Dissipative particle dynamics simulation of micro concentric eccentric annular flows

DISSIPATIVE PARTICLE DYNAMICS SIMULATION OF
MICRO-CONCENTRIC/ECCENTRIC ANNULAR FLOWS

PENGFEI CHEN

A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2004


ACKNOWLEDGEMENTS

I am very grateful to my supervisors, Professor Nhan Phan-Thien and Associate
Professor Yeo Khoon Seng for giving me the great opportunity to study in such an
interesting area. I would also like to express my sincere gratitude to them for their
constant guidance and encouragement throughout the course of this work.

I would like to thank Professor Fan Xijun, Dr. Dou Huashu, Dr. Chen Shuo, Dr. Lu
Zhumin, Dr. Shi Xing, Mr. Wu Tao, Ms. Luo Chunshan and Ms. Zhao Xijing for their
assistance and friendship.

My entire family deserves a special gratitude for their unlimited support,
encouragement and love throughout my stay in NUS.

Finally, I would like to give my acknowledgement to the National University of
Singapore for their Research Scholarship.

Thanks are also due to all others who have helped me in one way or another in this
effort.

I


Table of Contents
Acknowledgement..........................................................................................................I
Summary......................................................................................................................IV
Nomenclature................................................................................................................V
List of Figures...........................................................................................................VIII
List of Tables..............................................................................................................XII

Chapter 1

Introduction.......................................................................................1

1.1

Background...............................................................................……..1

1.2

Literature review.......................................................................……..4

1.3

Objective of this research project..............................................……..6

Chapter 2

Methodology.......................................................................................9

2.1

Basic equations of DPD method................................................……..9

2.2

Numerical Scheme..................................................................……...14

2.3

Initial

Particle

Models

in

concentric/eccentric

flow

field....................................................................................................16
Chapter 3

Steady Concentric/eccentric flows of simple DPD fluids at finite
Reynolds numbers..................................................................……..20

3.1

Implementation

of

non-slip

boundary

condition

in

DPD...................................................................................................21
3.1.1

Boundary condition used in this thesis…………………....24

3.1.2

Comparison of boundary conditions: Poisseuille flow…....26

3.1.3

Comparison of boundary conditions: Concentric rotating
cylinders flow……………………………………………..30
II


3.2

Effects of some DPD Parameters on simulation…………………...32
3.2.1

Effect of fluid particle density…………………….………33

3.2.2

Effect

of

conservative

force

factor

between

DPD

particles………………………………………….………...36
3.3

Steady Circular Couette flow and eccentric flow of simple DPD fluids
at finite Reynolds numbers………………..…………………….......43
3.3.1

Circular Couette flow of simple DPD fluids at finite

Reynolds numbers…………………………………….…..43

3.3.2

Eccentric flow ( ω out / ω in = 0 ) of simple DPD fluids at finite
Reynolds numbers…………………………………….…..48

Chapter 4

Steady Circular/eccentric flows of FENE Chain Suspension at
finite Reynolds numbers……………………………….…………60

4.1

Circular Couette flow of FENE chain suspension at finite Reynolds
numbers………………………………………………………….....62

4.2

Eccentric flow ( ω out / ω in = 0 ) of FENE chain suspension at finite
Reynolds numbers………………..………………………………...70

Chapter 5

Conclusion and future works..........................................……..….80

5.1

Conclusion…………..…………………………………………...…80

5.2

Future works…………………………….………………………….81

Appendix A…………………………………………….………………………...…...84
Appendix B……………………………………………………………………...…....87
Reference…………………………………………………………………………88-92

III


Summary

Dissipative Particle Dynamics (DPD) is a fairly new method for simulating complex
fluid flows and other colloidal phenomena. It is a mesoscopic method and offers the
possibility of capturing some degree of molecular-level detail while conforming to
continuum hydrodynamics at larger length scales. In this thesis, a new implementation
of the no slip boundary condition in the modeling of solid boundaries is studied. This
boundary is implemented to simulate the planar Poisseuille and circular Couette flow,
and the results compare excellently with similar results derived by more traditional
CFD methods. The effects of two important DPD parameters (particle density and
conservative coefficient) are studied. It is shown that these two parameters affect the
simulation accuracy considerably and should be carefully set. Furthermore, to confirm
the ability of DPD method to provide numerically accurate results in simulating
complex flow with rather complicated boundary conditions, the DPD method is
employed

to

study

the

flow

behavior of

three

dimensional

microscopic

concentric/eccentric flows at finite Reynolds numbers. A simple DPD fluid (made up
of simple DPD particles) and then bio-molecular suspensions (FENE chains are used to
model DNA macromolecules) are studied in detail respectively.

IV


Nomenclature
It is not practical to list all the symbols that have been used. Below the author list the
more important ones. Some of the symbols are defined as they are used. There are also
occasions where the same symbol is assigned a different meaning in a different context,
but the meaning should normally be clear from the usage.

English alphabets:
D

Diffusion constant

fij

Interparticle force on particle i by particle j

Fe

External force exerting on particle i

FijC

Conservative force on particle i by particle j

FijD

Dissipative force on particle i by particle j

FijR

Random force on particle i by particle j

H

Spring constant of FENE chain model

kB

Boltzmann constant

Lc

Length of one segment of a FENE chain

n

Particle density

N1, N2

The first and second normal stress

Nb

Number of beads in the FENE chain model

p

Constitutive pressure

Re

Reynolds number,

ρ Rin2 ω
η

V


Rin

Radius of inner cylinder

Rout

Radius of outer cylinder

ri

Position vector of particle i

rm

Maximum length of one chain segment of FENE chain model

Sc

Schmidt number

T

Temperature of the system

ui

Peculiar velocity of particle i

vi

Velocity vector of particle i

•

v i (t )

Acceleration of the particle i at the instant t

v i (t )

Prediction of the velocity of the particle i at the instant t

Greek alphabets:

α

ij

Maximum repulsion force between particles i and j

ω D (r )

r-dependent weighting function of dissipative force

ω R (r )

r-dependent weighting function of random force

γ

Coefficient controlling the strengths of the dissipative force

σ

Coefficient controlling the strengths of random forces

θ ij

Gaussian variable

α

f

α

f w

Maximum repulsive forces between fluid particles and wall particles

α

ww

Maximum repulsive forces between wall particles

f

Maximum repulsive forces between fluid particles

VI

ρ

Fluid density

η

Viscosity

δ

Displacement of inner cylinder axis from outer cylinder axis

κ

Radius ratio, Rin/Rout

c

Mean annular gap width, Rout-Rin

ε

Eccentricity ratio, δ / c

ω

Rotating angular velocity of the cylinder

ζ

ζ -coordinate of bipolar coordinate system

θ

θ -coordinate of bipolar coordinate system

Φ

Volume fraction of a dilute suspension

VII


List of Figures

Figure 1

Illustration of generating fluid particles in an annular area………….…18

Figure 2

Illustration of generating wall particles in angular direction……….…..19

Figure 3

Illustration of generating wall particles in radial direction………….….19

Figure 4

Illustration of the boundary reflection………………………….……….26

Figure 5

The fully developed velocity profile compares with the Navior-Stokes
solution in Poiseuille flow……………………………………….……...28

Figure 6

The temperature profile of a simple DPD fluid in Poiseuille flow….…..28

Figure 7

Shear stress distribution of a simple DPD fluid in Poiseuille flow….….29

Figure 8

Profiles of the first (N1) and second (N2) normal stress differences of a
simple DPD fluid in Poiseuille flow……………………….……….…...29

Figure 9

The Comparison plot of density profiles in Poiseuille flow…………….30

Figure 10

The comparison of pressure profiles in Poiseuille flow………………...30

Figure 11

The comparison of velocity profiles of a simple DPD fluid in circular
Couette flow………………………………………………………….…32

Figure 12

Illustration of the effect of fluid particle density on velocity of a simple
DPD fluid in circular Couette flow………………………………….….34

Figure 13

Comparison of temperature profiles of a simple DPD fluid in circular
Couette flow with different conservative force coefficient α f f …….…38

Figure 14

Comparison of density profiles of a simple DPD fluid in circular Couette
flow with different conservative force coefficient α f f ………………...38

VIII


Figure 15

Comparison of velocity profiles of a simple DPD fluid in circular Couette
flow with different conservative force coefficient α f f ……………..…39

Figure 16

Illustration of the effect of α ww on density of a simple DPD fluid in

circular Couette flow……………………………………….……….…..41

Figure 17

Comparison of temperature profiles of a simple DPD fluid in circular
Couette flow with different conservative force coefficient α ww …….…42

Figure 18

Comparison of velocity profiles of a simple DPD fluid in circular Couette
flow with different conservative force coefficient α ww …………..……42

Figure 19

Geometry for Circular Couette flow……………………………….…....43

Figure 20

Simulated Vx, Vz velocity and streamline contours of a simple DPD fluid
in circular Couette flow………………………………………………....45

Figure 21

The fully developed tangential velocity profile of a simple DPD fluid in
circular Couette flow………………………………………….………...46

Figure 22

Density and temperature profiles of a simple DPD fluid in circular Couette
flow…………………………………………………………………..…46

Figure 23

Pressure profile of a simple DPD fluid in circular Couette flow…….…47

Figure 24

Shear stress distribution of a simple DPD fluid in circular Couette
flow……………………………………………………………………..47

Figure 25

Profiles of the first and second normal stress differences of a simple DPD
fluid in circular Couette flow………………………………………..….48

Figure 26

Bipolar coordinate system and geometric parameters of the eccentric
annular region…………………………………………………………...50

IX


Figure 27

Streamline patterns of a simple DPD fluid in eccentric flow ( ω out / ω in = 0 )
with different Reynolds numbers…………………………….………….58

Figure 28

Comparison of polar angles of separation and reattachment points of
simple DPD fluids in eccentric flow ( ω out / ω in = 0 )…………….……....58

Figure 29

Comparison of tangential velocity profiles in the wide gap ( θ = π ) of
simple DPD fluids in eccentric flow ( ω out / ω in = 0 , Re=37.2)…….…….59

Figure 30

Comparison of tangential velocity profiles in the narrow gap ( θ = 0 ) of
simple DPD fluids in eccentric flow ( ω out / ω in = 0 , Re=37.2)……….….59

Figure 31

Comparison of velocity profiles of FENE chain suspensions of different
concentrations in the Circular Couette flow……………………….…….64

Figure 32

Comparison of pressure profiles of FENE chain suspensions of different
concentrations in the Circular Couette flow…………………….……….65

Figure 33

Comparison of Shear stress profiles of FENE chain suspensions of
different concentrations in the Circular Couette flow…………….……..65

Figure 34

The conformation of some typical FENE chains in circular Couette flow
withΦ=0.03883………………………………………………….……….67

Figure 35

The conformation of some typical FENE chains in circular Couette flow
withΦ=0.11650…………………………………………………….…….68

Figure 36

The conformation of some typical FENE chains in circular Couette flow
withΦ=0.34951……………………………………………….………….69

Figure 37

Streamline patterns of FENE chain suspension in eccentric flow

X


( ω out / ω in = 0 ) with different volume fractions……………….………….73
Figure 38

Comparison of tangential velocity profiles in the wide gap ( θ = π ) and
narrow gap ( θ = 0 ) of FENE chain suspension in eccentric flow
( ω out / ω in = 0 )……………………………………………………………74

Figure 39

Illustration plot of the development of the back flow of FENE chain

suspension ( φ = 0.11650 ) in the eccentric flow ( ω out / ω in = 0 )………...76

Figure 40

Snapshots showing the detailed backflow of one FENE chain in eccentric
flow……………………………………………………………………...79

XI


List of Tables

Table 1

Angles of separation, reattachment and eddy center of different volume
fractions………………………………………………………………...73

XII


Chapter 1

Introduction

1.1 Background
The numerical simulation of hydrodynamic interactions between suspended particles
and the surrounding fluid phase is of interest in many engineering applications
associated with particle transport, such as colloids, polymers, aerosols and
physiological system [1]. The properties of these systems are often determined by
their mesoscale structures, i.e. between the atomistic scale and the macroscopic scale,

thus endowing a complex fluid with unique and interesting features [2].

Dynamic simulation of these systems presents unique problems that are difficult to
address with established methods. On the one hand, the time and length scales of
interest make simulation by molecular dynamics (MD) impractical, a limitation that is
not likely to be solved in the near future, even with the rapid increases in
computational speed that are expected to develop. On the other hand, colloidal length
scales are often almost the same order as the flow domain size. (The ratio of the
colloidal length to the characteristic length of flow field can be taken as the equivalent
Knudsen number Kn of the flow. When Kn = O(1) , the flow may not be treated as a
continuum). So purely continuum approaches, such as conventional computational
fluid dynamics (CFD), are unacceptable, requiring as they do that one leaves out so
many details at the molecular level. Moreover, people still cannot make the full
connection from atomistic length-scale to the macroscopic world. Hence to obtain a
better understanding of the phenomena that occur in mesocale, some intermediate

1


simulation techniques are developed that are aimed at a length-scale larger than the
atomistic scale, but smaller than the macroscopic scale. Dissipative particle dynamics
(DPD) is a so-called mesoscopic simulation method that provides one possible means
of bridging the gap between purely molecular and continuum-level treatments.

Dissipative particle dynamics is a stochastic simulation technique introduced by
Hoogerbrugge and Koelman [3] in 1992 to simulate complex fluid dynamical
phenomena. DPD combines features from molecular dynamics and lattice-gas
automata (LGA) by introducing a LGA-type of time-stepping into MD schemes. In
contrast to molecular dynamics simulations, the particles are supposed to represent the
fluid on a mesoscopic level rather than a molecular level. For the simulation of

macroscopic fluid dynamic phenomena, this implies an advantage of computational
effort [4]. Though Brownian Dynamics Simulation (BDS), LGA and Lattice
Boltzmann (LB) are mesoscale simulation methods, it is difficult for BDS to deal with
complex flow field and for LGA and LB to cope with complex fluids and DPD has the
advantage of more flexibility.

The basic unit in DPD system is a set of discrete momentum carriers called particles
that move in continuous space and discrete time-steps. The momentum carriers are
coarse grained entities which are no longer regarded as molecules in a fluid but rather
representing the collective dynamic behavior of a large number of molecules (a fluid
“particle”). When first introduced by Hoogerbrugge and Hoelman [3], Dissipative

2


Particle Dynamics is such a method for simulation of the motion of this kind of “fluid
particles” often referred to in continuum mechanics. Conceptually this method
amounts to dividing up the molecules of a flow field into groups or packets, which
have dimensions many times larger than the mean free path of an individual molecule.
The mass m of each packet is localized to a point. As the points move, their interaction
is ‘soft’ as a result of the fact that the packets are deformable, and this deformation is
accompanied by a dissipation of energy. In addition, because of their small
colloid-like dimensions, the thermal motion of the molecules in the packets gives rise
to a random or Brownian contribution to their motion. A time step in a DPD
simulation therefore consists of summing interactions that consist of three terms: a
conservative repulsion that accounts for steric and energetic interactions between the
molecules of interacting packets and this repulsion force prevents particles from
overlapping [5], a dissipative interaction that accounts for energy that is lost due to
internal friction or viscosity within a packet, and a random step that arises from the
collective thermal motion of the molecules within a packet. The interactions are

summed over all pairs of particles, in a way that guarantees that linear and angular
momentum are conserved. Unless an explicit connection to molecular-level
interactions is needed, the parameters that govern these interactions are chosen in any
way that reproduces the continuum-level dimensionless groups that determine the
behavior of a particular system.

The last two terms, which account for dissipation and random motion, are necessarily

3


coupled by a fluctuation-dissipation theorem and the principle of equipartition of
energy. These two terms combine to create a continuous pseudofluid in which the
particles are suspended and free to interact hydrodynamically. The original algorithm
of Hoogerbrugge and Koelman [3] did not satisfy this requirement, leaving in doubt
whether a simulation could reach a true equilibrium, even at long times and in the
absence of bulk motion. A slight revision of the original algorithm, produced by
Espanol and Warren [6], remedied this problem, and has been used in most
applications since that time. The method has received considerable theoretical support
in other areas as well. Marsh et al. [7] make explicit connections between DPD and
the Navier-Stokes equations by deriving the Fokker-Planck-Boltzmann equation for
the single-particle distribution function, and solving it by the Chapman-Enskog
method. Their derivation includes expressions for transport properties such as the
shear viscosity that are valid in the limit of strong damping, where conservative
repulsive interactions are negligible. Flekkoy and Coveney [8] discussed creating
DPD particles by grouping molecules in MD simulations together.

1.2 literature review
Since DPD method is a fairly recent development and the method is still evolving,
only the fundamental and representative papers will be reviewed here. Literature

review about the flow field in concentric/eccentric flows will be noted in the section
on results and discussion.

4


A key and attractive feature of DPD is its ability to reproduce continuum-level fluid
mechanics over large enough length scales, even in the presence of inertia. However,
in spite of the growing literature on the applications of DPD to various problems,
there are still relatively few direct, quantitative comparisons between calculations
done with DPD and well-established analytical and numerical results. Those
comparisons that do exist pertain exclusively to low Reynolds numbers.

Simulation results obtained by DPD have also been reported for several problems of
interests. In their original paper, Hoogerbrugge and Koelman [3] calculated the
flow-induced drag on a cylinder in a periodic array, and compared it with result
reported by Sangani and Acrivos [9]. This comparison was made in the limit of low
Reynolds number, where inertial effects in the flow are negligible. Boek et al. [10, 11]
studied the rheological properties of colloidal suspensions of spheres and rods using
dissipative particle dynamics, and measured the viscosity as a function of shear rate
and volume fraction of the suspended particles. Furthermore Boek and van der Schoot
[12] used DPD to study fluid flow through a periodic array of cylinders as a model for
fluid filtration through a porous medium, and discussed the resolution effects in DPD.
Both the calculations of Hoogerbrugge and Koelman [3] and those of Boek et al. [10,
11, 12] were done with the original algorithm, without the modification proposed by
Espanol and Warren [6]; Boek et al [12] argued that the modification does not
significantly alter the calculations of suspension rheology. Model polymers have been

5

constructed by linking DPD particles together with springs, and polymer dynamics
and associated parameters have been studied by Kong et al. [13]. In addition,
polymer-surfactant aggregation and micelle formation have been simulated by Groot
[14], and an extension of DPD to non-Newtonian flows (Bosch [4]) has been proposed.
More recently, Fan et al. [15] presented their simulation results for macromolecular
suspension flows through microchannels. They also studied the Poiseuille flow of
simple DPD fluids and FENE (Finitely Extendable Nonlinear Elastic) chains
suspension and found that simple DPD fluids behave just like a Newtonian fluid while
FENE chains suspension can be fitted by dilute suspensions.

1.3 Objective of this research project
As noted above, DPD has its own advantages to numerically simulate the
hydrodynamic interactions of various complex systems: such as polymer suspensions
[13, 16, 17, 18], colloids [10, 11, 19] and multiphase fluids [20, 21, 22]. Although it is
a promising technique for complex fluids, there are very few microfluidic applications
of DPD in complex systems reported. The most recent literature that combines such a
microscopic application of DPD in complex system as well as giving a quantitative
simulation is noted in the paper of Fan et al. [15], but what is studied in their paper is
about the simple Poiseuille flow. In this thesis, a more complex flow problem is
studied: steady microscopic concentric/eccentric flow at finite Reynolds numbers.
This work is also stimulated by the recent innovations in MEMS devices, especially in
micro bearing applications of MEMS. This area is not fully investigated and

6


understood today.
Unlike the Poiseuille flow, firstly, the boundary condition of concentric/eccentric flow
is more complicated because it is a closed flow field and the implementation of the

boundary conditions will have a more significant impact on the simulation results.
Secondly, the flow patterns in concentric/eccentric flows are more intricate compared
with that of Poiseuille flow at low Reynolds numbers, especially for eccentric flow
(there is a clear recirculation region present in the flow field). Same with what are
presented in the paper of Fan et al., two kinds of fluids are studied here: simple DPD
fluids and FENE chains suspension. Some results on the deformation and migration of
FENE are also reported.

It should be noted that the purpose of this research program is not to propose DPD as
an instrument for use in computational fluid mechanics, but rather to test its ability to
capture continuum fluid mechanical effects in the presence of significant inertia. Even
for colloidal systems, in which length scales are shorter than 10 µ m , the ability to
capture inertial effects accurately is important. In their study of the Brownian motion
of interacting particles, for example, Hinch and Nitsche [23] incorporated O(Re)
corrections to the equations of motion for each frequency of oscillation. They found a
nonlinear force of interaction between the particles that, even for very small Reynolds
number, is of the order O(kT / R) . Here kT is the product of Boltzmann’s constant
and temperature, and R is the particle radius. Such an interaction has an effect on the
distribution of the particles, which in turn affects the thermodynamics and rheological

7


properties of a colloidal suspension.
Since fluid simulation by DPD is a fairly new development, the results presented in
this thesis are focused on a number of test problems. These problems include:
improving the dynamic behavior of DPD method, studying the effects of basic DPD
parameters ( α and n) on simulation results, and accessing the ability of DPD to
provide numerically accurate results in simulating complicated flows with
complicated boundary conditions. The simulation results presented in this work

confirm the ability of DPD to provide quantitatively accurate results for complicated
flows, provided conditions are such that the compression of the DPD fluid is not
significant.

In the present work, the DPD method is used to calculate three-dimensional flows at
finite Reynolds numbers, and examine conditions under which Newtonian fluid and
non-Newtonian fluid behaviour is reproduced quantitatively. First, the DPD method is
outlined. This is followed by the description of a new implementation of DPD
boundary condition, and then the effects of DPD parameters are studied, with some
simulation details presented. The simulation details for concentric/eccentric flows
with simple DPD fluids and FENE chains suspensions are presented next.

8


Chapter 2

Methodology

2.1 Basic equations of DPD method
The DPD system consists of a set of interacting “particles”, whose time evolution is
governed by Newton’s equation of motion. For a simple DPD particle i ,

dri
= vi ,
dt

dv i
= ∑ fij + Fe ,
dt

j ≠i

(1)

where ri and v i are the position and velocity vectors of particle i , and the unit of
mass is taken to be the mass of a particle, so that the force acting on a particle is
equal to its acceleration. Fe is the external force. fij is the interparticle force on
particle i by particle j , which is assumed to be pairwise additive. The dynamic
interactions between the particles are composed of two parts, dissipative and
stochastic, complementing each other to ensure a constant value for the mean kinetic
energy, kBT , of the system.

The force fij consists of three parts, a conservative force, FijC , a dissipative force,
FijD , and a random force, FijR :

f ij = ∑ FijC + FijD + FijR .

(2)

i≠ j

Here the sum runs over all other particles within a certain cutoff radius rc , which is
taken as the unit of length, i.e. rc =1 (see Appendix A).

Since the time average of the dissipative and fluctuation forces is zero, they do not
feature in the equilibrium behavior of the system, which is governed solely by
conservative forces. The conservative force, FijC , is a soft repulsion force acting
9

along the line of centers and is given by
α (1 − r / r )rˆ
ij c ij
 ij
F =
ij
0
C

where

(r < r )
ij c
,
(r ≥ r )
ij c

(3)

α ij is the maximum repulsion force between particles i and j ; and

rij = ri − r j , rij = rij , rˆij = rij / rij is the unit vector directed along j to i .

The dissipative force or drag force, FijD , on particle i by particle j , is given by
FijD = −γ wD (rij )(rˆij ⋅ v ij )rˆij ,

(4)

where ω D (r ) is an r-dependent weighting function that vanishes if r ≥ rc = 1 ,

vij = vi − vj . γ is a coefficient controlling the strengths of the dissipative force and
characterizing the extent of dissipation in a single simulation step. The negative sign
in front of γ indicates that the dissipative force is opposite to the relative
velocity v ij .

The random force or stochastic force, FijR , on particle i by particle j , is given by
FijR = σ wR (rij )θij rˆij ,

(5)

where ω R (r ) is also an r-dependent weight function that vanishes for r ≥ rc = 1 .

σ is the coefficient controlling the strengths of random forces, and θ ij is a
Gaussian variable with zero mean and variance equal to δ t −1 , where δ t is the time
step:

θ ij (t ) = 0 and θ ij (t )θ kl (t ' ) = (δ ikδ jl + δ ilδ jk )δ (t − t ' ) ,

(6)

with i ≠ j and k ≠ l . The strength of the random force is the integral correlation

10


over a time scale considerably larger than its correlation time scale:
2

σω R (rij )  =

+∞

∫

FijR (t ) ⋅ FijR (t + τ ) dτ .

(7)

−∞

The detailed balance condition, similar to a Fluctation-Dissipation theorem relating
the strength of the random force to the mobility of a Brownian particle, requires that

ω (r ) = ω R (r ) 
D

2

and γ =

σ2
2kBT

,

(8)

where kB is the Boltzmann constant and T the temperature of the system. This
will ensure that the temperature remains constant.

We use the following weight function to improve on the Schmidt number for the
system (Fan et al. [15]), instead of the quadratic function (1 − r / rc ) 2 that is usually
adopted,
2

 1 − r / rc
0

ω D (r ) = ω R (r )  = 

( r < rc )
.
( r ≥ rc )

(9)

This weight function yields a stronger dissipative force between particles than that
from the standard quadratic force for a given configuration of particles and
interaction strength.

It should be noted that in DPD, the random force between two particles, FijR ,
represents the results of thermal motion of all molecules contained in particles i
and j . It tends to “heat up” the system. The dissipative force, FijD , on the other
hand, reduces the relative velocity of two particles and removes kinetic energy from
their mass centre to cool the system down. When the detailed balance is reached, the
11


system temperature will approach the given value. The dissipative and random
forces act like the thermostat in molecular dynamics (MD).

When simulating complex fluids and flows, simple DPD particles, described above,
are used to model solvent or suspending fluid. The solid walls can be modeled by
frozen DPD particles. The Finitely Extendable Nonlinear Elastic (FENE) chain is a
model commonly used to model flexible polymer molecules in rheology; it is used in
this thesis to model bio-molecules. Beads of the polymer chain are replaced by DPD
particles. The intermolecular forces will act on these particles and should be added to
the right hand side of Eq. (1).

In the FENE chain [24], the force on bead i due to bead is j is

FijS = −

Hrij
,
1 − (rij / rm) 2

(10)

where H is the spring constant, and rm is the maximum length of one chain
segment. The spring force increases drastically with rij / rm and becomes infinity as
rij / rm =1.0. This model can capture the finite extensibility of the molecules and
predict a shear-rate dependent viscosity and finite elongational viscosity. The mass
of the beads is assumed to be unity, which is as same as that of other simple DPD
fluid particles.

12