A Molecular Dynamics Study of Flow Regimes with Eﬀects of Wall Properties · PDF...

A Molecular Dynamics Study of Flow

Regimes with Effects of Wall Properties in

2-D Nano-Couette Flows

by

David To

Thesis submitted for the degree of

Doctor of Philosophy

in

The University of Adelaide

School of Chemical Engineering

Faculty of Engineering, Computer & Mathematical Sciences

February 22, 2010

Chapter 1

Introduction

Fluid flow surrounds us everyday and occurs in a vast range of applications. Some common

examples are the blood flowing through our veins, water flowing through pipes, and air

flowing past an aircraft wing. A thorough understanding of fluid flow behaviour can provide

benefits across many fields and thus extensive research has been conducted in this area over

the past two centuries.

A significant discovery was found by Osborne Reynolds in 1880. He observed that fluids do

not always flow in the same manner. Experimental results showed that the flow was either

laminar or turbulent and that the type of flow was dependent on the ratio of inertial to

viscous forces. This ratio is now called the Reynolds number (Re). It is a dimensionless

number and can be used for determining dynamic similitude.

Fluid flow was first described mathematically by Euler. He developed equations that describe

an inviscid fluid flow. An extension of this was developed in 1840 by Navier and Stokes for

a viscous fluid.

1

Over the past twenty years, there has been an increase in the development of nano- and

micro-fluidic devices (Tabeling, 2005). They deal with fluids flowing through channels with

dimensions in the nano- and micro- metres. It has been shown that at these scales, the

behaviour of the fluid flow is no longer the same as at macroscopic scales. The Navier-Stokes

equation breaks down due to slip at the wall and there are fluctuations in temperature and

density across the channel. This has been studied mainly using Molecular Dynamics (MD).

MD is a computer simulation tool that allows individual molecules to interact with each

other through attractive and repulsive forces. After a period of time, the molecular paths

can be analysed (Haile, 1992). However, due to limitations in computational power, MD is

restricted to small systems and short simulation times.

Our understanding of the behaviour of fluid flow for nano- and micro-channel flows is limited.

There are experimental data available for the laminar to turbulent transition point and

transitional range for micro-channel flows (Wu and Little, 1983, 1984; Weilin et al., 2000;

Peng et al., 1994b), however the results are contradictory. A number of studies have been

conducted on the wall-slip and density fluctuations in nano-channel flows using MD (Xu

et al., 2004; Thompson and Troian, 1997; Travis and Gubbins, 2000), however no studies

have been conducted on flow regimes in nano-channel flows to our knowledge.

This thesis attempts to provide some data on flow regimes in nano-channel flows with em-

phasis on the laminar to turbulent transition point and transitional range. The MD set-up is

designed to simulate a simplified 2-D Couette flow in Cartesian coordinates. MD simulations

will be conducted at a range wall velocities. The effects of various wall properties on the

mixing will also be studied.

In the following sections, background information and a literature review covering nano-

and micro-channel flows are provided. Flow regimes and mathematical descriptions of flow

Chapter 1 Introduction 2

including MD are provided in Sections 1.1 and 1.2, respectively. Nano- and micro-channel

flows are discussed and reviewed in Section 1.3. Finally, the aims and structure of the thesis

are outlined in Section 1.4.

1.1 Characterisation of Flow Regimes

Fluid flow is classified into one of two regimes - laminar and turbulent flow. Laminar flow

occurs when the fluid flows smoothly and in layers parallel to each other. Turbulent flow, on

the other hand, is characterised by mixing and irregular movement of molecules. The flow

regime is an important consideration in the design of flowing systems because it affects the

amount of friction, pressure, heat transfer, and mixing.

Osborne Reynolds developed a dimensionless number that can predict the type of flow

(Reynolds, 1883). It is now called the Reynolds Number and is a ratio of inertial to viscous

forces, shown in Equation 1.1.

𝑅𝑒 =𝜌𝑣𝐿

𝜇(1.1)

where 𝜌 is the density of the fluid, 𝑣 is the bulk fluid velocity, 𝐿 is the characteristic length,

and 𝜇 is the fluid dynamic viscosity.

When viscous forces are dominant, the flow is usually laminar. For pipe flow, this occurs when

Re is below 2000. Conversely, when inertial forces are dominant, the flow is usually turbulent.

For pipe flow, this occurs when Re is above 3000. The range between 2000 and 3000 is where

both laminar and turbulent flows are possible and this is called the transition range. This

study is based on Couette flow instead of pipe flow for ease of parameter variations and

viscosity calculations. However macroscopically, it is assumed that the transition Re and

range for a Couette flow is the same as that for pipe flow, ie. 2000 and 1000 respectively. The


Reynolds number can be used to determine dynamic similitude between different conditions

and parameters.

1.2 Mathematical Descriptions of Fluid Flow

Fluid motion can be described mathematically by a continuum or molecular model. The

continuum model is based on infinitesimal fluid elements solved by the Navier-Stokes (N-S)

equations. The molecular model is based on solving individual molecular movement subject

to attractive and repulsive forces and is called Molecular Dynamics (MD).

1.2.1 Continuum Model

The continuum model assumes that the fluid is a continuous medium and disregards its

molecular structure. The N-S equations, developed in the 1840s, describes the continuous

fluid in terms of variations in density, velocity, pressure, temperature (Gad-el Hak, 1999).

The equations are based on Newton’s Second Law and can be derived by applying momentum

and energy conservation laws to an infinitesimal fluid element (Wendt and Anderson, 1992).

For an incompressible Newtonian single-phase fluid, the momentum balance is,

𝜌∂𝑣

∂𝑡+ 𝜌(𝑣.∇)𝑣 = −∇𝑃𝑟 + 𝜇∇2𝑣 + 𝜌𝑔 (1.2)

The Navier-Stokes equations are a set of partial differential equations. There are analytical

solutions for simple flows such as laminar flow, one dimensional flow, and Stokes flow since

they can be simplified to linear equations (Temam, 2001). However, the majority of flows are


turbulent and equations for these flows are too difficult to be solved analytically. This is due

to the non-linear convective term. Recently, improved computational power has allowed for

approximations to these equations to be solved using numerical methods. This field of study

is called Computational Fluid Dynamics (CFD) and uses techniques such as finite volume

and finite difference methods (Ferziger and Peric, 1999).

Direct Numerical Simulation (DNS) is a CFD method of solving the N-S equations com-

pletely without any simplifications (Moin and Mahesh, 1998). This approach is extremely

computationally expensive and not yet viable for studying turbulent flow. Therefore, ap-

proximations and simplifications to the N-S equations have been developed and since they

only apply to turbulent flows, they are called turbulent models.

Turbulence modelling was studied by Osborne Reynolds and he developed the Reynolds-

averaged Navier-Stokes equations (RANS) (Davidson, 2003). Reynolds rewrote the Navier-

Stokes equation as two equations, one for the mean velocity called the Reynolds stress, and

one for the fluctuations. The Reynolds stress adds more unknowns to the system and requires

closure which can be provided by various models.

The closure models are based on two approaches - the Boussinesq hypothesis and the

Reynolds Stress Model (RSM) (Petukhov et al., 1970). The Boussinesq hypothesis is based

on a simple relationship between the Reynolds stresses and the eddy viscosity. A number of

eddy viscosity models exist including k-𝜖, Mixing Length Model, and Zero Equation (Benocci

and Beeck, 2000). The Reynolds Stress Model directly manipulates the N-S equations for

the Reynolds stresses. Several equations are introduced and hence this approach is more

complicated and computationally expensive.

Other common approaches to solving turbulent flow include Large Eddy Simulation (LES),

Detached Eddy Simulation (DES), and Vortex Method (Sagaut, 2002). The details of these


methods are available in literature and will not be discussed here. They are added for

completeness.

A statistical approach to turbulence was developed by Kolgomorov in 1941 (Kolmogorov,

1941). It is based on Richardson’s original idea that turbulent flow is composed of eddies

of different sizes. The large unstable eddies break down into smaller eddies and the process

continues giving rise to smaller and smaller eddies. The kinetic energy is transferred down

from the large scales to smaller scales until it is dissipated into internal energy. Kolgomorov

stated that for very high Reynolds numbers, the small scales are universally and uniquely

determined by viscosity and the rate of energy dissipation. The dimensional number is called

Kolgomorov’s length scale (Eq.1.3). However, it is now recognised that Kolgomorov’s result

is not empirically correct.

𝜂 =

(𝜈3

𝜖

) 14

(1.3)

The N-S equations were also unable to be solved for practical problems such as calculating

the frictional shear force on a surface immersed in a flow. However, in 1904, Ludwig Prandtl

developed a boundary-layer concept (Prandtl, 1904). His theory was based on the assumption

that the effect of friction caused the fluid immediately adjacent to the surface to stick to the

surface, that is the no-slip boundary condition. The friction was only experienced by a thin

layer near the surface. Outside of this boundary layer, the flow was assumed to inviscid, and

this was solvable.

Although DNS is becoming more useful in studying the flow field with increasing computa-

tional power, the majority of studies on fluid flow still use various turbulence models or some

form of simplification such as Prandtl boundary layer equations. A large amount of success


has come from applying these models using CFD for macroscopic simulations (Anderson,

1995). However, fluid flow through nano- and micro-channels is different to macroscopic

flow. The behaviour cannot be predicted from the Navier-Stokes equations which assume

the fluid is a continuum and require a no-slip boundary condition at the fluid-solid interface.

The reasons for this will be discussed in the following sections.

1.2.2 Molecular Model

A more recent approach to describing fluid flow uses a molecular model called molecular

dynamics (MD), pioneered by Alder and Wainwright (Alder, 1960, 1959). It was tradition-

ally applied to study equilibrium systems but is now also applied to study non-equilibrium

systems. MD provides microscopic details of molecular movements as compared to the

macroscopic views using the N-S equations. An understanding at the molecular level is

becoming increasingly important in the field of information and biological technology. Re-

cently, with the increase in computational power, MD simulations have become one of the

most promising methods to provide a clear and fundamental understanding of microscopic

mass and heat transfer at the molecular level. However, its application is still restricted

mainly to the field of nano-channel flows.

MD is based on numerically solving Newton’s Second Law of Motion for individual molecules

subject to attractive and repulsive forces from other molecules. A pair of molecules separated

by long ranges have an attractive force (Van Der Waals force) and at short ranges have a

repulsive force (Pauli Repulsion). The most common model used to simulate these forces is

the Lennard-Jones potential.


𝑉 (𝑟) = 4𝜖

[(𝜎

𝑟

)12

−(𝜎

𝑟

)6]

(1.4)

where 𝜖 is the depth of the potential well, 𝜎 is the distance at which the inter-molecular

potential is zero, and 𝑟 is the distance between the molecules.

The(1𝑟

)12term describes the repulsion and the

(1𝑟

)6term describes the attraction. The

repulsion component has no theoretical justification. It is only an approximation and chosen

to be 𝑟12 for ease of computation. The attraction component is derived from dispersion

interactions.

1.3 Nano-Channel and Micro-Channel Flows

1.3.1 Introduction

Advances in technology are occurring in field of fluid dynamics. Nanofluidics are based on

devices with channels smaller than 1𝜇m and microfluidics are for channels smaller than 1mm.

The benefits of these devices are numerous and include the low cost for mass production,

reduced volume of fluid, and faster analysis of smaller fluid volumes (Whitesides, 2006).

Gas flow through nano- and micro-channels can be analysed with a number of models. The

Knudsen number is generally used to determine which model is appropriate. It is defined as

the gas mean free path divided by the channel dimension, and was developed by Gad-el-Hak

(Gad-el Hak, 1999), Table 1.1. For example, flows with a Knudsen number approaching zero

can be treated as inviscid and so the Euler equations are appropriate. On the other hand,

flows with a Knudsen number greater than ten require MD simulations.


Table 1.1: Gas flow regimes in microchannels (Gad-el Hak, 1999)

Liquid flow is not as well understood as gas flow. There is no model similar to the Knudsen

number and the condition at which the no-slip assumption becomes invalid is not clear

(Gad-el Hak, 1999). Previous experiments of liquid flow through micro-channels present

contradictory results (Pfahler et al., 1990a, 1991; Pfahler, 1992; Bau, 1994). At such small

scales, it is difficult to remove experimental errors.

Currently, a more appropriate method of analysis for nano-channel flows is to use MD.

Previous MD studies have revealed several key differences between nano- and macro-channel

flows. Nano-channel flows exhibit wall slip, density fluctuations, rheological changes, and a

possible lowering of the laminar-turbulent transition point and transition range. These will

be discussed in the following sections.

1.3.2 Literature Review

1.3.2.1 Wall-Slip

The no-slip boundary condition is a valid assumption for macro-scale flows. This states that

at the fluid-solid interface, the velocity of the fluid relative to wall is zero. However, it is

now well-established that at this interface, the fluid can slip, that is have a non-zero velocity

relative to the wall. The effect of slip can be significant for channels with large surface to

volume ratio. Some of the benefits of wall-slip include the reduction in energy requirements


a1172507

Text Box

NOTE: This figure is included on page 9 of the print copy of the thesis held in the University of Adelaide Library.

to pump liquids through a nano-channel (Holt et al., 2006), better lubrication (Chen et al.,

2006), and reduced wear in nano-scale gaps (Gao et al., 2000).

MD simulations are an ideal tool for investigating the behaviour of fluid molecules adjacent

to a surface. The boundary conditions are not specified, as in continuum mechanics, but

develop from wall-fluid interactions. Xu and Zhou showed that the velocity profile in nano-

channels deviates from that of the classical Poiseuille flow due to the wall-slip at the boundary

(Xu and Zhou, 2004). This has also been presented experimentally (Koplik et al., 1989; Neto

et al., 2005). In recent years, numerous MD studies have been conducted on the effect of

various wall and fluid parameters on the degree of slip at the wall-fluid interface.

Several studies have shown that slip is a function of the wall-fluid interaction strength.

Slip is increased for super-hydrophobic surfaces (Ou et al., 2004; Truesdell et al., 2006) and

hydrophobic surfaces with trapped nano-bubbles (Ishida et al., 2000; Tyrrell and Attard,

2001). Conversely, slip is reduced for hydrophilic surfaces (Cottin-Bizonne et al., 2005;

Vinogradova and Yakubov, 2006). The effect of shear rate on the degree of slip has also been

studied. Thompson and Troian presented MD simulations showing that slip increases with

increasing shear rate with a non-linear relationship (Thompson and Troian, 1997). Martini

et al. confirmed this and also showed that the slip-length asymptotes to a constant value

as shear rate increases (Martini et al., 2008). For increasing surface roughness, it has been

shown that there is a reduction in the degree of slip (Zhu and Granick, 2002; Schmatko et al.,

2006; Watanabe et al., 1999; Cottin-Bizonne et al., 2003)

Some studies have been conducted on a continuum and MD hybrid method for nano- and

micro-channel flow (Nie et al., 2004). This was developed due the high computational cost

of MD simulations. A hybrid approach combines the strengths of molecular and continuum

approaches (Hadjiconstantinou and Patera, 1997; Hadjiconstantinou, 1999; OConnell and


Thompson, 1995). They are based on the observation that the breakdown of the continuum

approach is primarily confined to regions near walls where slip occurs. An MD approach

is suitable in these regions, leaving the continuum equations for the remainder of the flow.

The main difficulty with this hybrid approach is in coupling the continuum and molecular

interface.

1.3.2.2 Density Fluctuations

It is well established that for macroscopic flows, the fluid density is temperature dependent

and does not vary significantly in space and time. This is the basis of the continuum

assumption. However, on the nano-scale, the density is a function of position and time

of the fluid molecules, and hence there is some variation in density. Travis and Gubbins

reported that the density oscillates along the flow direction with a wavelength of the order of

a molecular diameter in a channel of about 4 molecular-diameter width (Travis and Gubbins,

2000).

1.3.2.3 Viscosity

Flows through microfluidic devices have very high shear rates across the channel wall. Fluids

that are Newtonian at normal shear rates in macro-channels may become non-Newtonian at

extremely high shear rates (Xu et al., 2004). Israelachvili showed experimentally that the

viscosity of films thicker than 10 molecular layers or 5 nm has the same viscosity as the bulk

viscosity (Israelachvili, 1986; Gee et al., 1990). For thinner films, the viscosity depends on

the number of molecular layers and can be as higher as 105 times larger than the normal

bulk viscosity (Israelachvili, 1986; Gee et al., 1990).


One of the most challenging properties to determine using MD is viscosity, however two

methods have been used for nano-channel flows - equilibrium molecular dynamics (EMD)

and non-equilibrium molecular dynamics (NEMD). In equilibrium simulations, the viscosity

can be obtained from pressure fluctuations (Green-Kubo integration or Einstein relation) or

from momentum fluctuations (Palmer, 1994). These techniques are rigorous and require long

simulation times to obtain reliable results. Also, the computational requirements increase

dramatically as viscosity increases. There are a number of non-equilibrium methods that are

based on a steady-state shear. These use a periodic shear flow (Ciccotti et al., 1979), sliding

boundary conditions such as the commonly used SSLOD algorithm (Evans and Morriss,

1990), or a homogeneous-shear flow method. The equilibrium and non-equilibrium methods

mentioned above, except for the momentum fluctuation approach, are described in Allen and

Tildesley (Allen and Tildesley, 1989).

1.3.2.4 Laminar-Turbulent Transition and Transitional Range

In nano- and micro-channels, the flow is almost always laminar at common fluid velocities.

However, at high shear rates in micro-channels, it is possible for turbulent flow to occur and

this has been shown in a number of studies, discussed below. The behaviour of turbulent

flow is significantly different from laminar flow and so it is important to understand the

transitional Re and range.

In macro-channel flows, the transitional Re is known to be around 2000 and the transitional

Re ranges between 2000 and 3000. Numerous experimental and theoretical studies have

sought to determine whether effects which may become significant at the micro-scale influence

the transition from laminar to turbulent flow. However, they have provided contradictory

results.


Wu and Little (Wu and Little, 1983, 1984) measured friction and heat transfer of gases

in microchannels. They observed different experimental results of convective heat transfer

from that obtained in conventional macro-channels. They also found that friction factors

were larger than those obtained from the traditional Moody charts and indicated that the

transition from laminar to turbulent flow occurred much earlier at Re in the range of 400 to

900 for various configurations. Weilin et al. (Weilin et al., 2000) and Pfahler et al. (Pfahler

et al., 1990b) also found that the pressure gradient and flow friction in micro-channels were

higher than in macro-channels.

Peng et al. (Peng et al., 1994a,b) showed that the transitional Re occurred around 200 to

700 and the beginning of fully developed turbulent flow occurred at Re around 400 to 1500

for rectangular microchannels. They also found that the transitional Re and the transition

range became smaller as the microchannel dimensions decreased. This was also concluded

by Choi et al. (Choi et al., 1991).

Mala and Li (Mohiuddin Mala and Li, 1999) explored fluid flow in stainless steel and fused

silica microtubes of diameters in the range of 50 to 254𝜇m. Transition was observed to occur

at Re between 300 and 900, depending on the tube material.

Other studies concluded that there were no statistically observable differences in the tran-

sition Reynolds number between micro- and macroscale flows. Yu et al (Yu et al., 1995)

investigated the frictional pressure drop in circular fused silica microtubes using both nitro-

gen gas and water. The transition Re in the range 2000 to 6000 was reported. Xu et al

(Xu et al., 2000) explored laminar and transition flow in microchannels of hydraulic diame-

ter 30-344 𝜇m, and concluded that characteristics agreed with conventional behaviour of a

transition Re around 2000. Hetsroni (Hetsroni et al., 2005) compared the flow characteris-

tics of smooth and rough micro-channels. Results showed that for both smooth and rough


micro-channels, the transition for laminar to turbulent flow occur between 1800 and 2200,

in full agreement with conventional flows.

The contradictions among different studies indicate that the fluid flows in micro-channels

are still not well understood. Much of the contradictory results on the transtional Reynolds

number can be attributed to difficulties in minimising experimental uncertainty. Errors in

measurement, which are negligible on the macroscale, are amplified on the microscale (Sharp

and Adrian, 2004; Brutin and Tadrist, 2003).

No attempts to simulate laminar to turbulent transition in nano-scale flows have been con-

ducted to our knowledge. On the nano-scale, high shear rates are required to generate

turbulent flow. It does not seem currently feasible to conduct a study on this experimentally

due to the amount of errors that would be involved in measurements as well as the restric-

tions in the generation of high shear rates. Thus, an MD approach seems more appropriate.

Results from this thesis have been presented for review in two papers (To et al., 2009a,b).

1.4 Aims and Thesis Structure

The first aim of this study is to develop an MD method to characterise flow regimes based

on the degree of mixing (transverse molecular movement) which has not been previously

studied to our knowledge. The second aim is to investigate the effect of wall separation on

the laminar-turbulent transition point and transitional range. The third aim is to investigate

the effect of wall properties (wall interaction strength, wall wettability, wall density, and wall

roughness) on the laminar-turbulent transition point and transitional range.

The preceding sections identified that the Navier-Stokes equations are invalid and experi-

mental analysis involves a large amount of error in nano- and micro-scale flows. Thus, a


MD approach is chosen over CFD and experimentation. This study is confined to nano-scale

flows due to computational limitations which means that the shear rates simulated must be

extremely high to obtain turbulent flow. However, the average temperature is maintained

constant to allow for comparison with different shear rates. The flow behaviour is studied

over a wide range of shear rates in nano-channel flows.

The following chapters are divided into four parts: Parts I to IV. Part I addresses the

pre-simulation issues. This includes the MD simulation details and the flow regime charac-

terisation method.

Part II focuses on simulating flow in various flow regimes. Chapter 4 firstly looks at flow in

a single channel and discusses the details of the simulations. Chapter 5 expands to various

size nano-channels.

Part III is dedicated to measuring the effect of various wall properties on flow and mixing

behaviour. Chapter 6 focuses on wall interaction strength, Chapter 7 focuses on wall wetta-

bility, Chapter 8 focuses on wall density, and Chapters 9 and 10 focus on wall roughness.

Part IV summarises the results and presents future work.


Part I

PRE-SIMULATIONS

16

Chapter 2

MD Simulation Details

2.1 Calculation Domain

The MD set-up is designed to simulate 2-D Couette flow in Cartesian coordinates. The

calculation domain for a smooth wall channel is represented in figure 2.1. It contains 𝑁𝐹

fluid molecules within an area of width of 𝐿𝑥 and height 𝐿𝑦. Periodic boundaries are applied

in the 𝑥 direction. A molecule that flows past a side wall re-enters on the other side. Also,

molecules near the right boundary interact with molecules near the left boundary and vice

versa. This design allows the simulation to behave similar to infinite length parallel plates.

The molecules are initially distributed in their two-dimensional equilibrium positions which

is a regular triangular grid, as shown in figure 2.1. This corresponds to a face centred cubic

(fcc) lattice, which is the most common form of crystallised lattice structure for most noble

gases (Xue and Shu, 1999). The upper and lower walls consist of three layers, each consisting

of 𝑁𝑤 wall molecules.

For our simulations, the upper wall moves at a velocity between 0 nm/ps ≤ 𝑉𝑤 ≤ 1.5

17

Figure 2.1: Fluid and wall molecules set-up and initial distribution

nm/ps, Figure 2.2. A width, 𝐿𝑥, of 10nm is chosen because it has been verified that this a

sufficient length (Kim et al., October 2008). The height varies between 2.5nm ≤ 𝐿𝑦 ≤ 15nm.Of course, larger lengths would provide more accurate results however we are limited with

computational power.

2.2 Intermolecular Potential

A molecule 𝑖 interacts with all other fluid molecules 𝑗 and wall molecules 𝑗𝑤 except itself.

This part of the MD program is the most computationally expensive. The code was written in

High Performance Fortran (HPF) and was sent to a parallel computing centre, eResearchSA,

to achieve faster results.

The molecular interactions are based on the Lennard-Jones potential.

Chapter 2 MD Simulation Details 18

Figure 2.2: Couette flow simulation set-up (Shear rate not necessarily constant)

𝑉 (𝑟) = 4𝜖

[(𝜎

𝑟

)12

−(𝜎

𝑟

)6]

(2.1)

where 𝜖 and 𝜎 are the energy and length scales of the fluid phase, respectively and 𝑟 is the

separation distance between the molecules. To reduce the number of computations, a cut-off

distance of 2.5𝜎 is applied, ie. 𝑟 ≤ 2.5𝜎.

The force between molecules is

𝐹 (𝑟) =−∂𝑉 (𝑟𝑖𝑗)

∂𝑟𝑖𝑗= 24𝜖

[2𝜎12

𝑟13− 𝜎6

𝑟7

](2.2)

where 𝑟 ≤ 2.5𝜎

Note that this implies 𝐹 (𝑟) = 0, for all 𝑟 > 2.5𝜎. It has been shown that for 𝑟 > 2.2𝜎, the

force is negligible (Thompson and Troian, 1997).

The simulations used Argon as the fluid because the Lennard-Jones potential is the most

accurate for Argon. Its molecular mass is 𝑚 = 6.69𝑥10−26kg, molecular diameter is 𝜎 =


0.34nm and binding energy 𝜖 is 119.8𝑘𝑏 or 1.6539x10−21J.

2.3 Wall Simulation

Each wall is comprised of three layers of molecules in a similar equilateral triangular lattice

(fcc) to the fluid molecules, Figure 2.1. For fluid-wall interactions, a Lennard-Jones potential

similar to the one used for molecular interactions is applied,

𝐹 (𝑟) =−∂𝑉𝑤𝑓 (𝑟𝑖𝑗)

∂𝑟𝑖𝑗= 24𝜖𝑤𝑓

[2𝜎12

𝑤𝑓

𝑟13− 𝜎6

𝑤𝑓

𝑟7

](2.3)

where 𝜖𝑤𝑓 and 𝜎𝑤𝑓 are the wall-fluid parameters and 𝑟 is the separation distance between a

wall and fluid molecule. Similarly, 𝑟 ≤ 2.5𝜎 is applied.

The wall molecules do not interact with one another to reduce the computational load.

Hence, to simulate wall molecular movements, each wall molecule is attached by a stiff

spring to its lattice position (Cieplak et al., 2000). The wall springs have a potential of the

form

𝑉 (∣𝑟 − 𝑟𝑒𝑞∣) = 12𝑘𝑤(∣𝑟 − 𝑟𝑒𝑞∣)2 (2.4)

where 𝑘𝑤 is the stiffness of the spring and ∣𝑟 − 𝑟𝑒𝑞∣ is the distance between a wall molecularposition and its lattice site. A relatively soft wall spring with a stiffness of 𝑘𝑤 = 75𝜖𝜎

−2 is

used for our simulations.

Thus, the total force experienced by a fluid molecule, 𝑖, is

𝐹 (𝑟)𝑖 =𝑁𝑓∑

𝑗=1𝑗 ∕=𝑖

∂𝑉 (𝑟𝑖𝑗)

∂𝑟𝑖𝑗+

𝑁𝑤∑𝑗=1

∂𝑉𝑤𝑓 (𝑟𝑖𝑗)

∂𝑟𝑖𝑗(2.5)


and, the total force experience by a wall molecule, 𝑖, is

𝐹 (𝑟)𝑖 =𝑁𝑓∑𝑗=1

∂𝑉𝑤𝑓 (𝑟𝑖𝑗)

∂𝑟𝑖𝑗+ 𝑘𝑤(∣𝑟 − 𝑟𝑒𝑞∣) (2.6)

2.3.1 Wall Interaction

The wall-fluid interaction strength can be modified by changing 𝜖𝑤𝑓 . Numerous studies have

been undertaken using nano-channels with differing wall-fluid interaction strengths and its

effect on the level of wall-slip (Thompson and Robbins, 1990). In this study, 0.5𝜖𝑤𝑓 , 1.0𝜖𝑤𝑓 ,

and 2𝜖𝑤𝑓 were simulated. Although the interaction strength varies, the equilibrium position

for the fluid molecules from the wall remains identical as this position is not a function of

𝜖𝑤𝑓 , Equation 2.3.

2.3.2 Wall Wettability

For fluid-wall interactions that incorporate wettability variations, a modified L-J model is

used (Yang, 2006),

𝐹 (𝑟) = 24𝜖𝑤𝑓

[2𝜎12

𝑤𝑓

𝑟13− 𝛿

𝜎6𝑤𝑓

𝑟7

](2.7)

where 𝜖𝑤𝑓 and 𝜎𝑤𝑓 are the wall-fluid parameters, 𝑟 is the separation distance between a wall

and fluid molecule where 𝑟 ≤ 2.5𝜎, and 𝛿 is a convenient parameter to adjust the surface

wettability by controlling the strength of the attractive component. Variations of 𝛿 from

0.5 to 1.0, correspond to a contact angle at fluid-solid interface of 140𝑜 to 90𝑜, denoting

hydrophobic and hydrophilic surfaces, respectively (Barrat and Bocquet, 1999).


2.3.3 Wall Density

Normally, the wall density used for nano-channel simulations is identical to that of the fluid.

This is the case for the majority of simulations. However, for the chapter where variations

in wall densities are compared, the densities are modified to 1x, 2x, and 3x the fluid density.

Only wall densities greater than 1x the fluid density were investigated. This is because

fluid molecules would escape through the wall and thus require a different wall simulation

technique to keep the fluid molecules between the walls. The density of the wall atoms can

be adjusted by packing them at different distances from each other, such as in figures 2.3.3(a)

and 2.3.3(b).

With increasing in wall density, there is an increase in equilibrium distance for the fluid

molecules from the wall. However, to keep the wall separation constant, the wall positions

can not be changed and it is assumed that the slight compression on the fluid has a minor

effect on the outcome. However, an analysis of the flow behaviour of the various densities

will provide an indication of the effects of compression on the flow.

2.3.4 Wall Roughness

For rough walls, the position of the wall molecules are displaced in the y-direction by

Δ𝑦𝑅 = 𝐴𝑠𝑖𝑛(2𝜋𝑥

𝑃) (2.8)

where parameters 𝐴 and 𝑃 , which characterise the roughness, are, respectively, the ampli-

tude and period of the sinusoidal wall (Jabbarzadeh et al., 2000). However, this changes the

separation distances between the molecules where some may lie in the highly repulsive re-

gion. To remove this problem, the fluid molecules are positioned in an equilateral triangular


Figure 2.3: Wall Density (a) 2x and (b) 3x Fluid Density


distribution between walls of

𝑦𝑢𝑝𝑝𝑒𝑟 = 𝐿𝑦 + 𝐴𝑠𝑖𝑛(2𝜋𝑥

𝑃) (2.9)

𝑦𝑙𝑜𝑤𝑒𝑟 = 𝐴𝑠𝑖𝑛(2𝜋𝑥

𝑃) (2.10)

The upper and lower wall molecules are distributed in a similar fashion to create a thickness

of a minimum of three layers. Figures 2.4(a)(b)(c) show variations in period with the same

amplitude of 0.8nm. Figures 2.5(a)(b)(c) show variations in amplitude with a period of

3.0nm.

The length, 𝐿𝑥, of the simulation was maintained constant and hence, for some roughness

arrangements, the sinusoidal waves is not consistent with periodicity. This is ignored in our

simulations and assumed to increase the roughness slightly.

The average wall separation remains constant during the simulation and is measured from

the average position of the first layer of the sinusoidal wall as shown in the figure 2.6, (Jab-

barzadeh et al., 2000). However, the actual wall separation varies as the crests and valleys

pass each other as the upper wall moves. The wall separation varies between maximum and

minimum vales shown by 𝑍𝑚𝑎𝑥 and 𝑍𝑚𝑖𝑛, where

𝑍𝑚𝑎𝑥 = 𝑍𝑎𝑣𝑔 + 2𝐴

𝑍𝑚𝑖𝑛 = 𝑍𝑎𝑣𝑔 − 2𝐴


Figure 2.4: Wall roughness with Amplitude of 0.8nm and Period of (a) 1.5nm (b) 2.5nm and(c) 3.5nm


Figure 2.5: Wall roughness with Period of 3.0nm and Amplitude of (a) 0.2nm (b) 0.5nm and(c) 0.8nm


Figure 2.6: Wall roughness set-up (Jabbarzadeh et al., 2000)

2.3.5 Thermal Wall Model

The fluid molecules interact with the wall such that at a close distance, the wall molecule

will repel the fluid molecule away from the wall surface. However, it is still possible for some

fluid molecules to escape and pass through between the wall molecules, especially for lower

wall-fluid interaction strengths. Therefore, to be certain that the molecules remain between

the walls, a barrier is set. For smooth walls, this is achieved by using a thermal wall model,

described by Tenenbaum et al. (Tenenbaum et al., 1982). The two velocity components

after the liquid particles strike the wall surfaces are:

𝑣𝑥 =

√𝑘𝐵𝑇𝑤

𝑚𝜓𝐺

𝑣𝑦 = ±√−2𝑘𝐵𝑇𝑤

𝑚𝑙𝑛𝜓

The 𝑣𝑦 is positive for fluid molecules passing through the lower wall barrier, and negative for


a1172507

Text Box

NOTE: This figure is included on page 27 of the print copy of the thesis held in the University of Adelaide Library.

the fluid molecules passing through the upper barrier. 𝜓 is a uniformly distributed random

number in (0,1) and 𝜓𝐺, is Gaussian-distributed random numbers with zero mean and unit

variance (Alexander and Garcia, 1997).

2.4 Thermostat

Based on the kinetic theory of gases, the instantaneous temperature is directly proportional

to the average kinetic energy of the molecules. For two-dimensional flow,

𝑁𝑘𝑏𝑇 =1

2

𝑁∑𝑖=1

𝑚(𝑣2𝑖,𝑥 + 𝑣2𝑖,𝑦

)(2.11)

where 𝑁 is the total number of fluid molecules, 𝑘𝑏 is the Boltzmann constant, 𝑣𝑖,𝑥 and 𝑣𝑖,𝑦

are the fluctuating 𝑥 and 𝑦 velocity components of molecule 𝑖.

For a wall to behave as an ideal heat reservoir, it must consist of an large number of molecules

interacting with each other (Kim et al., October 2008). However, having wall molecules

interact with each other would increase the computational cost immensely.

For a rigid wall, where the wall molecules do not move from their lattice positions, there is

no heat transfer to or from the walls. The fluid temperature increases as the moving wall

adds energy to the fluid. Therefore it is necessary to dissipate work done by the moving

wall using a thermostat applied to the fluid molecules. For a flexible wall, although the wall

molecules do interact with the fluid molecules and are held together by a lattice spring, the

wall molecules do not interact with each other and so the heat transfer within the wall is not

correct. The first layer of the wall next to the fluid, would have most of the thermal velocity

distribution due to its close interaction with the fluid molecules. This would result in a


higher temperature for this layer compared with the other layers. Therefore, the thermostat

is applied to each wall layer separately at every time-step similar to Kim et al. (Kim et al.,

October 2008).

For initial conditions, the velocities for thermal oscillations were assigned from a Maxwell-

Boltzmann distribution consistent with the desired temperature, and then a thermostat is

applied to the wall and fluid molecules. The temperature of the simulations were set to

120𝐾 which is the temperature at which Argon is a liquid. A common thermostat used is

the Nose-Hoover Thermostat. The Nose equations of motion

𝑟𝑖 =𝑝𝑖𝑚𝑖

�̇�𝑖 = 𝐹𝑖 − 𝜁𝑝𝑖 (2.12)

𝜁 =1

𝑄

(𝑁∑𝑖=1

𝑝2𝑖𝑚−𝑁𝑘𝑏𝑇

)

where 𝑟𝑖 is the position of molecule 𝑖, 𝑝𝑖 is the momentum, 𝐹𝑖 is the force on the molecule, 𝑄 is

the heat bath “mass”, and 𝜁 is the thermodynamic friction coefficient. The third expression

in Eq. 2.12 shows the temperature control mechanism in the Nose-Hoover thermostat. The

term within the parentheses is the difference between the system instantaneous kinetic energy

and the kinetic energy at the desired temperature. The choice of 𝑄 is arbitrary but it affects

the performance of the thermostat. We chose a value of 1.0x107 eV fs2, which corresponds

to the typical atomic vibration frequency of the order of 1012 Hz (Hu and Sinnott, 2004).

In an actual laboratory experiment, shearing the fluid imparts viscous heat to the system

which is conducted through the walls. Several previous MD studies have maintained a con-

stant fluid temperature by using a Gaussian thermostat (Evans and Morriss, 1990), coupling

one of the components of the equations of motion to a heat bath (Thompson and Robbins,


1990), or by re-scaling the velocities of the fluid atoms (Manias et al., 1993, 1994). Other

studies have maintained only the wall at a constant temperature and not the fluid (Khare

et al., 1997; Kim et al., October 2008).

For this study, an isothermal fluid condition is desired since a change in fluid temperature

is normally accompanied by a change in viscosity. Therefore, the Nose-Hoover thermostat

is applied to the fluid at each time-step to remove viscous heating. The walls, as mentioned

above, also have the Nose-Hoover thermostat applied to it to remove any excess fluid tem-

perature. Although an isothermal condition is maintained, the viscosity is not assumed to

be constant with constant temperature. It is calculated for each simulation to observe if

there are any changes with shear rate or other parameters.

2.5 Numerical Integration Method

The Lennard-Jones potential is a stiff system and thus normally requires an implicit inte-

gration method. However, a compromise is needed between accuracy, ease of coding and

computational cost for molecule simulations. A number of methods for molecular simula-

tion have been used in the past including the Verlet, Velocity Verlet, Leapfrog, Beeman’s,

and Gear’s Predictor-Corrector methods (Fehske et al.). For this study, the Velocity Verlet

method was chosen. It is a second order method, with a computational cost of a first order

method. There is no numerical drift unlike some other implicit or explicit methods. Also,

the Verlet Velocity method is easy to code. The disadvantage of this method is that the

algorithm is of moderate precision.


The Velocity Verlet algorithm is:

𝑟(𝑡+ ∂𝑡) = 𝑟(𝑡) + ∂𝑡𝑣(𝑡) + 0.5∂𝑡2𝑎(𝑡) (2.13)

𝑣(𝑡+ ∂𝑡) = 𝑣(𝑡) + 0.5∂𝑡[𝑎(𝑡) + 𝑎(𝑡+ ∂𝑡)] (2.14)

where 𝑟 is the position, 𝑣 is the velocity, 𝑎 is the acceleration, and 𝑡 is the time.

The standard implementation scheme of this algorithm is:

1. Calculate: 𝑟(𝑡+ ∂𝑡) = 𝑟(𝑡) + ∂𝑡𝑣(𝑡) + 0.5∂𝑡2𝑎(𝑡)

2. Calculate: 𝑣(𝑡+ 0.5∂𝑡) = 𝑣(𝑡) + 0.5∂𝑡𝑎(𝑡)

3. Derive: 𝑎(𝑡+ ∂𝑡) from the interaction potential

4. Calculate: 𝑣(𝑡+ ∂𝑡) = 𝑣(𝑡+ 0.5∂𝑡) + 0.5∂𝑡𝑎(𝑡+ ∂𝑡)

2.6 Parameter Averaging

The Velocity Verlet algorithm require a suitable time-step for the simulations. The time-steps

used range from Δ𝑡=1𝑥10−15s to Δ𝑡=1𝑥10−17s depending on the shear velocity and the type

of wall simulation. The larger the shear force, the smaller the time-step. Also, the rougher

the walls, the smaller the time-step. Furthermore, smaller time-steps are used as a means to

determine whether there is a change in the flow field. Once there is no further change, this

is an indication of convergence, and the time-step is used for the specific simulation. The

time-step used for all simulations is smaller than the characteristic time of,

𝜏 =

√𝑚𝜎2

𝜖= 2.16−12s (2.15)


For statistical averaging, the flow system is divided into bins parallel to the flow direction.

Initially, the velocity of each layer increases until the system reaches a steady state. The

time it takes to reach steady state is dependent on shear rate and wall separation. The

higher the shear rate, the longer it takes to reach steady state. Similarly, the larger the wall

separation, the longer it will take. Various wall properties will also have an effect on the

length of time. The steady state time used is that for which there is less than 1% change

in velocity across the wall separation between time-steps. Once steady state is reached, the

velocity, temperature, and density fields, and degree of mixing were calculated for a further

time of 400ps (further details of mixing are discussed in Chapter 3). For analysis, a similar

approach to Xu et al. (Xu et al., 2004) was taken such that the function

𝐻𝑛(𝑦𝑖,𝑗) = 1 if (𝑛− 1)Δ𝑦𝑏 < 𝑦𝑖 < 𝑛Δ𝑦𝑏 (2.16)

otherwise 𝐻𝑛(𝑦𝑖,𝑗) = 0, where the subscript 𝑗 represents the 𝑗th time step.

The average 2-D dimensionless number density in the 𝑛th slab from time-step 𝐽𝑁 to time-step

𝐽𝑀 is

𝜌𝜎2 =𝜎2

𝐿𝑥Δ𝑦𝑏(𝐽𝑀 − 𝐽𝑁 + 1)

𝐽𝑀∑𝑗=𝐽𝑁

𝑁∑𝑖=1

𝐻𝑛(𝑦𝑖,𝑗) (2.17)

where 𝑦𝑖 is the coordinate of the mid-point of the 𝑛th slab, 𝐽𝑁 and 𝐽𝑀 are the start and

ending time step of the parameter averaging. The integrated time interval from 𝐽𝑁 time

step to 𝐽𝑀 time step is 400ps. The slab average velocity form 𝐽𝑁 time step to 𝐽𝑀 time step

is computed as


𝑣(𝑦𝑖) =1

(𝐽𝑀 − 𝐽𝑁 + 1)𝑁∑𝑖=1

𝐻𝑛(𝑦𝑖,𝑗)


𝑁∑𝑖=1

𝐻𝑛(𝑦𝑖,𝑗)𝑣𝑥𝑖,𝑗 (2.18)

where 𝑣𝑥𝑖,𝑗 is the velocity in 𝑥 component of particle 𝑖 at time step 𝑗.

Using Equation 2.18, the bin temperature is calculated as,

𝑇 (𝑦𝑖) =1

2𝑘𝑏(𝐽𝑀 − 𝐽𝑁 + 1)𝑁∑𝑖=1

𝐻𝑛(𝑦𝑖,𝑗)


𝑁∑𝑖=1

𝐻𝑛(𝑦𝑖,𝑗)[𝑣𝛼𝑖,𝑗 − 𝑣𝛼𝑦𝑖

]2(2.19)

where the subscript 𝛼 represent x or y. At steady state, 𝑣𝑦𝑦𝑖 , is zero for all bins.

The average temperature at a specific time is calculated as,

𝑇𝑎𝑣𝑔 =1

2𝑁𝑘𝑏

𝑁∑𝑖=1

𝑚(𝑣2𝑖,𝑥 + 𝑣2𝑖,𝑦

)(2.20)

Note that this approach is similar to that in Xu et al. (Xu et al., 2004).

2.7 Viscosity and Reynolds Number

Macroscopically, fluids are assumed to be a continuum and viscosity measurement is depen-

dent on whether it is a Newtonian or non-Newtonian fluid. If it is Newtonian, Newton’s

theory applies,

𝜇 = 𝜏𝑑𝑦

𝑑𝑉(2.21)


On the nano-scale, there are several methods for calculating the viscosity of discrete molecules

based on MD simulations. For our study, we use two methods to determine the viscosity.

The first method is the most frequently used one and imposes a constant shear rate, which

we use to calculate the Reynolds number (Martini et al., 2006).

The viscosity is calculated firstly by determining the shear stress which is the force of the

fluid molecules on the wall molecules divided by the area of the wall.

𝜎𝑤 =𝑁𝑤∑𝑖=1

𝑁𝐹∑𝑗=1

𝐹𝑖𝑗

𝐴𝑟

(2.22)

where 𝜎𝑤 is the wall shear stress, 𝐹𝑖𝑗 is the force on wall molecule 𝑖 from fluid molecule 𝑗,

and 𝐴𝑟 is the area of the wall, 𝑁𝑤 is the number of wall molecules, and 𝑁𝐹 is the number of

fluid molecules.

This method of shear stress calculation has been found to produce similar results to the

Method of Planes (Todd et al., 1995) if the plane is chosen to be at the position of the walls

(Zhang et al., 2001; Varnik et al., 2000). Also, the Irving-Kirkwood expression produces

similar results (Jabbarzadeh et al., 1998; Martini et al., 2006).

To determine the shear rate, a linear velocity profile is imposed between the top and bottom

walls.

�̇�𝑖𝑚𝑝 =𝑉𝑤

𝐿𝑦

(2.23)

where �̇�𝑖𝑚𝑝 is the imposed shear rate, 𝑉𝑤 is the velocity of the top wall, and 𝐿𝑦 is the

separation of the walls.

The bulk fluid viscosity is then calculated by,

𝜇𝑏 =𝜎𝑤

�̇�𝑖𝑚𝑝

(2.24)


This method assumes that the velocity of the fluid adjacent to the wall is the same, that is, no

slip. However, in previous studies, it has been shown that there is wall slip for nano-channel

flows and that the velocity profile is not linear (Thompson and Troian, 1997). However, this

method is still frequently used to determine the viscosity of thin films (Hu and Granick,

1998; Balasundaram et al., 1999; Jabbarzadeh et al., 1998; Zhang et al., 2001; Thompson

et al., 1992). The reason for this is because it is a simple method for MD simulations, and

also because its calculation method is consistent with that used in viscosity measurements

taken using a surface force apparatus (Zhang et al., 2001).

The Reynolds number is calculated using the bulk fluid viscosity,

𝑅𝑒 =𝜌𝑉𝑤𝐿𝑦

𝜇𝑏

(2.25)

The second method of calculating viscosity determines the shear rate at the wall surface,

which is used to compare with previous studies. The shear rate at the wall surface is

calculated from,

�̇�𝑤 =𝑉1 − 𝑉2

Δ𝑦𝑏(2.26)

where �̇�𝑤 is the shear rate at the wall surface, 𝑉1 is the fluid velocity adjacent to the top

wall, 𝑉2 is the fluid velocity of the next layer (bin) down, and Δ𝑦 is the bin size. Note that

since we are maintaining the same number of bins for different wall separations, the bin size

varies.

The local viscosity at the wall surface is then calculated by,

𝜇 =𝜎𝑤

�̇�𝑤(2.27)


The slip length is calculated by,

𝐿𝑠 =𝑉𝑤 − 𝑉1

�̇�𝑤(2.28)

The local viscosity, wall shear rates, and slip length are used for comparison with previous

works (Thompson and Troian, 1997).


Chapter 3

Flow Regime Characterisation

3.1 Characterisation Methods

A MD method to characterise flow regimes based on the degree of mixing has not been

previously studied to our knowledge. Various methods to quantify the degree of mixing were

examined. The method that is used for subsequent analysis of flow is based on calculating

the maximum distance each molecule moves in the y-direction over a specific time and then

averaging over the total number of fluid molecules, Eq. 3.1. This is a measure of net

positional shift instead of absoluate y-distance traversed.

Δ𝑦 =

𝑁∑𝑖=1

(𝑦𝑚𝑎𝑥,𝑖 − 𝑦𝑚𝑖𝑛,𝑖)

𝑁𝐹

(3.1)

where Δ𝑦 is the average mixing, 𝑦𝑚𝑎𝑥,𝑖 is the maximum y-position and 𝑦𝑚𝑖𝑛,𝑖 is the minimum

y-position, for molecule 𝑖.

37

This number is the average mixing and is not non-dimensionalised so that comparisons can

be made between different wall separations at a specific time. The percentage mixed is

defined as,

%(𝑚𝑖𝑥𝑒𝑑) =Δ𝑦

𝐿𝑦

100 (3.2)

3.2 Flow Regime Analysis

3.2.1 Mixing Validation

The mixing calculations were validated by simulating no mixing and complete mixing. To

simulate no mixing, the walls remained motionless, the temperature reduced to 0 K, and

the molecules were given an initial velocity of 0m/s. The degree of mixing was recorded and

showed that it remained at zero with time indicating the molecules did not move from their

initial positions.

Complete mixing was simulated by removing all molecular interactions. Each molecule was

assigned an initial velocity in accordance with the temperature and when a fluid molecule

reached a wall, it was reflected elastically. Again, the degree of mixing was observed and

showed that it increased with time reaching almost 100% after 50ps.

3.2.2 Dependence of Mixing on Time

Reynolds experiments on flow regimes were based on a constant length of pipe. This was

acceptable since molecular diffusion has a negligible effect on flow regimes in macroscale

Chapter 3 Flow Regime Characterisation 38

flows. However, the effect of molecular diffusion is enhanced for nano-scale flows, which

is described in detail below. This effect is taken into consideration in our simulations by

simulating with respect to a constant time-span instead of a constant length.

The laminar regime is described as flow with no mixing, except for molecular diffusion.

Molecules with a non-zero temperature will gradually mix with time due to molecular dif-

fusion, even when there is no flow. Based on the mixing characterisation method described

above, the average mixing for molecular diffusion or laminar flow will reach the channel

length much faster in a nano-channel than a macro-channel. This is illustrated by the red

line in figure 3.1. The mixing rate for turbulent flow is not much different to molecular

diffusion in nano-channels due to its small channel width. Whereas in macro-channel flows,

the difference is much greater. This is represented by the black line in figure 3.1. Note that

figure 3.1(a) 3.1(b) have different scales of length and time.

The average mixing based on a constant length and constant time are represented by two

arrows shown in figure 3.1(a) and figure 3.1(b). It illustrates the significance of molecular

diffusion in nano-channel flows and that it cannot be neglected. Based on the example in

figure 3.1(a), turbulent flow is achieved when the flow rate is 3m/s and laminar flow at 1m/s.

The time required for flow at 3m/s to travel a length of 0.5nm is 167ps whereas flow at 1m/s

requires 300ps. The average mixing after the flow has travelled the length of the pipe is 2nm

for both laminar and turbulent flow regimes. This comparison is obviously incorrect due

to the molecular diffusion and so a constant time method is adopted. At 500ps, turbulent

flow shows an average mixing of 2.5nm and laminar flow shows a mixing of 2nm. This

method takes molecular diffusion into consideration and allows for comparison of different

flow regimes at the nano-scale.

An example of a macro-channel flow is shown in figure 3.1(b). A similar analysis is conducted


Figure 3.1: Example of mixing as a function of time for (a) nano-channel (2.5nm) and (b)macro-channel (5cm) flows at 1 and 3 m/s


to show that molecular diffusion is negligible at this scale. For a 1m length pipe, laminar flow

at 1m/s requires 1s to flow through it whereas turbulent flow at 3m/s requires 0.33s. The

average mixing after this travelling this length is 4.8cm for turbulent flow and approximately

0cm for laminar flow. If a constant time method is used, the average mixing is 5cm and

0cm, for turbulent flow and laminar flow respectively. Therefore, either methods produce

similar results in macro-channels indicating molecular diffusion is negligible at this scale and

Reynolds experiments based on a constant length is valid.

In this study, molecular diffusion is calculated by maintaining the walls stationary, setting the

fluid temperature to 120K, and measuring the average of the maximum transverse movement

of the molecules as a function of time, Eq. 3.1. This is the non-sheared fluid mixing rate.

The mixing rate for sheared fluids are compared to that of the non-sheared fluid and if they

are similar, the flow is defined as in the laminar regime. Also, the average mixing after a

certain amount of time is similar for all flows in the laminar regime.

Flows in the transitional regime have a higher mixing rate than molecular diffusion and

increase with shear rate. For turbulent flows, the mixing rate is even greater than that of

transitional flows. However, further increases in shear rate do not increase the mixing rate

since the mixing rate of turbulent flow is at the maximum possible. Both transitional and

turbulent flows have mixing in addition to molecular diffusion.


Part II

SIMULATIONS: FLOW REGIMES

42

Chapter 4

Nano-Channel Flow Regimes

4.1 Introduction

This chapter provides an in-depth analysis of flow regimes in a single sized nano-channel.

Chapter 5 will focus on trends between various wall separations. Simulations were conducted

at various Reynolds numbers by modifying the velocity parameter between 0 nm/ps and 1.5

nm/ps for simulated Couette flow. The fluid parameters remained constant and the wall

separation used was 10nm. The flow and mixing behaviours were observed.

4.1.1 Program Validation

Before presenting results based on a new MD simulation set-up, it is important to validate

the model by reproducing and comparing results with previous MD simulations at similar

conditions. Figures 4.1 and 4.2 show that the velocity profiles obtained from our MD simu-

lations match the results in Thompson and Robbins (1990) (Thompson and Robbins, 1990)

43

Figure 4.1: Velocity profile at fluid-wall interaction strength of 𝜖𝑤𝑓=0.4𝜖. Comparison ismade with the results obtained by Thompson and Robbins(1990)

for fluid and wall interaction potential strengths of 𝜖𝑤𝑓=0.4𝜖 and 4𝜖. They predict the same

velocity-stick and velocity-slip behaviour. The simulations have both top and bottom walls

moving in opposite directions, however for the rest of our simulations, only the top wall is

moving.

Chapter 4 Nano-Channel Flow Regimes 44

Figure 4.2: Velocity profile at fluid-wall interaction strength of 𝜖𝑤𝑓=4𝜖. Comparison is madewith the results obtained by Thompson and Robbins(1990)


4.2 Results and Discussion

4.2.1 Flow Behaviour

The velocity profile, slip length, temperature profile, average temperature, and viscosity were

determined for each simulation to provide a thorough understanding of the flow behaviour,

make comparisons with previous studies, and determine whether there are any significant

effects and trends.

4.2.1.1 Velocity Profile & Slip Length

The velocity profiles with various wall velocities at steady state are shown in figure 4.3. The

slower wall velocities have a slightly more S-shaped profile than the higher wall velocities.

This suggests that at slower wall velocities, the molecules closer to the wall are slowed down

by the wall forces, whereas at higher wall velocities, the molecules overcome the attractive

forces of the walls and present more of a straight-line profile. The S-shaped profiles show

that the shear rate between the walls is not constant and that our imposed constant shear

rate for bulk viscosity calculations is not correct and includes errors.

As can be seen in figure 4.3, there is slip at the fluid-wall interface. As the wall velocity

increases, so does the slip. This is presented in figure 4.4. Although a flexible wall model

is applied, the results do not agree with that of Martini et al. (Martini et al., 2008). This

may be due to the difference in spring constants and fluid model. Our results do agree

with those of Thompson and Troian (Thompson and Troian, 1997) where the slip increases

exponentially with shear rate. This shows that our imposed constant shear rate for bulk

viscosity calculations is not accurate since we assumed no slip at the wall.


Figure 4.3: Velocity Profile for simulated Couette flow with wall separation of 10nm

Figure 4.4: Slip Length as a function of shear rate for simulated Couette flow with wallseparation of 10nm


4.2.1.2 Density Profile

It is well known that for nano- and micro-channel flows there are density fluctuations (Barrat

and Bocquet, 1999; Thompson and Robbins, 1990). Figure 4.5 show the density profile for

various wall velocities for simulated Couette flow at steady state. It can be seen that there is

a higher density closer to the wall surface. This is due to the attractive forces of the wall. The

molecules cannot get too close to the wall because it results in a highly repulsive force. This

explains the peak in the density profile at h/z of around 0.2 and 0.8. At further distances

from the wall (between 0.32 and 0.68), where the attractive forces of the wall are zero due

to a cutoff distance of 2.5𝜎, the density remains constant. At zero wall velocity, the density

is not constant. With increasing wall velocity, the density profile does not seem to change in

accordance with any trend. It was expected that with increasing wall velocity, the density

profile would even out, that is become more constant across the separation, however this

does not seem to be the case. An explanation for this may be that the wall-fluid interaction

strength is strong enough to keep the density profile structure, even for high wall velocities.

For macroscale flow, the wall-fluid interaction strength does not affect the majority of the

flow and thus the density is assumed constant.

4.2.1.3 Temperature

Few MD studies present temperature profiles, however it can be used as a good check that

the simulation is within the desired conditions, figure 4.6. Nagayama and Cheng presented

temperature profiles for pressure driven flow (Nagayama and Cheng, 2004). From our results,

no trend is observable of temperature profile with fluid velocity. However, it is useful to see

that the temperature throughout the channel fluctuates around the desired 120K between

110 and 130K.


Figure 4.5: Density Profile for simulated Couette flow with wall separation of 10nm

Figure 4.6: Temperature Profile for simulated Couette flow with wall separation of 10nm


Figure 4.7: Average Temperature as a function of time for simulated Couette flow with wallseparation of 10nm

The overall fluid temperature fluctuates between 119 and 121K which agrees with the desired

temperature of 120K, figure 4.7. The Nose-Hoover thermostat was applied to the fluid for

all wall velocities and so viscous heating was dissipated through the walls. This allows

for comparisons between viscosities at different wall velocities and wall properties discussed

later.

4.2.1.4 Viscosity

The viscosity as a function of shear rate is illustrated in figure 4.8 and shows that as the

local shear rate increases, the viscosity decreases. This is known as shear-thinning. When

the dimensionless shear rate reaches around 0.2, the dimensionless viscosity remains constant

at around 5. This corresponds to a Newtonian fluid. Thompson and Troian reported that

the dimensionless bulk viscosity of the fluid is constant at about 2.2 over a dimensionless

shear rate range of 0.001 to 1 (Thompson and Troian, 1997). Based on other studies of shear

flows, the bulk fluid becomes non-Newtonian for �̇�𝜏 ≥ 2 (Loose and Hess, 1989). However,


Figure 4.8: Viscosity as a function of shear rate for simulated Couette flow with wall sepa-ration of 10nm

Ashurst and Hoover presented results that the viscosity decreases with increasing shear rate

(Ashurst and Hoover, 1970).

For our simulations, the maximum dimensionless shear rate, as suggested by Loose and

Hess, is below 1. Therefore, it is expected that the viscosity remains constant since our

simulations do not exceed a dimensionless shear rate of 1. However, our results show a

dimensionless viscosity of approximately 5 decreasing to 3 with increasing shear rate. This

agrees with Ashurst and Hoover’s trend of dimensionless viscosity decreasing with increasing

dimensionless shear rate. The difference in results may be attributed to different models for

thermostat, fluid interaction, fluid density, and dimensions.


4.2.2 Mixing Behaviour

This section looks at the mixing as a function of time and the mixing as a function of

Reynolds number. The mixing as a function of dimensionless time for various fluid velocities

is presented in figure 4.9. It can be seen that when the bulk fluid velocity is zero, the mixing

still increases with time. This corresponds to molecular diffusion and reflects Brownian

motion. The non-zero temperature results in molecular movement. For nano-channels, the

effect of molecular diffusion is increased compared with macro-channels.

The increase in mixing with time may also be a result of errors involved with the length of

the time-step and the thermostat. However, with smaller time-steps, the resultant velocity

and density profiles were the same. The difference lies in the temperature fluctuations

because the time-step affects how often the thermostat is applied within a certain time

period. The smaller the time-step, the more often the thermostat is applied. Thus for higher

wall velocities, where the time-step is smaller, the temperature fluctuations are smaller.

Figure 4.9 illustrates the difference in rate of mixing for various flows. Theoretically, the

maximum mixing (Δ𝑦) is the separation of the walls (𝐿𝑦). For turbulent flow, the mixing

should start at zero and increase to approximately the maximum mixing much quicker than

for laminar flow. The mixing associated with laminar flow is only molecular diffusion. For

macrochannel flows, the difference in laminar and turbulent mixing is obvious, however in

nanochannel flows, they are closer. An example is shown in figure 3.1.

From figure 4.9, at 0.1nm/ps, the amount of mixing with time is similar to the diffusion

mixing rate. This suggests that the flow is laminar at 0.1nm/ps since there is no increase in

mixing. A further increase in velocity shows a significant increase in mixing which relates to

the transitional regime.


Figure 4.9: Mixing Profile for simulated Couette flow with wall separation of 10nm

The point where the flow becomes fully-turbulent is expected to be when the degree of mixing

with time does not increase any further with increasing wall velocity. However, there is not

a clear fully turbulent point from figure 4.9. This is because the simulation is not run for a

long enough time due to computational restrictions.

Theoretically, there should be two sections where mixing is independent of Reynolds number:

the laminar and turbulent regimes. The transition regime is where the mixing is dependent

on Reynolds number. A graph of the amount of mixing after 400ps versus the Reynolds

number is shown in figure 4.10. Figure 4.10(a) shows the degree of mixing including the

diffusion whereas figure 4.10(b) shows the degree of mixing without the diffusion. It is

clear from the graphs that the flow is laminar approximately between 0 ≤ Re ≤ 1. More

simulations are required at smaller velocity intervals to determine the exact transition point,

however, we can deduce that it will be approximately a Reynolds number of 1. From figure

4.10(a), the degree of mixing increases with increasing Reynolds number to an asymptote of


around 2.5nm of mixing. It can be approximated that the transitional regime lies between 1

≤ Re ≤ 15. Hence, fully turbulent flow is approximated to occur when the Reynolds numberis greater than 15. These results include a number of errors however provide an indication

of possible regimes in nanochannel flows. No previous work has been conducted on this

to our knowledge for us to compare. Further extensive simulation results for different wall

separations are presented in the next chapter.


Figure 4.10: (a) Mixing as a function of Reynolds Number and (b) Mixing minus moleculardiffusion as a function of Reynolds Number, for simulated Couette flow with wall separationof 10nm


Chapter 5

Flow Regimes in Various

Nano-Channel Separations

5.1 Introduction

This chapter looks at the flow and mixing behaviour of simulated Couette flows with different

wall separations. It follows on from the previous chapter and investigates the effect and

trends of variations in the wall separation on the flow. A large number of simulations were

conducted based on six wall separations, ranging between 2.5nm and 15nm, and each with

a further seven different flow velocities, ranging between 0 nm/ps and 1.5nm/ps, which

correspond to different Reynolds numbers.

56

5.2 Results and Discussion

5.2.1 Flow Behaviour

Similarly to the previous chapter, the section will look at the velocity profile, slip length,

temperature profile, average temperature, and viscosity, and observe the trends and effects

with variations in fluid velocity and wall separation.

5.2.1.1 Velocity Profile & Slip Length

The velocity profiles for six wall separations are presented in figure 5.1. The effect of the

attractive force of the wall on the fluid velocities adjacent to the wall is reduced with in-

creasing wall separation. This can be seen for the lower wall velocities, where the S-shape

velocity profile is pronounced in the 2.5nm wall separation flow, whereas the velocity profile

is almost straight in the 15nm wall separation flow. At high wall velocities, the velocity

profile is reasonably linear for all wall separations.

For all wall separations, the wall slip remains constant below a certain shear rate, figure 5.2.

Further increases in shear rate result in an exponential increase in wall slip. This result is

similar to that presented by Thompson and Troian (Thompson and Troian, 1983).

Another observation from figure 5.2 is that the shear rate at which the wall slip no longer

remains constant decreases with increasing wall separation. This suggests that for macroscale

flows, there is a shear rate where slip begins to occur. However, no evidence exists of this and

the shear rate at which this transition occurs may be extremely high. No previous studies

have been conducted on this to our knowledge for comparison.

Chapter 5 Flow Regimes in Various Nano-Channel Separations 57

Figure 5.1: Velocity Profiles for simulated Couette flow with wall separations (a) 2.5nm (b)5nm (c) 7.5nm (d) 10nm (e) 12.5nm (f) 15nm


Figure 5.2: Slip Length as a function of shear rate for various wall separations

5.2.1.2 Density Profile

Figure 5.3 shows the density profiles for various wall separations. It is should be noted that

as the wall separation increases, so does the bin size for parameter averaging. This is because

the number of bins is kept constant at seven to reduce computational costs, and thus needs to

be taken into account when analysing the data. Ideally, the bin sizes would remain the same

for flows through all wall separations with the larger wall separations having more bins. This

allows for better comparison between different wall separations however the computational

cost is increased.

For all wall separations simulated, the density of the middle section remains reasonably

constant. The region closer to the wall has a higher density than the middle section, however

due to differing bin sizes, it is difficult to deduce a conclusion for the difference in densities

of various wall separations.


Figure 5.3: Density Profiles for simulated Couette flow with wall separations ((a) 2.5nm (b)5nm (c) 7.5nm (d) 10nm (e) 12.5nm (f) 15nm


5.2.1.3 Temperature

Again, the temperature distribution across the pipe fluctuates around the desired 120K,

figure 5.4. This confirms that our thermostat is working correctly. No trends can be seen

between flows through the different wall separations.

The average temperature fluctuations depend on the constant used in the Nose-Hoover ther-

mostat and on the size of the time-step. The time-step used was a function of shear rate.

From figure 5.5, it can be seen that with increasing wall separation, the size of the aver-

age temperature fluctuation decreases. For a 2.5nm wall separation flow, the temperature

fluctuation was between 116 and 125K. For a 15nm wall separation flow, the temperature

fluctuation was between 119 and 121K. The decrease in fluctuation range at larger wall sepa-

rations is due to the smaller time-step used for the Nose-Hoover thermostat. The thermostat

maintains the temperature more frequently with smaller time-step.

5.2.1.4 Viscosity

The viscosity of the fluid is determined from the shear stress on the wall divided by the

imposed linear shear rate, equation 2.24. This may not provide an accurate viscosity because

it there is slip at the wall and the velocity profile is not linear. However, it can be used as an

approximation and the results of the viscosities in our simulations are shown in figure 5.6.

Note that the fluid temperature is maintained constant using by applying the Nose-Hoover

thermostat.

For all wall separations, at low wall velocities, the graph suggests the fluid may be Newto-

nian up to a certain shear rate when it becomes non-Newtonian. At this point, the fluid

becomes shear-thinning, that is, the viscosity decreases with increasing shear rate. How-


Figure 5.4: Temperature Profiles for simulated Couette flow with wall separations (a) 2.5nm(b) 5nm (c) 7.5nm (d) 10nm (e) 12.5nm (f) 15nm


Figure 5.5: Average Temperature as a function of Time for simulated Couette flow with wallseparations (a) 2.5nm (b) 5nm (c) 7.5nm (d) 10nm (e) 12.5nm (f) 15nm


Figure 5.6: Viscosity as a function of Shear rate for simulated Couette flow with various wallseparations

ever, the viscosities seem to plateau with increasing shear rate indicating a transition back

to a Newtonian fluid. From our results, there is slightly more evidence of the fluid being

Newtonian at higher shear rates than at lower shear rates and this will be discussed below.

Similar to our discussion in part 4.2.1.4, Thompson and Troian reported that the dimension-

less bulk viscosity of the fluid is constant at about 2.2 over a dimensionless shear rate range

of 0.001 to 1 (Thompson and Troian, 1997). Our viscosity results are slightly higher and do

not remain constant. This may be due to different simulation conditions. Other MD studies

of viscosity show the bulk fluid becomes non-newtonian for �̇�𝜏 ≥ 2 (Loose and Hess, 1989)however all our simulations are below 2. Our results agree with Ashurst and Hoover where

the viscosity decreases with increasing shear rate (Ashurst and Hoover, 1970).

To further compare with previous literature, a graph of the actual viscosity at which the fluid

is estimated to become Newtonian at high shear rates versus diameter, is shown in figure 5.7.

There is a slight curve in figure 5.7 suggesting that it may plateau for larger wall separations.


Figure 5.7: The Newtonian fluid viscosity at high shear rates for various wall separations

This would agree with macroscopic theory where the viscosity remains constant irrespective

of the wall separation. A rough estimate of the viscosity for larger wall separations, after it

has plateaued, is 6x10−4 Pa.s. This is slightly larger than the experimental Argon viscosity

extrapolated to 1x10−5Pa.s (Lemmon and Jacobsen, 2004). The difference may be attributed

to errors in the thermostat model, numerical integration, and two-dimensional simulation

domain.

The shear rate at which the fluid transitions from a Non-Newtonian to Newtonian seems

to decrease with increasing wall separation and is approximated in figure 5.8. The graph

suggests that the transitional shear rate will continue to decrease until it reaches a wall sep-

aration where the transitional shear rate is zero. This would correspond to the macroscopic

theory that the fluid remains Newtonian for all wall separations.


Figure 5.8: The shear rate at which the fluid transitions from non-Newtonian to Newtonian

5.2.2 Mixing Behaviour

This section looks at the mixing as a function of time and the mixing as a function of

Reynolds number for various wall separations. The mixing as a function of dimensionless

time for various fluid velocities is presented in figure 5.9. At 0nm/ps, mixing occurred for

all wall separations. As discussed in the previous chapter, this mixing can be attributed

to molecular diffusion. Due to the significance of diffusion at the nano-scale, a constant

mixing time was used to compare mixing at different wall velocities. Reynolds experiments

compared mixing based on a constant length of pipe however his experiments were on a

macroscopic scale where diffusion is negligible. On a micro and nano-scale, the effect of

diffusion becomes much more significant and thus to compare the degree of mixing requires

a constant mixing time. See figure 3.1 for further discussion on this.

Based on the assumption that the mixing rate cannot increase any higher than the turbulent

mixing rate, except with the addition of active mixing, a bunching of mixing rates at the

higher end suggest turbulent flow. Conversely, based on the assumption that the mixing rate


Figure 5.9: Mixing profiles for simulated Couette flow with wall separations (a) 2.5nm (b)5nm (c) 7.5nm (d) 10nm (e) 12.5nm (f) 15nm


cannot decrease any lower than the diffusive mixing rate, these flows suggest laminar flow.

For smaller wall separations, the majority of the lines are bunched together at higher mixing

rates which indicate turbulent flow, figure 5.9. Whereas for larger wall separations, the

majority of lines are bunched together at the lower end of mixing suggesting laminar flow.

The initial data point for each wall separation is the amount of diffusive mixing after 400ps,

figure 5.10(a). It can be seen that the amount of diffusive mixing is different for different

wall separations. This could be due to a number of reasons. The surface to fluid ratio may

have an effect. The wall thermostat and wall reflection model may also have an effect on

the amount of diffusive mixing. To compare flow regimes between different diameters, the

diffusion is removed, as shown in figure 5.10(b).

At smaller wall separations, there are only two regimes, transitional and turbulent, figure

5.10(b). For example, flow through a wall separation of 2.5nm shows a transitional range

between Re of 0 and 2.5 and a turbulent regime for Re higher than 2.5. Similarly, for

flow through wall separation of 5 and 7.5nm, there seems to be only 2 regimes. A possible

explanation is that the wall separation is so small that laminar flow does not exist. The effect

of the wall or diffusion is too large and thus, the flow is always at least in the transitional

regime. Another explanation could be that there are errors involved in the simulation and

so the laminar regime was not captured.

For wall separations of 10nm and above, there is more evidence of a laminar regime between

Re of 0 and 2. For Re higher than 2, the flow seems to be in the transitional regime. Higher

wall velocities were not possible due to computational limitations and so further simulations

are required to provide evidence of a turbulent regime for these wall separations.

Based on the limited number of data points in figure 5.10(b), a summary of approximate


Figure 5.10: (a) Mixing as a function of Reynolds Number and (b) Mixing minus moleculardiffusion as a function of Reynolds Number, for simulated Couette flow with various wallseparations


transitional Reynolds number and transitional range are plotted in figure 5.11. Figure 5.11(a)

shows that for wall separations smaller than 7.5nm, the flow is always in the transitional

regime since the transitional Re is close to zero.

For wall separations larger than 7.5nm, the transitional Reynolds number increases with

increasing wall separation. To support conventional theory, it is predicted that the increase

of transitional Reynolds number would gradually reach a plateau of around 2000. However,

not enough data support this conclusion. This would agree with the theory that for micro-

channel flows the transition Reynolds number ranges between 200 and 900 and for macro-

channel flows reaches a plateau of around 2000 (Peng et al., 1994a,b; Mohiuddin Mala and

Li, 1999).

The transitional range increases with increasing wall separation, figure 5.11(b). This graph

also suggests that transitional range will reach a plateau with increasing wall separation.

It would support the macroscopic evidence of the transitional range of approximately 1000

(between Re of 2000 and 3000) for all macroscopic wall separations.

Therefore from figure 5.11, there is some evidence that the transitional Reynolds number and

the transitional range increase with increasing wall separation. This agrees with previous

works in microfluidics (Peng et al., 1994a).


Figure 5.11: (a) Transitional Re for various wall separations (b) Transitional Range forVarious wall separations


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