Analysis of flow patterns and interfacebehavior in simulations of immiscible

liquid-liquid two phase-flow inmicrochannels using the conservative

level set method

By

M.A. van Iersel

Master of Science Thesis

Delft University of TechnologyFaculty of Applied Sciences

Dept. of Radiation Science and TechnologySect. Reactor Physics and Nuclear Materials

Supervisor: Dr.ir. M. Rohde, TNW, TU DelftMsc. Z. Liu, TNW, TU Delft

Committee: Dr.ir. M. Rohde, TNW, TU DelftDr.ir. D. Lathouwers, TNW, TU DelftDr.ir. D.A. Vermaas, TNW, TU DelftMsc. Z. Liu, TNW, TU Delft

Delft, June 2019

Abstract

Molybdenum-99 (Mo-99) is crucial for many medical procedures, for example cancer diag-nostics. Currently the majority of Mo-99 is produced by placing targets in a high neutronflux region of a nuclear reactor. After some time, the targets are removed and the Mo-99extracted. An alternative to this current process is a loop, filled with a solution containingU-235 flowing through the core of a reactor. As the solution leaves the high neutron fluxregion of the nuclear reactor, Mo-99 is present due to fission of U-235. The Mo-99 can thenbe extracted from the solution using continuous flow micro-scale unit operations. The ex-traction is based on diffusion of Mo-99 from the aqueous fluid to an organic solution, thetwo fluids are immiscible. The two fluids meet in a micro-channel with a height of 100 µm.There, stable parallel flow sustains a fluid-fluid interface for the length of the channel, whichallows for diffusion of Mo-99. At the end of the channel, separation of the aqueous andorganic phase is necessary. However, the interface is not always stable and leakage can occur.

Experiments focusing on flow regimes, sustaining parallel flow and leakage have beenperformed by Liu. Besides the experiments, modeling is done by Liu with the phase fieldmethod using COMSOL. One of the disadvantages of the phase field method is that it is verycomputationally demanding. A less demanding method is the conservative level set method.This thesis explores whether the conservative level set method is able to capture the dynam-ics of the flow and contact point behavior, while at the same time reducing the computationalcost of performing simulations of the micro-channel. The simulations with the conservativelevel set method are also carried out in COMSOL. The conservative level set method will becompared with both the experiments and the simulation results with the phase field method.

A benchmark of a spreading droplet is performed to test alterations to the conservativelevel set method. The phase field method and conservative level set method are both ableto quantitative capture the behavior. A comparison of the simulation run-times is, however,inconclusive. Simulations of the microchannels showed the phase field method outperformingthe conservative level set method. The conservative level set method is unable to qualita-tively reproduce the flow patterns found experimentally. Also, the phase field method hasconsiderably lower run-times than the conservative level set method, during simulations ofthe microchannels. It is concluded that for simulations of the microchannels, the phase fieldmethod outperforms the conservative level set method both in terms of results and run-times.

Contents

1 Introduction 31.1 Molybdenum-99 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Continuous Flow Micro-Scale Unit Operation . . . . . . . . . . . . . . . . . 51.3 Flow types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Theory 102.1 Pressure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Contact Angle Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Derivation of Dimensionless Navier-Stokes . . . . . . . . . . . . . . . 132.4 Conservative Level set method . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Boundary Conditions at the Contact Line: Slip . . . . . . . . . . . . 172.4.2 Boundary Conditions at the Contact Line: DIM . . . . . . . . . . . . 192.4.3 Implement DIM-model CLS Method . . . . . . . . . . . . . . . . . . 19

2.5 Force at the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Full Model: Conservative Level Set Method . . . . . . . . . . . . . . . . . . 21

2.6.1 Slip Condition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.2 Diffusive Interface Method . . . . . . . . . . . . . . . . . . . . . . . . 222.6.3 Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Phase Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7.1 Boundary Condition Phase Field Method . . . . . . . . . . . . . . . . 26

3 Model Equations 273.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Implementation in COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Angle at the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.2 Flipping Boundary Condition . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Adaptive Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1

4 Bird Benchmark: Spreading Droplet 354.1 Benchmark Description: Spreading Droplet . . . . . . . . . . . . . . . . . . . 354.2 Implementation: Spreading Droplet . . . . . . . . . . . . . . . . . . . . . . . 384.3 Results: Spreading Droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 Comparison with Phase Field Simulations . . . . . . . . . . . . . . . . . . . 44

5 Channel Experiment: Diffusive Interface Method 495.1 Benchmark Description: Micro-Channels . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Description of Experiments . . . . . . . . . . . . . . . . . . . . . . . 495.2 Model Description: Micro-channel Simulations . . . . . . . . . . . . . . . . . 50

5.2.1 Geometry and Materials . . . . . . . . . . . . . . . . . . . . . . . . . 505.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.3 Mesh: Diffuse Interface Method . . . . . . . . . . . . . . . . . . . . . 53

5.3 Results: Diffusive Interface Method . . . . . . . . . . . . . . . . . . . . . . . 565.3.1 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.2 Flow Regime Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.3 Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4 Comparison with Phase Field Method . . . . . . . . . . . . . . . . . . . . . . 60

6 Channel Experiment: Slip Boundary Condition 626.1 Implementation: Slip Boundary Condition . . . . . . . . . . . . . . . . . . . 62

6.1.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2 Results: Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2.1 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2.2 Flow Regime Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2.3 Scaling and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.3 Comparison with Phase Field Method . . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusions and Recommendations 717.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2

Chapter 1

Introduction

1.1 Molybdenum-99 Production

In the field of nuclear medicine, radioactive isotopes are utilized in order to diagnose andtreat diseases. Several diagnostic methods like positron emission topography (PET) andsingle photon emission computed topography (SPECT) require radioactive isotopes. Thesemethods are often used to detect behavior inside the body, examples are the allocationof glucose to detect tumors or tracking how blood flows through the heart. 80% of theseprocedures use Technetium-99m (Tc-99m) as the radioactive isotope. Tc-99m has severaladvantages including a convenient half-life and relatively low radiation dosage [2]. Moreover,it has characteristics that allow it to be chemically attached to molecules that have an affinityfor different parts of the body. Tc-99m is the radioactive decay product of Mo-99, the decaychain of Mo-99 is shown in figure 1.1. Almost all of the Tc-99m that is used in hospitals isproduced in this manner. Mo-99 has a half-life of 66 hours, and 88% decays into Tc-99m. AsMo-99 has a half-life time of 66 hours, a longer production process after the Mo-99 leaves thehigh neutron flux region will result in less Mo-99 reaching the hospital. Methods to shortenproduction time will directly result in more of the desired product.

Figure 1.1: Decay chain of Mo-99. Most of the Mo-99 decays into Tc-99m by β-decay.Tc-99m is an istope frequently used for medical procedures. [2]

3

Most Mo-99 is produced is by placing targets into a high neutron flux region inside a nu-clear reactor. Targets typically contain U-235 in an uranium-aluminum alloy contained in analuminum housing [2]. After 5-7 days the amount of Mo-99 in the targets has reached 70-80%of the theoretical maximum and the targets are removed. Afterwards, the irradiated targetsare left for a day to ensure the short half-life products are gone. Then the molybdenum-99needs to be extracted and purified for commercial sale. The targets are dissolved in sodiumhydroxide in a process called alkaline dissolution. Subsequently the products are separatedthrough various processes and shipped. The price of the molybdenum is set to the amountthat is still there 6 days after target processing, indicated as end of production (EOP) infigure 1.2. The activity after six days is reffered to as the 6-day curie, this amounts to about20-25% of the peak activity. [2]

Figure 1.2: Activity of Mo-99 over the production process. The activity increases as thetargets are placed near the reactor. The moment the targets are removed is referred to as theend of the bombartment (EOB). EOP stands for the end of production. The 6-day curie isindicated after the day the targets are set aside. [2]

One proposed method to improve the production of molybdenum-99 is to create a loopin the high neutron flux region. This loop is a tube filled with a solution containing uraniumsalts. As the uranium-235 undergoes fission the molybdenum-99 levels in the solution riseup to an equilibrium value. The solution is pumped through the loop at a constant rate.When the solution leaves the loop, the molybdenum-99 has to be extracted. Several studieshave been done with regard to the design and feasibility of such a loop [3] [4]. This thesiswill focus on the extraction of the isotopes using micro-scale unit operation.

4

Figure 1.3: Schematic representation of continues extraction loop. The loop will be placedinside a reactor after which the produced molybdenum-99 can be extracted contiously. [3]

1.2 Continuous Flow Micro-Scale Unit Operation

Micro-scale chemical processes are considered a promising alternatives to large scale chem-ical procedures. Micro-scale unit operations (MUOs) can be used to perform many stepsin these processes like mixing, extraction, reactions, ect. These processes are performes inmicrochannels. A typical height of such a channel is 100 µm. MUOs can be used in seriesor parallel in order to reach the same results as larger scale operations. The joining of sev-eral MUO can be done with continuous flow chemical processing (CFCP). Figure 1.4 showsseveral MUOs and a example of a system this could be applied to. The advantages of usingMUOs instead of large scale operations are varied and depend heavily on the goal of theoperation. For instance, MUOs CFCP are proposed in order to measure amphetamine-typedrugs in urine [5]. The major advantage is the gained mobility of the device and shorteranalysis time (20 minutes versus several hours). This allows instant, on-site analysis.

This thesis will focus on employing MUOs to the problem of medical isotope extraction,e.g. molybdenum-99. MUOs could shorten the extraction time of isotopes significantly. Sincethe isotopes have a relatively short half-life, a shorter extraction time directly results in moreof the desired product. This could be achieved by parallel flow microfluidic solvent extraction.The extraction process is based on an aqueous and an organic fluid that flow parallel to eachother but are immiscible, while the desired product diffuses between them. The interfacebetween the two fluids should be stable, however, this is not always the case [7]. Besidesstable parallel flow in the micro-channel, complete phase separation at the end of the micro-channel is also required. Several methods to stabilize parallel flow can be found in literature.These methods range from membranes [8], coatings [9], geometrical modifications [10] [11]and pressure models [12]. However, coatings and membranes are ill suited for this specificproblem, as radiation destroys the the coating/membrane over time. In summary, for thismethod to work in practice, stable parallel flow with complete phase separation at the endof the channel is required.

5

Figure 1.4: Several micro-scale unit operations (MUOs) are illustrated in the section on theleft. An example of a possible continuous flow chemical process (CFCP) is shown on theright-hand side. [6]

1.3 Flow types

There are several types of flow that can occur in immiscible liquid-liquid multi-phase flow.Two main groups can be identified, dispersed and non-dispersed flow [13]. These two flowtypes are illustrated in figure 1.5. Slug flow is dispersed flow where droplets travel throughthe channel, alternating between the fluids. Because of the hydrophobic behavior of theorganic phase with respect to the glass surface, the interface curves towards the aqueousphase. The size and speed of the droplets depends on the inlet velocities of the fluids andthe properties of the fluids. Non-dispersed flow has a continuous interface, when both fluidsare in contact with the channel wall it is called parallel flow. Stability of the interface isa necessity to maintain parallel flow. Besides the dynamics of the flow, movement of theinterface is very important. The point where the interface meets the boundary is called acontact point. Both fluids are present here as well as the boundary. The slug flow in figure1.5 contains several contact points.

6

Figure 1.5: Two most common flow types exhibited in a microchannel. Part A shows slugflow while part B shows parallel flow [14].

1.4 Previous Work

Experiments in microchannels have been performed by Liu [1]. One of the performed ex-periments focuses on the flow type observed in the channel. The type of flow changes fromslug flow to parallel flow as the inlet velocity is increased. The results from this experimentare shown in figure 1.6. The asterisk’s indicate slug flow, the circles a transition betweenthe two flow types and the pluses parallel flow. The experimental results are reproducedwith the phase field method by Liu. The phase field method is an often used method tosimulate two phase flow. These simulations are carried out in COMSOL, a finite elementpackage. However, because the phase field method is fairly advanced, it is computationallyvery demanding. This leads to long simulation run-times which reduces the amount of sim-ulations that can be carried out. The simulations of the flow in the channel for instance,are performed in two dimensions. Three dimensional simulations are not feasible, due to thehigh computational cost.

The conservative level set method is similar to the phase field method. The phase fieldmethod solves an extra energy conservation equation with respect to the conservative levelset method. Despite this simplification, the conservative level set method has been able toreproduce experimental results of wetting senerios [15]. Because the phase field method solvesan extra conservation equation, the conservative level set method is less computationallydemanding. This thesis will explore the possibility of simulating the flow dynamics andcontact point behavior in the microchannel with the conservative level set method. The

7

results produced with the conservative level set method will be compared with the phasefield method and the experimental results.

Figure 1.6: Graph showing different flow types based on flow rate. On the x-axis the flow rateof the organic fluid is shown while the y-axis shows the flow rate of the aqueous fluid. Theflow rate is given in µL per minute. The asterisk’s indicate slug flow, the circles a transitionbetween the two flow types and the pluses parallel flow. [1]

1.5 Research Goals

Three research questions are formulated. The research goals focus on the performance ofthe conservative level set method compared the the experimental results and the phase fieldmethod.

(I) Is the conservative level set method able to capture the different flow regimes in micro-channels, and how do the numerical parameters influence the results?

(II) Is the conservative level set method able to quantitative and qualitatively describecontact point behavior in the microchannel, and how do the numerical parametersinfluence the results?

8

(III) How do the results from the conservative level set method compare to the results fromthe phase field method, and is the conservative level set method able to reduce thecomputational cost with respect to the phase field method?

In order to properly answer the stated research questions, a good understanding is re-quired of how physical properties of the fluids influences the flow patterns in the microchan-nels. Moreover, the effect of the numerical parameters employed in the conservative level setmethod will be assessed in detail.

1.6 Outlook

In order to answer the research question described in section 1.5, the thesis will follow a clearstructure. Chapter 2 explores the theory of physics in microchannels as well as numericalmethods to describe this physics. These methods are the conservative level set method andthe phase field method. Three sets of equations describing the system are presented atthe end of chapter 2. Chapter 3 starts by discussing the discretization of the equations.Furthermore, implementation of the model in COMSOL is discussed in detail. In chapter 4,a benchmark is performed against a case of a spreading droplet, as described in literature.Chapter 5 and chapter 6 describe the results of simulation with the conservative level setmethod in the microchannels. The results are compared to both the experiments as wellas the simulation results produced with the phase field method. Chapter 7 contains theconclusions from this thesis and recommendations for future work.

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Chapter 2

Theory

This chapter explores the the physics in the microchannels and methods to numericallydescribe it. First, a simple pressure model that has been used in previous work will be dis-cussed followed by an analysis of contact angles at the contact line. Next, the Navier-Stokesequation and the continuity equation will be assessed and derived in their non-dimensionalform. Then, the conservative level set method is introduced as well as two different boundaryconditions that can be used to describe the movement of the interface at the boundary. Inthe next section the full model of equations is summarized and the physical and numericalparameters are discussed. Finally, a short theoretical description of the phase field methodis given.

2.1 Pressure Model

A model that is often used in literature balances the interfacial pressure in order to predictthe type of flow that will occur [7]. This model was proposed by Aota et al and is fairlyeasy to implement [12]. The model balances the hydrodynamic pressure with the Laplacepressure, this balance is illustrated in figure 2.1. If the hydrodynamic pressure is within acertain range, stable parallel flow should occur. Equation 2.1 is derived from the Young-Laplace equation. ∆PLaplace is caused by the interface bending towards the aqueous phase,due to the hydrophilicity of the glass surface. Here θ is the contact angle, d is the channelheight and σ is the interfacial tension.

∆PLaplace =2σsin(θ − 90)

d(2.1)

The Laplace pressure results in a force that needs to be balanced in order to have parallelflow. ∆Pflow can be used as this balancing force. The upper and lower limit for ∆Pflow can becalculated by looking at θadvancing and θreceding. The static contact angle for the organic fluidis restricted between the receding and advancing values, and thus a condition for ∆Pflow canbe determined. This is shown in equation 2.2. By evaluating this equation, the conditionsrequired to achieve parallel flow are found. If laminar flow is assumed and the chip hasno major geometric differences between the two channels, ∆PF will mainly depend on the

10

Figure 2.1: Illustrates pressure balance at the interface, ∆PL is the Laplace pressure, while∆PF is the pressure difference due to flow. The organic phase leads toward the aqueousdue to the hydrophilicity of the glass. θ is the contact angle between the fluids. θad is theadvancing contact angle, θre is the receding contact angle [6].

difference in the viscosity and inlet velocity. The theory predicts a higher and lower limitfor the inlet velocity.

2σsin(θre − 90)

d< ∆PF <

2σsin(θad − 90)

d(2.2)

This theory works well for a situation where the interface is fixed at the boundary, andthe bending towards the aquaous phase is observed in experiments [16]. However, this canonly be achieved by coatings and membranes [17] that are unavailable due to the specificnature of this problem. Such layers are destroyed by radiation. Consequently, this theory isa simplification that does not hold up for this real world application [13].

2.2 Contact Angle Theory

As mentioned in the previous paragraph the interface is not fixed at a specific position. Thismeans the position of the interface can move, such a problem is referred to as a movingcontact line problem. In three dimensions this is a line, in two dimentions it is a point.The position where the interface meets the wall is called the contact point. The movementof the contact point will be very important in order to understand the behavior in the mi-crochannel. The contact angle of the two phases with the solid surface of the microchannelis an important parameter to describe this behavior. The model in the previous section usesa static contact angle with an advancing and receding limit. In dynamic wetting scenariosthis description turns out to be insufficient [18]. Figure 2.2 illustrates there are three lengthscales with different contact angles. The largest scale contact angle is referred to as theapparent contact angle, θapp. At macro scale, the dominant forces are the surface tensionand the gravitational forces. These forces govern the shape of the droplet.

The second length scale is the micro-scale scale, this region is illustrated in the firstzoomed-in region indicated by (a) in figure 2.2. This region is characterized by a differentforce balance compared to the macroscopic level. The influence of gravity at the micro-scale versus viscous and surface tension forces is much less. This is due to the gravitationalforces scaling with volume while surface tension and viscosity scale with area. This new

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Figure 2.2: A schematic breakdown of the different contact angles at different scales. Thelargest scale is a macro setting where the angle θapp is the apperent angle between the dropletand the surface. The micro-scale is represented by the zooming done in (a). The contactangle θe is the microscopic contact angle. The situation at the molecular level is representedby area (b) [18].

force balance results in a smaller contact angle, θe. This effect is called the viscous bendingphenomenon. On the molecular level, the solid surface is obviously not perfectly flat anymoreand the fluid molecules can not be described by continuum mechanics anymore. Models existthat describe the individual molecules at this level in order to more accuracy describe fluiddynamics [19], e.g. contact point behavior. However, These methods fall outside the scopeof this thesis.

2.3 Navier-Stokes Equations

In order to model the physics encountered during a two-phase flow moving contact line prob-lem, the momentum equations needs to be solved for both fluids as well as the movement ofthe interface. Section 2.4 will further explore numerical methods to deal with the movementof the interface. Here, Navier-Stokes equation will be assessed, combined with the continuityequation.

∇ · ~u = 0 (2.3)

Equation 2.3 is the continuity equation for in-compressible Newtonian fluids. ~u is thevelocity field. The continuity equation conserves mass. If a control volume Ω is considered,all changes of mass that can occur to the mass of Ω need to be considered. In the liquid-liquidparallel flow problem with a moving interface, no mass can be added or lost by a source/sink.

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Furthermore, because the flow is in-compressible ρ will be constant which means the onlyway to change the mass of Ω is by convection, which is expressed in equation 2.3.

ρ(∂~u∂t

+ (~u · ∇)~u)

= −∇p+∇ ·(µ(∇~u+ (∇~u)T

))+ ρg~eg + ~Fsa (2.4)

Equation 2.4 is the Navier-Stokes equation with surface tension term, p is the pressure,Fsa is surface tension force, µ is the dynamic viscosity and ρ the density. The Navier-Stokes equation conserves momentum, similarly to the continuity equation for mass. Theways momentum can enter or leave the control volume Ω are more complex and numerous.Additionally, momentum can dissipate, for instance by diffusion. The left hand side containsa time dependent term and a convection term. The right hand side contains a pressure term,diffusion term, gravity term and a surface force term.

2.3.1 Derivation of Dimensionless Navier-Stokes

In practice, the dimensionless version is always used [20]. Non-dimensionalizing complexequations has several advantages. For instance, with the use of non-dimensional groups,the number of independent variables can be reduced [21]. Furthermore, the values of di-mensionless groups reveal a lot of information about the behavior in the channel. Section2.1 explored the relevant dimensionless groups in detail. When a scale is introduced for alldependent variables in equation 2.4, equation 2.5 is the result.

ρ[u]

[t]

∂~u

∂t+ρ[u]2

[~x](~u · ∇)~u = − [p]

[~x]∇p+

µ[u]

[~x]2∇ ·(µ′(∇~u+ (∇~u)T

))+ ρg~eg +

σ

[~x]2~Fsa (2.5)

An appropriate scale needs to be found for all dependent variables. First, [t] = [u]/[~x] isan obvious choice as a scale for [t]. This simplifies the left hand side of the equation and bydividing the left hand side becomes completely non-dimensional. This is done respectivelyin equation 2.6 and 2.7.

ρ[u]2

[~x]

(∂~u∂t

+ (~u · ∇)~u)

= − [p]

[~x]∇p+

µ[u]

[~x]2∇ ·(µ′(∇~u+ (∇~u)T

))+ ρg~eg +

σ

[~x]2~Fsa (2.6)

∂~u

∂t+ (~u · ∇)~u = − [p]

ρ[u]2∇p+

µ

ρ[~x][u]∇ ·(µ′(∇~u+ (∇~u)T

))+g[~x]

[u]2~eg +

σ

ρ[u]2[~x]~Fsa (2.7)

Several dimensionless numbers can be identified in the resulting expression [21]. Thescalar in front of the viscous term is one divided by the Reynolds number. The Reynoldsnumber is expressed in equation 2.8.

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Re =Inertia Forces

V iscous Forces=ρud

µ(2.8)

The Reynolds number is the balance between inertia and viscous forces. A low Reynoldsnumber indicates viscous dominated flow, i.e. laminar flow, while a large Reynolds numberpredicts inertia dominated flow, i.e. turbulent flow. It is important to consider the natureof these forces to properly assess them. Inertia forces are proportional to the volume of thechannel while the viscous forces are proportional to the area of the channel. In the caseof a microchannel the area to volume ratio is very large. The viscous forces are thus moreprominent on the micro scale versus the macro scale. This generally causes the Reynoldsnumbers in microchannels to be low, and flow laminar.

Then, the scalar in front of the gravitational term is one divided by the squared ofthe Froude number. The Froude number is expressed by equation 2.9. This dimensionlessnumber represents the relation between the inertia forces and the gravitational forces. Both ofthese forces are volume forces. The Froude number is typically fairly large in microchannels.This means gravity plays a very small role in the microchannels.

Fr =Inertia Forces

Gravity Forces=u2

gh(2.9)

Finally, the scalar in front of the surface tension force term is one divided by Reynoldsnumber times the Capillary number. The Capillary number is defined as the ratio betweenthe viscous forces and the surface tension force, expressed in equation 2.10. These forcesoften compete, e.g an oil droplet in water. Depending on the density of the oil, the dropletwill either rise or fall due to buoyancy effects. This will cause friction and thus the viscousforces attempt to deform the droplet, while the surface tension force will minimize the surfacearea. This will, for instance, affect the droplet’s ability to move through porous media. Inorder to do this, the droplet must be able to deform enough to pass through, which is resistedby the surface tension. A droplet with a very low Capillary number might not be able topass through a specific medium, while a droplet with a higher Capillary number would passthrough.

Ca =V iscous Forces

Surface Tension Forces=µu

σ(2.10)

It is worth noting ρ′ and µ′ are the dimensionless density and viscosity, while ~u is thedimensionless velocity field. ρ′ and µ′ are constants for each fluid. Lastly, in order to obtainequation 2.11 [p] = ρ[u]2 is chosen.

∂~u

∂t+ (~u · ∇)~u = −∇p

ρ′+

1

ρ′Re∇ ·(µ′(∇~u+ (∇~u)T

))+

1

Fr2~eg +

1

ρ′ReCa~Fsa (2.11)

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The next section will explore numerical methods to deal with the movement of the inter-face in order to obtain a full model.

2.4 Conservative Level set method

In order to fully describe the problem the position of the interface has to be numericallysolved as it is moving, along with numerically solving the Navier-Stokes equation for bothfluids. Such a problem is called a moving interface problem, and there are several numeri-cal methods to deal with such a problem. To limit the computational time required to dosimulations, only continuum methods are considered. Worner carried out a comprehensivereview of such models [22]. These methods can be divided in two groups; methods withsharp interface thickness(∼ nm), and methods with finite interface thickness(∼ µm). Thelatter group is discussed in this thesis. This section focuses on the conservative level setmethod as the method will be applied. First, the general idea and theoretical framework isdiscussed. Then, two different boundary conditions at the contact line are explored. Theforce at the interface is discussed next, after which the full model is summarized and morecontext is provided with regard to the numerical parameters.

To model the moving interface between the aqueous and organic fluids, the conservativelevel set method will be used [23]. The idea of the level set method is to not define theinterface as a surface or line, depending on the dimensionality of the problem, but as afunction throughout the domain [24]. This function then defines the interface implicitly.The way the function is constructed can best be explained with the use of an example, forinstance the problem of parallel flow in microchannels. The function φ is a distance functionto the interface, set to negative values in the aqueous fluid and to positive values in theorganic fluid. φ = 0 is the position of the interface. The difference in properties between thetwo phases can be expressed with a heavy-side function as shown in equations 2.12. In orderto improve numerical robustness, in practice a smeared out heavy-side function is often used,denoted by Hsm.

H(φ(~x)) = 0, φ < 0

H(φ(~x)) = 1, φ > 0(2.12)

Hsm(φ(~x)) =

0, φ < −ε12

+ φ2ε

+ 12π

sin(πφε

), −ε ≤ φ ≤ ε

1, φ > ε

Φ(~x) = Hsm(φ(~x)) (2.13)

15

Here, ε corresponds to half of the interface thickness. Hsm is also a level-set function,as is expressed in Equation 2.13. The interface has a finite thickness now and the middleof the interface is located at Φ = 1/2. Note that this method does require the position ofthe interface to be know at the beginning of the simulation. Defining the interface as aglobal function has several advantages, the computation of the several complex properties ofthe interface becomes much easier. For instance, the viscosity and density jump across theinterface is smoothed out over the thickness of the interface ε. Equation 2.14 and 2.15, givethe expression of the density and viscosity in the interface region. The smoothing of thesejumps in properties improve numerical robustness.

ρ = ρ1 + (ρ2 − ρ1) · Φ (2.14)

µ = µ1 + (µ2 − µ1) · Φ (2.15)

Additionally, calculation of the normal to the interface is quite simple. The curve isdefined implicitly by the value Φ = 1/2, so when moving along the curve the value of Φ doesnot change. Here, s is the curve/surface function, depending on the dimensionality of theproblem. The exact numerical formulation of this function is not required for the followingarguments. When Φs is assessed, the partial derivative of Φ with respect to s, the onlyconclusion can be that it has to be zero as Φ along s is constant. However, Φs can also berewritten as in equation 2.16.

Φs = Φxxs + Φyys =⟨∇Φ, ~T

⟩= 0 (2.16)

When ∇Φ is normalized, the first expression in equation 2.17 is obtained. This impliesorthogonality between the two vector which in turn implies the expression for ~N . Withthis expression the normal to the interface can easily be determined throughout the domain.Although this derivation was done for a two dimensional example, the three dimensionalcase has a very similar proof and the expression still holds. The direction of the normal,inwards and outwards, can be manipulated with a minus sign.⟨ ∇Φ

|∇Φ|, ~T⟩

= 0 =⇒ ∇Φ

|∇Φ|⊥ ~T =⇒ ~N =

∇Φ

|∇Φ|(2.17)

Another important property of the interface is the curvature, κ. The curvature is neces-sary to properly model the surface tension and movement of the interface. Just as before,the second derivative of Φ with respect to the curve s is zero, as Φ = 1/2 on s. Similarly tothe derivation of n, Φss can be written as in equation 2.18. Next, the equality in equation2.16 is used in the second step after which the derivative is brought inside of the brackets.

Φss =d

ds(Φxxs + Φyys) =

d

ds

⟨∇Φ, ~T

⟩=⟨ dds∇Φ, ~T

⟩+⟨∇Φ,

d

ds~T⟩

= 0 (2.18)

16

The curvature κ can be expressed as κ ~N = dds~T . An expression for the normal is already

obtained, which leads to the following step in the derivation.

−⟨ dds∇Φ, ~T

⟩=⟨∇Φ, κ ~N

⟩= κ

⟨∇Φ,

∇Φ

|∇Φ|

⟩= κ|∇Φ| (2.19)

κ = −∇ ·(∇Φ

|∇Φ|

)(2.20)

The movement of the interface now depends on the velocity field obtained from theNavier-Stokes equation. This relation is expressed in equation 2.21. Because the velocityfield is divergence free, the equality in equation 2.21 holds.

∂Φ

∂t+ ~u · ∇Φ =

∂Φ

∂t+∇ · (Φ~u) = 0 (2.21)

Equation 2.21 propagates the interface based on the calculated velocity field. However,in practice this equation will not produce an acceptable solution. This has to do with how ittranslates perturbations of the interface. If a delta peak is placed anywhere on the interface,the delta peak will travel with the same velocity as the rest of the interface. The interface ata later time will still have a delta peak perturbation. Numerical calculations will inevitablyintroduce perturbations. When these perturbations are allowed to built up, the interfacewill become increasingly distorted. In practice, a term is added that will artificially diffuseperturbations on the interface. Olsson and Kress [20] proposed a term for the artificialdiffusion, shown on the right hand side of equation 2.22.

∂Φ

∂t+ ~u · ∇Φ = γ1∇ ·

(ε∇Φ− Φ(1− Φ)

∇Φ

|∇Φ|

)(2.22)

ε still represents the thickness of the interface and γ1 is the amount of re-initialization orstabilization that will be done. γ1 needs to be tuned to a specific problem. If γ1 is too low,numerical robustness tends to decrease and the interface thickness is no longer conserved.However, when γ1 is too high the interface moves incorrectly. Typically, the highest velocityobserved in the problem is a good initial value for γ1. Numerically, solving equation 2.22consists of two steps. First, equation 2.21 is solved to get an initial profile for Φ. This initialvalue is used as input for equation 2.23, which is solved until steady state. In section 3.1this equation will be discussed in greater detail.

∂Φ

∂t= γ1

[∇ ·(ε∇Φ− Φ(1− Φ)

∇Φ

|∇Φ|

)](2.23)

2.4.1 Boundary Conditions at the Contact Line: Slip

The method described in section 2.4 fully covers multiphase flow problems without contactlines. For these types of problems the method works well, e.g. rising/falling bubble [15].In the case of the microchannels, contact lines are encountered as introduced in section 2.2.

17

When contact lines are involved, extra care needs to be taken with respect to the boundaryconditions at the contact line. The most common boundary condition used in computationalfluid dynamics is the no-slip boundary condition, see equation 2.24. This boundary conditionbasically states that the velocity at the wall is zero. The no-slip condition works very well forone phase flow but for a multi-phase flow situation a singularity is introduces at the contactline. This singularity is due to the fact that interface at the boundary is unable to move. Itis obvious this is not the case in reality.

u|boundary = 0 (2.24)

u|boundary = α∂u

∂n|boundary (2.25)

One way to deal with the issue of the interface being unable to move is introducing aslip condition at the boundary. This condition allows the fluid near the wall to have somevelocity that is controlled by a slip length parameter. The most basic slip boundary conditionis given by equation 2.25. Here α is the parameter that controls the amount of slipping nearthe boundary. n is the normal direction of the boundary. The parameter α mainly dependson the material and the roughness of that material [25]. This type of slip condition would bedifficult to implement as the interface velocity will not be constant, especially at the inlet andoutlet. Another type of slip length boundary condition is a condition where the slip velocityat the boundary is extrapolated based on the near wall velocity. Figure 2.3 illustrates howsuch a boundary condition would work. β is also called the slip length, but the slip is nowbased on the near wall velocity in stead of a predefined slip velocity.

Figure 2.3: Schemetic illustration of slip boundray condition implemented in COMSOL.(a) Shows the equilibribum angle between the organic and aquaous fluid. (b) Shows theextrapolated slip velocity at the boundary based on the slip lenght β [26].

σ(~n− ~nintcos(θe))δ −µ

β~u = Fst (2.26)

18

The way the interface moves as a result of the value of β and the wetting angle is expressedin equation 2.26. There are two driving forces that move the interface. If the angle betweenthe interface and the boundary is not equal to the microscopic contact angle, the interface willbe pushed toward the equilibrium angle. Then, the slip velocity that is assigned at the wallis based on the dynamic viscoity µ and the slip lenght β. The slip boundary condition hasseveral disadvatages, it has been reported to have diffulties accurately describing behaviorin capillary driven flow [27]. Moreover, the slip velocity at the boundary will be based onthe near wall velocity.

2.4.2 Boundary Conditions at the Contact Line: DIM

Another option is a diffusive interface method(DIM). This method still imposes a no-slipboundary condition but assigns a diffusion velocity to the interface at the boundary. Basicallythe contact line area is treated separately in a small boundary/interface area. If the anglebetween the interface and the boundary is not conform the equilibrium value, the contactline moves, by way of diffusion, until the equilibrium value is reached. The equilibrium valuewill depend on the size of the boundary/interface area, as different force balances prevail atdifferent scales [28]. In case of the microchannel, the microscopic contact angle is regardedas the equilibrium value. The calculation of this diffusion velocity involves solving an extradifferential equation. This can be an equation based on numerical and physical parameters.Different computational methods use different formulations. The most common example isthe Cahn-Hilliard equation coupled with the Navier-Stokes equation. This formulation isused in the phase field method. The Cahn-Hilliard equation calculates the movement ofthe interface based on the chemical potential. Chapter 2.7 expands on this topic. Thischemical potential is calculated by solving a conservation equation for energy, which makesit a computationally demanding method.

2.4.3 Implement DIM-model CLS Method

The conservative level set method as presented can be adjusted to implement a boundarycondition to model the movement of the contact line. A slip boundary condition is eas-ily implemented. However, implementing a diffusive interface boundary condition is moreinvolved. Zahedi and Kreiss suggested an alteration to the original conservative level setmethod to include contact line dynamics [27]. The approach is similar to the approachtaken in the Cahn-Hilliard/Navier-Stokes formulation. The interface is treated as a separateboundary-interface area approximately of size γ2 ∗ γ2 in which the contact line is moved bydiffusion and the interface reconstructed, illustrated in figure 2.4.

γ2 should be chosen such that γ2 << L, where L is the typical length for the system.However, the angle at the boundary is determined by imposing a normal vector for theinterface at the boundary, not by solving an energy equation. With ~nα at the boundary theangle will be αs at the boundary, which is the microscopic contact angle. In the remainder ofthe domain the normal is defined by the gradient of φ, see equation 2.17. The normal vector

19

Figure 2.4: Boundary Region in which the angle αs is imposed. αs is the micrsoscopic contactangle. The size of the boundary region is proportional to the numerical parameter γ2 [27].

field in the domain must be connected to the normal imposed at the wall. This results inthe regularized vector field n, which is calculated with equation 2.27.

n−∇ · (γ22∇n) =

∇φ|∇φ|

n|contact point = nαs (2.27)

The regularized vector field n is then used to reconstruct the interface by solving equation2.28. This equation replaces the re-initialization equation in the original formulation of theconservative level set method. ~n is the regularized vector field and ~t is the tangent to thatvector field. εn is the diffusion in the normal direction while εt is the diffusion in the tangentialdirection. The diffusion in tangential direction is very important for the contact line region,as it enables the interface to converge towards the prescribed angle with the boundary.

∂Φ

∂t= γ1

[∇(εn(∇φ · ~n ) ~n ) +∇(εt(∇φ · ~t ) ~t )−∇ · (φ(1− φ)~n)

](2.28)

2.5 Force at the Interface

Surface tension force is given by equation 2.29. The physical thickness of the interface is ofthe order of nm, which is much smaller than any feasible mesh element size. This is whycontinuum methods either assume the interface is sharp, or, assign a finite thickness to modelthe interface. A sharp interface means the surface tension is modeled as a surface force. Thesubscript sa stands for surface area. However in a simulation based on the conservative level

20

set method, the thickness of the interface will be finite and the size of the interface will belimited by grid size and numerical robustness. The surface tension force will need to besmeared out over the thickenss of the interface. In order to deal with this issue, the surfacetension is modeled as a three dimensional effect across the volume of the interface [29].

~Fsa = σκ(~xI)n(~xI) (2.29)

The combined force acting on the volume is equal to the combined force on the surface,it is only spread out across the finite width of the interface. If the thickness of the interfacegoes to zero, the two expressions become equal. It is important to note that the thicknessof the interface should be kept constant. The thickness of the interface is a delicate balance.If the interface is too wide, the approximation fails. If it is too thin however, numericalcomplications could occur in the calculation of the second derivative of Φ.

~Fsv(~x) = σ

(−∇ · ∇Φ

|∇Φ|

)∇Φ (2.30)

2.6 Full Model: Conservative Level Set Method

The full set of equations that will be solved using the formulation of the conservative levelset method are summarized in equation 2.31 - 2.40. The model using the slip boundarycondition is discussed in subsection 2.6.1 and the diffusive interface method in subsection2.6.2. Equation 2.31 - 2.34 give the slip condition model while 2.36 - 2.40 constitute thealtered formulation of the conservative level set method.

2.6.1 Slip Condition Model

The model with the slip boundary condition is summarized in equation 2.31 - 2.34. Thisincludes the continuity equation, the dimensionless Navier-Stokes equation, the advectionequation and the stabilization equation. The numerical parameters are very important inthe implementation of the method and as such they are discussed below.

∇ · ~u = 0 (2.31)

∂~u

∂t+ (~u · ∇)~u = −∇p

ρ′+

1

ρ′Re∇ ·(µ′(∇~u+ (∇~u)T

))+

1

Fr2~eg +

1

ρ′ReCa~Fsa (2.32)

∂Φ

∂t+∇ · (Φ~u) = 0 (2.33)

∂Φ

∂t= γ1

[∇ ·(ε∇Φ− Φ(1− Φ)

∇Φ

|∇Φ|

)](2.34)

21

ε controls the interface thickness which defines the width over which Φ changes, i.e. theinterface between the two fluids. The difference in properties are then smeared over theinterface, which improves numerical robustness. However, the smearing of the propertiesis not physical which presents a trade-off. A larger ε will reduce computational time andimproves the numerical robustness. On the other hand if ε becomes too large, the simulationwill not be an accurate description of the physics involved. In reality, the size of the interfaceis in the order of nm but is modeled as an interface in the order of µm. At some point thissimplification of reality does not hold anymore. The width of the interface should be muchsmaller than the characteristic size of the system. The Cahn number is a dimensionlessnumber that expresses the ratio between ε and the characteristic length of the system. Thisis expressed in equation 2.35.

Cn =ε

D(2.35)

γ1 is the parameter that sets the amount of stabilization done during the re-initializationstep. If γ1 is too small, the simulation might not converge. If γ1 is too large however, theinterface moves in a nonphysical way. An appropriate value for γ1 is the maximum velocityin the system.

β is the slip length and is part of the modeling of the wetting boundary condition. Theslip length defines how much the contact line is able to move. Introducing a slip velocityat the boundary is one way to deal with this issue. β determines how close the velocityat the boundary will be to the near wall velocity. COMSOL reference manual suggests anappropriate value for β is proportional to the mesh element size h.

2.6.2 Diffusive Interface Method

The diffusive interface method is summarized in equation 2.36 - 2.40. The continuity equa-tions and the dimensionless navier-stokes equation are obviously unchanged. The level setequations are changed, the altered normal vector field is calculated with equation 2.39. Lastlythe stabilization equation is changed to account for both tangential and orthogonal direc-tion. There are 4 numerical parameters in the models εn, εt, γ1 and γ2. These parameters arediscussed below. How these parameters are chosen is essential for the quality of the results.

∇ · ~u = 0 (2.36)

∂~u

∂t+ (~u · ∇)~u = −∇p

ρ′+

1

ρ′Re∇ ·(µ′(∇~u+ (∇~u)T

))+

1

Fr2~eg +

1

ρ′ReCa~Fsa (2.37)

∂Φ

∂t+∇ · (Φ~u) = 0 (2.38)

22

n−∇ · (γ22∇n) =

∇φ|∇φ|

n|contact point = nαs (2.39)

∂Φ

∂t= γ1

[∇(εn(∇φ · ~n ) ~n ) +∇(εt(∇φ · ~t ) ~t )−∇ · (φ(1− φ)~n)

](2.40)

εn is the parameter that controls the thickness of the interface, similarly to the ε in theslip condition model discussed in the previous subsection. However, εn also controls thediffusion in the normal direction with respect to the interface.

εt is the diffusion in the tangential direction. The diffusion in the tangential direction isessential to enable the interface to converge towards the imposed angle at the boundary. Alarger εt means the imposed angle will be reached more quickly. Also, a large εt will affectthe level-set function φ further away from the boundary-interface area. A low εt could resultin the imposed angle at the boundary not being reached.

γ1 performs exactly the same function in the slip condition model. γ2 is proportional tothe size of the interface-boundary area, this is the area where the normal of the interfaceis changed to impose a value at the boundary. γ2 should be chosen much smaller than thecharacteristic size of the system, typically εn ∼ γ2. Again, the decision for the appropriatevalue of γ2 is a trade-off. γ2 should be small enough to satisfy the two conditions statedpreviously, but a small γ2 will lead to large curvatures as the boundary-interface area is small.Large curvatures require a finer mesh to resolve, which makes the simulation computationallymore demanding. A γ2 that is too large on the other hand will lead to an unrealisticsimulation.

2.6.3 Physical Parameters

The dynamic viscosity µ is a material property that influence two main force balances insidethe microchannels. A large dynamic viscosity causes the Reynolds number to be smaller andviscous forces will be more dominant. Because there are two fluids in the microchannel it isimportant to think about the difference in the dynamic viscosity. A larger dynamic viscositywill cause the Capillary number to be larger. This will reduce the effect of the surface ten-sion force with respect to the viscous force. A larger dynamic viscosity will promote parallelflow [14].

The density of the fluids ρ is different for both fluids. Density affects the magnitude of thegravitational and inertial forces. The effect of density on the behavior of the microchannelis thought to be small as both the gravitational and the inertial forces scale with volume,where other force scale with surface [14].

The interfacial surface tension σ influences the way the interface behaves. The surfacetension force is very important on the micro scale. Moreover, in section 2.1 the Laplace

23

pressure is introduced and describes the curving of the interface due to the hydrophilicity ofthe aqueous phase with respect to the glass. The surface tension influences the magnitudeof the Laplace pressure. A larger σ will be decrease the value of the Capillary number andwill lead toward slug flow.

The micro-scale static contact angle θe defines the angle between the two fluids at thecontact point with the glass surface of the interface. This contact angle defines the wettingof the wall and will thus need to be assessed in order to model the boundary conditions ofthe system properly. Liu performed experiments to assess the microscopic contact angle [1].

The inlet velocity vinlet is a parameter that can be set to a wide range of values and theinlet velocity’s of the organic and aqueous inlets do not need to be equal. Higher flow rateswill result in parallel flow [14]. However, a higher flow rate also means a shorter contacttime between the two fluids resulting in lower diffusion of Mo-99.

2.7 Phase Field Method

One of the goals of this thesis is to compare the results and the run-time of the conservativelevel set method with the phase field method. The results from the phase field method arenot produced in this thesis, but are made available by Liu. Liu is carrying out a researchproject focused on understanding and reproducing experimental results from the microchan-nels with the phase field method [1]. Because part of the aim of this thesis is to comparethe two methods, it is important to understand the fundamental ideas behind the theory.These ideas are explored in this section. Only the theoretical framework is discussed, theimplementation and discretization in COMSOL is not.

The phase field method is a finite interface thickness method that propagates the interfaceby diffusion. There are a lot of similarities with the level set method. The main differencehowever, is the way the interface is propagated near the boundary points, i.e. the contactline. First, the similarities will be discussed.

u · ∇ = 0 (2.41)

ρ(~ut + (~u · ∇)~u

)= −∇p+∇ ·

(µ(∇~u+ (∇~u)T

))+ ρg~eg + ~Fsa (2.42)

Equation 2.41 and 2.42 are the continuity equation and the Navier-Stokes equation respec-tively. These equations are exactly the same as the level-set equations, with the exceptionof the smearing of the surface tension force over the finite width the interface. This is donein a different way, which will be discussed later in this section. Then, the level-set variableφls is also used in the phase field method, denoted φpf . The mapping of the interface issimilar to the level set method. The difference is that the value of the φpf varies from 1 forthe aquous fluid to −1 for the organic fluid. The interface is defined by the region where

24

−0.9 < φpf < 0.9, in which the physical properties µ and ρ are smeared out in the same wayas the level set method. The equations governing the smearing out are equation 2.43 and2.44.

ρφpf =1 + φpf

2∗ ρ1 +

1− φpf2

∗ ρ2 (2.43)

µφpf =1 + φpf

2∗ µ1 +

1− φpf2

∗ µ2 (2.44)

On the other hand, the way the interface is propagated is different. The phase fieldmethod is propagated by the Cahn-Hilliart equation, see equation 2.45. The left-hand sideof equation 2.45 is the same as the advection equation used to propagate the interface in theconservative level set method. The main difference between the two methods can be found inthe right-hand side of the Chan-Hilliart equation. The conservative level set method diffusesnumerical perturbation and artificially compresses the interface to its original thickness. Themovement of the interface in the phase field method is based on the gradient of the chemicalpotential in the system. The gradient of the chemical potential is denoted as G. χ is themobility parameter, one of the numerical parameter of the phase field method. The units ofχ are m3s/kg [1].

dφpfdt

+ ~u · ∇φpf = χ∇2G (2.45)

The gradient of the chemical potential G can be obtained by taking the derivative of thefree energy with respect to the phase field variable φpf . This is expressed in equation 2.46.E is the free energy in the system. E is the volume integral of the mixing energy density inthe system.

G =δE

δφpf(2.46)

E =

∫Ω

fmix dΩ (2.47)

The mixing energy density can be expressed in terms of φpf and ∇φpf , this relationis shown in equation 2.47. The function of ε is the same as in the conservative level setmethod. It is a numerical parameter of the phase field method as well. ε is proportionalto the thickness of the interface. λ is the mixing energy density parameter. The first termof equation 2.48 is the interface energy density and the second is the bulk energy density.When all these equations are put together, the gradient of the chemical potential G can beexpressed. The result is equation 2.49 [1].

fmix(φpf ,∇φpf ) =1

2λ|∇φpf |2 +

λ

4ε2(φpf

2 − 1)2 (2.48)

25

G =δE

δφpf=λ

ε2φpf (φ

2pf − 1)− λ∇2φpf (2.49)

However, the value of λ is not yet known. As it turns out, λ can be related to the valueof the interfacial surface tension. The interfacial tension parameter σ is equal to the integralof the free energy density over the interface. For a one-dimentinal scenario, this is expressedby equation 2.50 [18]. In equilibrium, this expression simplifies to equation 2.51.

σ = α

∫ ∞−∞

(dφpfdx

)2

dx (2.50)

σ =2√

2

3

λ

ε(2.51)

Equation 2.50 can be extended to include time dependent problems, the result is equation2.45. This extension was first done by Cahn and Hilliart and is thus appropriately named theCahn-Hilliart equation. Chan and Hilliart showed the diffusive flux can be equated to thegradient of the chemical potential [30] [31]. To propagate the interface, equation 2.45, 2.49and 2.51 are solved. This is the main difference between the conservative level set methodand the phase field method. The key idea of the phase field method is that the interface willexperience an equilibrium if G = 0, i.e. a local energy minimum is reached.

The surface tension force is modeled as a body force over the finite width of the interfacein a similar manner as done for the level set method, see section 2.5. This is expressed inequation 2.52. This is done in a way specific to the phase field method and thus differentthan the conservative level set method [1].

Fst = G∇φpf = λ

[−∇2φpf +

φ(φ2 − 1)

ε2

]∇φpf (2.52)

2.7.1 Boundary Condition Phase Field Method

The boundary condition applied to the contact line is extremely important, as it determinesthe ability to deal with the contact line. The boundary condition used at the wall in thephase field method is derived based on the idea that the free energy at the wall is zero. Thisis expressed in equation 2.53 [32].

ns · ∇φpf =

√2cos(θe)

2Cn

(1− φ2

pf

)(2.53)

26

Chapter 3

Model Equations

This chapter starts by describing the set of discretized equations that are solved in COMSOL.These equations are used in order to assess the conservation properties of the model. Next,the implementation of an adaptive mesh is discussed. Lastly, the main parameters of thesimulation of the flow in a microchannel, numerical and physical, are identified.

3.1 Discretization

This section provides the discretized form of the model equations and an overview of howCOMSOL solves the system of equations. However, the mathematical derivation of thediscretized equations are not provided. These derivations are very complex and fall outsideof the scope of this work. Each time step, COMSOL starts by solving the advection equation.The discretized form of equation 2.21, this equation is given below:∫

Ω

v∇Φn+1

∗ − Φn

∆tdx−

∫Ω

∇v · (Φn~un)dx = 0 (3.1)

Equation 3.1 is the result of a temporal discretization using forward Euler, this is de-scribed in detail in the paper by Ollson et al. [23] Solving this equation yields Φn+1

∗ , whichis the first estimate of the new position of the interface. v refers to a function in the finiteelement function space while ~v refers to the finite element vector space. For more informa-tion on how this space is defined, see [20]. With the initial estimate for Φ, Φn+1

∗ , an initialestimate for n is determined by solving equation 3.2. This equation follows directly fromequation 2.17. ∫

Ω

~v · ∇Φn+1∗

|∇Φn+1∗ |

dx =

∫Ω

~v · nn+1∗ dx (3.2)

The estimates calculated in the first two steps are used as input for the third step,the re-initialization step. Equation 3.3 is the discretized form of equation 2.23. Φn+1

∗ isthe initial value Φ0

c , after which the re-initilization equation is solved until steady state isreached, yielding Φn+1. Typically a numerical parameter δ is defined to determine whether

27

a satisfactory steady state is reached. δ functions as an acceptable error, this is expressed inequation 3.4. The m refers to the last time step before the error condition is met such thatk = 0, ....,m. Finally, Φn+1 = Φm+1

c .

∫Ω

v∇Φk+1

c − Φkc

∆τdx =

∫Ω

(Φkc + Φk+1

c

2−Φk+1

c Φkc

)∇v·nn+1

∗ −ε∇(Φk

c + Φk+1c

2

)·nn+1∗ (∇v·nn+1

∗ )dx

(3.3)

||Φm+1c − Φm

c ||∆τ

< δ (3.4)

It is important to note that this equation is not solved in time but with respect tothe artificial time τ . The re-initilization parameter γ1 is moved to the other side of theequation and disappears as τ is rescaled. The artifical time τ thus depends on the re-initilization parameter γ1. The next step in the algorithm is to globally determine thevalue of (∇Φ)n+1 and κn+1. This is achieved by respectively solving equation 3.5 and 3.6.The calculation of κ is more elaborate than presented here. In practice, κ is obtained bysolving 3.6 produces spurious oscillations. These oscillations are dampened using Fouriertransformation described in detail by Olsson [20].∫

Ω

(∇Φ)n+1 · ~v dx =

∫Ω

∇(Φn+1) · ~v dx (3.5)

∫Ω

vκn+1dx =

∫Ω

∇v · (∇Φ)n+1

|(∇Φ)n+1|dx (3.6)

Next, the Navier-Stokes equation will be assessed. There are several ways the Navier-Stokes equation can be discretized. The method used by Olsson is described by Guermond[33]. The results are equations 3.7, 3.8 and 3.9. All the terms of equation 2.11 are alsorepresented in the discretized equation 3.7. The scheme first determines an intermediatevelocity ~un+1

∗ . Equation 3.7 only uses the old pressure field, ~un+1∗ is used in equation 3.8 to

determine pn+1. Then, pn+1 is used to calculate un+1 in equation 3.9. Now that the position ofthe interface, the material properties throughout the domain and the Navier-Stokes equationare solved. This scheme is effective in situations where the Reynolds number is low. HighReynolds number cases require a fully coupled solution strategy.

1

∆t

∫Ω

(ρn+1~un+1∗ − ρn~un) · ~v dx−

∫Ω

(~un · ∇~v) · ρ~un+1∗ dx

∫Ω

(∇ · ~v)ρndx−

1

Re

∫Ω

µn+1∑i

∇vi · (∇un+1∗i + ~unxi)dx+

∫Ω

~v ·(ρn+1

Fr2~eg +

1

Weκn+1(∇Φ)n+1

)dx

(3.7)

− 1

∆t

∫Ω

v∇ · ~un+1∗ dx =

∫Ω

∇v · ∇(pn+1 − pn)

ρn+1dx (3.8)

∫Ω

~v · ~un+1 − ~un+1

∗∆t

dx−∫

Ω

~v · ∇(pn+1 − pn)

ρn+1(3.9)

28

3.2 Conservation

One of the disadvantages of the level set method is that mass is not well conserved. At everystep a little mass is gained or lost and these errors tend to accumulate. The addition of theintermediate step, which functions as an artificial compression, solves the mass conservationissue. Equations 3.1 and 3.3 hold for all v that are a part of the finite element functionspace, among them v = 1. If v = 1 equations 3.1 and 3.3 reduce significantly because the∇v = 0 in this case, this is expressed in equation 3.10 and 3.11.∫

Ω

Φn+1∗ dx =

∫Ω

Φndx (3.10)∫Ω

Φk+1c dx =

∫Ω

Φkcdx (3.11)

Using these expressions, the steps in equation 3.12 are straight forward. The m+1 step isthe artificial time step where steady state of the re-initialization equation is achieved. m = 0on the other hand, is the first guess for Φn+1 obtained by solving the advection equation. Itis thus justified to draw the conclusion expressed in the first part of equation 3.13. Becausethe density is governed by equation 2.14, conversation of Φ directly implies conservation ofdensity. Total mass will be completely conserved. Furthermore, Φ can be interpreted as thevolume fraction of the fluid defined by Φ = 1, as the fluid denoted as Φ = 0 will drop out ofthe integral. The total mass of the fluid denoted by Φ = 1 will then also be fully conservedbecause of the conserved density. It follow that the conservative level set method proposedby Olsson and Kreiss fully conserves the mass of the system [20].∫

Ω

Φn+1dx =

∫Ω

Φm+1c dx =

∫Ω

Φmc dx = . . . =

∫Ω

Φ0cdx =

∫Ω

Φn+1∗ dx =

∫Ω

Φndx (3.12)∫Ω

Φn+1dx =

∫Ω

Φndx =⇒∫

Ω

ρn+1dx =

∫Ω

ρndx (3.13)

3.3 Implementation in COMSOL

COMSOL multiphysics is a user friendly finite element software package that combines dif-ferent physics modules into a single simulation. The user defines a geometry, the system ofphysical equations, the materials and the boundary conditions. The user also defines themesh, the finite element solver and the order of the discretization. For a model based onthe equations introduced in section 2.6, the most important modules will be laminar flowand level set method. The different modules are connected in the multiphysics branch, somephysical effects require both the level set and the flow equations to be dealt with. For in-stance, the surface tension force. This is a force on the level set interface, but it also playsa role in the Navier-Stokes equation. COMSOL has tools to deal with the navier-stokesequation and the conservative level set formulations built in. The alterations to the level-setmethod on the other hand, need to be implemented manually.

29

3.3.1 Angle at the Boundary

The altered level set method imposes an angle at the boundary by imposing a normal at theboundary. This set boundary condition will yield a regularized normal vector field accordingto equation 3.14. This equation needs to be implemented in COMSOL such that the levelset module can use its vector field for the re-initialization step.

Figure 3.1: Normal and tangent vector field in microchannel. The curved line in the mi-crochannel represents the interface. The tangent field is shown in blue and the normal fieldis shown in red. The angle at the wall is imposed by a Dirichlet boundary condition.

The best way to implement this in COMSOL is to add a mathematical module called”coefficient from partial differential equation”. This module is a generic equation where thecoefficients can be specified, seen in 3.15. The coefficients are chosen as follows: ea = 0, da,α = 0, β = 0, δ = 0, a = I, c = γ2 ∗ I and f = ∇φ

|∇φ| . Where I is the 2 dimensional unitmatrix. Then, a normal nα is imposed at the boundary, corresponding to an angle α. Thisis done in the form of a Dirichlet boundary condition. Effectively, the normal vector fieldcomposed in this way differs from the normal vector field calculated with equation 2.17 ifthe angle at the boundary is not the imposed angle. The the regularized normal vector fieldis the blue print for the next position of the diffused interface. Figure 3.1 shows the normaland tangent fields for the advancing part of a droplet during slug flow.

n−∇ · (γ2∇n) =∇φ|∇φ|

n|contact point = nαs (3.14)

ea∂2n

∂t2+ da

∂n

∂t+∇ · (−c∇n− αn+ δ) + β · ∇n+ an = f (3.15)

30

The next step in altering the current level set formulation is changing the re-initializationequation. The original formulation first solves equation 2.21 and then re-initializes theinterface using equation 2.22. The equations are reiterated here. Equation 3.17 needs tobe adjusted to equation 3.18. The alteration is done to the discretized equations via theadvanced settings options. The alterations to this equation is fairly easy as the the diffusionterm in the normal direction can be copied. Then the normal field is changed to the tangentfield and the εn to εt.

∂Φ

∂t+ ~u · ∇Φ =

∂Φ

∂t+∇ · (Φ~u) = 0 (3.16)

∂Φ

∂t+ ~u · ∇Φ = γ∇ ·

(ε∇Φ− Φ(1− Φ)

∇Φ

|∇Φ|

)(3.17)

∂Φ

∂t= γ1

[∇(εn(∇φ · ~n ) ~n ) +∇(εt(∇φ · ~t ) ~t )−∇ · (φ(1− φ)~n)

](3.18)

3.3.2 Flipping Boundary Condition

One of the issues of the altered formulation is that the condition at the boundary is a normalvector as opposed to an equilibrium angle. This results in a need to change the boundarycondition for the advancing and receding part of the droplet. Figure 3.1 shows the advancingend of a droplet, the normal vector defines an angle between the organic and the aqueousfluid. The equilibrium angle between n-Heptane and water on the glass surface of the mi-crochannel is 46.58°. However, if the same condition was to be used for the receding partof the droplet the angle would be roughly 90° off. To determine if the interface is recedingor advancing the derivative of φ is assessed. The organic fluid is defined by φ = 0 and theaqueous fluid by φ = 1. This means that a transition from the organic to the aqueous fluidyields a positive derivative. A transition from the aqueous fluid to the organic fluid on theother hand, yields a negative derivative. This difference between the two transitions can beused to identify which situation is being dealt with. Figure 3.2 shows the receding part of adroplet in the channel. The angle between the organic and aqueous fluid is the same. Thisis achieved by flipping the boundary condition with respect to the x-axis. The function thatis implemented is an if-else condition formulated in equation 3.19. This operation has alsochanged the direction of the tangent field, but with respect to the y-axis. This is necessaryto facilitate one end of the interface being the receding end of a droplet while the other end isadvancing end of a droplet. An example of this would be a droplet that is only in contact withthe lower/upper boundary. Such conditions can be present in the entry region of the channel.

nα =

0, if dφ

dx> 0

1, otherwise(3.19)

31

Figure 3.2: Normal and tangent vector field in microchannel. The curved line in the mi-crochannel represents the interface. The tangent field is shown in blue and the normal fieldis shown in red. The angle at the wall is imposed by a Dirichlet boundary condition.

3.4 Solver

COMSOL solves the contact line multiphase flow problem in two different steps. The firststep is called the phase initialization step. In this step the steady state problem at t = 0 issolved and the interface is given a finite width proportional to ε. Then the time dependentproblem is solved using the steady-state solution produced in the phase initialization step.COMSOL has several possible solvers that each have their own advantages and disadvan-tages. These are discussed in the reference manuals provided for COMSOL version 5.2a [26].For the phase initialization the MUMPS, or Multifrontal Massively Parallel sparse directSolver, is chosen. The time dependent step is solved with the PARDISO, or Parallel DirectSparse Solver Interface, solver.

3.5 Adaptive Mesh

In order to obtain more accurate solutions, an adaptive mesh can be implemented. Theidea of an adaptive mesh is to add more grid points at a specific position in the simulation.For the liquid-liquid parallel flow in a micro-channel, more grind points are desired near theinterface. This allows tracking of the position of the interface more accurately. Also, thethickness of the interface ε can be smaller, as the grid size near the interface is much smaller.

32

The smearing of the difference in properties between the two liquids over the interface isdone in a smaller range. As the smearing of the properties is unphysical, a simulation witha small ε is a better representation of reality. An example of an adaptive mesh implementedwith COMSOL is given in figure 3.3.

Figure 3.3: Example of an adaptive mesh. The thick line is the position of the interface, thegrid cells become increasingly smaller as the interface is approached.

The refined mesh is based on the the derivative of Φ, which very accurately contains theposition of the interface. As the interface moves, the adaptive mesh needs to be re-initializedin order to adequately refine the mesh around the interface. The moment the adaptive meshis re-initialized is based on an error function. Figure 3.4 shows how the mesh moves witha rising bubble at several times, the area around the interface clearly has a more refined mesh.

33

(a) t=0 [s] (b) t=0.10 [s] (c) t=0.14 [s]

(d) t=0.17 [s] (e) t=0.20 [s] (f) t=0.22 [s]

Figure 3.4: Adaptive mesh used to simulate an oil droplet rising in water due to buoyancyforces. The mesh adapts several times to keep up with the oil-water interface. The timeseach mesh was used are shown in the captions.

34

Chapter 4

Bird Benchmark: Spreading Droplet

In this chapter a benchmark of a bubble spreading is treated. First, the benchmark isdescribed in detail as well as simulation results produced with the phase field method. Theimplementation in COMSOL is discussed next. This includes the chosen geometry, materials,meshes and boundary conditions. Then, the results of simulation with the conservative levelset method are discussed. In the final section the conservative level set method results arecompared with results produced with the phase field method.

4.1 Benchmark Description: Spreading Droplet

In this benchmark, dynamic wetting of a droplet is investagated. The benckmark is based onan experiment conducted by Bird et al. [34]. In the experiments a small water droplet with adiameter 1.64 mm is produced using a needle with a diameter of 0.6 mm. The droplet growstowards a spherical shape by the quasi-static addition of fluid through the needle. The needleis positioned a distance equal to the diameter of the droplet above a surface, such that thedroplet will make contact with the surface when it has reached the desired size. The surfacethat the droplet will encounter is a silicon wafers that is coated using a silanization process.Different silan compositions are used to create different equilibrium angles. The surfacesused in the experiment have equilibrium angles of 3°, 43°, 117° and 180°. The describedexperiment is illustrated in figure 4.1. This figure shows the droplet for the different equilib-rium angles. The droplet deforms when contact is made as the interface converges towardsthe equilibrium angle. A smaller equilibrium angle will cause a larger deformation in thesame time-span. This is observable in figure 4.1. In a couple of milliseconds the equilibriumvalue is reached. This behavior is captured using a Phantom V7 camera able to record 67000frames per second, or 67 frames per millisecond. The moment of contact is defined as the firstframe in which visible changes are observed, t = 0 is the frame before any change is observed.

Carlson performed a simular benchmark reproducing the results from the experimentswith the phase field method [32]. Carlson uses the phase field formulation proposed byJacmin [28], which is discussed in chapter 2.7. However, although this formulation is able tocapture the experimental results qualitatively, it is not very accurate. This is illustrated in

35

Figure 4.1: Illustration of the experiments performed by Bird et al [34]. There are fourdifferent equilibrium angles, from top to bottom: 3°, 43°, 117° and 180°. The deformation ofthe droplet is clearly visible for different times.

figure 4.2 by the dotted line, where the different colors represent different equilibrium angles.The open circles, squares and diamonds represent the experimental data. The deviationbetween the experimental results and the predictions made by the phase field method isclear. To bridge this divide, Carlson made a change to the condition the phase field methodimposes at the boundary. The boundary condition introduced in equation 2.53 is replacedby equation 4.1.

D∗w∂φpf∂t

= σε∇φpf · ~n+ (σsg − σsl)w′(φpf ) (4.1)

Dw is introduced in equation 4.1. Dw is a dynamic wetting parameter and contains athird numerical parameter µf . The dynamic wetting parameter is the prefactor to a newtime dependent term, this term controls how fast the equilibrium angle is reached. Withthis alteration to the boundary conditions the results are much more closely approximated,as can be see in figure 4.2. The full lines indicate the simulation results using the phase fieldmethod with the altered boundary condition.

36

The goal of this benchmark is to reproduce the results from the experiment performedby Bird et al. with the conservative level set method. The benchmark has similarities anddifferences with respect to the flow in the microchannels. The benchmark features dynamicbehavior on the micro-scale and contact point behavior is very important in this case. Thistype of behavior is also important in the microchannel. On the other hand, the dimensionsof the problem are larger than the microchannel. This causes the inertial forces to be moreimportant. The formulation described in subsection 2.6.2 will be used. First the results fromthe simulations will be assessed, then the computational run-times will be compared to thesame simulation using the phase field method. Both simulations will be carried out withCOMSOL. The next section will discuss the implementation in COMSOL.

Figure 4.2: Results of simulations with the phase field method performed by Carslon [32]compared with the experimental results by Bird et al. [34]. On the y-axis the spreading radiusis plotted against the time on the x-axis. The blue lines/points correspond to an equilibriumangle of 117°, the black lines/points to an angle of 43° and the red lines/points to an angleof 3°. The open symbols represent experimental data and the closed symbols simulation data.The dashed lines are the results without the dynamic wetting condition. The full lines arethe results with the dynamic wetting condition. [32]

37

4.2 Implementation: Spreading Droplet

The spreading droplet benchmark is modeled using COMSOL, the structure of the modelis discussed in chapter 3. The modeling of the materials, geometry, mesh and boundaryconditions are discussed. The droplet is made of water, the surrounding space is filled withair. The density of water is 988 kg∗m−3 and the viscosity 10−3 Pa∗s. Air has a density of1.2 kg∗m−3 and a viscosity of 1.6∗10−5 Pa∗s. The surface tension between air and water is0.0727 N∗m−1. The equilibrium angle depends on the type of silan composition.

The geometry created in COMSOL includes the needle, the droplet and some space forthe droplet to spread. The created geometries are equal to the point defined as t = 0 in theexperiments. Figure 4.3 shows the geometries tested during this benchmark. It also showsa corresponding mesh to each of the geometries.

Geometry 1 is the most basic geometry looking at the experimental description. However,the corresponding mesh has an concentrated distribution with heavy refinement around thecontact points near the needle and the surface. This is due to the very small dimensions nearthe contact points which translate into very small mesh elements near these points. Oneof the mesh parameters is the maximum element growth rate, which controls the area ratiobetween two adjacent mesh elements. Unless a very high maximum element growth rate isused, the small elements near the contact points will lead to a lot of refinement near thosepoints. However, a very high maximum element growth rate could lead to convergence andaccuracy issues. Another way to prevent this refinement is to change the geometry slightly.Geometry 2 in figure 4.3 has the needle slightly inserted into the droplet, such that thesmall dimensions near the contact point are prevented. This could be explained as being awetting effect also observable in figure 4.1. Then, geometry 3 has the droplet slightly off thesurface. This again prevents major mesh refinement near the contact point. The physicaljustification for this change, is the fact that during the experiments the situation just beforecontact is deemed t = 0. In order to see if these changes affect the results, some simulationswere performed. The results of these simulations are summarized in table 4.1.

Twelve separate simulations were carried out: four different equilibrium angles in allthree geometries. For each simulation the radius after 2 milliseconds and the run-time of thesimulation is recorded. The change made in the second geometry had no effect on the resultsbut did reduce the computation times to a modest degree. The change made in the thirdgeometry, slightly lifting the droplet, reduced the simulation time even further. However,it also changed the droplet radius after two milliseconds, effectively delaying the spreading.For this reason geometry 3 was not used for the generation of the results.

Several meshes are tested in the simulation. COMSOL has predefined mesh sizes forfluid dynamics that range from extremely coarse to extremely fine. These predefined meshesare used to see what mesh is suitable for this benchmark. Table 4.2 lists the specificationsof these predefined meshes. Choosing a mesh is always a trade-off between accuracy and

38

(a) Geometry 1 (b) Mesh 1

(c) Geometry 2 (d) Mesh 2

(e) Geometry 3 (f) Mesh 3

Figure 4.3: Geometries and corresponding meshes used during simulations. The altering ofthe geometry results in a simpler mesh and shorter simulation times. In the second mesh theneedle is lowered into the droplet. In the third case, the droplet is also slightly lifted off thesurface.

39

Table 4.1: Results of simulations testing different geometries. Four different equilibriumangles are tested in all three geometries, being 3°, 43°, 117° and 180°. For each simulationthe droplet radius after 2 milliseconds and the run-time is recorded. In this simulation thefollowing parameters are set as follows; εn = 0.04∗D, γ1 = 0.1m/s, εt = 3∗εn and γ2 = 5∗εn.

Equilibriumangle

Measure Geometry 1 Geometry 2 Geometry 3

θ = 3° Radius after 2 ms 1230 µm 1230 µm 1183 µmRun-time 10:07 [min:sec] 11:07 [min:sec] 05:30 [min:sec]

θ = 43° Radius after 2 ms 1100 µm 1100 µm 1070 µmRun-time 04:51 [min:sec] 08:13 [min:sec] 04:33 [min:sec]

θ = 117° Radius after 2 ms 485 µm 485 µm 420 µmRun-time 06:27 [min:sec] 05:33 [min:sec] 02:36 [min:sec]

θ = 180° Radius after 2 ms 0 µm 0 µm 0 µmRun-time 12:20 [min:sec] 08:26 [min:sec] 04:47 [min:sec]

Table 4.2: Specification of several mesh sizes. The maximum element size, minimum ele-ment size and maximum element growth rate are shown. These pre-set meshes are used todetermine the required mesh size.

Type Max Element Size Min Element Size Maximum Element Growth RateCoarser 170 µm 7.8 µm 1.25Coarse 131 µm 5.85 µm 1.2Normal 87.8 µm 3.9 µm 1.15Fine 68.3 µm 1.95 µm 1.13Finer 54.6 µm 0.78 µm 1.1Extra Fine 25.4 µm 0.293 µm 1.08Extremely Fine 13.1 µm 0.039 µm 1.05

simulation times. For all simulations, a fine mesh was used as it converged for all cases.Implementing a finer mesh did not change the results. Table 4.3 shows a set of simulationsused to determine the required mesh.

There are several boundaries in the geometry that all need to be assigned a specificcondition. The circle that indicates the water-air interface is set to the initial interface.Knowing the initial interface position is necessary in order to initialize the interface. Theinitial interface is sharp and needs to be given a finite width that is proportional to εn for themethod to work. The lower surface of the geometry represents the silicon wafer. For the flow,a no-slip boundary condition is set at the wafer. For the interface on the other hand, theboundary condition at the wafer is implemented as described in section 3.3.1. Dependingon the silan composition, the contact angle will be different. This means that for everycontact angle a different normal vector is imposed. As the direction of the interface is very

40

Table 4.3: Several meshes that are used to simulate the spreading of the droplet. The equi-librium angle for this series was 117°. The final droplet radius after 2 ms has stabilized withthe increased mesh density.

Type of mesh implemented Droplet Radius after 2 ms Run-timeCoarse dncNormal 284 µm 03:38 [min:sec]Fine 302 µm 05:02 [min:sec]Finer 305 µm 07:18 [min:sec]Extra Fine 305 µm 08:25 [min:sec]

predicable, flipping of the boundary condition is not necessary as it is in the microchannels.This means the right hand side of the surface has a different condition than the left handside. Implementing the flipping of the condition as described in section 3.3.2 also works andyields exactly the same result. However, the run-times do go up slightly as an extra step isinvolved. The rest of the outer geometry is set to a no-slip boundary condition. The needlehas a boundary condition similar to the silicon waver, with an equilibrium angle set to 90°.

4.3 Results: Spreading Droplet

In order to obtain decent results from the simulations, the choice of numerical parametersis very important. Subsection 2.6.2 provides context as to how these parameters are cho-sen. The initial values for the parameters are εn = 0.04∗D, γ1 = 0.1 m∗s−1, εt = 3∗εnand γ2 = 5∗εn. A parameter study is performed to asses the effect of the parameters onthe simulation results and run-times, to further fine-tune the choice of parameters. Table4.4 details the different values to which the parameters are set and the corresponding results.

The first parameter that is varied is εn. The choice of εn is related to the Cahn number.The characteristic length of this problem is the droplet diameter D, which means the Cahnnumber in the parameter analysis is 0.03, 0.04 and 0.05 respectively. A lower εn is expectedto lead to longer simulation times. This is due to the interface thickness being proportionalto εn. A thinner interface is more expensive to resolve with the same amount of mesh ele-ments. A finer mesh on the other hand, leads to longer simulation times. It is also observedthat a lower εn leads to smaller droplet radius after 2 ms.

Changing the value of γ1 has similar effects. γ1 is the amount to which the interfaceis stabilized. Considering the equilibrium position has a much larger radius than the ini-tial position, it makes sense that more stabilization will lead to an interface that travelsfaster. A very low γ1 leads to longer run-times as numerical distortions of the interface arenot diffused as much. This leads to poorer convergence. A high γ1 however will lead to lessaccurate results, as too much stabilization causes the interface to move in a non-physical way.

41

Table 4.4: Simulations results where the numerical parameters are set to a different valuein order to investigate the way they influence the results. The simulations are done with anequilibrium angle of 43°. The base values are εn = 0.04∗D, γ1 = 0.1 m∗s−1, εt = 3∗εn andγ2 = 5∗εn. The droplet radius after 2 milliseconds and the run-time are recorded.

Parameter Value Droplet Radius after 2ms Run-Timeεn 0.03 ∗D [m] 1040 µm 08:52 [min:sec]εn 0.04 ∗D [m] 1070 µm 04:33 [min:sec]εn 0.05 ∗D [m] 1110 µm 04:41 [min:sec]γ1 0.03 [m∗s−1] 895 µm 12:26 [min:sec]γ1 0.06 [m∗s−1] 1005 µm 05:47 [min:sec]γ1 0.1 [m∗s−1] 1070 µm 04:33 [min:sec]γ1 0.2 [m∗s−1] 1135 µm 04:39 [min:sec]γ2 2.5 ∗εn [m] 1050 µm 05:26 [min:sec]γ2 5 ∗εn [m] 1070 µm 04:33 [min:sec]γ2 10 ∗εn [m] 1110 µm 05:21 [min:sec]εt 1 ∗εn [m] 930 µm 06:16 [min:sec]εt 3 ∗εn [m] 1070 µm 04:33 [min:sec]εt 6 ∗εn [m] 1140 µm 04:51 [min:sec]

The effect of γ2 on run-time is relatively small. A larger boundary region will lead tofaster interface propagation. This makes sense as a larger boundary region in which theinterface is forced towards the equilibrium angle allows the interface to move away from thecenter more quickly. εt has a large influence on the speed with which the interface travels.A large εt leads to quicker convergence towards the equilibrium angle, which leads to fasterinterface propagation.

The analysis of the parameters leads to an understanding of how these numerical param-eters influence the results produced with the simulations. Figure 4.4 shows simulations withthe following numerical parameters; εn = 0.04∗D, γ1 = 0.1 m∗s−1, εt = 3∗εn and γ2 = 5∗εn.The simulation results for the equilibrium angles 3° and 43° deviate from the experimentalresults. The simulation results predict the droplet to spread faster than observed in theexperiment. For the equilibrium angle of 117° on the other hand the simulation results agreequite well with the experiments.

Based on table 4.4, a second set of numerical parameters can be chosen in order to bet-ter approximate the experimental result. The spreading of the droplet is slower in realitythan the simulations predict and thus the numerical parameters are altered to give a betterrepresentation of the spreading of the droplet. This results in a set of parameters as follows;εn = 0.04∗D, γ1 = 0.06 m∗s−1, εt = 1∗εn and γ2 = 2.5∗εn. The corresponding simulationresults are shown in figure 4.5. The radius of the droplet is significantly lower comparedto the first simulation. This is the case for all the equilibrium angles. Now the simulations

42

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time(s) 10-3

0

100

200

300

400

500

600

700

800

900

1000

1100

Dro

plet

Rad

ius

Rel

ativ

e to

Sym

met

ry A

xis(

um)

e = 3

e = 43

e = 117

Figure 4.4: Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird et al. [34]. On the y-axis thespreading radius is plotted against the time on the x-axis. The green lines correspond to anequilibrium angle of 117°, the blue lines to an angle of 43° and the red lines to an angleof 3°. The closed lines represent experimental data and the dashed lines the simulationdata. The simulations are performed with the following numerical parameters; εn = 0.04∗D,γ1 = 0.1 m∗s−1, εt = 3∗εn and γ2 = 5∗εn.

with an equilibrium angle of 3° and 43° match the experimental results much better. The sit-uation with an equilibrium angle of 117° on the other hand, is now spreading much too slowly.

It appears the dynamics of the 117° case cannot be captured by the same numericalparameters as the 3° and 43° case. This can also be observed for the phase field methodwithout the changed boundary condition, see figure 4.2. The 117° case is approximated wellbut simulations of the other two angles predict a droplet radius larger than the experimentalsituation. This is similar to the results obtained with the conservative level set method. Thechange Carlson made to the phase field method allowed it to more accurately describe theexperimental results. However, it did so with different numerical parameters for the 117°case as apposed to the 3° and 43° case. The numerical parameter controlling the behaviorat the boundary is given a different value. With this fact in mind, the same is done for thelevel set condition where εt and γ2 mainly control the behavior near the boundary. Figure

43

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Time(s) 10-3

0

100

200

300

400

500

600

700

800

900D

ropl

et R

adiu

s R

elat

ive

to S

ymm

etry

Axi

s(um

)

e = 3

e = 43

e = 117

Figure 4.5: Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird. [34]. On the y-axis the spreadingradius is plotted against the time on the x-axis. The green lines correspond to an equilibriumangle of 117°, the blue lines to an angle of 43° and the red lines to an angle of 3°. The closedlines represent experimental data and the dashed lines the simulation data. The simulationsare performed with the following numerical parameters; εn = 0.04∗D, γ1 = 0.06 m∗s−1,εt = 1∗εn and γ2 = 2.5∗εn.

4.6 shows the results of this simulation. All simulations are done with εn = 0.04∗D andγ1 = 0.06 m∗s−1. The 3° and 43° case are performed with εt = 1∗εn and γ2 = 2.5∗εn whilethe 117° is performed with εt = 3∗εn and γ2 = 5∗εn. The results of these simulations moreclosely approach the experimental results. None the less, there is still a difference betweenexperiment and simulation.

4.4 Comparison with Phase Field Simulations

An important goal of this thesis to is to assess if the conservative level set method can beused to reduce simulation run-times and produce the same quality in terms of results withrespect to the phase field method. The previous section discussed the results of the con-servative level set method compared to experimental results. This section will compare theconservative level set method with the phase field method. Figure ?? contains the results of

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

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Figure 4.6: Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird. [34]. On the y-axis the spreadingradius is plotted against the time on the x-axis. The green lines correspond to an equilibriumangle of 117°, the blue lines to an angle of 43° and the red lines to an angle of 3°. Then,the closed lines represent experimental data and the dashed lines the simulation data. Thesimulations are performed with the following numerical parameters; εn = 0.04∗D, γ1 =0.06 m∗s−1. The simulations with an equilibrium angle of 3° and 43° are performed withεt = 1∗εn and γ2 = 2.5∗εn, while the 117° case has a εt = 3∗εn and γ2 = 5∗εn.

the experiment, the conservative level set method and the phase field method without thealtered boundary condition. The phase field simulations are carried out in COMSOL withthe same simulation settings, mesh and solver configurations. The green lines correspond toan equilibrium angle of 117°, the blue lines to an angle of 43° and the red lines to an angle of3°. The empty symbols represent the experiment, the closed symbols the conservative levelset method and the yellow symbols the phase field method. Both methods are not able toperfectly reproduce the experimental results. The phase field method is more accurate in thefirst 0.5 ms, afterwards the level set method seem to approximate the results better. Over-all the phase field method and conservative level set method are both able to qualitativelydescribe the benchmark, but not quantitatively.

The run-time of both methods are compared by performing the same simulations withboth methods. Exactly the same mesh configuration is used by both methods. Also, the

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Figure 4.7: Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method. The experimentalresults by Bird et al. [34] are also plotted in the graph. On the y-axis the spreading radiusis plotted against the time on the x-axis. The equilibrium angle is 3°. The open symbolsrepresent experimental data, the closed symbols the conservative level set results and theyellow symbols the phase field method results.

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Figure 4.8: Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method. The experimentalresults by Bird et al. [34] are also plotted in the graph. On the y-axis the spreading radiusis plotted against the time on the x-axis. The equilibrium angle is 43°. The open symbolsrepresent experimental data, the closed symbols the conservative level set results and theyellow symbols the phase field method results.

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Figure 4.9: Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method. The experimentalresults by Bird et al. [34] are also plotted in the graph. On the y-axis the spreading radiusis plotted against the time on the x-axis. The equilibrium angle is 117°. The open symbolsrepresent experimental data, the closed symbols the conservative level set results and theyellow symbols the phase field method results.

solver is exactly the same. The run-time of 24 simulation are summarized in table 4.5. Thetable includes simulations with all three equilibrium angles with the set of numerical pa-rameters that produced the best results. Both simulations are performed on the TU Delftcluster hpc11. The run-times of both methods are fairly similar and it is observed that thesame simulation does not always yield the same simulation time. This could be due to a lotof traffic on the cluster. For this project, the COMSOL simulations on the cluster are alwaysrun on a single node. However, the amount of processors allocated to the simulation caneasily be changed. The run-time of these simulation is not dependent on the amount of pro-cessors, which was unexpected. A possible explanation of the run-time not being dependenton the amount of processors is that splitting the work between the processors is inefficientas the simulation time is very short.

Furthermore, on average the run-time of the level set method seems to be slightly longerthan the phase field method. Due to the independence of the run-time on the amount ofprocessors, it is not straightforward why this is the case. It is also possible the level setmethod is slower because it contains the extra step of calculating the generalized normalvector field.

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Table 4.5: Simulations where the phase field simulation is compared to the level set method.The numerical parameters for the phase field methods are; εpf = 0.04∗D. The numericalparameters for the phase field methods are; εn = 0.04∗D, γ1 = 0.06 m∗s−1, εt = εn andγ2 = 2.5∗εn

Method # Processors θ = 3° θ = 43° θ = 117°Level Set 1 07:33 [min:sec] 06:39 [min:sec] 03,37 [min.sec]Level Set 4 04:55 [min:sec] 06:31 [min:sec] 05,35 [min.sec]Level Set 8 07:21 [min:sec] 05:36 [min:sec] 05,36 [min.sec]Level Set 12 08:15 [min:sec] 07:56 [min:sec] 03,58 [min.sec]

Phase Field 1 03:57 [min:sec] 05:20 [min:sec] 04,44 [min.sec]Phase Field 4 05:01 [min:sec] 04:45 [min:sec] 03,11 [min.sec]Phase Field 8 03:57 [min:sec] 03:36 [min:sec] 03,16 [min.sec]Phase Field 12 03:33 [min:sec] 03:14 [min:sec] 07,49 [min.sec]

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Chapter 5

Channel Experiment: DiffusiveInterface Method

Simulations with the level set method are compared with experimental results form themicrochannels and simulations with the phase field method. First, the experiments aredescribed followed by the implementation of the situation in COMSOL. Then, the resultsare discussed and compared with the phase field method.

5.1 Benchmark Description: Micro-Channels

The experiments are preformed with the chip shown in figure 5.1. The image labeled 5.1(a) shows the experimental set-up with a scale in centimeters. The inlets and outlets areclearly visible on both sides of the chip. The Y-shaped channel can be seen in the image.The length of the section where the two fluids are in contact with each other is 2 cm long,this is the section from the Y-inlet to the Y-outlet. Label (b) is a schematic cross section ofthe channels. This chip has a width of 100 µm and a depth of 40 µm.

The length of the inlet tubes is long, which means the velocity profile is fully developed.One of the inlets is for the aqueous fluid, which is water. The other inlet is for the organicfluid. Although both n-heptane and toluene were used during the experiments, the focus ofthe simulation is on the n-heptane experiments. The aqueous fluid is water with a density of998 kg∗m−3 and a dynamic viscosity of 10−3 Pa∗s. The organic fluid is n-heptane. n-heptanehas a density of 680 kg∗m−3 and a viscosity of 3.86∗10−4 Pa∗s. The interfacial tensionbetween the n-heptane and water is 50.2∗10−3 N∗m−1. The equilibrium angle between thetwo fluids and the glass surface of the microchannel is 46.58°.

5.1.1 Description of Experiments

Several experiments are performed by Liu [1]. Most notably is the experiment where theflow rate of both the aqueous and organic fluids are changed between measurements. Acombination of organic and aqueous flow rates leada to either slug flow, parallel flow or a

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Figure 5.1: Pictures of the microchip used in the experiments. (a) Experimental set-up whenin use, the tubes are used to feed the inlets/outlets. A blue methylene solution is flowingthrough the chip for the purpuse of highlighting the channels. The scale provided at the topof the image is in centimeters. [1] (b) A schematic cross section of the microchannel.

transition between the two flow types. The results of this experiment are shown in figure5.2. [1] The asterisk’s indicate slug flow, the circles a transition between the two flow typesand the plusses parallel flow. Reproducing the results of this experiment would indicatethe simulations with the conservative level set method is able to capture the physics of theproblem. Simulations with the phase field method have successfully reproduced the type offlow observed in the experiments [1].

5.2 Model Description: Micro-channel Simulations

5.2.1 Geometry and Materials

For the implementation into COMSOL a geometry needs to be defined. As the schematiccross-section in figure 5.1 (b) shows, the height of the channel is 100 µm with a maximumdepth of 40 µm. The geometry can include the entire chip, as is show in figure 5.3. Thelength of the channel is the full 2 cm as in reality. At the y-junction, the two inlet channelsmeet. Both inlet and the outlet channels are 0.5 cm long. This is less than the real chip.However, the inlet flow can be set to fully developed at the inlet. The angle between theupper and the higher inlet/oulet is 26°. Thus both make a 13° angle with the main part ofthe channel. The flow taking place in the channel moves from the left to the right.

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Figure 5.2: Graph showing different flow types based on flow rate. On the x-axis the flow rateof the organic fluid is shown while the y-axis shows the flow rate of the aqueous fluid. Theflow rate is given in µL per minute. The asterisk’s indicate slug flow, the circles a transitionbetween the two flow types and the plusses parallel flow [1].

Figure 5.3: Geometry 1 as implemented in COMSOL. The width of the channel is 100 µmand the depth is 40 µm. The chip is symmetrical and the inlet/outlet sections are 0.5 cmwith an angle of 13° with respect to the main channel. The main channel is 2 cm long.

The second geometry that is used is significantly smaller. This is mainly done to reducerun-time of the simulations. Only the first y-junctions remains and the size of the mainchannel is reduced to 0.5 cm. The size of the inlet channels is also reduced to 0.25 cm. Theoutlet channels are removed as well. The right hand side of the main channel functions asan outlet. The resulting geometry is shown in figure 5.4. The initial position of the interfaceis set to line number 2, located in the lower inlet. The line marked as 1 in the lower inletand line number 3 in the upper inlet indicate the start of a finer mesh. Before these linesonly one fluid is present and thus a coarser mesh is adequate to resolve the equations.

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Figure 5.4: Geometry 2 as implemented in COMSOL. The width of the channel is 100 µmand the depth is 40 µm. The inlet sections are 0.25 cm with an angle of 13° with respectto the main channel. The main channel is 0.5 cm long. The right hand side of the mainchannel is the output in this geometry.

During initial simulations, a problem was observed at the sharp point between the inlets.More specifically, the moment the interface reaches this point the simulation shuts down andconvergence is not achieved. To solve this issue both geometry one and two were slightlymodified. Figure 5.5 illustrates this change, the sharp edge of the inlets is removed and aflat surface of about 4 µm is created. The real chip will also not have a sharp edge at thatlocation due to the uncertainty in producing the chip. This modification aides the transitionbetween the boundary conditions applied at the different walls of the chip.

Figure 5.5: Change made to geometry to improve convergence.

The simulations start in a scenario where the the upper inlet, main channel and theoutlets are filled with the aqueous fluids. The lower inlet contains the organic fluid up untilthe initial interface position. The aqueous fluid is water and the organic fluid n-heptane.The properties of these fluids are listed in section 5.1 and implemented in COMSOL.

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5.2.2 Boundary Conditions

Several boundary conditions are implemented via the COMSOL interface. The inlets are thesame for both geometries. An inlet boundary condition is applied in the form of a set flowrate. The entrance length is set to 1 m such that the flow is fully developed. The outlets areset at the end of the channel, either at the outlets in geometry 1 or at the end of the mainchannel for geometry 2. The wall of the chip is set to a no-slip boundary condition. Then, atthe same boundary a normal vector is imposed such that the equilibrium angle is satisfied,as described in section 3.3.1. The interface is then able to move by way of diffusion. Thissection also discusses the flipping of the normal vector for the receding/advancing part of thedroplet, as specified in section 3.3.2. Figure 5.6 shows the flipping of the boundary conditionduring a simulation. Subfigure 5.6a shows the phase variable φ. It can be seen that twodroplets have formed in the channel. Subfigure 5.6b shows the corresponding derivative andsubfigure 5.6c the resulting function that is used to flip the boundary condition. The recedingend of the droplet has the shape that would be expected with the boundary condition thathas been implemented.

5.2.3 Mesh: Diffuse Interface Method

The mesh is very important to contain run-time but still produce decent results. The phasefield method uses a relatively coarse mesh with refinement near the interface, by using theadaptive mesh function built into COMSOL. Initially, a similar strategy is used for the con-servative level set method. A predefined mesh for fluid dynamics is used ranging from normalto extremely fine. On top of that the adaptive mesh is implemented, the amount of refine-ment can be increased or decreased. This is controlled by two main settings, the maximumelement growth rate and the maximum element refinement. Three different combinations areused ranging from a max element refinement of 3 to 10 and a maximum element growth rateof 1.3 to 1.7. Convergence with this type of mesh proved difficult. Short simulations havebeen performed in three stages, to test convergence and fine-tune the method. The first simu-lation covers a time-span of 1 ms. This features the initialization of the interface and a movetoward a situation where the equilibrium angle is satisfied. The second step is a simulationof 10 ms where the interface travels to the end of the inlet. The simulations are performedin the slug flow regime so the next simulation would feature the break-up of a droplet. Thisis captured with a simulation time of 33 ms. All simulations took longer than expected andproved very difficult to converge. The influence of numerical parameters is assessed at thisstages to investigate whether changing the numerical parameters could improve convergence.

The fine-tuned set of numerical parameters with an adaptive mesh are able to simulatethe droplet breakup. However, the run-time of this configuration far exceeds a similar simu-lation with the phase field method. Changes to the mesh, either basic mesh settings or thesettings of the adaptive mesh, have not reduce run-times. Simulations with a coarser meshdo not converge. A finer mesh on the other hand, only increases the run-time.

To reduce the run-time the adaptive mesh feature in COMSOL is not used anymore

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(a) Snapshot of simulation, 21 ms after the initial state. φ = 0 is indicated by the color red, φ = 1by the blue. The transition from red to blue indicates the interface. Slug flow is observed.

(b) Value of derivative of φ with respect to x at the same time and the same settings as (a). Thevalue is calculated at the lower boundary of (a). The advancing part of the droplet results in apositive derivative while the receding part in a negative derivative. On the x-axis the x coordinateused in the simulation is plotted and on the y-axis the value of the derivative.

(c) With the use of an if-else condition, the region where the boundary condition needs to be flippedcan be determined. On the x-axis the x coordinate used in the simulation is plotted and on the y-axisthe value of the if-else condition, either 1 or -1. 1 corresponds to the advancing condition while -1corresponds to the receding condition

Figure 5.6: Implementation of the flipping of the boundary condition. (a) shows the value ofφ. (b) value of the derivative of φ with respect to x on the lower boundary. (c) result of theif-else condition.

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and replaced by a static mesh with predefined refinement. The mesh is then split into twosections, a section where one phase flow occurs and a section where two phase flow occurs.This is done for the entire geometry. The inlet channels are predictable and do not need tobe changed per situation. Then, the question for the main channel and the outlet channelis more difficult. A prediction is made using the flow rate boundary conditions. Up to thepoint two phase flow is predicted, a very fine mesh is implemented. In the rest of the domaina much coarser mesh is used. During the initial trial simulations, it was observed a meshthat is too coarse leads to dissipation of mass. This results is shown in figure 5.7b. Inorder to determine the required mesh, a mesh that becomes increasingly less fine is used, seefigure 5.7a. The point where the mass of the droplet starts to dissipate determines when themesh has become too coarse. This leads to a usable mesh where the mass of the droplet isconserved, see sub-figure 5.7c. The settings of this mesh in the two-phase flow region are amaximum element size of 6 µm, a minimum element size of 1 µm and a maximum elementgrowth rate of 1.05. In the remainder of the domain a predefined mesh is used, which ismuch coarser. A finer mesh based on a maximum element size of 4 µm produced the sameresults.

(a) Illustration of mesh used to find required mesh size. In reality the change in mesh size betweenthe different regions is much smaller.

(b) Example of simulation where the mesh is too coarse. Dissipation of the mass of the droplet isobserved, eventually the droplet disappears entirely.

(c) Example of simulation where the droplet mass is properly conserved.

Figure 5.7: (a) Mesh used to find required mesh size.(b) Example of simulation with a meshthat is to fine. (c) Example of simulation with the required mesh size.

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5.3 Results: Diffusive Interface Method

5.3.1 Parameter Study

In order to understand the influence of all parameters a series of simulations are performedwhere the numerical parameters are varied with respect to a base value. The effect on therun-time and behavior of the droplet is recorded. The channel is simulated from the initialposition for 0.05 s. During this time, the first droplet is formed and travels along the mainchannel. Depending on the settings, a second droplet can also be formed. The base valuesof the numerical parameters are εn = 0.06∗D, γ1 = 0.1 m∗s−1, εt = 3∗εn and γ2 = 6∗εn. Bothchannels are set to a flow rate of 5 µL/min and a simulation time of 0.05 s. Table 5.1 showshow the numerical parameters are individually varied. Table 5.1 also records the dropletbreakup and the run-time.

Both increasing and decreasing the value of γ1 and εn results in very similar behavior.The run-time of the simulations all go up by changing γ1. The fact that a lower γ1 leadsto a higher run-time is likely due to a more distorted interface. A higher γ1 on the otherhand, could result in unrealistic behavior due to too much re-initialization. The breakupof the droplet is different than every other case, which could be explained by too muchre-initialization. As expected a lower interface thickness increases the run-time, in otherrespects the results are close to identical. A thicker interface decreases run-time. When εnis changed, the parameters expressed in terms of εn are kept at the same absolute value. εtcontrols the speed with which the interface moves by way of diffusion toward the equilibriumposition. Increasing εt slightly increases the run-time while decreasing εt leads to a shorterrun-time. This could be due to the fact that lower εt means less diffusion. The moment thedroplet breaks up does not change.

The most influential numerical parameter is γ2. As explained in subsection 2.6.2, γ2

controls the the size of the boundary region. The boundary region should be in the orderof the interface thickness. In the series of simulations, γ2 varies from 2 ∗ εn to 8 ∗ εn. Thesimulation where γ2 has a value of 2 ∗ εn did not converge. A small boundary layer thicknessleads to higher curvatures which caused the computation to not converge. A value of 4∗εn didproduce a result, but the corresponding run-time greatly increased. This limits the feasibilityof the method. Further increasing the value of γ2 to 8 ∗ εn also significantly increases therun-time. The boundary region is now almost half the channel width which is unlikely toproduce physical results. When γ2 has a value of 6 ∗ εn the run-time is lowest. However theboundary layer thickness is still a significant part of the domain. Reducing the value of γ2

on the other hand, will lead to unacceptably long calculation times. One of the reasons forusing the level set method is to reduce the run-times of the simulations.

5.3.2 Flow Regime Chart

The most striking conclusion that can be drawn from the parameter study is that decreasingγ2 leads to unacceptably long calculation times. However, simulation results performed with

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Table 5.1: Simulations where the numerical parameters are set to a different value in orderto investigate the way they influence the results. The flow rate of both fluids is 5 µL/min. Thebase values are εn = 0.06∗D, γ1 = 0.1 m∗s−1, εt = 3∗εn and γ2 = 6∗εn. The time of thebreakup of the first droplet is recorded as well as the run-time.

Parameter ValueDropletBreakup

Run-Time

εn 0.04 ∗D [m] 0.032 [s] 22:41:09 [hr:min:sec]εn 0.06 ∗D [m] 0.032 [s] 09:28:41 [hr:min:sec]εn 0.08 ∗D [m] 0.032 [s] 06:43:45 [hr:min:sec]γ1 0.04 [m∗s−1] 0.032 [s] 15:19:48 [hr:min:sec]γ1 0.1 [m∗s−1] 0.032 [s] 09:28:41 [hr:min:sec]γ1 0.25 [m∗s−1] 0.036 [s] 11:33:45 [hr:min:sec]γ2 2 ∗εn [m] dnc dncγ2 4 ∗εn [m] 0.032 [s] 57:25:39 [hr:min:sec]γ2 6 ∗εn [m] 0.032 [s] 09:28:41 [hr:min:sec]γ2 8 ∗εn [m] 0.033 [s] 19:56:20 [hr:min:sec]εt 1 ∗εn [m] 0.032 [s] 04:45:54 [hr:min:sec]εt 3 ∗εn [m] 0.032 [s] 09:28:41 [hr:min:sec]εt 5 ∗εn [m] 0.032 [s] 10:55:35 [hr:min:sec]

a higher γ2 did not agree with the flow types observed in the experiments. Simulations wereperformed at different flow rates, both inlet velocities are assigned the same value. Thevalues at which simulations are performed are 3.3 µL/min, 5 µL/min, 10 µL/min and 15 µL/min.All simulations produced slug flow instead of the type of flow that is expected based on theexperimental results.

In order to find out why the simulations and experiments are not matching, the physicalparameters are scaled such that parallel flow should be observed. The idea of this scaling isthat if the model is physically correct, changing the balance of forces should have an effectand either change the flow type towards slug flow or towards parallel flow. In this case, thegoal of the scaling is to achieve parallel flow. In order to do this the value of the Capillarynumber is changed by a factor 100. The expression for the capillary number is reiterated inequation 5.1. The scaling is achieved by increasing µ by a factor 10 and reducing σ by afactor 10, respectively the dynamic viscosity and interfacial surface tension. The density isalso increased by a factor 10 in order to keep the Reynolds number at the same value.

Ca =V iscous Forces

Surface Tention Forces=µv

σ(5.1)

This scaling did not lead to parallel flow, this means changing balance of forces did notproduces the expected results.

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5.3.3 Vector Field

The experimental results were not reproduced and scaling the capillary number did not signif-icantly influence the results. This means the model with these parameters does not producephysically accurate results. The most obvious cause for the discrepancies between the resultsand the experiments is the value of γ2. γ2 should be much smaller than the dimensions ofthe system, which is not the case in the simulations. A large boundary layer will effect thenormal vector field further away from the boundary. In order to further investigate this,figure 5.8 shows the regularized normal field and tangent field of two situations. Sub-figure5.8a shows a stable droplet in the channel and the corresponding regularized normal vectorfield and tangent field. Sub-figure 5.8b shows the break-up of a droplet at the begin of themain channel and the corresponding regularized normal vector field and tangent field. thenormal fields are represented by the red arrows while the tangent field is represented by theblue arrows. The situation show in sub-figure 5.8a shows how the model should function.The normal and tangent fields correspond to the interface and the angle of the interfacematches the Dirichlet boundary condition imposed at that boundary. At the receding partof the droplet the boundary condition is flipped. The situation illustrated in sub-figure 5.8bon the other hand, shows a mismatch between the shape of the interface and the vector fields.The size of the boundary region is too large which causes the vector fields to be influencedtoo far away from the boundary. This mismatch could be the cause for the droplet breakingat the start of the channel instead of parallel flow being established.

To validate this conclusion, the same situation is simulated with a smaller value of γ2.The simulations presented in figure 5.8 are done with a boundary region 40% of the heightof the channel. Figure 5.9 shows the same situation simulated with a boundary region thatis 20% of the height of the channel. This is still a substantial part of the domain but muchcloser to the value of εn. Sub-figure 5.9a shows the breakup of the first droplet in the mainchannel, similarly to sub-figure 5.8b. Although a droplet is still formed, it happens at a latertime and further in the main channel. This is not parallel flow, but decreasing γ2 does movethe results towards a transition to parallel flow. Sub-figure 5.9b illustrates the normal andtangent vector field corresponding to sub-figure 5.9a. It is clear that these vector fields matchthe interface much better than is the case with a larger γ2. However, the experiments showparallel flow should be found which is not the case. The reason γ2 was given a large valuein the first place, was to reduce the run-time of the simulation. Given the results presentedin section 5.3.1, it is not surprising the run-time of the simulation went up significantly.Changing γ2 from 40% to 20% increased the run-time from 7 to 40 hours. This increase inrun-time reduces the feasibility of the conservative level set method. Trying to further reduceγ2 did not yield better results. Long simulations did not converge and a short simulation of0.01 s took 32 hours.

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(a) Snapshot of simulation, a stable droplet in the channel. The corresponding regularized normalvector field and tangent field. The red arrows represent the normal field while the blue arrowsrepresent the tangent field. The fields correspond reasonably well to the situation.

(b) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The corre-sponding regularized normal vector field and tangent field. The red arrows represent the normal fieldwhile the blue arrows represent the tangent field. The fields do not match what should be expectedin this situation.

Figure 5.8: The numerical parameters have the following values: εn = 0.06∗D, γ1 =0.1 m∗s−1, εt = 1∗εn and γ2 = (6 + 2/3)∗εn. The flow rate of the aqueous fluid is 10 µL/minwhile the organic flow rate is 15 µL/min. (a) Droplet in microchannel. The level set variableφ and the regularized normal vector and tangent field are shown. (b) Droplet breaking at thestart of the microchannel. The level set variable φ and the regularized normal vector andtangent field are shown.

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(a) Snapshot of simulation, the breakup of a droplet at the start of the main channel.

(b) The regularized normal and tangent vectorfield corresponding to the breaking up of the dropletin (a). The red arrows show the regularized normal field. The blue arrows to the regularized tangentfield. The regularized normal and tangent vector field do not correspond to the interface.

Figure 5.9: The numerical parameters have the following values: εn = 0.06∗D, γ1 =0.1 m∗s−1, εt = 1∗εn and γ2 = (10/3)∗εn. The flow rate of the aqueous fluid is 10 µL/minwhile the organic flow rate is 15 µL/min. (a) Droplet breaking at the start of the microchan-nel. The level set variable φ is plotted. (b) The regularized normal vector and tangent fieldscorresponding to the situation in (a) are shown.

5.4 Comparison with Phase Field Method

One of the goals of the thesis is to compare the conservative level set method with the phasefield method. The conservative level set method with a diffusive interface boundary conditiondid not reproduce the results from the experiments. The transition from slug flow to parallelflow found in the experiments is not reproduced with the simulations. Increasing the inletvelocities or scaling the capillary number did not yield parallel flow. A boundary region thatis too large interferes with the interface far from the boundary. A smaller boundary regionon the other hand, leads to unacceptably high run-times. The phase field method is able to

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quantitatively describe the flow and qualitatively find the transition observed experimentally.

In order to compare the run-times of simulations preformed with the conservative levelset method with a diffusive boundary condition versus the phase field method, two situationsare assessed. The geometries, meshes, solvers and physical input parameters are identicalfor both methods. The first situation is used for the parameter study. This means the flowrate for both the organic and aqueous fluids are set to 5 µL/min. The simulation time is set to0.05s. The experiments predict the flow to be in transition between slug and parallel flow.The conservative level set method is set to the following numerical parameters: εn = 0.06∗D,γ1 = 0.1 m∗s−1, εt = 1∗εn and γ2 = 6∗εn. With these settings slug flow is found and the run-time is 4 hours and 45 minutes. The numerical parameters that are part of the phase fieldmethod are set to the following values: ε = 0.06∗W , χ = 1 s∗m∗kg−1 and µf = 0. With thesesettings, slug flow was found with a simulation run-time of 2 hours and 18 minutes. Breakupof the first droplet occurs at 0.032s for both simulations. The run-time with the conserva-tive level set method took about twice as long as the simulation with the phase field method.

The second situation is used for the assessment of the vectorfield. This means the flowrate for the organic fluid is set to 15 µL/min and aqueous fluid is set to 10 µL/min. The sim-ulation time is set to 0.03s. The experiments predict parallel flow at these inlet velocities.The conservative level set method is set to the following numerical parameters: εn = 0.06∗D,γ1 = 0.1 m∗s−1, εt = 1∗εn and γ2 = (6 + 2/3)∗εn. With these settings slug flow is foundand the run-time is 7 hours. If γ2 is reduced to 10/3 ∗ εn, the run-time is 40 hours. Thenumerical parameters that are part of the phase field method are set to the following values:ε = 0.06 ∗W , χ = 1 s∗m∗kg−1 and µf = 0. With these settings, parallel flow was found witha simulation run-time of 2 hours and 43 minutes. The level set method simulation found slugflow, which does not match the experimental results. Additionally, the phase field methodrun-time for this simulation is much shorter.

The phase field method outperforms the conservative level set method with a diffusiveboundary condition, both in terms of quality and run-time.

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Chapter 6

Channel Experiment: Slip BoundaryCondition

The description of the experiments is presented in section 5.1. This description is also validfor this chapter and therefore it is not reiterated here. This chapter starts by discussing theimplementation in COMSOL in section 6.1. Section 6.2 presents the simulation results afterwhich the results are compared with phase field method in section 6.3.

6.1 Implementation: Slip Boundary Condition

In Chapter 5 a diffusive interface boundary condition is used. In this chapter on the otherhand, a slip condition is employed at the boundary. The boundary condition is described insubsection 2.4.1.

6.1.1 Mesh

The mesh used to simulate the flow in the micro channel is very similar to the mesh usedin chapter 5. Initially, an adaptive mesh is used. The base mesh is a free triangular meshwith a maximum element size ranging from 10 µm to 20 µm and an element growth rate of1.05. The adaptive mesh is set to an element refinement of 3 and an element growth rateof 1.7. These simulations are compared to a static mesh build in the same way that wasdone in chapter 5. The refinement is applied a short distance before the two fluids meet atthe start of the channel. The maximum element size is set from 8 µm and 10 µm. Again,it is found that an adaptive mesh causes the simulations to have very long run-time. Asimulation using the full channel geometry with a flow rate of 3.3 µL/min takes 10 to 5 timeslonger with an adaptive mesh compared to the same simulation without an adaptive meshusing static refinement. The results are tabulated in table 6.1.

To determine the appropriate mesh size, the same simulation is performed with severalmesh settings. The results are shown in table 6.2. All meshes are refined up until 2000 µminto the main channel. If the mesh is too coarse mass dissipation can occur, as discussed in

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Table 6.1: The numerical parameters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1

and β = 2∗h. The flow rate of both fluids are set to 3.3 µL/min. Mesh settings per simulationsare shown. The simulation times is 0.05 s.

Max. elementsize

Max. elementgrowth rate

Adaptivemesh

Breakup Run-Time

8 [µm] 1.05 [−] No 0.045 [s] 15:46:13 [hr:min:sec]10 [µm] 1.05 [−] No 0.046 [s] 09:42:09 [hr:min:sec]10 [µm] 1.05 [−] Yes 0.044 [s] 98:04:35 [hr:min:sec]15 [µm] 1.05 [−] Yes 0.044 [s] 92:34:41 [hr:min:sec]20 [µm] 1.05 [−] Yes 0.044 [s] 86:54:36 [hr:min:sec]

section 5.2.3. At a maximum mesh size of 8 µm, the size of the droplet is conserved. Breakupof the droplet did occur slightly earlier when the mesh was refined. For this reason a meshsize of 6 µm is used to refine the two-phase flow region. A simulation with a maximumelement size of 4 µm produced exactly the same result.

Table 6.2: The numerical parameters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1

and β = 2∗h. The flow rate of both fluids are set to 3.3 µL/min. Mesh settings per simulationsare shown. The simulation time is 0.07 s.

Max. elementsize

Max. elementgrowth rate

Adaptivemesh

Breakup Run-Time

4 [µm] 1.05 [−] No 0.044 [s] 13:21:12 [hr:min:sec]6 [µm] 1.05 [−] No 0.044 [s] 10:54:24 [hr:min:sec]7 [µm] 1.05 [−] No 0.045 [s] 07:34:24 [hr:min:sec]8 [µm] 1.05 [−] No 0.045 [s] 10:34:49 [hr:min:sec]10 [µm] 1.05 [−] No 0.046 [s] 03:06:19 [hr:min:sec]

6.2 Results: Slip Boundary Condition

6.2.1 Parameter Study

The numerical parameters influence the behavior in the channel, the way the parametersindividually influence the results is discussed in section 2.6.1. In order to better understandthe consequences of these parameters in the micro channel, they are individually increasedor decreased. Details of these simulations are given in table 6.3. The base parameters for thesimulation are ε = 0.06∗D, γ1 = 0.2 m∗s−1 and β = 12 µm. Both the organic and aqueousfluid are set to an inlet velocity of 3 µL/min. Geometry 2 is used and the mesh is describedin section 6.1.1.

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ε controls the thickness of the interface. A smaller interface thickness leads to longersimulation times, this is confirmed by the results from the simulations. A much longer sim-ulation time is observed. However, the break-up of the first droplet is later than the othersimulations. This can be explained by the fact that ε also partly controls amount of artificialdiffusion that is applied. The time it takes to run a simulation with a very small ε is notfeasible compared to the phase field method. The run-time is also increased if ε is increased,which is somewhat surprising. It could be due to more difficult compression of the interfaceor the thickness of the interface being to close too the characteristic size of the system.

Table 6.3: Simulations where the numerical parameters are set to a different value in order toinvestigate the way they influence the results. The numerical parameters have the followingvalues: ε = 0.06∗D, γ1 = 0.2 m∗s−1 and β = 12 µm. Both the organic and aqueous fluidare set to an inlet velocity of 3 µL/min, the length of the simulation is 0.08 s. The time ofbreakup of the first droplet is recorded as well as run-time.

Parameter ValueDropletBreakup

Run-Time

ε 0.04 ∗W [m] 0.075 [s] 98:50:25 [hr:min:sec]ε 0.06 ∗W [m] 0.051 [s] 11:26:04 [hr:min:sec]ε 0.08 ∗W [m] 0.045 [s] 32:29:23 [hr:min:sec]γ1 0.1 [m∗s−1] dncγ1 0.2 [m∗s−1] 0.051 [s] 11:26:04 [hr:min:sec]γ1 0.3 [m∗s−1] 0.049 [s] 25:56:50 [hr:min:sec]β 2h/3 [m] 0.051 [s] 21:00:49 [hr:min:sec]β 2h [m] 0.051 [s] 11:26:04 [hr:min:sec]β 6h [m] 0.050 [s] 14:14:55 [hr:min:sec]

γ1 controls the amount of re-initialization in the form of the artificial compression andthe diffusion of numerical perturbations. If γ1 is too small, the simulation will not convergedue to the perturbations. This was the case for the simulation where γ1 = 0.1 m∗s−1. Alarge γ1 can lead to non-physical behavior of the interface. For this reason the value of 0.2is used for the rest of the simulation. The longer run-time observed when the value of γ1

is increased could be due to a more artificial compression and diffusion. The effect on thebreak-up of the droplet is small.

β controls the amount of slip the interface is allowed. The slip length has little effect onthe moment of breakup. COMSOL manual suggest the slip length should be proportionalto the size of the mesh, which is defined by h.

6.2.2 Flow Regime Guide

In order to validate the model, the experimental results should be reproduced. Figure 6.1shows the different flow regimes plotted versus the flow rate of the organic fluid on the x-axis

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Figure 6.1: Graph showing different flow types based on flow rate. On the x-axis the flowrate of the organic fluid is shown while the y-axis shows the flow rate of the aqueous fluid.The flow rate is given in µL per minute. The asterisk’s indicate slug flow, the circles a tran-sition between the two flow types and the plusses parallel flow. [1] The red squares representsimulation points. The line shows the path that is followed, as the flow rate is increased thetype of flow should change from slug flow to parallel flow.

and the flow rate of the aqueous fluid on the y-axis. The aim is to reproduce the graphwith the model. Figure 6.1 show the same flow regime chart. However, the points that aresimulated are added by red squares. The red line is the path followed. The simulations areperformed in geometry 2 with the mesh described in section 6.1.1. The mesh is refined forthe first 2000 µm of the main channel. The numerical parameters are set to the followingvalues: ε = 0.06∗D, γ1 = 0.2 m∗s−1 and β = 14 µm. As the flow rate is increased thetype of flow should change from slug flow to parallel flow. This change is not observed, allsimulations resulted in slug flow. The results are summarized in table 6.4. The table showsthe flow rates of the organic and aqueous fluid, the flow rates are equal to each other withthe exception of the last simulation. Then, the simulation time is shown in the next column.This is the time length of the simulation. The simulation time is based on the expectedmoment the two phase flow will reach the point where the refined mesh ends, i.e. 2000 µmin the main channel. The higher the flow rate the shorter the simulation time. Table 6.4also features the flow type and run-time.

In order to make sure β does not greatly influence the results, the value of β is changedsignificantly at a point where parallel flow should be observed. The flow-rate at this pointis equal for both fluids and is 15 µL/min. The value of β is set to respectively h/10, h, h ∗ 10and h ∗ 100. The value of h is 6 µm. ε and γ1 are not changed and set to ε = 0.06∗D,

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Table 6.4: The numerical parameters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1

and β = 12 µm. The flow rate of the aqueous fluid is varied per simulation. As the flow rateis increased, breakup occurs later in the channel. However, slug flow is still the type of flow.

FlowRate:AQ

FlowRate:ORG

SimulationTime

Flow Type Run-Time

1 [µL/min] 1 [µL/min] 0.24 [s] Slug 57:50:32 [hr:min:sec]2 [µL/min] 2 [µL/min] 0.12 [s] Slug 19:44:12 [hr:min:sec]3 [µL/min] 3 [µL/min] 0.08 [s] Slug 11:26:04 [hr:min:sec]5 [µL/min] 5 [µL/min] 0.05 [s] Slug 19:24:11 [hr:min:sec]7 [µL/min] 7 [µL/min] 0.038 [s] Slug 07:29:12 [hr:min:sec]9 [µL/min] 9 [µL/min] 0.027 [s] Slug 05:36:29 [hr:min:sec]11 [µL/min] 11 [µL/min] 0.022 [s] Slug 07:15:13 [hr:min:sec]13 [µL/min] 13 [µL/min] 0.019 [s] Slug 04:57:05 [hr:min:sec]15 [µL/min] 15 [µL/min] 0.016 [s] Slug 05:59:10 [hr:min:sec]15 [µL/min] 20 [µL/min] 0.016 [s] Slug 09:34:26 [hr:min:sec]

and γ1 = 0.2 m∗s−1. It is not possible to perform a similar analysis for the other numericalparameters because the simulation would then not converge, as demonstrated by the resultsin table 6.3. The results are shown in table 6.5. A larger β allows for more slip. This can beobserved as a larger β leads to earlier breakup. There is no difference between a value for βof 10 ∗ h and 100 ∗ h. This makes sense as the near wall velocity will basically become thevelocity at the wall. A value of h and h/10 results in slightly later breakup. However, theinfluence of β on the type of flow is small. Although the time of breakup is slightly different,the type of flow is still slug.

Table 6.5: Simulations where the value of β is set to different values in order to investigatethe way they influence the results. The other numerical parameters have the following basevalues: ε = 0.06∗D, and γ1 = 0.2 m∗s−1. Both the organic and aqueous fluid are set to aninlet velocity of 15 µL/min, the length of the simulation is 0.016 s. The time of the breakupof the first droplet is recorded as well as the run-time.

Parameter Value Breakup Flow Type Run-Timeβ h/10 [m] 0.011 [s] Slug 05:59:10 [hr:min:sec]β h [m] 0.011 [s] Slug 03:47:36 [hr:min:sec]β 10 ∗ h [m] 0.0098 [s] Slug 05:03:17 [hr:min:sec]β 100 ∗ h [m] 0.0098 [s] Slug 07:28:27 [hr:min:sec]

Although all simulations resulted in slug flow, the point where the droplets broke offshifted further into the main channel. Figure 6.2 shows the point of breakup for half thesimulated points from figure 6.1. In sub-figure 6.2a to 6.2c the breakup occurs at the actualstart of the channel while in sub-figure 6.2d and 6.2e the breakup occurs inside the channel

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itself. These flow rates should result in parallel flow. However, the shifting of the breakuppoint further into the channel is an indicator that a transition to parallel flow is startingto occur. Scaling the dimensionless numbers or ramping up the inlet velocities should yieldparallel flow. This way it is shown the model is quantitatively physically sound.

6.2.3 Scaling and Velocity

The scaling that is applied is the same as the scaling done in chapter 5. This boils down tochanging the capillary number by a factor 100. The other dimensionless numbers are keptthe same. This is achieved by changing the physical parameters in the following way: µ∗10,ρ ∗ 10 and σ/10. The second way parallel flow should be achieved, is increasing the inletvelocities for both fluids. Figure 6.3 features the results of both simulations. The numericalparameters have the following values: ε = 0.06∗D, γ1 = 0.2 m/s and β = 14 µm. Sub-figure6.3a shows the simulation where the capillary number is scaled. Parallel flow is observed. Byscaling the capillary number, the force balance in the channel is different. The fact that thescaling resulted in parallel flow shows that the model is quantitatively correct. By changingthe magnitude of the forces, the results can be influenced in a predicable way. Sub-figure6.3b shows the results of a simulation where both inlet velocities are set to 100 µL/min. Thisalso results in parallel flow. Again, this is an indication the model is quantitatively correctas increased velocity should lead to parallel flow. The question that remains is; what causesthe mismatch between the experimental results and the simulation?

6.2.4 Diffusion

An explanation can be found when the peclet number (Pe) is assessed. The peclet numberis the ratio between the diffusive transport and the convective transport. The diffusivetransport is artificial and is determined by the product of γ1 and ε. The convective transportis determined by the speed in the channel and the characteristic length in the channel, whichis the height of the channel. This relation is expressed in equation 6.1.

Pe =γ1ε

vL(6.1)

If the flow-rate of both fluids is around 3 µL/min, the Pe number will be one. A pecletnumber equal to one means the transport caused by the artificial diffusion is just as importantas the convective transport caused the flow. This could explain why the behavior of theinterface does not match the experimental results. Unfortunately, as the artificial diffusionis controlled by the aforementioned numerical parameters, it is not feasible to decrease theamount of artificial diffusion. If ε is decreased, the run-time is increased significantly. Also,as ε is reduced the interface thickness becomes smaller and a finer mesh is required. Thiswill also increase the run-time. Choosing a smaller value for γ1 will cause the simulation notto converge, as demonstrated in table 6.3.

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(a) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The flow rateis 3 µL/min for both the organic and the aqueous fluid.

(b) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The flow rateis 7 µL/min for both the organic and the aqueous fluid.

(c) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The flow rateis 11 µL/min for both the organic and the aqueous fluid.

(d) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The flow rateis 15 µL/min for both the organic and the aqueous fluid.

(e) Snapshot of simulation, the breakup of a droplet at the start of the main channel. The flow rateis 15 µL/min for the aqueous fluid, while the flow rate is 20 µL/min for the organic fluid

Figure 6.2: The numerical parameters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1

and β = 14 µm. The flow rate of the aqueous fluid is varied per simulation. As the flow rateis increased, breakup occurs later in the channel. However, slug flow is still the type of flow.

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(a) Snapshot of simulation, parallel flow in the start of the main channel. The flow rate is 15 µL/minfor the aqueous fluid, while the flow rate is 20 µL/min for the organic fluid

(b) Snapshot of simulation, stable parallel flow in the main channel. The flow rates are 100 µL/minfor both the aqueous and the organic fluid.

Figure 6.3: The numerical parameters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1

and β = 14 µm. (a) Results of simulation where the Capillary number is scaled. (b) Resultsof simulations where the flow rate of both fluids is significantly increased

6.3 Comparison with Phase Field Method

Although the conservative level set method with a slip length boundary condition is quanti-tatively correct, the results from the experiments cannot be reproduced. The transition fromslug flow to parallel flow found in the experiments is not reproduced with the simulations.The phase field method on the other hand is able to qualitatively find the transition observedexperimentally.

In order to compare the run-times of simulations preformed with the conservative levelset method with a slip length boundary condition versus the phase field method, two sit-uations are assessed. The geometries, meshes, solvers and physical input parameters areidentical for both methods. The first situation is used for the parameter study. This meansthe flow rate for both the organic and aqueous fluid are set to 3 µL/min. The simulationtime length is 0.08 s. The experiments predict slug flow. The conservative level set methodis set to the following numerical parameters: ε = 0.06∗D, γ1 = 0.1 m∗s−1 and β = 2 ∗ h.With these settings slug flow is found and the run-time is 11 hours and 26 minutes. Thenumerical parameters that are part of the phase field method are set to the following values:ε = 0.06 ∗W , χ = 1 s∗m∗kg−1 and µf = 0. With these settings, slug flow was found with asimulation run-time of 3 hours and 4 minutes. Breakup of the first droplet did occur at thesame time in both methods, at t = 0.051 s. The conservative level set method simulationrun-time is much longer than the phase field equivalent.

The second situation is used for the assessment of influence of β and part of the flow

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regime simulation. This means the flow rate for both the organic and aqueous fluid are setto 15 µL/min. The simulation time length is 0.02 s. The experiments predict parallel flowat these inlet velocities. The conservative level set method is set to the following numericalparameters: ε = 0.06∗D, γ1 = 0.1 m∗s−1 and β = h. With these settings slug flow is found,although breakup occurs later in the channel. The run-time of this simulation is 3 hoursand 47 minutes. The numerical parameters that are part of the phase field method are setto the following values: ε = 0.06 ∗W , χ = 1 s∗m∗kg−1 and µf = 0. With these settings,parallel flow was found with a simulation run-time of 1 hours and 3 minutes. The level setmethod simulation resulted in slug flow. The conservative level set run-time is almost fourtime longer. It should be noted that simulating slug flow typically takes longer than parallelflow. If χ is set to 100, the simulation results in slug flow and took 2 hours and 3 minutes.This is still less than the conservative level set method.

The phase field method outperforms the conservative level set method with a slip lengthboundary condition, both in terms of quality and run-time.

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Chapter 7

Conclusions and Recommendations

7.1 Conclusions

Section 1.5 introduced three main goals for this thesis. The research goals are reiteratedhere:

(I) Is the conservative level set method able to capture the different flow regimes in micro-channels, and how do the numerical parameters influence the results?

(II) Is the conservative level set method able to quantitative and qualitatively describecontact point behavior in the microchannel?

(III) How do the results from the conservative level set method compare to the results fromthe phase field method, and is the conservative level set method able to reduce thecomputational cost with respect to the phase field method?

The first research question focuses on whether the conservative level set method is able todescribe the different flow regimes in the microchannels. The conservative level set methodwith the diffusive interface method has four numerical parameters: εn, εn, γ1 and γ2. Theinterface is moved by diffusion. This method is unable to find the transition from slug flow toparallel flow found experimentally. Tweaking the numerical parameters, with inlet velocitiesthat experimentally led to parallel flow, did not result in parallel flow in the simulation. Thephysical parameters were scaled such that the Capillary number was increased by a factor100. This effectively reduces the effect of the surface tension force with respect to the viscousforces. Moreover, the inlet velocities were increased significantly from the point of transitionbetween slug and parallel flow was found experimentally. Both the scaling and the increasedvelocities did not yield parallel flow. The discrepancy between simulation and experimentis due to the numerical parameter γ2 being too large. This parameter controls the size ofthe boundary region. A large γ2 causes the interface to be effected by diffusion far from theboundary, resulting in nonphysical behavior. Reducing this parameter leads to very longsimulation run-times which are not feasible.

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The conservative level set with a slip boundary condition at the contact line was alsoused to reproduce the experimentally observed transition from slug flow to parallel flow.The interface now moves by an extrapolated speed based on a slip length parameter β.The other numerical parameters are γ1 and ε. Again, the transition was not reproducedin the simulations. Tweaking the numerical parameters was insufficient to bridge the dis-crepancy. Although the conservative level set method with a slip boundary condition at thecontact point was unable to qualitatively reproduce the experimental results, quantitativelythe model is correct. This was demonstrated by scaling the Capillary by a factor 100, whichresulted in parallel flow. Moreover, increasing the inlet velocities also resulted in parallelflow. This shows the model responds to the physical parameters the way that is expectedbased on physical arguments. Qualitatively reproducing the results was not possible. Thiscan only be achieved by lowering the amount of artificial diffusion in the system, whichmeans lowering either ε or γ1. Lowering ε increases run-time and requires a finer mesh toresolve. Decreasing γ1 on the other hand, leads to the simulation not converging.

The second research question focuses on whether the conservative level set method isable to capture the physics at the contact point. A benchmark is performed where a initiallyspherical droplet makes contact with a surface. The droplet spreads out towards an equi-librium position. The diffusive interface method was used as boundary condition for thisproblem. The simulation reproduced the behavior experimentally observed. However, thesimulated droplet radius at a set time slightly exceeded the radius observed during the exper-iments. Despite this fact, the simulation reflected the experimentally observed behavior well.

The last research question focuses on how the conservative level set method compareswith the phase field method. Both methods are able to quantitatively describe the spreadingdroplet benchmark. The run-times of both methods are comparable. In the microchannel onthe other hand, the phase field method was able to quantitatively and qualitatively capturethe flow regimes observed experimentally. In terms of quality, the phase field method out-performs the conservative level set method in simulations of the microchannel. Additionally,for simulations in the microchannel, the phase field method is faster than the conservativelevel set method. This was unexpected as the conservative level set method is generallyfound to be less computationally demanding in literature.

7.2 Recommendations

For simulations of the microchannel, the phase field method outperformed the conservativelevel set method. Therefore, it is more suitable for this specific situation. The treatmentof the boundary and the interface is different for both methods. The phase field methodpredicts leaking at the outlet that does not correspond to the experimentally observed be-havior. The conservative level set method could be used to assess the leakage at the endof the channel. Additionally, the conservative level set method performed better at higherflow rates. This causes the artificial diffusion transport to be smaller with respect to theconvective transport. If higher flow rates are investigated the conservative level set method

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could be used.

Also, the spreading droplet benchmark showed that in a geometry with less emphasis onthe boundary the conservative level set method performed equally well with respect to thephase field method. The comparison of the simulation run-times was inconclusive due to theshort simulation times. If a longer simulation is carried out in a similar geometry/setting,the conservative level set method could be used in an attempt to reduce the simulationsrun-times.

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[25] Minyao Zhou. Immiscible-fluid displacement: Contact-line dynamics and the velocity-dependent Capillart pressure. Physical Review, 45(8), 1992.

[26] COMSOL. COMSOL Multiphysics Reference Manual 5.2a. COMSOL Multiphysics,2016. Chapter 19.

[27] Sara Zahedi, Katarina Gustavsson, and Gunilla Kreiss. A conservative level set methodfor contact line dynamics. Journal of Computational Physics, 228(17):6361–6375, 2009.

[28] D. Jacmin. Contact-line dynamics of a diffuse fluid interface. Journal of Fluid Mechan-ics, 402:57–88, 2000.

[29] J. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modeling surfacetension. Journal of Computational Physics, 100(2):335–354, 1992.

[30] John W Cahn and John E Hilliard. Free Energy of a Nonuniform System .I. InterfacialFree Energy. Journal of Chemical Physics, 258(July), 1958.

[31] John W Cahn and John E Hilliard. Free Energy of a Nonuniform System .III. Nucleationin a Two Component Incompressible Fluid. Journal of Chemical Physics, 688(February),1959.

[32] Andreas Carlson. Capillarity and dynamic wetting. PhD thesis, Royal Institute ofTechnology, Stockholm Sweden, March 2012.

[33] J.-L. Guermond and L. Quartapelle. Calculation of Incompressible Viscous Flows by anUnconditionally Stable Projection FEM. Journal of Computational Physics, 132(1):12– 33, 1997.

[34] James C Bird, Shreyas Mandre, and Howard A Stone. Short-Time Dynamics of PartialWetting. Physical Review Letters, 234501(June):1–4, 2008.

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List of Figures

1.1 Decay chain of Mo-99. Most of the Mo-99 decays into Tc-99m by β-decay.Tc-99m is an istope frequently used for medical procedures. [2] . . . . . . . . 3

1.2 Activity of Mo-99 over the production process. The activity increases as thetargets are placed near the reactor. The moment the targets are removed isreferred to as the end of the bombartment (EOB). EOP stands for the endof production. The 6-day curie is indicated after the day the targets are setaside. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Schematic representation of continues extraction loop. The loop will be placedinside a reactor after which the produced molybdenum-99 can be extracted con-tiously. [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Several micro-scale unit operations (MUOs) are illustrated in the section onthe left. An example of a possible continuous flow chemical process (CFCP)is shown on the right-hand side. [6] . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Two most common flow types exhibited in a microchannel. Part A shows slugflow while part B shows parallel flow [14]. . . . . . . . . . . . . . . . . . . . . 7

1.6 Graph showing different flow types based on flow rate. On the x-axis theflow rate of the organic fluid is shown while the y-axis shows the flow rate ofthe aqueous fluid. The flow rate is given in µL per minute. The asterisk’sindicate slug flow, the circles a transition between the two flow types and thepluses parallel flow. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Illustrates pressure balance at the interface, ∆PL is the Laplace pressure, while∆PF is the pressure difference due to flow. The organic phase leads toward theaqueous due to the hydrophilicity of the glass. θ is the contact angle betweenthe fluids. θad is the advancing contact angle, θre is the receding contact angle [6]. 11

2.2 A schematic breakdown of the different contact angles at different scales. Thelargest scale is a macro setting where the angle θapp is the apperent anglebetween the droplet and the surface. The micro-scale is represented by thezooming done in (a). The contact angle θe is the microscopic contact angle.The situation at the molecular level is represented by area (b) [18]. . . . . . . 12

2.3 Schemetic illustration of slip boundray condition implemented in COMSOL.(a) Shows the equilibribum angle between the organic and aquaous fluid. (b)Shows the extrapolated slip velocity at the boundary based on the slip lenghtβ [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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2.4 Boundary Region in which the angle αs is imposed. αs is the micrsoscopiccontact angle. The size of the boundary region is proportional to the numericalparameter γ2 [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Normal and tangent vector field in microchannel. The curved line in the mi-crochannel represents the interface. The tangent field is shown in blue and thenormal field is shown in red. The angle at the wall is imposed by a Dirichletboundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Normal and tangent vector field in microchannel. The curved line in the mi-crochannel represents the interface. The tangent field is shown in blue and thenormal field is shown in red. The angle at the wall is imposed by a Dirichletboundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Example of an adaptive mesh. The thick line is the position of the interface,the grid cells become increasingly smaller as the interface is approached. . . . 33

3.4 Adaptive mesh used to simulate an oil droplet rising in water due to buoyancyforces. The mesh adapts several times to keep up with the oil-water interface.The times each mesh was used are shown in the captions. . . . . . . . . . . . 34

4.1 Illustration of the experiments performed by Bird et al [34]. There are fourdifferent equilibrium angles, from top to bottom: 3°, 43°, 117° and 180°. Thedeformation of the droplet is clearly visible for different times. . . . . . . . . 36

4.2 Results of simulations with the phase field method performed by Carslon [32]compared with the experimental results by Bird et al. [34]. On the y-axis thespreading radius is plotted against the time on the x-axis. The blue lines/pointscorrespond to an equilibrium angle of 117°, the black lines/points to an angleof 43° and the red lines/points to an angle of 3°. The open symbols representexperimental data and the closed symbols simulation data. The dashed linesare the results without the dynamic wetting condition. The full lines are theresults with the dynamic wetting condition. [32] . . . . . . . . . . . . . . . . 37

4.3 Geometries and corresponding meshes used during simulations. The alteringof the geometry results in a simpler mesh and shorter simulation times. Inthe second mesh the needle is lowered into the droplet. In the third case, thedroplet is also slightly lifted off the surface. . . . . . . . . . . . . . . . . . . . 39

4.4 Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird et al. [34]. On they-axis the spreading radius is plotted against the time on the x-axis. The greenlines correspond to an equilibrium angle of 117°, the blue lines to an angle of43° and the red lines to an angle of 3°. The closed lines represent experimentaldata and the dashed lines the simulation data. The simulations are performedwith the following numerical parameters; εn = 0.04∗D, γ1 = 0.1 m∗s−1, εt =3∗εn and γ2 = 5∗εn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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4.5 Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird. [34]. On the y-axisthe spreading radius is plotted against the time on the x-axis. The green linescorrespond to an equilibrium angle of 117°, the blue lines to an angle of 43° andthe red lines to an angle of 3°. The closed lines represent experimental dataand the dashed lines the simulation data. The simulations are performed withthe following numerical parameters; εn = 0.04∗D, γ1 = 0.06 m∗s−1, εt = 1∗εnand γ2 = 2.5∗εn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Results of simulations using the conservative level set method performed withCOMSOL compared with the experimental results by Bird. [34]. On the y-axisthe spreading radius is plotted against the time on the x-axis. The green linescorrespond to an equilibrium angle of 117°, the blue lines to an angle of 43° andthe red lines to an angle of 3°. Then, the closed lines represent experimentaldata and the dashed lines the simulation data. The simulations are performedwith the following numerical parameters; εn = 0.04∗D, γ1 = 0.06 m∗s−1.The simulations with an equilibrium angle of 3° and 43° are performed withεt = 1∗εn and γ2 = 2.5∗εn, while the 117° case has a εt = 3∗εn and γ2 = 5∗εn. 45

4.7 Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method.The experimental results by Bird et al. [34] are also plotted in the graph. Onthe y-axis the spreading radius is plotted against the time on the x-axis. Theequilibrium angle is 3°. The open symbols represent experimental data, theclosed symbols the conservative level set results and the yellow symbols thephase field method results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.8 Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method.The experimental results by Bird et al. [34] are also plotted in the graph. Onthe y-axis the spreading radius is plotted against the time on the x-axis. Theequilibrium angle is 43°. The open symbols represent experimental data, theclosed symbols the conservative level set results and the yellow symbols thephase field method results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 Results of simulations with the phase field method performed by Carslon [32]compared with results of simulations with the conservative level set method.The experimental results by Bird et al. [34] are also plotted in the graph. Onthe y-axis the spreading radius is plotted against the time on the x-axis. Theequilibrium angle is 117°. The open symbols represent experimental data, theclosed symbols the conservative level set results and the yellow symbols thephase field method results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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5.1 Pictures of the microchip used in the experiments. (a) Experimental set-upwhen in use, the tubes are used to feed the inlets/outlets. A blue methylenesolution is flowing through the chip for the purpuse of highlighting the chan-nels. The scale provided at the top of the image is in centimeters. [1] (b) Aschematic cross section of the microchannel. . . . . . . . . . . . . . . . . . . 50

5.2 Graph showing different flow types based on flow rate. On the x-axis the flowrate of the organic fluid is shown while the y-axis shows the flow rate of theaqueous fluid. The flow rate is given in µL per minute. The asterisk’s indicateslug flow, the circles a transition between the two flow types and the plussesparallel flow [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Geometry 1 as implemented in COMSOL. The width of the channel is 100 µmand the depth is 40 µm. The chip is symmetrical and the inlet/outlet sectionsare 0.5 cm with an angle of 13° with respect to the main channel. The mainchannel is 2 cm long. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 Geometry 2 as implemented in COMSOL. The width of the channel is 100 µmand the depth is 40 µm. The inlet sections are 0.25 cm with an angle of 13°with respect to the main channel. The main channel is 0.5 cm long. The righthand side of the main channel is the output in this geometry. . . . . . . . . . 52

5.5 Change made to geometry to improve convergence. . . . . . . . . . . . . . . . 525.6 Implementation of the flipping of the boundary condition. (a) shows the value

of φ. (b) value of the derivative of φ with respect to x on the lower boundary.(c) result of the if-else condition. . . . . . . . . . . . . . . . . . . . . . . . . 54

5.7 (a) Mesh used to find required mesh size.(b) Example of simulation with amesh that is to fine. (c) Example of simulation with the required mesh size. . 55

5.8 The numerical parameters have the following values: εn = 0.06∗D, γ1 =0.1 m∗s−1, εt = 1∗εn and γ2 = (6 + 2/3)∗εn. The flow rate of the aqueousfluid is 10 µL/min while the organic flow rate is 15 µL/min. (a) Droplet in mi-crochannel. The level set variable φ and the regularized normal vector andtangent field are shown. (b) Droplet breaking at the start of the microchannel.The level set variable φ and the regularized normal vector and tangent fieldare shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.9 The numerical parameters have the following values: εn = 0.06∗D, γ1 =0.1 m∗s−1, εt = 1∗εn and γ2 = (10/3)∗εn. The flow rate of the aqueousfluid is 10 µL/min while the organic flow rate is 15 µL/min. (a) Droplet breakingat the start of the microchannel. The level set variable φ is plotted. (b) Theregularized normal vector and tangent fields corresponding to the situation in(a) are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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6.1 Graph showing different flow types based on flow rate. On the x-axis the flowrate of the organic fluid is shown while the y-axis shows the flow rate of theaqueous fluid. The flow rate is given in µL per minute. The asterisk’s indicateslug flow, the circles a transition between the two flow types and the plussesparallel flow. [1] The red squares represent simulation points. The line showsthe path that is followed, as the flow rate is increased the type of flow shouldchange from slug flow to parallel flow. . . . . . . . . . . . . . . . . . . . . . . 65

6.2 The numerical parameters have the following values: ε = 0.06∗D, γ1 =0.2 m∗s−1 and β = 14 µm. The flow rate of the aqueous fluid is varied persimulation. As the flow rate is increased, breakup occurs later in the channel.However, slug flow is still the type of flow. . . . . . . . . . . . . . . . . . . . 68

6.3 The numerical parameters have the following values: ε = 0.06∗D, γ1 =0.2 m∗s−1 and β = 14 µm. (a) Results of simulation where the Capillarynumber is scaled. (b) Results of simulations where the flow rate of both fluidsis significantly increased . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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List of Tables

4.1 Results of simulations testing different geometries. Four different equilibriumangles are tested in all three geometries, being 3°, 43°, 117° and 180°. For eachsimulation the droplet radius after 2 milliseconds and the run-time is recorded.In this simulation the following parameters are set as follows; εn = 0.04∗D,γ1 = 0.1m/s, εt = 3∗εn and γ2 = 5∗εn. . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Specification of several mesh sizes. The maximum element size, minimumelement size and maximum element growth rate are shown. These pre-setmeshes are used to determine the required mesh size. . . . . . . . . . . . . . 40

4.3 Several meshes that are used to simulate the spreading of the droplet. Theequilibrium angle for this series was 117°. The final droplet radius after 2 mshas stabilized with the increased mesh density. . . . . . . . . . . . . . . . . . 41

4.4 Simulations results where the numerical parameters are set to a different valuein order to investigate the way they influence the results. The simulations aredone with an equilibrium angle of 43°. The base values are εn = 0.04∗D, γ1 =0.1 m∗s−1, εt = 3∗εn and γ2 = 5∗εn. The droplet radius after 2 millisecondsand the run-time are recorded. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Simulations where the phase field simulation is compared to the level set method.The numerical parameters for the phase field methods are; εpf = 0.04∗D.The numerical parameters for the phase field methods are; εn = 0.04∗D,γ1 = 0.06 m∗s−1, εt = εn and γ2 = 2.5∗εn . . . . . . . . . . . . . . . . . . . . 48

5.1 Simulations where the numerical parameters are set to a different value inorder to investigate the way they influence the results. The flow rate of bothfluids is 5 µL/min. The base values are εn = 0.06∗D, γ1 = 0.1 m∗s−1, εt = 3∗εnand γ2 = 6∗εn. The time of the breakup of the first droplet is recorded as wellas the run-time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1 The numerical parameters have the following values: ε = 0.06∗D, γ1 =0.2 m∗s−1 and β = 2 ∗ h. The flow rate of both fluids are set to 3.3 µL/min.Mesh settings per simulations are shown. The simulation times is 0.05 s. . . 63

6.2 The numerical parameters have the following values: ε = 0.06∗D, γ1 =0.2 m∗s−1 and β = 2 ∗ h. The flow rate of both fluids are set to 3.3 µL/min.Mesh settings per simulations are shown. The simulation time is 0.07 s. . . . 63

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6.3 Simulations where the numerical parameters are set to a different value inorder to investigate the way they influence the results. The numerical param-eters have the following values: ε = 0.06∗D, γ1 = 0.2 m∗s−1 and β = 12 µm.Both the organic and aqueous fluid are set to an inlet velocity of 3 µL/min, thelength of the simulation is 0.08 s. The time of breakup of the first droplet isrecorded as well as run-time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 The numerical parameters have the following values: ε = 0.06∗D, γ1 =0.2 m∗s−1 and β = 12 µm. The flow rate of the aqueous fluid is varied persimulation. As the flow rate is increased, breakup occurs later in the channel.However, slug flow is still the type of flow. . . . . . . . . . . . . . . . . . . . 66

6.5 Simulations where the value of β is set to different values in order to investi-gate the way they influence the results. The other numerical parameters havethe following base values: ε = 0.06∗D, and γ1 = 0.2 m∗s−1. Both the organicand aqueous fluid are set to an inlet velocity of 15 µL/min, the length of thesimulation is 0.016 s. The time of the breakup of the first droplet is recordedas well as the run-time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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