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Technische Universität München
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
2