Numerical modelling of fluid-structure interaction with application in hemodynamics

The thesis deals with numerical algorithms for fluid-structure interaction problems with application in blood flow modelling. It starts with a short introduction on the mathematical description of incompressible viscous flow with non-Newtonian viscosity and a moving linear viscoelastic structure. The mathematical model consists of the generalized Navier-Stokes equation used for the description of fluid flow and the generalized string model for structure movement. The arbitrary Lagrangian-Eulerian approach is used in order to take into account moving computational domain. A part of the thesis is devoted to the discussion on the non-Newtonian behaviour of shear-thinning fluids, which is in our case blood, and derivation of two non-Newtonian models frequently used in the blood flow modelling. Further we give a brief overview on recent fluid-structure interaction schemes with discussion about the difficulties arising in numerical modelling of blood flow. Our main contribution lies in numerical and experimental study of a new loosely-coupled partitioned scheme called the kinematic splitting fluid-structure interaction algorithm. We present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Here, we assume both, the nonlinearity in convective as well is diffusive term. We analyse the convergence of proposed numerical scheme for a simplified fluid model of the Oseen type. Moreover, we present series of experiments including numerical error analysis, comparison of hemodynamic parameters for the Newtonian and non-Newtonian fluids and comparison of several physiologically relevant computational geometries in terms of wall displacement and wall shear stress. Numerical analysis and extensive experimental study for several standard geometries confirm reliability and accuracy of the proposed kinematic splitting scheme in order to approximate fluid-structure interaction problems.

Fourier-transform rheology on anionically synthesised polymer melts and solutions of various topology

Aim of this thesis was to further extend the applicability of the Fourier-transform (FT) rheology technique especially for non-linear mechanical characterisation of polymeric materials on the one hand and to investigated the influence of the degree of branching on the linear and non-linear relaxation behaviour of polymeric materials on the other hand. The latter was achieved by employing in particular FT-rheology and other rheological techniques to variously branched polymer melts and solutions. For these purposes, narrowly distributed linear and star-shaped polystyrene and polybutadiene homo-polymers with varying molecular weights were anionically synthesised using both high-vacuum and inert atmosphere techniques. Furthermore, differently entangled solutions of linear and star-shaped polystyrenes in di-sec-octyl phthalate (DOP) were prepared. The several linear polystyrene solutions were measured under large amplitude oscillatory shear (LAOS) conditions and the non-linear torque response was analysed in the Fourier space. Experimental results were compared with numerical predictions performed by Dr. B. Debbaut using a multi-mode differential viscoelastic fluid model obeying the Giesekus constitutive equation. Apart from the analysis of the relative intensities of the harmonics, a detailed examination of the phase information content was developed. Further on, FT-rheology allowed to distinguish polystyrene melts and solutions due to their different topologies where other rheological measurements failed. Significant differences occurred under LAOS conditions as particularly reflected in the phase difference of the third harmonic, Ħ3, which could be related to shear thinning and shear thickening behaviour.

Synthesis of hexa-peri-hexabenzocoronenes - from building blocks to functional materials

Discotic hexa-peri-hexabenzocoronene (HBC) derivatives have attracted intensive scientific interest due to their unique optoelectronic properties, which depends, to a large extend, upon the attached functional groups. The presented work covers the synthesis of novel HBC building blocks and new HBC derivatives as functional materials. The traditional preparation of HBC derivatives requires elaborate synthetic techniques and tremendous effort. Especially, more than 10 synthetic steps are usually necessary to approach HBCs with lower symmetries. In order to simplify the synthetic work and reduce the high costs, a novel synthetic strategy involving only four steps was developed based on 2,3,5,6-tetraphenyl-1,4-diiodobenzene intermediates and palladium catalyzed Suzuki cross coupling reactions. In order to introduce various functionalities and expand the diversity of multi-functionalizations, a novel C2v-symmetric dihalo HBC building block 2-47, which contains one iodine and one bromine in para positions, was prepared following the traditional intermolecular [4+2] Diels-Alder reaction route. The outstanding chemical selectivity between iodo and bromo groups in this compound consequently leads to lots of HBC derivatives bearing different functionalities. Directly attached heteroatoms will improve the material properties. According to the application of intramolecular Scholl reaction to a para-dimethoxy HPB, which leads to a meta-dimethoxy HBC, a phenomenon of phenyl group migration was discovered. Thereby, several interesting mechanistic details involving arenium cation intermediates were discussed. With a series of dipole functionalized HBCs, the molecular dynamics of this kind of materials was studied in different phases by DSC, 2D WAXD, solid state NMR and dielectric spectroscopies. High charge carrier mobility is an important parameter for a semiconductive material and depends on the degree of intramolecular order of the discotic molecules in thin films for HBC derivatives. Dipole – dipole interaction and hydrogen bonds were respectively introduced in order to achieve highly ordered supramolecular structure. The self-assembly behavior of these materials were investigated both in solution and solid state. Depending upon the different functionalities, these novel materials show either gelating or non-linear optical properties, which consequently broaden their applications as functional materials. In the field of conceivable electronic devices at a molecular level, HBCs hold high promise. Differently functionalized HBCs have been used as active component in the studies of single-molecular CFET and metal-SAMs-metal junctions. The outstanding properties shown in these materials promise their exciting potential applications in molecular devices.

Multifunctional polyether architectures : versatile materials for surface attachment and functionalization

Chapter 1 of this thesis comprises a review of polyether polyamines, i.e., combinations of polyether scaffolds with polymers bearing multiple amino moieties. Focus is laid on controlled or living polymerization methods. Furthermore, fields in which the combination of cationic, complexing, and pH-sensitive properties of the polyamines and biocompatibility and water-solubility of polyethers promise enormous potential are presented. Applications include stimuli-responsive polymers with a lower critical solution temperature (LCST) and/or the ability to gel, preparation of shell cross-linked (SCL) micelles, gene transfection, and surface functionalization.rnIn Chapter 2, multiaminofunctional polyethers relying on the class of glycidyl amine comonomers for anionic ring-opening polymerization (AROP) are presented. In Chapter 2.1, N,N-diethyl glycidyl amine (DEGA) is introduced for copolymerization with ethylene oxide (EO). Copolymer microstructure is assessed using online 1H NMR kinetics, 13C NMR triad sequence analysis, and differential scanning calorimetry (DSC). The concurrent copolymerization of EO and DEGA is found to result in macromolecules with a gradient structure. The LCSTs of the resulting copolymers can be tailored by adjusting DEGA fraction or pH value of the environment. Quaternization of the amino moieties by methylation results in polyelectrolytes. Block copolymers are used for PEGylated gold nanoparticle formation. Chapter 2.2 deals with a glycidyl amine monomer with a removable protecting group at the amino moiety, for liberation of primary amines at the polyether backbone, which is N,N-diallyl glycidyl amine (DAGA). Its allyl groups are able to withstand the harsh basic conditions of AROP, but can be cleaved homogeneously after polymerization. Gradient as well as block copolymers poly(ethylene glycol)-PDAGA (PEG-PDAGA) are obtained. They are analyzed regarding their microstructure, LCST behavior, and cleavage of the protecting groups. rnChapter 3 describes applications of multi(amino)functional polyethers for functionalization of inorganic surfaces. In Chapter 3.1, they are combined with an acetal-protected catechol initiator, leading to well-defined PEG and heteromultifunctional PEG analogues. After deprotection, multifunctional PEG ligands capable of attaching to a variety of metal oxide surfaces are obtained. In a cooperative project with the Department of Inorganic and Analytical Chemistry, JGU Mainz, their potential is demonstrated on MnO nanoparticles, which are promising candidates as T1 contrast agents in magnetic resonance imaging. The MnO nanoparticles are solubilized in aqueous solution upon ligand exchange. In Chapter 3.2, a concept for passivation and functionalization of glass surfaces towards gold nanorods is developed. Quaternized mPEG-b-PqDEGA diblock copolymers are attached to negatively charged glass surfaces via the cationic PqDEGA blocks. The PEG blocks are able to suppress gold nanorod adsorption on the glass in the flow cell, analyzed by dark field microscopy.rnChapter 4 highlights a straightforward approach to poly(ethylene glycol) macrocycles. Starting from commercially available bishydroxy-PEG, cyclic polymers are available by perallylation and ring-closing metathesis in presence of Grubbs’ catalyst. Purification of cyclic PEG is carried out using α-cyclodextrin. This cyclic sugar derivative forms inclusion complexes with remaining unreacted linear PEG in aqueous solution. Simple filtration leads to pure macrocycles, as evidenced by SEC and MALDI-ToF mass spectrometry. Cyclic polymers from biocompatible precursors are interesting materials regarding their increased blood circulation time compared to their linear counterparts.rnIn the Appendix, A.1, a study of the temperature-dependent water-solubility of polyether copolymers is presented. Macroscopic cloud points, determined by turbidimetry, are compared with microscopic aggregation phenomena, monitored by continuous wave electron paramagnetic resonance (CW EPR) spectroscopy in presence of the amphiphilic spin probe and model drug (2,2,6,6-tetramethylpiperidin-1-yl)oxyl (TEMPO). These thermoresponsive polymers are promising candidates for molecular transport applications. The same techniques are applied in Chapter A.2 to explore the pH-dependence of the cloud points of PEG-PDEGA copolymers in further detail. It is shown that the introduction of amino moieties at the PEG backbone allows for precise manipulation of complex phase transition modes. In Chapter A.3, multi-hydroxyfunctional polysilanes are presented. They are obtained via copolymerization of the acetal-protected dichloro(isopropylidene glyceryl propyl ether)methylsilane monomer. The hydroxyl groups are liberated through acidic work-up, yielding versatile access to new multifunctional polysilanes.

Numerical simulation of some viscoelastic fluids

Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.

Analysis and numerical solution of the Peterlin viscoelastic model

Liquids and gasses form a vital part of nature. Many of these are complex fluids with non-Newtonian behaviour. We introduce a mathematical model describing the unsteady motion of an incompressible polymeric fluid. Each polymer molecule is treated as two beads connected by a spring. For the nonlinear spring force it is not possible to obtain a closed system of equations, unless we approximate the force law. The Peterlin approximation replaces the length of the spring by the length of the average spring. Consequently, the macroscopic dumbbell-based model for dilute polymer solutions is obtained. The model consists of the conservation of mass and momentum and time evolution of the symmetric positive definite conformation tensor, where the diffusive effects are taken into account. In two space dimensions we prove global in time existence of weak solutions. Assuming more regular data we show higher regularity and consequently uniqueness of the weak solution. For the Oseen-type Peterlin model we propose a linear pressure-stabilized characteristics finite element scheme. We derive the corresponding error estimates and we prove, for linear finite elements, the optimal first order accuracy. Theoretical error of the pressure-stabilized characteristic finite element scheme is confirmed by a series of numerical experiments.