presence of surfactant using the Lattice Boltzmann approach Interfacial Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Skartlien, R.
Sollum, E.
Kjeldby, T.B.
Meakin, P.
Furtado, K.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: surfactant
adsorption kinetics
interfacial stability
hydrodynamic simulation
 Notes
Abstract: The hydrodynamic stability of a vertical liquid oil jet in water and the formation of buoyant oil droplets in the presence of oil-soluble surfactant were investigated using lattice Boltzmann simulations. The purpose of the work was to evaluate the potential application of our lattice Boltzmann model to flowing oil/water/surfactant systems under conditions where hydrodynamic instabilities occur and the interface topology changes. The model is based on the amphiphilic Lattice Boltzmann model of Nekovee et al. (2000), and it has been implemented in MPI/Fortran90 for efficient parallel 3D simulations. The jet is unstable due mainly to capillary and shear forces. We confirm that in the linear regime, the unstable modes grow faster with surfactant due to the lower average interfacial tension. In the subsequent non-linear regime, we find that the surfactant is swept up into the crests of the waves that develop on the surface of the jet, and this slows down wave growth due to the accompanying Marangoni stress. For single low-density oil droplets released from an inlet at the bottom of a layer of water, the dynamics is controlled by gravity acting on the density difference between the oil and water, and capillary forces. The liquid filament that forms between the rising droplet and the inlet is subject to larger deformation and stretching rates with surfactant. The model constants (coupling strengths between the fluids and the surfactant) were calibrated to Exxsol D80 oil and oil soluble Span 80 surfactant (Skartlien et al. 2009), using pendant drop experiments to measure the dynamic interfacial tension. The adsorption kinetics is diffusion controlled in this system (e.g., Ferri and Stebe 2000).
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00067
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 271-Skartlien-ICMF2010.pdf

Full Text



7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Simulation of oil/water interfacial dynamics in the presence of surfactant,
using the Lattice Boltzmann approach


R. Skartlien E. Sollum T. B. Kjeldby 1 P. Meakin K. Furtado *

Institute for Energy Technology (IFE), Kjeller, Norway
t Center for the Physics of Geological Processes, University of Oslo, Norway
Idaho National Laboratory, Center for Advanced Modeling and Simulation, USA
SNTNU, Trondheim, Norway
roar.skartlien@ife.no
Keywords: Surfactant, adsorption kinetics, interfacial stability, hydrodynamic simulation




The hydrodynamic stability of a vertical liquid oil jet in water and the formation of buoyant oil droplets in the
presence of oil-soluble surfactant were investigated using lattice Boltzmann simulations. The purpose of the work
was to evaluate the potential application of our lattice Boltzmann model to flowing oil/water/surfactant systems under
conditions where hydrodynamic instabilities occur and the interface topology changes. The model is based on the
amphiphilic Lattice Boltzmann model of Nekovee et al. (2000), and it has been implemented in MPI/Fortran90 for
efficient parallel 3D simulations.
The jet is unstable due mainly to capillary and shear forces. We confirm that in the linear regime, the unstable modes
grow faster with surfactant due to the lower average interfacial tension. In the subsequent non-linear regime, we find
that the surfactant is swept up into the crests of the waves that develop on the surface of the jet, and this slows down
wave growth due to the accompanying Marangoni stress. For single low-density oil droplets released from an inlet at
the bottom of a layer of water, the dynamics is controlled by gravity acting on the density difference between the oil
and water, and capillary forces. The liquid filament that forms between the rising droplet and the inlet is subject to
larger deformation and stretching rates with surfactant.
The model constants (coupling strengths between the fluids and the surfactant) were calibrated to Exxsol D80 oil
and oil soluble Span 80 surfactant (Skartlien et al. 2009), using pendant drop experiments to measure the dynamic
interfacial tension. The adsorption kinetics is diffusion controlled in this system (e.g., Ferri and Stebe 2000).


1 Introduction

Surface active agents are used to control the breakup of droplets and jets in many industrial applications (e.g., Liao et
al. 2006). Indigenous crude oil surfactants play a similar role in pipelines and separators, although droplet generation,
breakup and coalescence is usually not well-controlled. Surfactant lowers the average interfacial tension and it takes
less energy to deform and break up the interface in sheared laminar flow or turbulent flow. Thus, surfactant serves
to promote emulsification in oil/water pipeline flows. Furthermore, once the emulsion is formed, surfactant inhibits
droplet coalescence and the net result is a stabilized emulsified system. In order to study the fundamental fluid dy-
namics of such systems, and to transfer this knowledge into improved closure relations for use in engineering flow
simulators, it is crucial to have CFD tools that can account for surfactant effects on hydrodynamic stability, breakup
and coalescence of interfaces.
The breakup of a sheared interface can be idealized as a three step process (e.g., Marmottant and Villermaux 2004):
1) non-linear steepening of wave crests and formation of sheets, 2) breakup of sheets into "fingers"/liquid columns
and 3) breakup of fingers into droplets. The simple jet mimics to some extent breakup of a liquid "finger" into
droplets. A single jet can be used to study the fundamentals of shear driven interface instability, both numerically
and experimentally. It provides a controllable experimental system that can be used as a benchmark for surfactant
CFD simulations. In this study we will focus on the "jetting mode" (as opposed to the dripping mode), where Kelvin-
Helmholtz shear driven instability is the important mechanism. We examine the results from numerical simulations







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


of liquid jets both with and without surfactant, and make qualitative comparisons with the standard surfactant-free
Kelvin-Helmholtz analysis.
The effect of surfactant on jet stability has been studied experimentally (e.g., Skelland and Walker 1989), via
linear Kelvin-Helmholtz theory (Funada et al. 2004), and via Orr-Sommerfeldt analysis including surfactant. Kwak
and Pozrikidis (2001) and Kwak et al. (2001) studied non-linear effects of insoluble surfactant on the breakup of
fluid threads and filaments, and found that the formation of satellite droplets (and therefore the resulting droplet
size distribution) is sensitive to surfactant diffusivity, activity and density in the interface. Liao et al. (2006) studied
similar effects of insoluble surfactant on the breakup of thin liquid filaments, both experimentally and numerically, and
Davidson and Harvie (2007) studied the effect of insoluble surfactant on the deformation of rising droplets using the
volume-of-fluid (VOF) method. Johnson and Borhan (2000) used boundary integral methods to study the distortions
in the wake-side interface of rising surfactant laden droplets.
The amphiphilic Lattice Boltzmann method of Nekovee et al. (2000) is a "Shan-Chen" model, that explicitly in-
cludes the effects of intermolecular forces on a mesoscopic level (Furtado and Skartlien 2009). The rotational degree
of freedom of the amphiphile is accounted for, and the HLB (hydrophobic-lypophihilic balance) and solubility of the
amphiphile can be controlled by the strength of the interaction forces. Furthermore, the coupling between surfac-
tant density and interfacial tension, and the associated Marangoni-dynamics is accounted for. The model constants
(coupling strengths between the fluids and the surfactant) were calibrated to Exxsol D80 oil and oil soluble Span 80
surfactant, by matching the dynamic interfacial tension of a pendant drop (Skartlien et al. 2009).


2 Kelvin-Helmholtz linear analysis

Rayleigh-Plateau (RP) inviscid analysis in the limit of negligible axial velocity and zero gravity, shows that capillary
instability is possible for wavelengths larger than the circumference of the liquid column, A > 7Di = Ac (or for
wavenumbers less than the critical wavenumber kc = 2.0 normalized to the jet diameter Di). This instability is
triggered by Laplacian pressure perturbations due to the imposed perturbations in interfacial curvature. The most
unstable (fastest growing) wavelength for this kind of instability is Am = 4.51D, (or km = 1 :' Ii. In contrast, when
the effects of shear at the interface dominate capillary effects, a Kelvin-Helmholtz (KH) analysis accounting for the
difference in the velocities of the fluids on opposite sides of the interface must be performed. It is then apparent that
wavelengths that are smaller than the critical RP wavelength (wDi) may be unstable.
Homma et al. (2006) define six different jet instability modes. For a Weber number less than unity, gravity and
capillarity are igli ik.itll and the jet is in the dripping mode (I). For larger Weber numbers, a connected liquid column
that break up into droplets beyond a characteristic pinch-off length that increases with the Reynolds number is formed.
In this jetting mode, the Weber and capillary numbers are important, and gravity plays only a secondary role. The
jetting mode is separated into three regimes: jetting with uniform droplets (II), jetting with non-uniform droplets (III),
and jetting without breakup (IV). Homma et al. (2006) show that these regimes separate well in a Weber number -
viscosity ratio diagram. We will consider only the jetting mode (IV) in the following, where KH-analysis is applicable.

Range of unstable axisymmetric modes. The stability properties are in general influenced by the following
non-dimensional numbers. The Weber number

We p (1)

the Capillary number
Ca p (2)

and with gravity, the Bond number (or EOtvos number)

Bo D- .Ap (3)

The Froude number Fr = We/Bo and the Reynolds number Re = We/Ca are often used as auxiliary dimensionless
numbers. Here, p, Uo, cr, v, Ap, and g are the density, average inlet velocity, interfacial tension, kinematic viscosity,
density difference between the jet (oil) and the surrounding liquid (water), and the gravitational acceleration. The
subscript "o" refers to the oil phase. For a vertical jet, standard Kelvin Helmholtz (KH) analysis with a "plug flow"







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


A2 A3


No surfactant Surf.: 2e-4


Surf.: 5e-4 Surf.: le-3


Figure 1: Fluid jets with progressively increasing surfactant concentration (left to right). Left panel: Density map of
a vertical cut through a 3-dimensional fluid jet for case AO (no surfactant) at t=5400. The jet is unstable with
a wavelength of about 3 Di. The remaining panels: The surfactant density field for cases A1-A3 at t=5400.
The jet has a progressively larger growth rate for the unstable mode, and non-linearities develop faster. The
surfactant is swept up in the wave crests (most visible for A3). With the largest amount of surfactant (A3) the
wavelength above the inlet is about 2.5 Di, smaller than 3 Di for AO, as predicted by standard KH-theory.
For A3, the symmetry of the jet is eventually broken, and it forms a liquid volume with an irregular shape.


velocity profile can be adopted if the effects of buoyancy and the accompanying stretching) in the axial direction can be
ignored (small Bond number / large Froude number) and the effects of viscous momentum diffusion into the ambient
fluids. It is well known that the thickness of the interfacial boundary layer alters the stability properties. Thus, a KH
analysis serves only as a rough guide.
In general, the wavenumber range of the unstable axisymmetric modes increases with increasing Weber number.
Therefore, a larger inlet velocity Uo or a smaller interfacial tension a gives a larger range 0 < k < kc allowing
progressively smaller wavelengths to become unstable. The smallest unstable wavelength can therefore be smaller
than the smallest unstable wavelength predicted by the Rayleigh-Plateau instability analysis (rDi) for a liquid column
of the same diameter. Indeed, for k greater than about ~ 5, the asymptotical expression for the critical wavenumber is
(Funada et al. 2004),
kc + We, (4)
(1+ m)2
where the density ratio is
n= (5)







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Table 1: LBM force coupling parameters and Bhatnagar-Gross-Krook (BGK) relaxation times. The same param-
eters were used for both the jet and the droplet runs. Subscripts "o", "w", "s" refer to oil, water and surfactant,
respectively.
I00 G_ G. G G,. SO G8 80 T TW TS
surfactant -1.0 -3.0 3.07 -1.0 x 103 -1.3 x 103 -1.0 x 103 0.98 0.73 0.8
no surfactant -1.0 -3.0 3.07 0 0 0 0.98 0.73 -


Table 2: Non-dimensional numbers for the jet and the droplet. The surfactant density is given at the inlet, relative
to the oil density. Uo is the jet inlet velocity. Lower limits for the Weber, capillary and Bond numbers (We,
Ca and Bo) are given for those systems that contain surfactant. Bond numbers in parentheses refer to the
outer fluid density (using pw instead of Ap = pw po in the Bond number). The Reynolds numbers were
calculated from the inlet velocity for the jet, and for the terminal velocity for the droplet.
Case (label) ps/po Ug [m/s] We Ca Bo Fr Re
Jet AO 0.0 0.58 6.29 0.11 0.37 16.6 55.7
Jet Al 2 x 10- 0.58 >6.29 >0.11 >0.37 16.6 55.7
JetA2 5 x 10T 0.58 >6.29 >0.11 >0.37 16.6 55.7
JetA3 1 x 10T 0.58 >6.29 >0.11 >0.37 16.6 55.7
Drop AO 0.0 0.018 1.5 (4.5) -
Drop Al 1.0 x 10 >0.018 > 1.5 (4.5) 16 (terminal)
Drop A2 1.5 x 10 >0.018 > 1.5 (4.5) 18
Drop A3 2.0 x 10 >0.018 > 1.5 (4.5) 20


and the viscosity ratio is
m = = n-. (6)
Po o Vo
For the jet we have simulated, kc ~ 0.88We ~ 5.5, or Ac = 1.14 Di (compared to 7Di for the RP-instability).

The most unstable mode. We expect that the most visible wavelength on the jet interface is also the most
unstable or fastest growing wavelength. The most unstable wavenumber always satisfies k, < kc (the most unstable
wavelength is larger than the critical wavelength ). The most ,igmiik.nil interfacial tension and velocity effect is
(Holmhs 2008): The most unstable wavelength is decreased, and its growth rate is increased for lower interfacial
tension and/or higher jet injection velocity. Therefore, lower interfacial tension and higher velocity have qualitatively
the same effect of making the jet more unstable at shorter wavelengths. Consequently, it is reasonable to expect that the
addition of surfactant would have an effect similar that observed when the surface tension is decreased or the injection
velocity is increased if the effects of the Marangoni stress can be ignored. However, the growth rate is almost constant
above a characteristic interfacial tension, and varies significantly below this value. Finally, for larger jet viscosities
(smaller viscosity ratios m), the growth rate of the most unstable wavelength decreases, without a significant change
in most unstable wavelength1.


3 Simulations with surfactant

3.1 Surfactant modelling

The lattice Boltzmann model for multiphase fluid behavior with a surfactant is given in e.g., Nekovee et al. (2000).
It is based on the Shan and Chen approach, where the fluid forces are incorporated into an approximation of the
Maxwell-Boltzmann distribution function. The new ingredient in the surfactant model by Nekovee et al. (2000), is the
introduction of forces between the two ordinary fluids and between the fluids and the surfactant, taking into account
'There is a range of viscosity ratios mrr < m < m, where the jet is stable (Holmas 2008). The extent of the range decreases for increasing
Weber number, and the jet is unstable for all m above a "critical" Weber number where zm = mn, (for the density ratio n, used in this work,
the jet is unstable for all m, for We > 1.8).







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010



S7 OIL





I0 0
,0 0

m d o o o WATER



Figure 2: Left figure: The surfactant related forces considered in the model, shown by arrows. The amphiphiles
are represented as dipoles of length d, with a rotational degree of freedom. The oil and water molecules are
represented as monopoles. The fluid-surfactant forces (monopole-dipole forces) and the surfactant-surfactant
(dipole-dipole) forces are shown. Right figure: A cartoon showing surfactant located at an interface, with
some surfactant molecules in the oil. A few force pairs are indicated.


the rotational degree of freedom of the surfactant. It can be shown (Furtado and Skartlien 2009) that averaging over
the associated model molecules in a kinetic theory framework generates a mesoscopic model equivalent to that of
Nekovee et al. (2000). The surfactant molecules are treated as a simple dipoles (representing molecules that have
distinctly different "heads" and "tails"), where one end of the surfactant attracts the oil molecules, and the other end
attracts the water molecules. The forces are described qualitatively via figure 2, for brevity. The forces between the
fluids composed of molecules i andj are assumed to be of the classical form Fj = , ./R2, where the Gij are the
coupling constants. This simple model mimics the reduction of interfacial tension and Marangoni effects, since the
LBM scheme includes a surfactant fluid that diffuses through the other two fluids, and on the interface. In addition, the
HLB balance and solubility of the surfactant in the oil and water phases can be controlled via the coupling constants
Gij.

3.2 Simulation setup
The code is implemented in Fortran90 with MPI for parallel processing. The LBM algorithm is extremely well suited
for parallel processing since data is transferred between neighbor nodes only (by advection at the lattice velocities).
The jet simulations were performed on a 3D Cartesian grid with 100 x 100 nodes in the horizontal directions, and 400
in the vertical direction, giving a total of 4 million grid points. We split the domain in N equal horizontal slabs, one for
each processor. The domain size was chosen to be 1.4 x 1.4 x 5.6 cm. The inlet diameter was Di = 0.42 cm. These
computations take about one day on a normal workstation with N=4 microprocessors. We used a D3Q19 velocity
quadrature (a set of 19 3-dimensional basis vectors that ensure isotropy of the 4th order viscosity tensor) with a second
order approximation to the Maxwell-Boltxmann equilibrium distribution function. For the droplet simulations we used
a 200 x 400 2D grid (2.8 x 5.6 cm), with a D2Q9 quadrature. The spatial resolution was set to Ax = 0.14 mm and
the temporal resolution to At = 0.07 ms
We used open inflow/outflow boundary conditions (a fixed pressure at the boundary) along the vertical edges and
at the top of the domain, so that the mass in the box remained essentially constant (to allow the water to move out of
the box when the oil was pumped in). We performed a test simulation for the jet with a box-width that was twice as
large (200 x 200 nodes) with no appreciable difference, confirming that the open boundary emulates, to a reasonable
approximation, an "infinite domain". The no-slip (standard bounce-back) condition at the lower wall and on a circular
pipe segment at the inlet, was used. The contact angle with the wall depends on the fluid-wall coupling constants.
These were set equal for the oil and water to generate a 90 degree contact angle. The pipe segment was implemented
by using bounceback-conditions along a "staircase" approximation to the curved wall. The oil phase and surfactant
with different concentrations in solution, was injected through the pipe.
The coupling constants and the BKG relaxation times are given in table 1. The BKG relaxation times were chosen so
that the viscosity ratio between oil and water was m =0.7, and the the density ratio was n = 1.5, both representative of







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


AO A2


t=4800 t=5000


t=5200 t=5400


Figure 3: Time sequence for cases AO and A2 showing the propagation and growth of the non-linear waves. The
horizontal lines mark particular wave crests, that can be followed. The amplitude close to the inlet is larger
for case A2 than for the surfactant free case AO, demonstrating larger growth rate with surfactant.


a water/ExxsolD80 system. Gravity was set equal to 9.8m/s2 (by imposing an external force in the vertical direction).
The surfactant temperature parameter was set to Ts 1/30, and the rotational relaxation time for the amphiphile to
Td = 5.0 (see Nekovee et al. (2000) for an explanation of these parameters).
van der Waals type coupling strengths Gi within the fluids was chosen such that the equation of state determines
the desired density ratio between oil and water. We choose interaction potentials of the form

Pa= P -j(1 P/P). (7)

This form gives good numerical stability due to the limiting behavior for densities larger than, .,,. A desirable property
for analytical reasons is that ya is proportional p" for smaller densities.
We tuned the "repulsive" coupling constant Gij between oil and water so that the surfactant-free interfacial tension
was ao 25 mN/m. The surfactant related constants were tuned so that the equilibrium interfacial tension with
surfactant was amo 10 mN/m, and so that the surfactant (representative of Span 80) was essentially insoluble in the
water phase (Skartlien et al. 2009). The compressible nature of the LBM scheme results in a density gradient against
gravity and the density dependent fluid forces would therefore be decrease with increasing height. In order to maintain
an approximately constant interfacial tension, we scaled (amplified) the repulsive force with height, using the form
A p'(y O0)/p'(y), where p, is the initial density profile of the outer fluid, and A is a constant (here set to
unity).


AO A2


AO A2


AO A2







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


action lover crest)


Figure 4: Surfactant density profiles for case A2 at t=5400, near the black horizontal line in the right hand panel
of figure 3. The full thick line is the density profile where the surfactant concentration at the interface is
maximum. The other curves are profiles above (the wider part of the crest) and below (near the narrow
neck). The maximum surfactant concentration on the interface is elevated by about 50% compared to the
interfacial values above and below.


3.3 Non-linear jet dynamics with surfactant

The surfactant concentrations and non-dimensional numbers for the four jet-cases studied are given in table 2. The
surfactant free jet (AO) should be unstable according to KH-theory, with a critical wavelength of 1.14 Di. Unstable
modes should be found at longer wavelengths, and the simulations show that this is indeed the case (figure 1 left hand
side). The most unstable wavelength appears to be about 3 Di. With surfactant (cases A1,A2,A3), the Weber number
based on the average interfacial tension is reduced, and the jet becomes more unstable (a larger growth rate of the
most unstable mode) at shorter wavelengths. The most unstable wavelength for case A3 is observed at about 2.5 Di
(table 3). This trend certainly corresponds to KH linear theory, that predicts shorter wavelengths for lower interfacial
tensions (excluding Marangoni effects).
It can be seen (figure 1) that the surfactant density is elevated at the interface relative to the bulk density in the jet,
as expected. This is the energetically most favorable configuration (in terms of free energy) due to the binding energy
of the amphiphile to the interface. Even though the model surfactant was near immiscible in the water phase for the
pendant drop simulation (Skartlien et al. 2009), a small amount of surfactant becomes dissolved in the water phase
(figure 4). The local build-up of surfactant on the interface is to some degree counteracted by diffusion back into the
oil.
As the waves propagate upwards, they become more complex (figure 3). As the amount of surfactant is increased, the
characteristic of non-linear behavior develops closer to the inlet due to the larger growth rate. The Z-shaped interface







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


















Z- -


-- =_==== .... .. ... =.--.-.-.-----







Figure 5: Interfacial surfactant distribution and velocity field for case A2 at t=5400. The oil enters to the left in
the figures, and gravity acts to the right. The darker shade in the 3D visualization in the upper figure shows
increased surfactant concentration in the wave crests. A number of flow lines in a vertical cut through the
center of the jet are shown in the bottom figure, on top of the interface contour.


Table 3: Wavelengths on the jet. As > Ac denotes the most unstable wavelength in the simulations, and Ac the critical
(minimum) wavelength for instability from KH-theory.
Case As/D, Ac/Di
Jet AO -3 1.14
Jet Al 3 < 1.14
Jet A2 < 3 < 1.14
JetA3~ 2.5 < 1.14


profile that develops are qualitatively similar to the interface profiles observed in simple shear flow (considering that
the jet moves upwards, faster than the surrounding fluid). The surfactant is swept into the wake of the wave crests
(figures 4 and 5). Figure 5 shows a 3D visualization of the surfactant concentration at the interface (darker shade), and
demonstrate the sweeping up of surfactant in the wave crests. A number of flow lines in a the center plane of the jet
are shown in the bottom figure, on top of the interface contour, and these reveal an internal vortex ring in the jet. This
vortex ring forms in response to the shear set up by the flow of the outer fluid over the crest.
The associated Marangoni stress in the interface acts in a stabilizing manner. The stress is directed away from the
region of elevated surfactant concentration (and along the interface), reducing the growth rate of the steepening wave.
This effect is stronger for more surfactant. The KH-theory predicts larger amplitudes as the amount of surfactant
is increased (as the average interfacial tension decreases). This is not observed, and we suggest that the stabilizing
Marangoni stress is responsible for this deviation from the behavior predicted by KH-theory. It is noteworthy that
suppressed growth due to the Marangoni effect has also been reported in the linear regime (Behroozi et al. 2007) for
insoluble surfactant (restricted to the interface only). This explains the "calming effect" of an oil layer on capillary
waves on a gas-liquid interface.
At the inlet in the current simulations, the surfactant initially diffuses fast enough from the bulk to the interface to
maintain a uniform distribution on the interface, and the results are in agreement with linear theory. Further away from








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


t= 2300 t= 3100 t= 3900
40 400 400
0.05

0 0 0 0.04

0.04


0.02

10oo 1o 1o -0.01

0.00
o o 0 2000 4000 6000 8000

t= 4700 t= 5500 t= 6300
400 400 40'
0.12







0.04
10 10 10
S0.04


0.00 /
a 0 0 2000 4000 6000 8000
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200Time


Figure 6: Time sequence for a rising droplet: The surfactant density of case A2 for times t= 2300, 3100, 3900, 4700,
5500, 6300. Right upper panel: the position of the droplet (the apex) as a function of time step (the full
dimensional time span corresponds to 0.7 seconds). There are four stages (separated by symbols): i) fluid
injection ii) filament stretching, iii) droplet detachment and relaxation of shape, iv) steady, translating droplet
at terminal velocity. Full line: case Al, dashed line: case A2, and dash-dotted line: case A3. Right bottom
panel: The corresponding velocities.


the inlet, the bulk concentration is reduced due to adsorption at the interface, and the hydrodynamic effect of sweeping
up the surfactant becomes significant, leading to a suppression of further wave growth.
Pinch-off into droplets was not observed within the computational domain in the simulations. For even larger Weber
numbers (or Reynolds number), we expect that the non-linear growth would continue and eventually lead to folded
sheets that would break up into fingers, and then into droplets (Marmottant and Villermaux 2004). In particular,
the increased surfactant concentration in the steepening wave crest would lead more easily to droplet formation ("tip
streaming") during later stages as compared to the surfactant-free case.


3.4 Buoyant surfactant laden droplets

Two-dimensional simulations of droplets rising from an orifice were performed. Cases Al through A3 have succes-
sively higher surfactant concentration, with injection densities of ps/Po 1 x 103, 1.5 x 10 3,2 x 10 3. The
different surfactant concentrations and non-dimensional numbers for the four droplet simulations are given in table
2. The Reynolds numbers were calculated for the rising droplet at late times. The coupling constants were the same
as those used for the jet simulations (table 1). Case AO was run without surfactant, and the droplet did not detach
from the inlet and remained as a pendant drop. With surfactant, the average interfacial tension is reduced, and the
Bond numbers becomes large enough for the droplets to detach. Figure 6 shows a time sequence for case A2 on the
full computational domain. The surfactant concentration is shown, with larger concentrations indicated by lighter grey
shades. The surfactant is swept back on the interface and the local reduction of interfacial tension induces a Marangoni
stress.








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Al
1.0


0.8


0.4

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0


Al
1.0 -

0.8 '\

0.6 4-,\ 4 1

0.4 <, -

0.2 ,

0.0
0.0 0.2 0.4 0.6 0.8 1.0


A2
-- -- -- ^


K>


/ r


0.4 ,

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0


A2
1.0 .



0.6

0.4

0.2 "

0.0
0.0 0.2 0.4 0.6 0.8 1.0


A3


0.8

0.6

0.4

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0


A3
1.0

1

0.6 4 ,i

0.4 -

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0


Figure 7: Upper row: Velocity field at the time of detachment. A1-A3: t=4100, 2700, 2300 respectively. Lower row:
Velocity field after detachment, when the droplets have reached the same height. A1-A3: t=4200, 2800,
2300. Only a sub-domain of 100 x 100 cells are shown of the full domain of 200 x 400 cells.


Rise velocities. Figure 6 (right hand side panels) shows the position and vertical velocity of the droplet apex as
a function of time (1000 time steps corresponds to 72 ms of dimensional time). The symbols on the curves mark
the transitions between the following four evolutionary stages: i) fluid injection, ii) filament stretching driven by
gravity, iii) droplet detachment and relaxation of droplet shape, and iv) steady, translating droplet at constant (terminal)
velocity. The velocity differences between the cases for stage iv) are not significant, although a lower terminal velocity
is expected for more surfactant, due to the well known suppression of internal circulation by surfactant. This point is
discussed further below.


Filament stretching. The main surfactant-effect in the current simulations is that the stretching rate of the filament
(in stage ii) is larger for more surfactant. This is seen as a larger average upwards velocity during stage ii) in figure
6. With surfactant, the liquid volume stretches faster under the influence of gravity acting on the density difference
between the water and oil phases since the interfacial tension is on the average lower, and it takes less energy to deform
the interface. The upper row in figure 7 shows all three cases near the pinch-off time (beginning of stage iii). The
filament between the rising drop and the inlet pipe pinches off sooner when more surfactant is present. The length
of the filament is slightly larger for more surfactant, and the instantaneous rise velocity after pinch-off is also slightly
larger. The lower row in figure 7 shows the cases immediately after pinch-off. For cases Al and A2, the time delay
from pinch-off is the same, and it can be seen that the filament retracts faster towards the inlet for smaller amount of
surfactant/larger interfacial forces.


Droplet shape and internal circulation. The internal circulation in the droplet is shown in figure 8. The lower
row shows the droplets after detachment (in phase iii, near maximum upward velocity). As expected, a simple double
vortex develops. The droplets are flatter on the front/top that in the back, and this is also observed in experiments
(Myint et al. 2007) when the viscosity in the droplet is larger than the viscosity in the surrounding fluid. The 2D two-
dimensional droplets are similar to cuts through oblate spheroids like those observed in experiments at the same Bond








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Al
1.0o- . . 1.


0.0 0.2 0.4 0.6 0.8 1.0


Al
1.0

0.8

0.6

0.4

0.2

0.0 .
0.0 0.2 0.4 0.6 0.8 1.0


A2
1.0o


0.0 0.2 0.4 0.6 0.8 1.0


0.8

0.6

0.4

0.2

0.0 ....
0.0 0.2 0.4 0.6 0.8 1.0


A3
1.0



0.6

0.4 '

0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0


A3


0.8 r


0.0 1
0.0 0.2 0.4 0.6 0.8 1.0


Figure 8: Upper row: Internal circulation for the droplets when they have reached the same height and when they
have reached terminal velocity (in stage iv). Al: t=8000, A2: t=6600, A3: t=6400. Lower row: Internal
circulation near maximum rise velocity at earlier times, in stage iii). Al: t=5400, A2: t=4000, A3: t=3700.


and Reynolds numbers (Clift et al. 1978). The upper row shows the internal circulation at later times, when the droplet
shapes have stabilized (in phase iv). The asymmetric interfacial shape becomes more pronounced as the amount of
surfactant is increased. The surfactant is swept back under the droplet, and this reduces the interfacial tension there
with an accompanying larger curvature, giving a cone shape for the larger surfactant concentration. There is also a
slight expansion of the droplet at later times, due to the compressible nature of the lattice Boltzmann fluid. The internal
circulation pattern is very complex with eight visible vortices, four on each side of the centerline. The RMS amplitude
of the circulation velocity decreases as the droplet rises (table 4), indicating that the circulation is suppressed by the
surfactant, or by viscous damping.
Figure 6 (velocities shown in the right hand side panels) indicates that at later times the droplet rises with a terminal
velocity that is smaller than the maximum rise velocity. This observation is consistent with the reduced internal
circulation at later times. The rise velocity is reduced by a factor of almost 2 relative to the maximum velocity (see
figure 6), while the RMS circulation velocity is reduced by a factor of about 3 (table 4).
The classical literature (e.g., Clift et al. 1978) reports that the internal circulation pattern is expected to be a double
vortex (or vortex ring in 3D) that is shifted forwards, while a stagnant region should develop in the rear of the droplet
due to the opposing Marangoni stress. The internal flow patterns generated during the early stages of the simulations
(figure 8 lower row), with a double vortex that sits in the upper half of the droplet, are consistent with these expecta-
tions. At later times (upper row), multiple vortices develop with a smaller RMS velocity, and there are no clear signs
of stagnant regions or a dominating double vortex.
The terminal velocity for surfactant laden droplets that are near spherical in shape is given approximately by (Griffith
1962),
UT UT(1 + Z/2) (8)

where UTS is the lower limit, hard-sphere Stokes value, and

S4(Y(Bo') 1)
2 + 3/m







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Table 4: RMS internal circulation velocities normalized to Uc. "Low" refers to the droplets in the lower row in figure
8, and "high" to the upper row.
Case VRMs(low)(x10 ) VRM^hi
VRMS(low)
Drop Al 2.9 0.38
Drop A2 3.2 0.39
Drop A3 3.1 0.50


accounts for the effect of surfactant via a smooth function Y(Bo') of the modified Bond number Bo' that rises
smoothly from Y 1.0 to Y 1.5 as Bo' increases from zero. The modified Bond number, Bo', is defined in
terms of the change Ao in interfacial tension relative to the interfacial tension in a surfactant free system. The modi-
fied Bo' numbers were 5 or lower, so that Y 1. Although these formulae provide only a rough guide in the current
case, we do not expect a Nigniiik.iiI effect of surfactant on the terminal velocity. In order to see a clear effect, we will
perform simulations with larger Bond numbers in a follow-up study.


4 Conclusions

We demonstrate increased linear growth of axisymmetric waves on buoyant oil jets in water, due to surfactant. This
is explained by the standard Kelvin-Helmholtz analysis, which predicts that the growth rate of the shear instability
increases with decreasing interfacial tension. However, the subsequent non-linear wave steepening is suppressed by
Marangoni stresses induced by surfactant that is swept into the wave crests. This is similar to Marangoni induced
suppression of capillary waves on oil-laden air-water surfaces (Behroozi et al. 2007).
The rise of single buoyant oil droplets in water with surfactant was simulated in 2D. The dynamics of the stretching
filament between the inlet pipe and the rising droplet, and the internal circulation in the free droplets were investigated.
The filament stretching rate is increased with larger amount of surfactant, since it takes less energy to deform the
interface. Without surfactant, the droplet did not detach and remained pendant. After detachment, the droplets are
oblate spheroids with a pronounced fore/aft asymmetry. They are flatter on the top, with larger curvature in the
bottom. This is consistent with experimental work (e.g., Myint et al. 2007) when the droplet viscosity is larger than
the external viscosity. The internal circulation is initially a simple double vortex structure, but develops into a multi-
vortex system when the droplet relaxes to a steady shape. The surfactant is swept back on the interface and the local
reduction of interfacial tension induces a Marangoni stress. The current results indicate that the Bond numbers in
the simulations were too small (Griffith 1962) to observe a ,igili.iiii reduction of the terminal velocity with larger
amount of surfactant.
To summarize, we have demonstrated that the current modelling approach captures the following surfactant related
effects:

The interfaces are more deformable with more surfactant. We observe this via 1) larger growth rate for Kelvin-
Helmholtz unstable linear waves on the jet, 2) larger filament stretching rate during droplet formation, and 3)
slower retraction of the liquid filament after droplet detachment.

We observe Marangoni stabilized growth of the non-linear waves on the jet, due to sweeping of surfactant into
the wave crests.

Marangoni-stabilization of non-linear waves may delay the entrainment of droplets from the interface of a jet at
higher Reynolds numbers. Later stages of the non-linear evolution of the interface (with breakup) will have to be
considered as well to obtain a more complete picture of how surfactant influences the droplet generation. We have
previously calibrated the amphiphilic interaction strengths and other factors in the Lattice Boltzmann model to Exxsol
D80 oil and Span 80 oil soluble non-ionic surfactant using measurements of dynamic interfacial tension on a pendant
drop. The model accounts for diffusion controlled adsorption of surfactant between the bulk and the interface. The
solubility of the surfactant can be controlled to make it oil soluble and essentially water insoluble (Skartlien et al.
2009). In upcoming work, we will compare simulation results directly with experiments using similar fluids and
surfactants.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Acknowledgements This work was performed by the FACE centre a research cooperation between IFE, NTNU
and SINTEE The centre is funded by The Research Council of Norway, and by the following industrial partners:
StatoilHydro ASA, Norske ConocoPhillips AS, Vetco Gray Scandinavia AS, Scandpower Petroleum Technology AS,
FMC, CD-adapco, ENI Norge AS, Shell Technology Norway AS. K. Holmas provided material for the linear stability
analysis of the liquid jet, T. E. Unander, I.E. Smith, and F. Krampa provided preliminary experimental data and
additional analysis, and CD-adapco through S. Lo provided initial VOF simulation results without surfactant. T. B.
Kjeldby performed initial jet simulations as a "summer student" at IFE.

References

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2007

Davidson M R. and Harvie D. J. E., Predicting the effect of interfacial flow of insoluble surfactant on the deformation
of drops rising in a liquid, ANZIAM 48, pp. 661-676, 2007

Funada T., Joseph D.D., Yamashita S., Stability of a liquid jet into incomressibe gases and liquids, Int. Journal of
Multiphase Flow, 30, pp.1279-1310, 2004

Ferri, J. K., and Stebe K. J., Which surfactants reduce surface tension faster? A scaling argument for diffusion-
controlled adsorption, Advances in Colloid and Interface Science, 85, pp. 61-97, 2000

Furtado K. and Skartlien R., Derivation and thermodynamics of a lattice Boltzmann model with soluble amphiphilic
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Clift R., Grace J.R. and Weber M.E., Bubbles, drops, and particles, Academic Press Inc., 1978

Griffith, R.M., Chem. Eng. Sci., 17, pp. 1057-1070, 1962

Holmas, K., Linear stability analysis of liquid jets, FACE report IFE/KR/F-2008/274, 2008

Homma et al., Breakup mode of an axisymmetric liquid jet injected into another immiscible fluid, Chem. Eng. Sci.,
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Physics of Fluids, 12, pp. 773-784, 2000

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Science and technology, 2, pp. 184-195, 2007

Nekovee M., Coveney P.V., Chen H., Boghosian B.M., Lattice Boltzmann model for interacting amphiphilic fluids,
Physical Review, E62, pp. 8282-8294, 2000

Skartlien R., Furtado K., Sollum E., Meakin P., Lattice Boltzmann simulation of surfactant adsorption kinetics on a
pendant drop interface, submitted to Journal of Colloid and Interface Science, 2009

Skelland A. H. P. and Walker P. G., The effects of surface active agents on jet breakup in liquid-liquid systems, Can.
J. Chem. Eng., 67, pp.762-770, 1989




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