1 

Diffusion in Liquids: Equilibrium Molecular Simulations and Predictive Engineering Models
The aim of this thesis is to study multicomponent diffusion in liquids using Molecular Dynamics (MD) simulations. Diffusion plays an important role in mass transport processes. In binary systems, mass transfer processes have been studied extensively using both experiments and molecular simulations. From a practical point of view, systems consisting more than two components are more interesting. However, experimental and simulation data on transport diffusion for such systems are scarce. Therefore, a more detailed knowledge on mass transfer in multicomponent systems is required. The presence of multiple components in a system introduces difficulties in studying diffusion in experiments. Investigating the concentration dependence of diffusion coefficients seriously increases the required experimental effort. In this thesis, we will use MD simulation based on classical force fields to study multicomponent diffusion in liquids. Diffusion can be described using both Fick and Maxwell Stefan (MS) diffusion coefficients. Experiments provide Fick diffusion coefficients while simulations usually provide MS diffusion coefficients. Fick and MS diffusivities are related via the matrix of thermodynamic factors. A brief survey on methods for studying liquid diffusion and their limitations is presented in chapter 1
In chapter 2, we study the diffusion in the ternary system nhexanecyclohexanetoluene. The existing models for predicting MS diffusivities at finite concentrations (i:e: the Vignes equation) as well as the predictions at infinite dilution (i:e: predictions of Ðxk!1 i j using the socalled WK, KT, VKB, DKB and RS models) are tested using MD simulations. We find that (1) the Vignes equation only results in reasonable predictions for MS diffusivities yielding differences of 13% compared to the actual diffusion coefficients; (2) the best predictive model (the KT model) for calculating MS diffusivities at infinite dilution results in differences of 8% compared to the actual diffusion coefficients. It is important to note that the differences of 8% can be a coincidence since KT model is empirical and does not have a theoretical basis. This limitation makes KT model unreliable for other systems.
To overcome the difficulties in predicting ternary MS diffusivities at infinite dilution (i:e: Ðxk!1 i j ), we derive the socalled LBV model based on the Onsager relations. MS diffusivities at infinite dilution can be expressed in terms of binary and pure component selfdiffusivities and integrals over velocity crosscorrelation functions. By neglecting the latter terms, we obtain the LBV model. In chapter 3, the LBV model is validated for WCA fluids and the ternary systems nhexanecyclohexanetoluene and methanolethanolwater. We find that: (1) for ideal mixtures i:e: the WCA system, as well as the nhexanecyclohexanetoluene system, the LBV model is accurate and superior compared to the existing models for predicting ternary MS diffusivities at infinite dilution (i:e: the WK, KT, VKB, DKB and RS models); (2) in mixtures containing associating components, i:e: the ethanolmethanolwater system, the LBV model indicates that in this system the integrals over velocity crosscorrelation functions are important and cannot be neglected. Moreover, the LBV model provides an explanation why the MS diffusivity describing the friction between adsorbed components in a porous material is usually very large.
In chapter 4, we focus on describing the values of MS diffusivities at finite concentration. A multicomponent Darken model for describing the concentration dependence of MS diffusivities is derived from linear response theory and the Onsager relations. In addition, a predictive model for the required selfdiffusivities in the mixture is proposed leading to the socalled predictive DarkenLBV model. We compare our novel models to the existing generalized Vignes equation and the generalized Darken equation. Two systems are considered: (1) ternary and quaternary WCA systems; (2) the ternary system nhexanecyclohexanetoluene. Our results show that in all studied systems, our predictive DarkenLBV equation describes the concentration dependence better than the existing models. The physicallybased DarkenLBV model provides a sound and robust framework for prediction of MS diffusion coefficients in multicomponent mixtures.
In chapter 5, diffusion in more complex ionic liquid (IL) systems are investigated. Previous research reported in literature has largely focused on selfdiffusion in ILs. For practical applications, mutual (transport) diffusion is by far more important than selfdiffusion. We compute the MS diffusivities in binary systems containing 1alkyl 3 methylimidazolium chloride (CnmimCl), water and/or dimethyl sulfoxide (DMSO). The dependence of MS diffusivities on mixture composition are investigated. Our results show that: (1) For solutions of ILs in water and DMSO, selfdiffusivities decrease strongly with increasing IL concentration. For the system DMSOIL, an exponential decay is observed for this; (2) For both waterIL and DMSOIL, MS diffusivities vary by a factor of 10 within the concentration range which is still significantly smaller than the variation of the self diffusivities; (3) The MS diffusivities of the investigated IL are almost independent of the alkyl chain length; (4) ILs stay in a form of isolated ions in CnmimClH2O mixtures, however, dissociation into ions is much less observed in CnmimClDMSO systems. This has a large effect on the concentration dependence of MS diffusivities; (5) The LBV model for predicting the MS diffusivity at infinite dilution described in chapter 3 suggests that velocity crosscorrelation functions in ionic liquids cannot be neglected and that the dissociation of ILs into ion pairs has a very strong influence on diffusion.
In experiments, Fick diffusion coefficients are measured and molecular simulation usually provides MS diffusivities. These approaches are related via the matrix of thermodynamic factors which is usually known only with large uncertainties. This leaves a gap between theory and application. In chapter 6, we introduce a consistent and efficient framework for the determination of Fick diffusivities in liquid mixtures directly from equilibrium MD simulations by calculating both the thermodynamic factor and the MS diffusivity. This provides the missing step to extract Fick diffusion coefficients directly from equilibrium MD simulations. The computed Fick diffusivities of acetonemethanol and acetonetetrachloromethane mixtures are in excellent agreement with experimental values. The suggested framework thus provides an efficient route to model diffusion in liquids based on a consistent molecular picture.
In chapter 7, we validate our method for computing Fick diffusivities using equilibrium MD simulations for the ternary system chloroform  acetone  methanol. Even though a simple molecular model is used (i:e: rigid molecules that interact by LennardJones and electrostatic interactions), the computed thermodynamic factors are in close agreement with experiments. Validation data for diffusion coefficients is only available for two binary subsystems. In these binary systems, MD results and experiments do agree well. For the ternary system, the computed thermodynamic factors using Molecular Dynamics simulation are in excellent agreement with experimental data and better than the ones obtained from COSMOSAC calculations. Therefore, we expect that the computed Fick diffusivities should also be comparable with experiments. Our results suggest that the presented approach allows for an efficient and consistent prediction of multicomponent Fick diffusion coefficients from MD simulations. Now, a tool for guiding experiments and interpreting multicomponent mass transfer is available.

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2 

Depth and Motion Information for 2D Image Sequences
Diffusion curves as defined by Orzan et al. (2008) have so far been only applied to single, static images. We propose a system to construct diffusion curves over a sequence of images. These curves can then be used to construct depth and motion information to augment the image sequence.

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3 

Further analysis of sand concentration distributions based on a diffusion concept
Morphological changes of coastal areas are caused by sediment transport. Sediment transport is one of
the most important subjects in coastal engineering research. Various models are available to predict
the sediment transport rate as a function of wave height, current velocity and grain size. The reliability
of these models is unknown, because data under field conditions are scarce. Also the sediment
transport processes are not fully understood. Considering that in the coastal areas the waves are
superimposed on a current (longshore current), the problem becomes more complicated.
In this study a sediment transport model based on a diffusion concept is used. The sediment transport
is based on the product of a velocity profile u(z) and a sediment concentration profile c(z) over the
depth, resulting in a sediment transport profile over the water depth. The concentration profile can be
described by a bed concentration and a shape of a concentration distribution over the water depth. To
calculate the concentration distribution over the depth, a diffusion (or mixing) coefficient distribution
can be used. To predict the sediment transport it is important to understand the influence of wave
height, current velocity, water depth and grain size on the diffusion coefficient distribution (&8distribution).
It is tried in this study to determine the reliability of the diffusion model for predicting the sediment
concentration over the height. Nine existing diffusion coefficient methods (methods to describe the
shape of the diffusion coefficient distribution) are used, on several sets of experiments, to determine
the performance of these diffusion coefficient methods. Each ofthe methods can be determined with 2,
3 or 4 parameters (degrees of freedom). For the five best performing diffusion coefficient methods the
influence of the boundary conditions on the involved parameters of the diffusion coefficient methods
is determined. It shows that the diffusion coefficient method with 4 degrees of freedom performs best.
However, the difference with the diffusion coefficient methods with 3 degrees of freedom is not very
large. The influence of the boundary conditions on the involved parameters (degrees of freedom) of
the diffusion coefficient methods is not always what one expects. One parameter does not show very
much consistency with the wave height or current velocity. There can not be made a distinction
between the five best performing methods, since these methods perform approximately the same for
calculating the sediment concentration over the height.
With the calculated diffusion coefficient distributions, the velocity profiles of certain experiments
(experiments with current wave angle of 90°) are determined. The measured velocity profiles are
compared with the calculated velocity profiles. It shows that for most experiments the calculated
velocity profiles do not fit to the measurements very well. The resulting bottom shear stresses for the
experiments are discussed. It appears that the bottom shear stress decreases for increasing wave height (for most experiments). This result is unexpected. To try and fit the velocity measurements better, three diffusion coefficient methods are restricted by fixing one of their degrees of freedom on a certainvalue. The result ofthis restriction is still an acceptable fit of the calculated concentration profiles with the measured concentrations. However, the fit of the calculated velocity profiles with the velocitymeasurements increases substantially in quality. Nevertheless, the unexpected result of a decreasing bottom shear stress with increasing wave height remains.
For the best performing method (of the current wave angle 90° experiments) the calculations are
carried out on other experiments (current and wave angle 180°). The results for these experiments
(current and wave angle 180°) calculated with the best performing method of the current wave angle
90° experiments for the concentration profiles, are very poor, The calculated concentration profiles do
not fit the measurements very well.

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4 

A 3D Atlas of MR Diffusion Parameters in the Preterm Neonatal Brain
Every year fifteen million infants are born prematurely in the whole world. The maturation of the brain occurs mainly during the third trimester (2440 weeks of gestational age). Therefore extremely preterm infants (infants born before 25 weeks of gestational age) are vulnerable to brain injuries (perinatal asphyxia and perinatal stroke being the most common) and are more likely to develop behavioral problems, learning disabilities or functional consequences later in life. Magnetic resonance imaging (MRI) has good spatial resolution and softtissue contrast, providing a highly detailed anatomical image of the developing brain. Additionaly, Diffusion Tensor Imaging (DTI) allows visualization and quantification of the microarchitecture of brain structures in vivo.
Abnormal values of diffusion tensor measures in preterm infants can indicate abnormal brain development caused by e.g. haemorrhage. To use MRI and DTI effectively in clinical practice, it is necessary to have an atlas of the neonatal brain, consisting of a structural MRI template segmented in anatomically relevant regions and providing reference values for diffusion parameters in healthy preterms (showing normal development). Most existing atlases either include only a structural MRI template, without DTI information, or, when they do include a DTI atlas, they are based on term infants rather than preterm infants. Therefore, we conclude that there is no atlas that can be used as a reference for the diffusion parameters in the preterm developing brain. We constructed an atlas consiting of a structural template and a diffusion template encoding mean and standard deviation of DTI parameters in a population of healthy preterms. The method to construct such a template is explained. For the construction of the structural template we performed intersubject pairwise image registration and for the construction of the diffusion template intrasubject registration was additionally used. Image registration depends on several settings and parameters.
Those parameters were optimized to minimize registration errors. Manual segmentation is used to segment several areas of the brain, to allow for regionbased analysis. In addition, we present an example of how the template can be used in clinical practice. The parameter optimization was performed using 9 preterm infants without evidence of focal lesions on conventional magnetic resonance imaging. The template is based on 19 preterm infants without evident brain abnormalities. An independent set of 16 premature born infants (12 with evident abnormalities and 4 without evident abnormalities) is used for evaluation. Based on the results, we conclude that the proposed interactive atlas of diffusion MR parameters may aid in early recognition of subjects with a high risk of abnormal development.

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5 

Analysis of concentration and grain size distributions based on a diffusion concept
Gradients in sediment transport due to waves and currents cause coastal changes and sediment transport is an important subject in coastal engineering research. It is therefore necessary to predict the sediment transport rates in the coastal region. Various formulae are available to predict the sediment transport rates, for example the sediment transport
formulae of Bijker (1971) and Van Rijn (1993). These formulae use a diffusion concept for the calculation of the concentration and the velocity distribution over the water depth. In order to calculate the concentration and velocity distribution, the diffusion coefficient distribution has to be known. This is not always the case. Therefore, the
main objective of this study is to increase our knowledge of the diffusion coefficient distribution. In many coastal areas, nonuniform grain size distributions of bottom material and suspended sediment occur. It is thought that the diffusion coefficient distribution does not or hardly depends on the grading of the sediment and will be mainly dominated by
the hydraulic conditions. From measurements (Jacobs & Dekker (2000) and Sistermans (2000)) higher concentrations were found for experiments with wellgraded bottom material when compared to experiments with uniform bottom material. From these
measurements the diffusion coefficient distribution can be calculated. Most calculations in this study have been executed with the program CALCEPS. The program CALCEPS calculates several possible diffusion coefficient distributions from measured
concentration distributions. When a calculation is done with the so called 'fraction method', more or less the same diffusion coefficient distributions are found for experiments with the same hydraulic conditions. The vertical distribution of the diffusion coefficient can be calculated from measured concentration distributions. It is assumed that the diffusion coefficient distribution as has been determined is correct only when both the concentration and grain size distribution over the water depth are being calculated correctly. The grain size distribution was extensively measured in the above mentioned experiments. In the first analysis the diffusion coefficient distribution described both concentration and grain size distribution very well. From these calculations it has been concluded that a diffusion concept can be used to calculate the concentration and grain size distribution of each experiment. In practical applications, measurements of the grain size distributions over the water depth are not available. Sometimes, the grain size distribution of the bottom material is
known. Therefore, it is tried to impose the bottom material at a certain height and to describe the concentration and grain size distribution over the water depth as good as possible. It was found that the height above the bed where the bottom material can be imposed, should be in the order of magnitude of the measured ripple height. It can be concluded that the diffusion concept can be used for the modelling of concentration distributions of (well) graded sediment.

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6 

A model of turbulent diffusion
Mathematical description of turbulent diffusion with constant mean velocity gradient.

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7 

Evaluation of Isothermal ChemicalVapor Infiltration with LangmuirHinshelwood Type Kinetics

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8 

Separating convection from diffusion in a model of dispersion in fluid injection in forward flow
Recent research (John et al., 2008) has shown that the usual way of measuring dispersion in flow through permeable layers doesn’t distinguish between dispersion caused by convective spreading and mixing. Dispersion is used to mean both the variation of arrival times of material at a well and the molecular mixing of components in a reservoir. John et al used flow reversal to distinguish between the two, and quantified dispersion using particletracking.
To make it more clear, in this thesis the case with convection and diffusion is called “diffusion”, and the case with convectiveflow only is called “convectiveflow”.
Comparing results of forward movement with the results of flowreversing allows one to distinguish the effects of convection alone from convection and diffusion. John et al. came to the conclusion that the difference can be seen only in flowreversal. They compared the positions of “diffusion” particles and the mean of the positions of the “convectiveflow” particles. This is possible in a computer simulation, where diffusion can be excluded from the process.
Part of the difficulty in distinguishing between convection and diffusion arises because convection alone gives rise to a variation of particle positions that on the large scale looks scattered and random. On the small scale the convection particles trace a surface that is deterministic, based on the distribution of permeability in the medium. We test whether it is possible to distinguish convection from diffusion if one compares the positions of "diffusion" particles to the surface of “convectiveflow” particles rather than the mean position of the convectiveflow particles.
If such a difference could be found it would give more insight in the movement of particles during forward movement.
To achieve this, one would have to interpolate the convection surface between the “convectiveflow” particles. This was done in Matlab and the features of this surface were closely examined. I conclude that the amount of data (number of particles) was not sufficient to give an accurate representation of the features of this surface. Unfortunately I conclude that, with this amount of data, one cannot distinguish quantitatively between convection and diffusion in forward flow, because more data are needed to make a representative surface for the convection particles. The idea of plotting a surface to give a more accurate difference between diffusion and convection might in principle still work. To find out if it would really work, a larger amount of particles is needed to represent the convection surface.

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9 

Relationship between the evolution of an old and a new technological innovation
Nowadays many innovative technologies are emerging in the different markets. The relationship between an innovation which is already established in the market and a new budding innovation is not well known in every kind of market, especially when two innovative technologies are competing in different markets. This is the case of the PMP (Portable Media Player) and the smart phones market.
For this thesis we are interested in the study of how earlier technological innovations are related to new technological innovations. Currently, there are already a lot of researches on specific innovations, but not on the relationship between these particular innovations. For this reason, it would be interested for practice and science to explore it, in order to extend the knowledge in this field.
Therefore, analyzing the relationship between an old and already established technological innovation and a new technological innovation in the last phases of its diffusion process will be the objective of this thesis. In order to reach this objective, a pair of technologies has been chosen and will be analyzed. The old technology is the MP3 and the new technological innovation is the smart mobile phone.
To reach the mentioned objective, the main research question that will be answered is: What is the relationship between market stabilization phase of an old technological innovation and market adaptation and stabilization phases of a new technological innovation (in different markets)?
To answer the research question and other sub research questions, a review of the relevant literature was conducted. Thus, based on this literature, the main elements which may have an influence within the relationship between technologies were selected, consequently analyzed and then applied to our case analysis: MP3, iPod and iPhone. Then, the necessary data was collected in order to look for relationships between the two technological innovations. Gathered data was mainly based on: the internet, Apple annual reports, analysis reports and company press releases.
The main result showed us that, some of the major elements which are related to the two technologies are kind of innovation, prices, organizational issues, product cycle life, market actors and strategies. Other important findings which were not expected are that other important aspects such as technological changes are also influencing the relationship between technologies in their specific phases. These changes within technologies let the new technology learn from the old one.
Furthermore, limitations on this research were also found. This research has been carried out from the productive and development perspective. However, regarding the iPod and iPhone cases, platform perspective and its implications are an extensive approach which could be considered for further investigations. Among others, it is also important to mention the explorative and qualitative character of the research as limitations of this study.
Although this research has its limitations, and has left many issues to investigate, this study has achieved its aim of examining the relationships between technological innovations. It also made a contribution for filling the gap in existing literature on the relation between patterns of diffusion within the innovation field.

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10 

CVA calculation, an extended marked branching diffusion approach
The marked branching diffusion algorithm as proposed by (HenryLabordere, 2012), based on the particle diffusion introduced by (McKean, 1975), is extended upon to include stochastic interest rate models. This extended branching diffusion algorithm is used to solve pricing PDE's for equity derivatives including CVA, using the two types of default conventions as in (Brigo and Morini, 2011). Analytical results are then used to evaluate the performance of the algorithms in the case of one sided payos at maturity in a constant and Hull White interest rate world. An implementation of the lower and upper bounds, as suggested by (HenryLaborere et al., 2013), for the error introduced by the algorithms polynomial approximation is also given. The main results are the mismatches in price introduced in far out of the money derivatives having value close to zero and the extension to the stochastic interest rate framework.

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11 

Commercialization of HighTech Products: Niche Market Applications Prior to LargeScale Diffusion
The process by which radically new hightech products are commercialized and diffused in the massmarket can be long and unpredictable. We propose a conceptual model in which managerial decisions, regarding new product introductions, are presented with their possible outcomes.
We analyzed multiple case studies regarding radically new hightech products quantitatively and qualitatively to answer our research questions. We found two reliable categorizations for hightech products which can be used for statistical analysis: industry and product complexity. Regarding the types of niche market applications, we adapted a wellknown categorization introduced by the National Bureau of Statistics. The three types of niche market applications are government, business and consumer. Using ANOVA and chisquare tests, we found the most common types and the average number of niche market applications which appear prior to largescale diffusion. Moreover, the types and average number of niche markets change depending on the two product categorizations mentioned above.
We also studied the sequences of niche market applications which appear prior to largescale diffusion. We found the most common sequences of niche market applications with appear in general, and depending on the industry to which the product belongs. Finally, we provide general reasons for which such sequences of niche market applications appear when a radically new hightech product is introduced to the market.

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12 

Effect of Diffusion on Foam Bubble Size Distribution and Gas Mobility: an Idealized 2D Model
In this thesis the effect of gas diffusion between bubbles on the bubblesize distribution and capillary resistance to foam flow in a bubble train is investigated using an idealized 2D pore model. First the shape of the model pore is discussed and how a lamella moves through it. Then the physical forces (lamella curvatures and pressure differences between bubbles) in the pores are explained and how they affect capillary resistance to foam flow. Next the parameters of the dimensionless model are related to measured fluid properties. Under certain conditions (large film permeability to gas, large surface tension, low pressure, small pores, and low velocity of flowing gas) it is possible that characteristic diffusion rate can be greater than imposed convection rate, and that all gas transport is from diffusion across lamellae, not bubble movement.
We present model results for different ratios of characteristic diffusion to convection rates. If bubbles are smaller than pores, diffusion reduces the number of bubbles and increases average bubble size, whether convection is imposed on the foam or not. If convection is imposed, lamellae disappear not in pore throats but after first colliding in jumps across pore bodies. If bubbles are larger than pores, diffusion does not increase average bubble size. Diffusion increases the capillary resistance to flow; the increase is greatest when the characteristic rate of diffusion is close to the convection rate. Diffusion increases capillary resistance to flow because lamellae spend more time in positions of greater curvature than in the absence of diffusion. For characteristic diffusion rates much greater or much less than the imposed convection rate the effect of diffusion on capillary resistance to foam flow is modest.
These results suggest that by itself, an increase in diffusion rate through lamellae does not make foam flow with less resistance. With diffusion lamellae spend more time in pore throats where capillary resistance to flow is greatest.

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13 

Predicting the Diffusion of Technology in the Market Adaptation Phase
Introduction of a new product in the market brings new opportunities to a company but also involves many risks. Many companies cease to exist in this turbulent period, even before the technology reaches massdiffusion. In this study we look at the processes in the period of prediffusion and in particular in the market adaptation phase.
With this thesis, we try to identify the set of most important variables at play and use this set to model these processes. We use this to provide an answer on how to successfully predict the length of the market adaptation phase, with the data available at/or prior to the start of the phase.
Three different approaches were tried: the simple audit, extended audit and Monte Carlo simulation. Our findings indicate that there is no single/basic, minimum and universal set of variables suitable for prediction/modeling. Rather, the minimum set of variables depends on the type of model, type of industry, complexity and the number of available data. Furthermore, we show that the performance of the models heavily depends on the amount of the data available.
The main conclusion of this thesis is that a successful prediction of the length of the market adaptation phase is indeed possible but that it requires a careful consideration of quality and availability of the data, the choice of variables for modeling, initial conditions of the model and the choice of the model itself. In our case, the implementation of the Monte Carlo algorithm in the simulation of the process proved to provide the best results.

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14 

The effect of the cellReynolds number in the numerical solution of the convectiondiffusion equation
Internal memo.
In the numerical solution of convectiondiffusion equations a difficulty is commonly met in the occurrence of "standing waves" or spatial oscillations of a numerical origin. This has given rise to a considerable number of methods designed to avoid oscillations, often by means of upstream differencing or a similar approach in finite elements. In the present report, a short analysis of the phenomenon and some methods of remedy are given. The discussion is by no means complete, but it gives some general trends. For simplicity, the analysis is restricted to steadystate solutions of a linear convectiondiffusion equation with constant coefficients and in one space dimension

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15 

Diffusion and dispersion of numerical schemes for Hyperbolic problems
In the following text an overview is given of numerical schemes which can be used to solve hyperbolic partial differential equations. The overview is far from extensive and the analysis of the schemes is limited to the application on the simplest hyperbolic equation conceivable, namely the so called 'one dimensional simple wave equation'. In practical applications these schemes will be used to solve differential equations which are too complicated to analyze the quality of the numerical scheme. The author prefers not define the term quality as it strongly depends on kind of problem which is to be solved. Given the requirements for a certain problem the reader should however realize that the analysis given will show an optimal perfortnance of the numerical scheme. There are several reasons which support this statement. Firstly the simple differential equation for which the analysis is made. Secondly the fact that it is a one dimensional problem. And finally the fact that the analyzed discretization is assumed to be on an equidistant computational grid. This text can therefore not be used to choose the best scheme when for instance a three dimensional nonlinear hyperbolic differential equation is to be solved on a curvilinear grid. It can be used however, to eliminate numerical schemes which do not meet the requirements for the simple problem analyzed here. Applied on a more complicated problem the results acquired with the scheme would certainly be disappointing.

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16 

Jet diffusion in stagnant ambient fluid
Submarine outfall disposal of domestic and industrial sewage is a method of disposal of steadily growing importance. The flow from an ocean outfall is essentially that of a submerged horizontal or vertical jet. Thus a study of the hydrodynamics of such jets is needed to evaluate the dilution of the sewage flow. A study of this kind is presented for the case of jets issuing vertically upwards and for jets issuing horizontally into heavier homogeneous ambient fluid. The results are summarized in Fig. 23, 25, 37 and 314, which show how the dilution along the axis of the jet depends on the initial difference in density between the jet and the ambient fluid and on the velocity of the jet at the nozzle. In.case of jets issuing vertically upwards into lighter ambient fluid, the fluid of the jet cannot penetrate in a vertical direction beyond a certain ceiling level. For the case of homogeneous lighter ambient fluid the ceiling level is given by Eqs (413) and (424). For the case of stratified lighter ambient fluid the ceiling level is given by Eqs (511) and (520). Knowledge of the ceiling level is required to predict possible submergence of the sewage field, which may occur when lighter warm surface water overlies colder heavier bottom water.

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17 

Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid
A theoretical treatment of diffusion as function of turbulence, by working out the dispersion of small particles in turbulent fluid. The method of Kolmogoroff is used to decribe the movement in a statistical ways

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18 

Direct and comparative visualization techniques for HARDI Data
DWI is an MRI imaging technique used to gain information concerning the diffusion process in tissue. Using DTI techniques, a diffusion profile can be constructed for fiber tract analysis. Recently developed HARDI techniques increase the detail to visualization on the process of diffusion. While HARDI reconstruction methods are used to model the underlying diffusion process, the HARDI signal attenuation data can be used for a better understanding of noise in DWI data.
This project addresses the direct visualization of HARDI data without any intermediate processing steps between acquisition and visualization. We present new glyph shapes for direct and comparative visualization of HARDI data using the signal attenuation or ADC and a multiple linked views layout. We developed new difference metrics to create a complete comparative visualization pipeline to identify and explore areas of interest.
Evaluation of our developed methods by means of a case study, indicates the techniques to be a valued addition. The comparative visualization allows for quick identification of areas of interest. The glyph representation allows for rapid exploration of local diffusion data.

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19 

Effect of Bath Condition on the Diffusion of Contrast Agents Across Articular Cartilage
In the present study, the effect of concentration, osmolality and charge of xray contrast agents on their diffusion and equilibrium distribution across different zones of cartilage was investigated. Fullthickness cartilage discs (Ø = 8.5 mm, n = 3) were extracted from healthy equine femoral condyle (n = 2). The diffusion of four different contrast agent baths (Condition A: Visipaque 320 mg/ml, 290 mOsmol/kg; Condition B: Visipaque 320 mg/ml, 600 mOsmol/kg; Condition C: Visipaque 160 mg/ml, 290 mOsmol/kg; Condition D: Hexabrix 320 mg/ml, 600 mOsmol/kg) was allowed only through the articular surface. Samples were imaged with a micro computed tomography scanner (microCT) before the contrast agent bath was applied, and after 5, 10, 20, 30 minutes and 1, 2, 3, 4, 5, 6, 7, 10, 12, 24, 30, 36, and 48 hours. Findings show that osmolality and concentration do not have a pronounced effect on diffusion. However, concentration influence on diffusion is seen on zonal curves. Moreover, the diffusion coefficient of Hexabrix was between 2.9 and 8.6 times lower than that of Visipaque that reflects the important effect of solute’s charge on the transport through charged hydrated tissue such as articular cartilage. Slightly different diffusion coefficient observed within dilute and concentrated Visipaque baths suggested deviation from ideal Fickean behavior within articular cartilage. However, close diffusion coefficients of cartilage exposed to low and high osmolality baths confirmed the minor effect of osmolality on the transport of neutral solutes.

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20 

Numerical modelling on superdiffusion of carbondioxide in coal
Case II (super)diffusion is characterized by a diffusion like process that goes faster than the square root of time. Superdiffusion can occur in polymeric structures if a stress gradient enhances the diffusion process. In such a case the part of the polymer invaded by the penetrant, e.g. CO2, undergoes a transition in structure from the glass state to a rubber like state. In ideal cases the transition zone can propagate proportional to time. The Thomas & Windle model, a model that describes anomalous diffusion behavior in polymers, was also proposed to describe diffusion of CO2 in coal. The proposal was due to the similarities in structure between coal and glassy polymers. Coal has a molecular structure of carboncarbon bonds with ‘open holes’ between the chains. If these holes are smaller in volume than the volume of a CO2molecule, the uptake of CO2 leads to a geometric problem. The CO2 molecules will squeeze themselves into the structure, resulting into a swollen rubbery coal. This phenomenon is happening due to the fact that a carboncarbon dioxide bond is thermodynamically favored above the carboncarbon bond and is more stable than the other. Analysis of the Thomas & Windle model shows there is a time where diffusion is much faster than conventional diffusion. The results also show that superdiffusion is only a transitional effect, i.e., there is an initial phase where relaxation dominates, a transitional phase that exhibits anomalous diffusion and a long time phase that shows the typical square root of time behavior of an ordinary diffusion process.
This study gives details of a numerical model used to solve the stress induced diffusion equation using the Finite Element Method implemented in Matlab. The numerical results show that for some favorable set of parameters diffusion can indeed be faster than Fickean diffusion during an intermediate stage. For other parameters the diffusion can be slower. A comparison to a laboratory result shows an example where diffusion appears to be slower.

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