Mixing driven by Rayleigh -Taylor Instability in the ...icsr.agh.edu.pl/~dzwinel/files/publications/R-T.pdf · macroscopic models incorporating the two microstructural mechanisms

Dzwinel, W. and D.A. Yuen, "Mixing Driven by Rayleigh-Taylor Instability in the Mesoscale Modeled with Dissipative Particle Dynamics", International J. of Modern Physics C, Vol. 12, No. 1, 91-118, 2001.

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Mixing driven by Rayleigh-Taylor Instability in the Mesoscale Modelled with Dissipative Particle Dynamics

Witold Dzwinel AGH Institute of Computer Science, al. Mickiewicza 30, 30-059, Kraków, Poland

David A.Yuen1

Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55415-

1227, USA

Abstract

In the mesoscale mixing dynamics involving immiscible fluids is truly an outstanding problem in many fields, ranging from biology to geology, because of the multiscale nature, which causes severe difficulties for conventional methods using partial differential equations. The existing macroscopic models incorporating the two microstructural mechanisms of breakup and coalescence do not have the necessary physical ingredients for feedback dynamics. We demonstrate here that the approach of dissipative particle dynamics (DPD) does include the feedback mechanism and thus can yield much deeper insight into the nature of immiscible mixing. We have employed the DPD method for simulating numerically the highly nonlinear aspects of the Rayleigh-Taylor (R-T) instability developed over the mesoscale for viscous, immiscible, elastically compressible fluids. In the initial stages we encounter the spontaneous, vertical oscillations in the incipient period of mixing. The long term dynamics are controlled by the initial breakup and the subsequent coalescence of the microstructures and the termination of the chaotic stage in the development of the R-T instability. In the regime with high capillary number breakup plays a dominant role in the mixing, whereas in the low capillary number regime the flow decelerates and coalescence takes over and causes a more rapid turnover. The speed of mixing and the turnover depend on the immiscibility factor, which results from microscopic interactions between the binary fluid components. Both the speed of mixing and the overturn dynamics depend not only on the mascrocopic fluid properties, but also on the breakup and coalescent patterns, and most importantly on the nonlinear interactions between the microstructural dynamics and the large-scale flow .

Keywords: Rayleigh-Taylor mixing, mesoscopic fluids, immiscible fluids, breakup and

coalescence morphology, dissipative particle dynamics (DPD)

1 corresponding Author, Minnesota Supercomputing Institute, University of Minnesota, 1200 Washington Av. South,

Minnesota, 55415-1227, USA e-mail: [email protected] fax: 612 624 8861


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INTRODUCTION

The problem of mixing and dispersion of heterogeneities in viscous and immiscibel fluids is found

universally in many fields in science and engineering [1-4]. In spite of the recent advances in

understanding the mixing of homogeneous fluids in the macroscale [1,3], realistic mixing

problems of heterogeneous multicomponent fluids are inherently difficult due to the complex,

multiple-scale nature of the flow fields, the intrinsic rheological complexity, and the cross-scaling

nature of the nonlinear physics, such as grain-boundary processes, chemical reactions and

geological processes with complex rheologies [2,4]. For these reasons, mixing problems have

traditionally been treated on a case-by-case basis. Classical modelling with partial-differential

equations [5] becomes intractable, if one wishes to examine all of the details simultaneously on a

cross-scale basis. As shown in [1-3], current models incorporate two competing processes in

mixing; breakup and coalescence. These two mechanisms involve one-sided interactions without

any feedback from the microstructural dynamics back to the global flow structure. This problem of

feedback of microstructural dynamics is very important in many applications, such as dispersion

of solid nano-particles through a polymer blend and dynamical mixing of crystals in magmatic

flows. In order to address some of the issues raised above, we will focus our attention on the

canonical problem of viscous mixing driven by the Rayleigh-Taylor instability in immiscible fluids.

Numerous factors, which influence the development of the Rayleigh-Taylor instability in a

simple macroscopic fluid, such as the surface tension, viscosity, compressibility, effects of

converging geometry, three-dimensional effects, the time dependence of the driving acceleration,

shocks, and variety of forms of heterogeneities, are being investigated for many years

theoretically, experimentally and by computer simulations employing computational fluid

dynamics techniques (CFD) [5-20]. The studies of coupling the breakup and coalescence

processes in the context of the R-T mixing in immiscible fluids in the mesoscale by employing

continuum hydrodynamics equations are still difficult and not reliable.

In mixing problems dealing with immiscible fluids one needs to monitor closely the

interface separating the different components. One fundamental problem in simulating the motion

of sharp interfaces is a proper description of transition phenomena, such as merging and

reconnection of interfaces in wave dynamics. These tough numerical problems can be solved in

continuum fluid models by using the so-called front-tracking method [21-23]. In mesoscopic

systems in which thermal fluctuations cannot be neglected, the internal boundary structure

remains unknown, which means that the dynamical equations of the interface cannot be solved.

This point has stressed in numerical simulations of jets in the nanoscale [24].

The processes of breakup and coalescence in mixing can in fact be viewed as population

dynamics of microstructural structures [1,3], which is driven and maintained by the chaotic flow.

Such an idea has inspired us to employ the discrete-particle approach for coupling the breakup

and coalescencing processes present in the mixing of immiscible binary fluids.


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The microscopic molecular dynamics (MD) approach can be successfully employed in

large-scale simulations of realistic microscale flows over an area of 106 Å2. They have revealed

the formation of bubbles and spikes for the R-T mixing [25-28] and breakup of nanojets created

by injection of fluid through nanoscale convergent gold nozzles [24]. However, the MD

simultations take up too much computer time in extending the same technique for simulating

complex fluids over the meoscopic length scale of 100 to 1000 Å (angstroms).

The extension of discrete-particle method to larger spatial scales can be realized by

changing only the notion of the inter-particle interaction potential by treating a large-sized particle

as a cluster of computational molecules. This idea of up-scaling has already been adapted in the

dissipative particle dynamics (DPD) method [29-39]. This method is based similarly on the

principles of discrete particle mechanics connected by two-body potentials, like MD. The

dissipative particles can be viewed as "droplets" of MD atoms. The DPD method is intrinsically

mesoscopic in nature, because it can only resolve the center-of-mass motion of the droplet and

avoids the description of its internal state. It can potentially link the microscale at tens of angstrom

scale with the macroscopic flow at the millimeter level. The key factor, which dictates the use of

the DPD method in simulating the R-T mixing of immiscible fluids is its recent successful

application in simulating complex fluids and phase separation [33-35]. By using a two-level MD-

DPD technique, we have successfully simulated the fragmentation, agglomeration and ordering in

colloidal arrays [35-39].

In this paper we will employ DPD simulations to direct our attention on the characteristic

microstructures developed in the course of mixing of immiscible fluids. This study would entail

monitoring simultaneously both the break -up and coalescencing processes and also the mutual

interaction between the microstructures and the global flow.

First, we introduce the DPD model, the numerical method and the simulation conditions.

Then we investigate the mixing driven by the R-T instabilities in a rectangular box. Next, we

present and discuss the simulation results. We focus then on the temporal evolution of the

interface between two fluids and its dependence of the elastic compressibility, immiscibility and

kinematic viscosities of the two superimposed fluids. Finally, we will discuss both the advantages

and disadvantages of the DPD model in coupling in one single model the processes of breakup

and coalescence with the macroscopic fluids, at least, up to the millimeter scale.

THE DPD MODEL AND SIMULATION CONDITIONS

Numerical model

In the DPD model [29,33-35] the discrete particles move about within the confines of a

rectangular box with a height h and a length Lxy . Periodic boundary conditions are imposed along

the x-direction and reflecting boundary conditions are employed in the y-direction. We have


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divided the box into two parts, with the upper part of the box filled up with heavy fluid particle (H)

and the lower part filled with lighter fluid particles (L). An external gravity field G pointing

downwards is present. The particles are defined by the mass Mi, position ri, and momentum pi.

We use the two-body, short-ranged DPD force FT= FC+FB+FD as in [39]. This type of force

consists of conservative FC, dissipative FD and Brownian FB components. The value of FT=

FC=FB=FD =0 for rij>rc. Otherwise, we give the following definitions:

( ) ( ) ( ) ( ) ijijij

BijijijijDijijC rt

rMr eFeveFeF ⋅⋅∆

⋅=⋅⋅⋅⋅=⋅⋅= 121 , , ω

θσωγωπ o (1)

where:

ω1() and ω2() - are the weight functions defined such that ( ) 10

=⋅ ∫ rdrncr

mD ω for m=1,2.,

rij - the distance between particles i and j,

rc - a cut-off radius, for which ω1(r) = ω2(r)= 0,

nD - an average particle density in D-dimensional system (D – dimension of the system),

e ij - a unit vector pointing from particle i to particle j,

π - the scaling factor for the conservative part of collision operator,

γ - the scaling factor for the dissipative force,

σ - the scaling factor for the Brownian motion,

θij - a random variable with a zero mean and actually normalized variance.

We assume that the normalized weight functions ω1(rij) and ω2(rij) are linear (see Eqs.(5)) as it is

in [30-35]). According to the fluctuation-dissipation theorem they are chosen such that

ω2(rij)=ω1(rij)2 [30-33].

The temporal evolution of the particle ensemble obeys the Newtonian equations of

motion. For integrating the equations of motion we employ the “leap-frog” algorithm in timesteping

for the particle positions rin, and the Adams-Bashforth scheme for the particle velocities vi

n and

momenta pin. For the two-component fluid, where k=g(i) and l=g(j) denote the types of particle i

and j (k,l∈H,L), the equations of motion in 2-D space can be represented in the following

discretized form.

( ) ( ) ( ) ( )p p e v ei

n

i

n

kl ijn

kl kl ijn

ijn

ijn kl ij

ijn

j iijnr M r

tr t

+ −

≠= + ⋅ − ⋅ ⋅ +

⋅

⋅∑

1

2

1

21 2 1π ϖ γ ϖ

σ θϖo ~

∆Λ (2)

r rp

in

in i

n

iMt+

+

= + ⋅1

12

∆ pp p

in i

n

i

n

=+

+ −12

12

2 (3)


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( )~ ~ ~ ~v p p v v v r r er r

in

ii

n

i

n

ijn

in

jn

ijn

in

jn

ijn i

njn

ijnM

rr

=⋅

⋅ −

= − = − =

−− −12

312

32

2, , , (4)

( ) ( )( )( )

≠+⋅

==

−⋅=

−⋅= )( if

2)( if

,16

,13

2

2221 jgigMM

MMjgigM

Mr

r

nrr

r

r

nrr

ji

ji

i

ijc

nij

c

nij

c

nij

c

nij π

ωπ

ω (5)

where:

∆t – the timestep,

Mkl - the mass of DPD particle; for interactions between particles of different kind (k≠ l) the center

of mass motion mean is computed.

In calculating the total force acting on each particle at a given timestep we have

employed an improved algorithm based on sorting out the linked-cells [40], which has O(N)

complexity (where N is the number of particles). The method is very efficient for serial

computations giving linear increase of CPU time with the number of particles and is more than

two times faster than the standard linked-cells method [41]. A typical simulation time for

N=1.8×105 particles in two dimensions in 1.2×105 timesteps for a particle density of 20 particles

within a cut-off radius rc is about 5 hours on a single R12000/300 processor of the SGI/Origin

2000 system.

Input data

In Table 1 we present the properties of the input data, which define physically the DPD particle

fluid model. The density ρk of the k th particle system is equal to:

Dkk nmS ⋅⋅=ρ (7)

where, mk is the mass of atom (or molecule) of fluid k, S - scaling factor of the spatio-temporal

scale simulated, Mk=S⋅mk is the mass of DPD particle and nD is a molecular density. For 2-D

particle system (D=2) n2 =λ2 where 33n=λ , n3 - particle density in 3-D, and λ - the average

distance between the neighboring particles. We assume that n2 is the same for both heavy (H)

and light (L) fluids and the density contrast ρH/ρL=5 is constant. It determines the Atwood number,

an important parameter governing the density jump in the R-T problem:

LH

LH

ρρρρ

α+−

= (8)


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to be equal to 2/3. In the R-T instabilities found in nature the density contrast ranges from O(1) for

geological systems to O(10) or greater in astrophysics.

Similar to the MD simulations, in [29, 33-34] dimensionless units are employed to define

the DPD particle system. However, in the former case, the temporal scale (and the timestep) is

determined by the temperature of the system related to the Lennard-Jones potential well depth,

which together with the size of atom couples spatial and temporal scales in MD simulation. For

this reason, the dimensionless units define precisely spatio-temporal scale for MD particle

system. In DPD model the interparticle force is employed instead of potential function (see Eq.(1))

[29-39]. Thus the reference level of potential energy, similar to the depth of potential well for the

L-J interactions, is lacking. The size of the DPD particle is also not precisely defined and should

be matched to the spatio-temporal scale S under interest. Moreover, the Brownian part is

dependent on the timestep ∆t (see Eq.(1)). Therefore, the values of coefficients of DPD collision

operator, thus the physical properties of the system, will strongly depend on the scaling factor S

for a given range of spatial-temporal scales.

To control the actual scale of simulation, we have matched (see [35]) the DPD particle

system to the saturated liquid Ar thermodynamic properties by using kinetic theory equations [29-

32]. We employ this spatio-temporal scale as a reference point to our earlier 2-D MD simulations

of the R-T mixing in the microscale [26-28]. From the physical properties of H (heavy) and L

(light) fluids such as: the temperature T of the system, kinematic viscosities νH, νL and sound

velocities cH and cL, we can calculate the DPD interparticle force coefficients πkl,γkl, σkl (see

Eq.(1,2)) by employing the following equations from kinetic theory [29,31] and in the continuum

approximations:

klBklkl MTk ⋅⋅= γσ 22 (9)

D

rnP klD

kl ⋅=

2

π , P ck k= ⋅

12

2 ρ (10)

vk TMk

T B

k

= (11)

( )221

2

+⋅=

DD

rnkl

kl

γν (12)

where the average moment values in Eqs.(10,12) ( )∫⋅=cr

mm drrnr0

rω .


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We define also the following values, which describe the dynamical regimes of the DPD model

[42]:

Tc Dvr /γ=Ω , r sc = ⋅ λ,

c

xy

r

Lfr = (13)

where Lxy is the length of computational box in x and y directions. We assume that Ω=100, to

obtain realistic kinematic viscosities for the particle fluids system. However, the real viscosities of

the L and H fluids can be then about 70% higher than these presented in Table 1 [31]. As shown

in [42], for Ω<15, fr>30 and s>2.5 the transport coefficients values are predicted by the kinetic

theory with relatively high accuracy. Otherwise, we can detect quantitatively the differences

between the theoretical predictions and the numerical simulations.

Table1. Input data

a) Macroscopic data

H H-L interface L

Atomic mass m

DPD part. mass M

Density ρ [kg/m3]

Temperature T [K]

Viscosity ν [m2/s]

P in 2-D (Eq.(10)):

a) high_c [N/m]

b) low_c [N/m]

Compressibility

a) high_c [m/N]

b) low_c [m/N]

Gravitational accel.

[m/s2]

40

1000

1400

84

2.06×10-7 (for Ω=100)

0.233

0.0233

31.5

57.3

1011

84

3.59×10-7 (for Ω=100)

0.233-0.500

0.0233-0.055

8

200

280

84

4.64×10-7 (for Ω=100)

0.233

0.0233

38.0

66.0

1011

b) Microscopic data

λ [×10-9 m]

s=rc [in λ]

Ω [dimensionless]

h [nm]

Lxy/h

Lxy/h [in rc units]

1.06

2.5

12,5 50 100

224

2:1 or 8:1

84:42 or 336:42

1.06

2.5

12,5 50 100

1.06

2.5

12,5 50 100

224

2:1 or 8:1

84:42 or 336:42

Number of particles

No. part. in rc sphere

22,500-90,000

20

22,500-90,000

20


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No. of timesteps

∆ t [×10-13 s]

120,000

1 or 2

120,000

1 or 2

The kinetic theory equations have been developed in the limit without any conservative

forces. Therefore, the transport coefficients computed from the theory can be used as a first

approximation. For example, the kinematic viscosity is underestimated theoretically but its

approximate dependence on the density is similar in both cases [31]. For a better matching, one

can apply an iterative procedure as in [43]. The contrast in dynamic viscosities p=µL/µH is chosen

to be close to 1 (0.6-2.2). As suggested in [1], for such the viscosity ratio wide spectrum of

breakup patterns are observed in laboratory experiments.

The simulations were carried out for DPD particle fluids with low and high sound

velocities c (low_c and high_c systems, respectively). Bec ause the pressure Pk in fluids H and L

should be the same [44], the difference in c comes from the density contrast between H and L

fluids.

Let us define the value of

∆ =PHL-Pk, (14)

which is equal to the difference between the partial pressures computed by using Eq.(10) for k≠ l

(PHL) and for k=l (P k=PH=PL). The value of ∆ - called here immiscibility factor – governs the

immiscibility of both the L and H fluids. For ∆≤0 the fluids are miscible. Otherwise, they separate

one from another [33-35,39]. We have conducted R-T mixing simulations for several values of ∆

for both low_c and high_c DPD particle systems (see table1).

As shown in [35], the DPD particle system defined in Table 1 (S≈25) reveals liquid

ordering in spatial scales larger than the simple L-J fluid (see Fig.1). While for greater S we can

observe the Kirkwood-Alder fluid-solid transition [45] (see, the RDF plot for S=100 in Fig.1). For a

lower density n2, lower sound velocities and shorter cut-off radius rc, the value of S, in which DPD

fluid does not freeze, can be greater. Because the complex fluid has a longer length-scale

structure than the ordinary molecular scales, we may assume that the DPD particle system may

simulate a colloidal-like suspension with nanoscale colloidal particles, about 2nm. As shown in

[37, 39], defining the colloidal and solvent types of particles and assuming the short-ranged

attractive forces between them, we can obtain a more sophisticated ordering in colloidal fluids by

including the effects due to the creation of micelles and colloidal arrays.

Due to the limited range of the spatial-temporal scales of the system, the gravitational

acceleration G (or an other external shock) should be very large for observing the mixing process

for reasonable number of timesteps. The value of G assumed here is an order of magnitude

smaller than the value of G we used for the Rayleigh-Taylor mixing in the microscale (S=1) [26-


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28]. Therefore, we may expect that for S=109 mixing will take place for G values of order 102 m/s2

- the range of acceleration commonly used in mixers. The model requires some modifications at

larger spatial scales, than those in our DPD model. For example, instead of using a linear weight

function, we can apply a more realistic types of ω(r), e.g., the first derivative of DLVO potential

used for colloids [45-46] or a Gaussian-like function (see [47]).

In spite of the problems dealing with matching the DPD model to the larger spatial scales,

we show below that in the present form this model can be used as a viable first approximation for

investigating the dynamic properties of mesoscopic fluids. From a computational point of view,

DPD approach has an advantage in that it is less demanding than using MD. In the simulation of

the system defined in Table 1 for a fixed spatio-temporal scale (e.g., 1µm in 10 ns) we find

astonishingly that MD needs roughly 10 times more memory and 100 times more CPU time than

DPD. These ratios can be even higher for greater values of S.

SIMULATION RESULTS

As shown in [26-28], there are basically four stages in the temporal evolution of the microscale R-

T instability in miscible Lennard-Jones fluids:

1. The incipient regime, during which the system oscillates vertically after the gravity field

has been turned on. The horizontal perturbations are created due to the thermal

fluctuations.

2. The perturbations, which correspond to the most unstable wavelength λm, start growing

exponentially.

3. The large scale regime enters in, when the effects from the initial conditions have been

erased. The thickness of the mixing layer increases as t2.

4. The final chaotic regime with many length-scales participating.

For immiscible and viscous R-T mixing of very thin fluid layers, the final chaotic regime may not

be resolved, due to the small-scale coalescent process involving droplets. Below we report the

corresponding stages of immiscible mixing in mesoscale on the basis of simulation results

obtained by using the DPD model.

Incipient regime

In macroscopic continuum models, the R-T instability is initiated by introducing incipient 2-D

harmonic perturbations (e.g. [8,9]). In the linear regime disturbances larger than a critical


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wavelength λcrit grow exponentially [6] in the initial stages of the R-T mixing. For some types of

initial perturbation, dominating wavelengths can depart from the characteristic one. We note that

in 3-D circumstances the fastest growing wavelength may be different [59]. This could lead to

characteristic mixing patterns in the final stages such as diapiric spacing close to the

characteristic wavelength [9].

On the other hand, in the world of discrete particles the R-T instability emerges

spontaneously [26-28] without any preference for wavelength selection. In the microscale, over

the nanometer length scale, both thermal fluctuations and the fluid elastic compressibility cause

principally the process of mixing. Because of the limited depth h of the computational box, initial

compression of lighter fluid induces the vertical oscillations of the particle system. Assuming that

the oscillations are harmonic, the period of these oscillations τ can be roughly estimated to be:

ch~πτ = (15)

where c~ is the average sound velocity of the system. The energy of the one-dimensional

oscillations is transferred by the amplification of the thermal fluctuations.

This situation resembles the initiation of fingering observed in experiments from thin liquid

films flowing down an incline [36,48], where 2-D periodic adjacent wave fronts precede the 3-D

instabilities. As show laboratory experiments [48], when entrance flow rate is perturbed at

adequate frequency by applying small sinusoidal pressure variations, after some start-up time

the straight 2-D wave fronts brake spontaneously resulting in 3-D wave production and then in

chaotic flow. The pressure fluctuations enable the energy of 2-D oscillation be transferred in

horizontal direction. We show that because of the large thermal fluctuations in the modeling of

thin films flowing down a slope, the 2-D waves are broken shortly after the initiation.

Together with the thermal fluctuations, the elastic compressibility constitutes the very

important factor impacting on the start-up time of the R-T fingering. For the high_c system, i.e.,

the fluid with lesser elastic compressibility, where Hc~ =971m/s and h=224nm (h=42 in rc units),

the period of oscillation τ≈0.72 ns, that is, about 3600 timesteps. As shown in Figs.2a,b, the

value of τ for the high_c system is about 3 times shorter than for the low_c system. This comes

from the Eq.(15), because the sound velocity Lc~ =307m/s is approximately 3 (√10) times less

than Hc~ . Likewise for a computational box 4 times higher, τ increases proportionally according

to Eq.(15) (compare Fig.2a and Fig.2c). This means that for the macroscopic systems with an

infinite depth h, the oscillations do not appear as τ→∞. For finite values of h, due to the lack of a

physical mechanism for converting the vertical oscillations into horizontal perturbations in the

continuum models, they will not be able to excite the R-T instability.


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As shown in Fig.3, the start-up time is distinctly shorter for more compressible fluids, i.e.,

for the low_c system. For this system, the initial one-dimensional oscillations of larger amplitudes

and longer oscillation period τ appear (see Eq.(15) and Fig.2) than for less compressible fluid

high_c. Therefore, there is a higher probability for the creation of the vertical perturbations, which

correspond to the most unstable wavelength λm.

Linear regime

During the evolution of the R-T instability, vertical oscillations last for a long time, thus coupling

nonlinearly with the large thermal fluctuations at the density interface. It is indeed difficult to

identify the linear regime in Fig.3, in which the perturbations corresponding to the most unstable

wavelength start to grow exponentially. In this regime, the capillary number is small. We define

here the capillary number to be the ratio of viscous forces µHV/δv to capillary forces θ/δl [49] ,

where V/δv is the characteristic shear rate, θ is the surface tension and δl is the striation

thickness, respectively. The density interface wrinkles and small ripples grow into fingers. Their

size and separation depend, among other things [6-20], on the surface tension, viscosity and

elastic compressibility. Below we will estimate approximately whether some theoretical

predictions from the macroscopic model are valid for compressible, viscous DPD fluids in the

linear regime of the R-T mixing.

For incompressible, viscous fluids with the surface tension θ=0, the system becomes

unstable for disturbances at all wavelengths. For a macroscopic fluid the mode of maximum

instability appears at the wavelength [6-7]:

3/12

4

≈

Gmαν

πλ (16)

where: LH

LH

ρρµµ

ν++

= is the average kinematic viscosity, µH, µL and ρH, ρL are respectively the

dynamic viscosities and densities of heavy and light fluids. Let us define additionally

m

xym

LN

λ= (17)

which is approximately equal to the number of fingers at the fluids interface at the beginning of

the R-T mixing. When the surface tension θ is taken into account, unstable modes occur only for

λ>λcrit where [6]:

( )[ ]LHcrit G ρρπλ −⋅= ?2 (18)


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and λcrit is independent on viscosity. For inviscid fluids λm=√3λcrit . For viscous fluids λm increases

with the surface tension θ and with fluid viscosity [6].

The surface tension θ at the density interface for DPD system depends on the

immiscibility of the two superimposed fluids. In the DPD system the immiscibility is defined by ∆

value (see Eq.(14)). For ∆=PHL-PH≤0 (see Eqs.(10,14)), the fluids H and L are miscible, while for

∆>0 they are immiscible. From Fig.3, we can see that the speed of mixing decreases rapidly with

∆ value both for the high_c and for the low_c systems. Therefore, we may expect the surface

tension θ at the interface between two immiscible fluids to be a strong function of ∆. In order to

find θ (∆) dependence we have employed the Laplace scaling law in the same way as in [33-35].

As shown in Fig.4, where we display two plots of θ (∆) for two different kinematic viscosities

(given in Ω units) of the particle system, the surface tension is clearly a non-linear monotonically

increasing function of ∆ and ν . In particular, in the miscible case, (i.e., ∆=0) the surface tension θ

is equal to 0 (see Fig.4).

Employing Eqs.(16-17) to the values from Table 1 for the high_c system with ∆=0 and

with box aspect ratios f=Lxy/h=2 and f=8, we obtain that Nm1=1.8 and 7.2, while for small surface

tension θ ≈1×10-11 N (∆=0.03 N/m, see Fig.4) it comes out from Eqs.(18,17) that Nm2=2.2 and 8.9

for the respective aspect ratios. The values of Nm2 will be even smaller in the viscous case.

Because the Nm values are very small (λm value is equal to the thickness of the L and H fluids)

and boundary effects influence the flow, this situation is very different from the macroscopic

models, in which the total thickness of the two fluid layers is taken to be infinite.

Surprisingly, the number of fingers (Figs.5,6) resulting from DPD simulations is very close

to the Nm values computed from the formulae derived from the continuum hydrodynamical

equations. The finite-thickness correction for λm≈h reduces the effective Atwood number by only

less than 2% [12]. As one would expect, the number of fingers is greater in the more

compressible (low_c) case in which the surface tension is an order of magnitude smaller than the

high_c system [35] (see Figs.5,6).

Regimes with large-scale features and the breakup of microstructures

In the turbulent R-T instability, the sustained acceleration causes the dominant spatial scales to

evolve self-similarly and the initial scales are forgotten. The mixing layer depth hT increases with

time as hT≈αATGtk (α - the Atwood number (see Eq.(8)), G - gravitational acceleration, AT –

proportionality constant) as shown by many numerical [7,8,11] and experimental results [18-20].

For the two end-members:

1. two fluid layers of equal depths,


13

2. very thin heavy fluid layer placed at the surface of lighter fluid,

the values of k=2 and k=1 are obtained respectively. In [28] we show that for two superimposed

microscopic Lennard-Jones fluids of equal depth the value of k=2. Moreover, the proportionality

constant AB for bubble mixing layer contribution is approximately the same as these obtained in

laboratory experiments and macroscopic simulations. A similar scenario can be found in DPD

fluids. As shown in Fig.3a, the growth of interface length with time for two fluid layers of equal

depth can be well approximated by a + bt +ct2 polynomial in the period of time from the beginning

of the mushroom structures formation to the breakup of stems. For the cases without the erosion

by the mushrooms (∆>0) the t2 dependence can be attributed to the propagation speed of the

mixing front. We can conclude from DPD simulation results depicted in Fig.3b that in the case of

thinner region with heavy fluid, the time exponent k changes from 2 to 1. Thus the same behavior

is found for both the macroscopic and mesoscopic DPD particle system.

The capillary number increases with time due to the increase of the shearing rate, which

also influences the viscosity. This shows the non-linear dependence of the viscosity on the

shearing rate. Because the box is very thin, growing fingers and the bubbles formed after fingers

breakup (Figs.5,6) saturate almost completely the lighter fluid from the bottom. As shown in Fig.

3, mixing is more vigorous for more compressible system and slows down with increasing surface

tension (and ∆). For fully miscible DPD fluids (∆=0), due to the local decrease of viscosity and its

dependence on local shearing on the density interface, the fingers and mushroom structures of

lighter fluid erode forming plume like patterns (see Fig.5). For small values of the surface tension

(∆=0.03 N/m) in the low_c system, secondary fingers and thin needles appear instead of plume

(see Fig.5c, and Fig.6a). In [39] we inspected this phenomena for larger particle system (2×106

particles) and more compressible fluids. It turns out that the secondary fingers appear after

decompression of the lighter fluid, i.e., when the less viscous fluid pushes the more viscous fluid.

This dynamical effect is very similar to the Saffman-Taylor instability [50]. For the high_c system

and low_c fluid with greater surface tension, the bubble erosion occurs only locally in selected

places with low viscosity caused by the greatest rate of shear (see Fig.6b,c and Fig.7a).

The erosion patterns presented in Figs.5,6a change the flow radically in comparison with

situation depicted in Figs.6b,7. The kinetic energy of the light particles overpowers that for the

heavy phase (see Fig.8a). During miscible mixing (∆=0) the fingers disappear and light particles

disperse completely in the heavy fluid (see Fig.5b). The density interface vanishes and chaotic

motion of particles can be observed. Other type of coupling between flow and breakup occurs for

the low_c system with small ∆. The clustering of the thin fibers initiated by the secondary fingers

and needles into thicker threads shown in Fig.9, dissipate the energy from the system. The

coalescencing process becomes to be the dominant type of flow resulting in rapid overturn (see

Figs.6a).


14

We observe the fast mixing of the thin mesoscopic fluid layers at the density (light -heavy)

interface at the onset, which is caused by the erosion of the fingers and mushroom structures.

This is to be contrasted with the flow patterns shown in Figs.7,10, in which relatively stable

bubbles appear. The process of detachment of mushroom structures is responsible for the

change in flow type from growing fingers to the dynamics of free bubbles. For the high viscosity

cases (Ω=100) the stems of mushroom structures are saturated. They become thinner and

approach a zero thickness without any breakup of the stem (see Figs.6,7a, 9,10). By decreasing

the viscosity of L and H liquids eight times (Ω=12.5), thus resulting also in more than twofold

increase of the surface tension for ∆=0.03 N/m (see Fig.4), we find a radical change in

mechanism of stem disintegration. The four principal mechanisms, the same as those responsible

for droplets breakup in macroscale [51], can be observed in DPD simulation of the R-T instability.

As shown in [1,51], moderately extended drops for capillary number close to a critical value

breakup by a necking mechanism due to a concentration of low viscosity from the high-rate of

shear. Due to the elongation flow (see Fig.11 - velocity field) in mushroom stems and p=µL/µH ≈1

assumed, according to [1], the necking mechanism cause the breakup in Fig.7b. In the two

rectangles in Fig.7b (1 and 2, marked by thick line) one can see the two phases of this process.

After breakup (rectangle 1) two daughter droplets are created (rectangle 2) and the stem remnant

contracts producing the spherical droplet (rectangle 3).

Another mechanism responsible for the stem breakup is shown in the same figure in

dashed rectangle 4 in Fig.7b. The mushroom structure detaches thus producing from the stem tip

a cluster of small bubbles, which can overtake the large bubble (see pictures 1 and 3 in Fig.11, 3

is the reverse of 2 image). This process is similar to the tipstreaming mechanism, in which small

drops break off from the tips of moderately extended pointed drop [51]. The conditions at which

tipstreaming occurs are not well known [1]. From Fig.7b one can see that the stem is distinctly

larger and thicker than those in the necking case. Moreover, the remnant pieces do not relax

back so quickly to spherical droplet as in the case of necking.

A supercritically extended drop in the presence of low shear rate and surface tension

causes a breakup of the end-pinching type. When the drop is stretched to a highly extended

thread the microstructure arises from short-wavelength capillary instabilities . The thread becomes

unstable to small fluctuations and will eventually disintegrate into a number of drops consisting of

secondary droplets between the larger primary drops. Because the end-pinching and capillary

instabilities are only observed when [1]:

1. the length of filament is more than 15 times the initial radius of the drop,

2. the flow is halted,

we cannot found these instabilities in thin film flows and vigorous flow situations, as is in Fig.6,7.

In Fig.12 we show the zoomed-in fragments of snapshots from DPD simulations of the R-T

instability in a deeper computational box and with a heavy fluid layer (gray structures in Fig.12)


15

ten times thinner than the light phase. As shown in Fig.12, the three mechanisms: necking, end-

pitching and capillary instabilities can be observed on different portions of the same extended

thread. The same situation was encountered previously by Tjahjadi and Ottino [52] in laboratory

experiments.

Mixing can be severely influenced by viscosity stratification in the system, as shown by

the work of Ten et al. [53]. A large enough viscosity contrast would inhibit mixing [53,54]. Lower

contrast in viscosity p=µL/µH results in much better mixing due to critical erosion of mushroom

structures (Fig.13a). Whereas, non-monotonic dependence of interfacial length as a function of

average viscosity of L-H (light-heavy) fluids for high_c system (see Fig.13b), reflects the change

in the flow character for mushroom patterns.

Coalescence

Due to the extremely thin width of the box, after breakup of stems the flow speed after reaching a

maximum slows down (see Fig.8). The capillary number decreases and the two fluids overturn.

As shown in Fig.13, the density interface begins to shrink, due to the overturn and appearance of

coalescing microstructures.

In the course of mixing of immiscible fluids the coalescence process is competitive with

the tendency toward breaking-up of the microstructures. In the case of zero gravity, when the

immiscible fluids are initially mixed and ∆ is sufficiently small, the fluids separate and average

domain size R growths with time obeying algebraic scaling low R(t)=tβ. In 2-D space the value of

β is equal to ½ in the beginning and 2/3 for longer simulation times. As a result, the two fluids

separate completely after same time [33-35]. Otherwise (large ∆ value), at the onset, immiscible

fluids separate out into stable droplets of various sizes, thus creating emulsion-like substance

[35,39] (see Fig.14a).

Coalescence consists of three phases: collision, drainage of the thin liquid between the

two drops and the eventual rupture of the thin film [1]. The time for coalescence depends mostly

on the drainage time and the collision frequency depending on the dispersed phase volume

fraction.

In the case of no erosion, i.e., high viscosity and high surface tension (large ∆, see

Fig.10), the overturning of the two fluids evolves smoothly without any microscopic breakup

morphology. However, the thin film of the heavy fluid persists for a long time. This is because the

drainage time decreases with the drop size [1] thus larger drops are less likely coalescing than

smaller droplets. The similar behavior is observed for the low viscosity and high surface tension

case (see Fig.7b), in which the breakup patterns occur but the bulk of matter is confined in the

bubbles, thus resulting in more than 90% overturn within a very short period of time. Because of

the coalescence of the larger structures and the agglomeration of the remaining droplets to the


16

bulk evolve slowly, the overturning takes very long time. It is even longer in the case of a higher

viscosity of the lighter fluid (see Fig.13a), which reduces greatly the mobility of the interface.

In the case of erosion the coalescing microstructures can be seen only for ∆>0. As shown

in Fig.6a, in which the erosion is high, it occurs also for larger shearing rate resulting in clustering

of thin fibers into thicker threads. However, coalescence is more likely for a low capillary number

at the end of mixing process. In the places of the lower shearing rate the threads coalesce

trapping heavy fluids in "holes" in the bulk of light fluid (Fig.5c, Fig.6b). Simultaneously, the

droplets of light fluid coalesce in the bulk of the heavy fluid (Fig.6b, the last snapshot). Both

processes are of much finer granularity as compared to the cases without erosion and much

faster because the coalescing droplets are orders of magnitude smaller. Large erosion and then

fast coalescence result in rapid stop of vertical flow and emulsification of one or both fluids [1].

Thus a very long overturn time is observed. For the extreme cases, e.g., large differences in

density and elastic compressibility contrasts of the two fluids and a very large external field, the

creation of stable foam-like structures (see Fig.14b) may completely stop the overturning process.

CONCLUSIONS

In this work we have employed the dissipative particle dynamics approach to attack the problem

of immiscible Rayleigh-Taylor mixing at the mesosopic scale. The significant advances of DPD

can be delineated as follows:

1. Conceptually the DPD method is simple and is based on fundamental physical principles of

Newtonian mechanics.

2. Unlike the lattice-gas models [55,56], DPD particles evolve over continuum space in real

time, thus allowing for realistic visualization and understanding.

The method can produce quantitative results, when constructed within the context of cross-scale

systems [43, 57-58] in conjunction with microscopic (MD) or/and macroscopic particle models

(e.g., smoothed particle dynamics - SPH [47]).

The use of DPD in the simulation of complex fluids in the mesoscale has many distinct

advantages, such as

1. Fluid granularity effects can be investigated, e.g., thermal fluctuations influence on incipient

of mixing,

2. The microstructures are coupled with the macroscopic flow.

3. Other processes, e.g. microscopic forces between dispersed phase and solvent, can be

included (see a two-level model presented in [37-39]).

4. Sophisticated boundary conditions with complicated multi-phase interfaces can be easily

simulated.


17

The interactions between DPD particles influence strongly the speed of mixing. The surface

tension θ is the effect arising from the immiscibility of two superimposed fluids and is an

increasing function of ∆. We show that in the case of high surface tension mainly larger

mushroom structures appear and carry the bulk of the fluid. The small microstructures do not

exert any influence on the flow. For lower viscosity, breakup patterns of mushroom stems appear.

One can distinguish in DPD simulation all four types of droplet breakup [1]: necking, tipstreaming,

end-pinching and capillary instabilities. This microscopic morphology changes the flow type from

rising mushrooms to bubble dynamics.

For small values of θ and viscosity contrast p=µL/µH ≈1 the R-T mixing occurs due to the

erosion of larger mushroom structures. This morphological breakup, other than droplet breakup

mechanisms described in [1], produces fine grained mixing, which results in fast coalescence of

small droplets. The coalescing droplets dissipate energy from the system and decelerate the flow,

thus influencing the flow structure.

The DPD approach suffers also some drawbacks in the simulation of the R-T instability.

As in the other discrete-particle methods, DPD takes a lot of computer time. The 2-D simulations

presented here were carried out for small particle systems, which correspond to the fluid samples

of 1 µm in size. Less than 20 ns can be simulated for very thin liquid layers. For such systems the

gravitational acceleration needs to be very large.

The system can be also matched to the actual spatio-temporal scales by rescaling, i.e.

increasing scaling factor S. However, we show in [35] that matching DPD system to the spatio-

temporal scale under interest is difficult. For relatively small scaling factor S and for given

physical properties of the liquid, the DPD system "freezes". We can easily "melt" it by assuming

unrealistically small sound velocities in the particle system. For non-equilibrium systems like the

R-T mixing, such an assumption may produce erroneous results. Therefore, we may surmise that

the DPD model involves the change of the shape of ω(r) weight function. For larger scales S we

suppose that it should take the form of a Gaussian function, the same as it is used for the SPH

method [47]. This is reasonable; since the DPD method can be treated as a special, mesoscopic

case of the SPH technique [32]. We have shown here that the DPD method can be successfully

employed for simulating Rayleigh-Taylor mixing in mesoscopic complex fluids with microscopic

ordering for scales greater than 10 angstroms.

Acknowledgments

Thanks are due to Dr Arkady Ten and Dr Dan Kroll from the Minnesota Supercomputing Institute

for inspiring discussions. Support for this work was provided by the Energy Research Laboratory

Technology Research Program of the Office of Energy Research of the U.S. Department of


18

Energy under subcontract from the Pacific Northwest National Laboratory and partly by the Polish

Committee for Scientific Research (KBN) Grant No. 8T11C00615.

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23

0.0 0.5 1.0 1.5 2.0 2.5DISTANCE

0

1

2

3R

DF

RDF

MD L-J liquid

DPD S=25

DPD S=100

Fig.1. Radial distribution functions for DPD fluid for two different values of S. Distance is given in λ units.

The RDF for the Lennard-Jones fluid (dashed line) is presented to show the contrast in the spatial scale

between L -J and DPD fluid for S=25.


24

0 40000 80000 120000 160000TIME (*2.0E-13)

0.20

0.21

0.22

0.23

0.24

PRES

SUR

E [N

/m]

Fig.2. The variations of thermodynamic pressure (computed from the virial law [33-35]) with time for three

DPD simulations describing the R-T instability. Upper line (dark) corresponds to the L fluid, bottom one

(gray) to the H fluid. a) high_c fluid, b) low_c fluid, sound velocity c is about 3 times smaller than that for a),

c) high_c fluid but the box height H is 4 times greater than that for a) and b). The thick line in Figs a) and b)

is the moving average fit of the pressure.

a b c


25

a)

10000 20000 30000 40000 50000 60000TIMESTEP

0

1000

2000

3000

4000

5000

INC

RE

ASE

OF

INT

ER

FA

CE

LE

NG

TH

(*2.

24E

-9 m

)Sonic Speed

High

Low

sqr(t) fit

Delta =0.013 N/m

Delta=0.0

Delta =0.033 N/m

b)

2 3 4 5 6 7 8 9 2 3 4 5100 1000

Time step

2

3

4

5

6

7

8

9

2

1

10

Wid

th o

f mix

ing

laye

r

Thickness ratio (light f luid)

0.80 (power 1.06)

0.75 (power 1.64)

0.50 (power 2.02)

Fig.3. a) The temporal evolution of interface length between H and L fluids for high_c and low_c 2-D DPD particle systems for various values of the immiscibility factor ∆. The t2 regime is approximated by 2nd order polynomials. b) The mixing layer growth with time for 3-D DPD particle system for various fluid thickness ratios. The numbers on the left in the legend show the light fluid thickness in correspondence to the unit box height. The numbers in parentheses correspond to the fit of the time exponent k.


26

0.00 0.01 0.10 1.00

IMMISCIBLITY FACTOR (N/m)

2

6

10

0

4

8

SUR

FA

CE

TE

NSI

ON

(*E

-11

N)

Viscosity

lower ( Om ega =12.5)

higher (Omega = 100)

Fig.4. The surface tension dependence on the immiscibility factor ∆ for the high_c fluid. For lower kinematic

viscosity (Ω=12.5) the surface tension is higher. The lines are drawn as guides for visual purposes.


27

a)

b)

C)

Fig.5. The snapshots from DPD simulation of the R-T mixing of two DPD fluids L and H a) high_c fluid (∆=0,

Ω=100) b) low_c fluid (∆=0, Ω=100) c) low_c fluid (∆=0.03, Ω=100). The box dimensions in rc units are

84×84.

6 ns 10 ns

4 ns 10 ns

6 ns 10 ns


28


29

T=3.2×10-9 sec.

T=4.8×10-9

sec.

T=6.4×10-9

sec.

T=7.6×10-9

sec.

T=1.0×10-8

sec.

T=1.2×10-8

sec.

Fig.6. The snapshots from DPD simulation of the R-T mixing of two immiscible fluids L and H (∆=0.033 N/m,

Ω=100) a) low_c and b) high_c systems. We emphasize the initiation of bubbles erosion by using two

rectangles. In c) the velocity field for the fragment (in black rectangle) of the last snapshot from b) is shown.

a) b)

c)


30

The highest shearing rate corresponds with the places of maximum bubble erosion. The box dimensions in

rc units are 84×336.


31

T=3.2× 10-9 sec.

T=4.8× 10-9

sec.

T=6.4× 10-9

sec.

T=8× 10-9

sec.

T=6.4× 10-9 sec.

T=8.0× 10-9

sec.

T=1.0× 10-8

sec.

T=1.2× 10-8

sec.

Fig.7. The snapshots from DPD simulation of the R-T mixing of two immiscible fluids L and H a) low_c fluid

(∆=0.041 N/m, Ω=100) b) high_c fluid (∆=0.033 N/m, Ω=12.5).

0 2 0000 4000 0 60 000 80000 1000 00

TIMESTEP (*2.0E-13 sec.)

40

80

1 20

1 60

2 00

KIN

ET

IC E

NE

RG

Y (K

)

Fig.8. The evolution of the kinetic energy of DPD particle systems with time a) low_c, ∆=0 b) low_c, ∆=0.04

N/m c) high_c, ∆=0.3 N/m.

b)

L

H L

H

a) b)

a)

1 2 3

1 2

3

4

L

H

32

Fig.9. The zoomed-in snapshots from simulations presented in Fig.6a. The clustering of the thin fibers

initiated by the secondary fingers and needles into thicker threads

Fig.10. The snapshots from DPD simulation of the R -T mixing for a large immiscibility factor (∆=0.31 N/m).

Fig.11. The zoomed-in snapshots from simulations presented in Fig.7b The mushroom stem severs and

produces a cluster of small bubbles, which can overtake the large bubble (see 1 and 3, 3 is the reverse of 2

image). The velocity field is shown for situation from picture 1.

1

2 3 4

10 ns

15 ns 35 ns

33

Fig.12. Three breakup microstructures observed on different portions of the same extended thread obtained

from DPD simulation of turnover of thin heavy layer in the bulk of lighter fluid.

0 40000 80000 120000TIMESTEP

0

500

1000

1500

2000

2500

INC

RE

AS

E O

F IN

TE

RF

AC

E L

EN

GT

H (

*2.2

4E-9

m)

Viscosity ratio

0.6

2.2

0 40000 80000 120000TIMESTEP

0

100

200

300

400

500

INC

RE

AS

E O

F I

NT

ER

FA

CE

LE

NG

TH

(*2

.24E

-9 m

)

Omega

1 00

5 0

1 2.5

Fig.13. The temporal evolution of interface length between H and L fluids for DPD system for different p and

Ω contrasts.

end-pinching

capillary instabilities

necking necking

a) b)

34

Fig.14. The large differences in density and compressibility contrasts for the two DPD fluids, results in

emulsification (a) or creation of stable foam like structures (b).

Mixing driven by Rayleigh -Taylor Instability in the ...icsr.agh.edu.pl/~dzwinel/files/publications/R-T.pdf · macroscopic models incorporating the two microstructural mechanisms

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