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Fakultät für Mathematik

Detection of particles transported in weakly

compressible ﬂuids: mathematical models,

analysis, and simulations

Thomas Georg Amler

Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität München

zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Gero Friesecke, Ph. D.

Prüfer der Dissertation: 1. Univ.-Prof. Dr. Dr. h.c. mult. Karl-Heinz Hoffmann

2. Univ.-Prof. Dr. Herbert Spohn

ˇ3. Prof. Dr. Pavel Krejcí, Czech Academy of Sciences,

Prag / Tschechien

(schriftliche Beurteilung)

Die Dissertation wurde am 23.11.2010 bei der Technischen Universität München eingereicht und

durch die Fakultät für Mathematik am 07.01.2011 angenommen.Abstract

In this thesis, the problem of detecting small particles dispersed in air is considered. A method

for the quantitative measurement of the particles, which is studied here, was developed at the

research institute CAESAR in the course of the European integrated project NANOSAFE2. We

investigate two issues: the transport of particles by air to a washing ﬂask where the particles are

being immersed in water and motion of in water ﬂowing through a wet cell having an

active boundary part responsible for the measurement.

For the transport of particles, a mathematical model that describes the evolution of the ﬂow, the

motion of dispersed and the interaction between particles and air is derived. Thus, this

model is related to a two-component ﬂow problem. Under certain assumptions, the existence and

uniqueness of weak solutions to the governing initial-boundary value problem on a non-empty

time interval is shown. This result is established using a ﬁxed-point technique.

For the measurement of particles, we ﬁrst derive a coupled initial-boundary value problem

that describes the evolution of the ﬂow, particle density, and surface mass density of measured

particles, and the interaction between particles and water. The surface mass of

particles is described via a boundary condition of hysteresis type on the particle density posed on

the active part of the wet cell. To investigate the derived model theoretically, the inﬂuence of the

particles on the water is neglected. Thereby the whole problem is divided into two sub-problems,

the ﬂow problem and the evolution of particle density, so that the velocity and pressure can be

found independently of the particle density. The existence of weak solutions is proved on a non-

empty time interval determined by the data of the ﬂow problem. The uniqueness is proved under

the assumption that the divergence of the velocity ﬁeld is essentially bounded. The existence and

uniqueness of weak solutions to the evolution of the particle density can be shown in the case of

arbitrary ﬁnite time intervals, provided that the velocity ﬁeld is sufﬁciently regular.

Finally, the numerical simulation of the model of measurement in the case of full coupling

is described. We propose a scheme for the numerical solution of the model equations using

the ﬁnite element method. The numerical behavior of the proposed scheme is discussed for

some selected examples. First simulations of the measurement in the wet cell in two and three

dimensions are presented.

iiiAcknowledgements

This thesis could not have been realized without the support of several people. My gratitude goes

to all of them who directly or indirectly contributed to the completion of this work.

I want to thank my adviser Prof. Karl-Heinz Hoffmann for giving me the possibility to work on

this topic, and for his guidance since the diploma thesis. Besides his help in functional respects, I

would like to mention in particular the encouragement to apply for a scholarship, and the support

during the time between my diploma and the beginning of the scholarship.

My deep gratitude goes to Dr. Nikolai Botkin for his dedicated help. I could learn a lot during

discussions with him and proﬁt from his motivation. His support in all issues regarding the

modeling, theory, numerics, and improvement of my English was important for the completion

of this thesis.

It is a pleasure to thank Prof. Pavel Krejcíˇ for writing a report on the thesis and helping me

with the “evolution of the particle density problem”. His insights in anisotropic embeddings

were essential to show the uniqueness of weak solutions to this problem.

My special thank goes to my colleague and carpool partner Jürgen Frikel for proofreading parts

of the manuscript, and for his patience in discussing about open questions with me. Thereby,

some problems could be solved on the way to the university or back.

I am thankful to Dr. Lope A. Flórez Weidinger and Dr. Luis Felipe Opazo from Göttingen for

providing me with several overview papers about aptamers and their possible applications.

Furthermore, I want to thank Florian Drechsler for proofreading parts of the manuscript and

correcting some of my mistakes in English.

My thank also goes to Prof. Hans Wilhelm Alt for the time he spent in discussions with me

and his helpful orientation.

I also want to thank the Chair of Mathematical Modeling at the Technical University of Munich

for providing a stimulating atmosphere, and the hard- and software I could use to complete the

thesis.

Moreover, I would like to acknowledge the support from the Foundation of German Business

(Stiftung der Deutschen Wirtschaft, sdw); I was granted a scholarship for doctoral candidates.

I am deeply thankful to my family for their support, patience and encouragement, when it was

needed.Contents

Abstract iii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Description of the detection procedure . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Brief overview of conventional models and methods . . . . . . . . . . . . . . . . 4

1.4 Objectives and description of the results obtained . . . . . . . . . . . . . . . . . 7

1.4.1 Derivation of mathematical models . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Theoretical investigations . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.3 Numerical computations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Derivation of mathematical models 12

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Presentation of the mathematical models . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Motion of weakly compressible ﬂuids . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Transport of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Motion of a single particle . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2 Averaged motion of particles . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.3 Interaction with the liquid . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.4 A simpliﬁed model for the particle transport . . . . . . . . . . . . . . . . 25

2.5 Measurement of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Mathematical description of the active part of the wet cell . . . . . . . . 26

2.5.2 Evolution of the particle density . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3 Inﬂuence of the particles on the liquid . . . . . . . . . . . . . . . . . . . 31

2.6 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.A The Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.B The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.C Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.D The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.D.1 Connection to macroscopic quantities . . . . . . . . . . . . . . . . . . . 42

3 Theoretical investigations 44

3.1 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Used methods and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 The transport problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Representation of the pressure and the particle density . . . . . . . . . . 48

3.3.2 The convective term and the regularity of the right-hand side . . . . . . . 50

3.3.3 Existence and uniqueness of solutions to the auxiliary problem . . . . . . 51

3.3.4 Fixed-point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

viContents

3.4 The decoupled measurement problem . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1 The ﬂow problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4.1.1 Construction of approximate solutions . . . . . . . . . . . . . 62

3.4.1.2 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.1.3 Passage to the limit and additional regularity . . . . . . . . . . 65

3.4.1.4 Regularity of the right-hand side . . . . . . . . . . . . . . . . 67

3.4.1.5 Fixed-point method . . . . . . . . . . . . . . . . . . . . . . . 71

3.4.2 Evolution of the particle density . . . . . . . . . . . . . . . . . . . . . . 75

3.4.2.1 Construction of approximate solutions . . . . . . . . . . . . . 77

3.4.2.2 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.2.3 Passage to the limit . . . . . . . . . . . . . . . . . . . . . . . 87

3.4.2.4 Representation of the trace . . . . . . . . . . . . . . . . . . . 90

3.4.2.5 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.4.2.6 An anisotropic embedding . . . . . . . . . . . . . . . . . . . . 95

3.A Elementary inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.B Gronwall type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.C Hilpert’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.D Convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.E Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.F Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.G Results on the solvability of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.G.1 Elliptic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.G.2 Monotone operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.H The conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Numerical Simulations 114

4.1 Discretization scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.1.1 Discretization of the particle system . . . . . . . . . . . . . . . . . . . . 116

4.1.2 of the ﬂow problem . . . . . . . . . . . . . . . . . . . . . 117

4.2 Computation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2.1 Regularization of the hysteresis boundary condition . . . . . . . . . . . . 120

4.2.2 Comparison of geometries . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.3 Simulation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 127

4.2.4 in three . . . . . . . . . . . . . . . . . . . . . . . 130

Conclusion 132

vii1 Introduction

The present thesis is devoted to the detection of small particles. This problem has been studied at

the research institute CAESAR in the course of the European integrated project NANOSAFE2 –

Safe production and use of nanomaterials. Within the project, CAESAR investigated the possi-

bility of capturing and detecting nanoparticles immersed in aqueous solvents by means of tech-

niques based on speciﬁcally binding peptides, see [69]. We will focus on two components of the

developed technique and consider mathematical modeling and analysis of the derived models.

We will then present results of numerical simulations of one of these models.

We will distinguish the problems of detection and pure transport of particles in a ﬂowing

medium. The reason for this distinction is that the detection procedure developed at CAESAR is

divided into two sub-processes: the washing out of particles and their measurement. These two

sub-processes explained in Section 1.2 are modeled in different ways.

This chapter is structured as follows: the motivation is given in Section 1.1, the detection

method developed at CAESAR is described in Section 1.2. Some conventional models and

methods are reviewed in Section 1.3, and the objectives of the thesis and the contents of the

following chapters are summarized in Section 1.4.

1.1 Motivation

The detection of particles gained special interest in the last decades when the possibilities of

nanotechnology were discovered. The ability to tailor material properties at nanoscale enabled

the engineering of novel materials that have entirely new properties, which led to new research ar-

eas and to the development of new commercially available products. With only a reduction of size

the fundamental characteristics of substances such as electrical conductivity, colour, strength, and

melting point – properties which are usually considered constant for a given material – can all

change. Therefore, nanomaterials show promising application potentials in a variety of ﬁelds

such as chemistry, electronics, medicine, cosmetics or the food sector. For example, metal oxide

nanopowders have found already increasing applications in commercial products like sunscreens,

cosmetics, catalysts, functional coatings, medical agents, etc.

However, not only its large potential was recognized but also sceptical voices concerning

nanotechnology could be heard in public. One of the sharpest critics of industrial nanoparticle

applications is the non-governmental organisation ETC Group. However, the fear of risk asso-

ciated with nanoparticle use was mainly caused by limited scientiﬁc knowledge about potential

side effects of nanoparticles in the human body and the environment due to their special proper-

ties. They may, for example, penetrate into body cells and break through the blood-brain barrier

[42]. See also [68].

The objectives of the European project NANOSAFE were to assemble available information

from public and private sources on chances and possible hazards involving industrial nanopar-

ticle production, to evaluate the risks to workers, consumers and the environment, and to give

recommendations for setting up regulatory measures and codes of good practice to obviate any

11 Introduction

danger [42]. The research on nanoparticles was continued in a second project, NANOSAFE2.

Within NANOSAFE2, 25 partners from industry, research centers and universities work on four

sub-projects: detection and characterization techniques, health hazard assessment, and develop-

ment of secure industrial production systems and safe applications, societal and environmental

aspects.

As one of the participants, the research institute CAESAR has developed a peptide based

biosensor for detecting nanoparticles. Besides this approach, other detection methods have been

investigated, for example, light scattering-based techniques or techniques based on different

physical principles such as electrostatics, thermophoresis, bubbling, vapour condensation, etc.

[70].

In the present thesis, two steps of the detection procedure developed at CAESAR will be

considered from the mathematical point of view. We will derive mathematical models to describe

the physical processes, analyse the solvability issue, and present simulation results for one of the

derived models. Before describing the detection plant we are going to model, we mention some

“classical applications” of nanomaterials.

The particular properties of small particles have been exploited by humans since prehistory

but without speciﬁc knowledge. Famous and perhaps surprising examples are objects made of

clay, a highly stable blue pigment the Mayas used to paint their ﬁgures, Damascus blades, or the

brilliant red colour of some church windows.

Clay largely consists of the mineral kaolinite, which has the structure of thin platelets, only a

few tens of nanometers thick. These slide readily over each other when the mineral has absorbed

water whereby clay becomes smeary and easily shapeable. From the eighth century on the Mayas

were able to paint their clay ﬁgures with a blue pigment that could resist the ravages of time.

They synthesized an inorganic-organic nanocomposite consisting of palygorskite, another clay

mineral also known as “mountain leather”, and an organic indigo pigment. This highly stable

old pigment is now again being produced in the USA by MCI Mayan Pigments, Inc. Damascus

blades were renown in the Middle Ages for their ﬁligree markings, their sharpness, and their

fracture toughness. For a long time modern metallurgy could not ﬁnd a scientiﬁc explanation for

these properties. Only at the end of 2006, carbon nanotubes could be found in the blades. This

nanowire reinforcement at least explains their fracture toughness. In the Middle Ages church

windows were coloured using an extremely ﬁne, nano-scale dispersion of gold. This causes a

brilliant red colour that endures for centuries [53].

1.2 Description of the detection procedure

Figure 1.2.1 shows schematically a device developed at CAESAR for the detection of particles.

The considered device was constructed in particular to speciﬁcally detect large organic molecules

in air. An organic molecule is a chain consisting of many links connected by ﬂexible bonds so

that the molecule can assume different conﬁgurations.

Before such particles can be detected, they are prepared as shown in Figure 1.2.1.b. Air

containing particles is injected into a vessel. The air ﬂow from the inlet to the outlet transports

the particles into the water quench of a washing ﬂask. Concurrently a loudspeaker generates

acoustic waves to prevent the particles from the deposition on the bottom of the vessel. In the

washing ﬂask, the particles are washed out of the air into the water when the air bubbles rise to

the water surface. After a certain time, the water contains a signiﬁcant amount of particles, and

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