On Undecidable Dynamical Properties of Reversible One-Dimensional Cellular Automata

Cellular automata are models for massively parallel computation. A cellular automaton consists of cells which are arranged in some kind of regular lattice and a local update rule which updates the state of each cell according to the states of the cell's neighbors on each step of the computation. This work focuses on reversible one-dimensional cellular automata in which the cells are arranged in a two-way in_nite line and the computation is reversible, that is, the previous states of the cells can be derived from the current ones. In this work it is shown that several properties of reversible one-dimensional cellular automata are algorithmically undecidable, that is, there exists no algorithm that would tell whether a given cellular automaton has the property or not. It is shown that the tiling problem of Wang tiles remains undecidable even in some very restricted special cases. It follows that it is undecidable whether some given states will always appear in computations by the given cellular automaton. It also follows that a weaker form of expansivity, which is a concept of dynamical systems, is an undecidable property for reversible one-dimensional cellular automata. It is shown that several properties of dynamical systems are undecidable for reversible one-dimensional cellular automata. It shown that sensitivity to initial conditions and topological mixing are undecidable properties. Furthermore, non-sensitive and mixing cellular automata are recursively inseparable. It follows that also chaotic behavior is an undecidable property for reversible one-dimensional cellular automata.

Non-Markovianity and Quantum Correlations in Qubit-Systems

In this Thesis I discuss the exact dynamics of simple non-Markovian systems. I focus on fundamental questions at the core of non-Markovian theory and investigate the dynamics of quantum correlations under non-Markovian decoherence. In the first context I present the connection between two different non-Markovian approaches, and compare two distinct definitions of non-Markovianity. The general aim is to characterize in exemplary cases which part of the environment is responsible for the feedback of information typical of non- Markovian dynamics. I also show how such a feedback of information is not always described by certain types of master equations commonly used to tackle non-Markovian dynamics. In the second context I characterize the dynamics of two qubits in a common non-Markovian reservoir, and introduce a new dynamical effect in a wellknown model, i.e., two qubits under depolarizing channels. In the first model the exact solution of the dynamics is found, and the entanglement behavior is extensively studied. The non-Markovianity of the reservoir and reservoirmediated-interaction between the qubits cause non-trivial dynamical features. The dynamical interplay between different types of correlations is also investigated. In the second model the study of quantum and classical correlations demonstrates the existence of a new effect: the sudden transition between classical and quantum decoherence. This phenomenon involves the complete preservation of the initial quantum correlations for long intervals of time of the order of the relaxation time of the system.

MCMC Analysis for Optimization of Stochastic Models

In any decision making under uncertainties, the goal is mostly to minimize the expected cost. The minimization of cost under uncertainties is usually done by optimization. For simple models, the optimization can easily be done using deterministic methods.However, many models practically contain some complex and varying parameters that can not easily be taken into account using usual deterministic methods of optimization. Thus, it is very important to look for other methods that can be used to get insight into such models. MCMC method is one of the practical methods that can be used for optimization of stochastic models under uncertainty. This method is based on simulation that provides a general methodology which can be applied in nonlinear and non-Gaussian state models. MCMC method is very important for practical applications because it is a uni ed estimation procedure which simultaneously estimates both parameters and state variables. MCMC computes the distribution of the state variables and parameters of the given data measurements. MCMC method is faster in terms of computing time when compared to other optimization methods. This thesis discusses the use of Markov chain Monte Carlo (MCMC) methods for optimization of Stochastic models under uncertainties .The thesis begins with a short discussion about Bayesian Inference, MCMC and Stochastic optimization methods. Then an example is given of how MCMC can be applied for maximizing production at a minimum cost in a chemical reaction process. It is observed that this method performs better in optimizing the given cost function with a very high certainty.

Application of Computational Fluid Dynamics in Modeling Blood Flow in Human Thoracic Aorta

The aim of this study was to simulate blood flow in thoracic human aorta and understand the role of flow dynamics in the initialization and localization of atherosclerotic plaque in human thoracic aorta. The blood flow dynamics in idealized and realistic models of human thoracic aorta were numerically simulated in three idealized and two realistic thoracic aorta models. The idealized models of thoracic aorta were reconstructed with measurements available from literature, and the realistic models of thoracic aorta were constructed by image processing Computed Tomographic (CT) images. The CT images were made available by South Karelia Central Hospital in Lappeenranta. The reconstruction of thoracic aorta consisted of operations, such as contrast adjustment, image segmentations, and 3D surface rendering. Additional design operations were performed to make the aorta model compatible for the numerical method based computer code. The image processing and design operations were performed with specialized medical image processing software. Pulsatile pressure and velocity boundary conditions were deployed as inlet boundary conditions. The blood flow was assumed homogeneous and incompressible. The blood was assumed to be a Newtonian fluid. The simulations with idealized models of thoracic aorta were carried out with Finite Element Method based computer code, while the simulations with realistic models of thoracic aorta were carried out with Finite Volume Method based computer code. Simulations were carried out for four cardiac cycles. The distribution of flow, pressure and Wall Shear Stress (WSS) observed during the fourth cardiac cycle were extensively analyzed. The aim of carrying out the simulations with idealized model was to get an estimate of flow dynamics in a realistic aorta model. The motive behind the choice of three aorta models with distinct features was to understand the dependence of flow dynamics on aorta anatomy. Highly disturbed and nonuniform distribution of velocity and WSS was observed in aortic arch, near brachiocephalic, left common artery, and left subclavian artery. On the other hand, the WSS profiles at the roots of branches show significant differences with geometry variation of aorta and branches. The comparison of instantaneous WSS profiles revealed that the model with straight branching arteries had relatively lower WSS compared to that in the aorta model with curved branches. In addition to this, significant differences were observed in the spatial and temporal profiles of WSS, flow, and pressure. The study with idealized model was extended to study blood flow in thoracic aorta under the effects of hypertension and hypotension. One of the idealized aorta models was modified along with the boundary conditions to mimic the thoracic aorta under the effects of hypertension and hypotension. The results of simulations with realistic models extracted from CT scans demonstrated more realistic flow dynamics than that in the idealized models. During systole, the velocity in ascending aorta was skewed towards the outer wall of aortic arch. The flow develops secondary flow patterns as it moves downstream towards aortic arch. Unlike idealized models, the distribution of flow was nonplanar and heavily guided by the artery anatomy. Flow cavitation was observed in the aorta model which was imaged giving longer branches. This could not be properly observed in the model with imaging containing a shorter length for aortic branches. The flow circulation was also observed in the inner wall of the aortic arch. However, during the diastole, the flow profiles were almost flat and regular due the acceleration of flow at the inlet. The flow profiles were weakly turbulent during the flow reversal. The complex flow patterns caused a non-uniform distribution of WSS. High WSS was distributed at the junction of branches and aortic arch. Low WSS was distributed at the proximal part of the junction, while intermedium WSS was distributed in the distal part of the junction. The pulsatile nature of the inflow caused oscillating WSS at the branch entry region and inner curvature of aortic arch. Based on the WSS distribution in the realistic model, one of the aorta models was altered to induce artificial atherosclerotic plaque at the branch entry region and inner curvature of aortic arch. Atherosclerotic plaque causing 50% blockage of lumen was introduced in brachiocephalic artery, common carotid artery, left subclavian artery, and aortic arch. The aim of this part of the study was first to study the effect of stenosis on flow and WSS distribution, understand the effect of shape of atherosclerotic plaque on flow and WSS distribution, and finally to investigate the effect of lumen blockage severity on flow and WSS distributions. The results revealed that the distribution of WSS is significantly affected by plaque with mere 50% stenosis. The asymmetric shape of stenosis causes higher WSS in branching arteries than in the cases with symmetric plaque. The flow dynamics within thoracic aorta models has been extensively studied and reported here. The effects of pressure and arterial anatomy on the flow dynamic were investigated. The distribution of complex flow and WSS is correlated with the localization of atherosclerosis. With the available results we can conclude that the thoracic aorta, with complex anatomy is the most vulnerable artery for the localization and development of atherosclerosis. The flow dynamics and arterial anatomy play a role in the localization of atherosclerosis. The patient specific image based models can be used to diagnose the locations in the aorta vulnerable to the development of arterial diseases such as atherosclerosis.

Dispersal and metacommunity dynamics in a soft-sediment benthic system : how well is the seafloor connected?

Connectivity depends on rates of dispersal between communities. For marine soft-sediment communities continued small-scale dispersal as post-larvae and as adults can be equally important in maintaining community composition, as initial recruitment of substrate by pelagic larvae. In this thesis post-larval dispersal strategies of benthic invertebrates, as well as mechanisms by which communities are connected were investigated. Such knowledge on dispersal is scarce, due to the difficulties in actually measuring dispersal directly in nature, and dispersal has not previously been quantified in the Baltic Sea. Different trap-types were used underwater to capture dispersing invertebrates at different sites, while in parallel measuring waves and currents. Local community composition was found to change predictably under varying rates of dispersal and physical connectivity (waves and currents). This response was, however, dependent on dispersal-related traits of taxa. Actively dispersing taxa will be relatively better at maintaining their position, as they are not as dependent on hydrodynamic conditions for dispersal and will be less prone to be passively transported by currents. Taxa also dispersed in relative proportions that were distinctly different from resident community composition and a significant proportion (40 %) of taxa were found to lack a planktonic larval life-stage. Community assembly was re-started in a large-scale manipulative field experiment over one year across several sites, which revealed how patterns of community composition (α-, β- and λ-diversity) change depending on rates of dispersal. Results also demonstrated that in response to small-scale disturbance, initial recruitment was by nearby-dominant species after which other species arrived from successively further away. At later assembly time, the number of coexisting species increased beyond what was expected purely by local niche requirements (species sorting), transferring regional differences in community composition (β-diversity) to the local scale (α-diversity, mass effect). Findings of this thesis complement more theoretical studies in metacommunity ecology by demonstrating that understanding how and when individuals disperse relative to underlying environmental heterogeneity is key to interpreting how patterns of diversity change across different spatial scales. Such information from nature is critical when predicting responses to, for example, different types of disturbances or management actions in conservation.

JNK, a versatile regulator of kinesin-1 transport and cytoskeleton dynamics

C-Jun N-terminal kinase (JNK) is traditionally recognized as a crucial factor in stress response and inducer of apoptosis upon various stimulations. Three isoforms build the JNK subfamily of MAPK; generally expressed JNK1 and JNK2 and brain specific JNK3. Degenerative potency placed JNK in the spotlight as potential pharmacological option for intervention. Unfortunately, adverse effects of potential drugs and observation that expression of only JNK2 and JNK3 are induced upon stress, restrained initial enthusiasm. Notably, JNK1 demonstrated atypical high constitutive activity in neurons that is not responsive to cellular stresses and indicated existence of physiological activity. This thesis aimed at revealing the physiological functions of JNK1 in actin homeostasis through novel effector MARCKS-Like 1 (MARCKSL1) protein, neuronal trafficking mediated by major kinesin-1 motor protein and microtubule (MT) dynamics via STMN2/SCG10. The screen for novel physiological JNK substrates revealed specific phosphorylation of C-terminal end of MARCKSL1 at S120, T148 and T183 both ex vivo and in vitro. By utilizing site-specific mutagenesis, various actin dynamics and migrations assays we were able to demonstrate that JNK1 phosphorylation specifically facilitates F-actin bundling and thus filament stabilisation. Consecutively, this molecular mechanism was proved to enhance formation of filopodia; cell surface projections that allow cell sensing surrounding environment and migrate efficiently. Our results visualize JNK dependent and MARCKSL1 executed induction of filopodia in neurons and fibroblast indicating general mechanism. Subsequently, inactivation of JNK action on MARCKSL1 shifts cellular actin machinery into lamellipodial dynamic arrangement. Tuning of actin cytoskeleton inevitably melds with cell migration. We observed that both active JNK and JNK pseudo-phosphorylated form of MARCKSL1 reduce actin turnover in intact cells leading to overall diminished cell motility. We demonstrate that tumour transformed cells from breast, prostate, lung and muscle-derived cancers upregulate MARCKSL1. We showed on the example of prostate cancer PC-3 cell line that JNK phosphorylation negatively controls MARCKSL1 ability to induce migration, which precedes cancer cell metastasis. The second round of identification of JNK physiological substrates resulted in detection of predominant motor protein kinesin-1 (Kif5). Mass spectrometry detailed analysis showed evident endogenous phosphorylation of kinesin-1 on S176 within motor domain that interacts with MT. In vitro phosphorylation of bacterially expressed kinesin heavy chain by JNK isoforms displayed higher specificity of JNK1 when compared to JNK3. Since, JNK1 is constitutively active in neurons it signified physiological aspect of kinesin-1 regulation. Subsequent biochemical examination revealed that kinesin-1, when not phosphorylated on JNK site, exhibits much higher affinity toward MTs. Expression of the JNK non-phosphorable kinesin-1 mutant in intact cells as well as in vitro single molecule imaging using total internal reflection fluorescence microscopy indicated that the mutant loses normal speed and is not able to move processively into proper cellular compartments. We identify novel kinesin-1 cargo protein STMN2/SCG10, which along with known kinesin-1 cargo BDNF is showing impaired trafficking when JNK activity is inhibited. Our data postulates that constitutive JNK activity in neurons is crucial for unperturbed physiologically relevant transport of kinesin-1 dependant cargo. Additionally, my work helps to validate another novel physiological JNK1 effector STMN2/SCG10 as determinant of axodendritic neurites dynamics in the developing brain through regulation of MT turnover. We show successively that this increased MT dynamics is crucial during developmental radial migration when brain layering occurs. Successively, we are able to show that introduction of JNK phosphorylation mimicking STMN2/SCG10 S62/73D mutant rescues completely JNK1 genetic deletion migration phenotype. We prove that STMN2/SCG10 is predominant JNK effector responsible for MT depolymerising activity and neurite length during brain development. Summarizing, this work describes identification of three novel JNK substrates MARCKSL1, kinesin-1 and STMN2/SCG10 and investigation of their roles in cytoskeleton dynamics and cargo transport. This data is of high importance to understand physiological meaning of JNK activity, which might have an adverse effect during pharmaceutical intervention aiming at blocking pathological JNK action.

Memory Effects in the Dynamics of Open Quantum Systems

In this Thesis various aspects of memory effects in the dynamics of open quantum systems are studied. We develop a general theoretical framework for open quantum systems beyond the Markov approximation which allows us to investigate different sources of memory effects and to develop methods for harnessing them in order to realise controllable open quantum systems. In the first part of the Thesis a characterisation of non-Markovian dynamics in terms of information flow is developed and applied to study different sources of memory effects. Namely, we study nonlocal memory effects which arise due to initial correlations between two local environments and further the memory effects induced by initial correlations between the open system and the environment. The last part focuses on describing two all-optical experiment in which through selective preparation of the initial environment states the information flow between the system and the environment can be controlled. In the first experiment the system is driven from the Markovian to the non- Markovian regime and the degree of non-Markovianity is determined. In the second experiment we observe the nonlocal nature of the memory effects and provide a novel method to experimentally quantify frequency correlations in photonic environments via polarisation measurements.

Dynamics of coherent vortices in mixing layers using direct numerical and large-eddy simulations

Coherent vortices in turbulent mixing layers are investigated by means of Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES). Subgrid-scale models defined in spectral and physical spaces are reviewed. The new "spectral-dynamic viscosity model", that allows to account for non-developed turbulence in the subgrid-scales, is discussed. Pseudo-spectral methods, combined with sixth-order compact finite differences schemes (when periodic boundary conditions cannot be established), are used to solve the Navier- Stokes equations. Simulations in temporal and spatial mixing layers show two types of pairing of primary Kelvin-Helmholtz (KH) vortices depending on initial conditions (or upstream conditions): quasi-2D and helical pairings. In both cases, secondary streamwise vortices are stretched in between the KH vortices at an angle of 45° with the horizontal plane. These streamwise vortices are not only identified in the early transitional stage of the mixing layer but also in self-similar turbulence conditions. The Re dependence of the "diameter" of these vortices is analyzed. Results obtained in spatial growing mixing layers show some evidences of pairing of secondary vortices; after a pairing of the primary Kelvin-Helmholtz (KH) vortices, the streamwise vortices are less numerous and their diameter has increased than before the pairing of KH vortices.

Application of the center manifold theory to the study of slewing flexible Non-ideal structures with nonlinear curvature: a case study

In this paper is Analyzed the local dynamical behavior of a slewing flexible structure considering nonlinear curvature. The dynamics of the original (nonlinear) governing equations of motion are reduced to the center manifold in the neighborhood of an equilibrium solution with the purpose of locally study the stability of the system. In this critical point, a Hopf bifurcation occurs. In this region, one can find values for the control parameter (structural damping coefficient) where the system is unstable and values where the system stability is assured (periodic motion). This local analysis of the system reduced to the center manifold assures the stable / unstable behavior of the original system around a known solution.

Market integration, return and volatility dynamics; empirical evidence from African stock markets

Frontier and Emerging economies have implemented policies with the objective of liberalizing their equity markets. Equity market liberalization opens the domestic equity market to foreign investors and as well paves the way for domestic investors to invest in foreign equity securities. Among other things, equity market liberalization results in diversification benefits. Moreover, equity market liberalization leads to low cost of equity capital resulting from the lower rate of return by investors. Additionally, foreign and local investors share any potential risks. Liberalized equity markets also become liquid considering that there are more investors to trade. Equity market liberalization results in financial integration which explains the movement of two markets. In crisis period, increased volatility and co-movement between two markets may result in what is termed contagion effects. In Africa, major moves toward financial liberalization generally started in the late 1980s with South Africa as the pioneer. Over the years, researchers have studied the impact of financial liberalization on Africa’s economic development with diverse results; some being positive, others negative and still others being mixed. The objective of this study is to establish whether African stock-markets are integrated into the United States (US) and World market. Furthermore, the study helps to see if there are international linkages between the Africa, US and the world markets. A Bivariate- VAR- GARCH- BEKK model is employed in the study. In the study, the effect of thin trading is removed through series of econometric data purification. This is because thin trading, also known as non-trading or inconsistency of trading, is a main feature of African markets and may trigger inconsistency and biased results. The study confirmed the widely established results that the South Africa and Egypt stock markets are highly integrated with the US and World market. Interestingly, the study adds to knowledge in this research area by establishing the fact that Kenya is very integrated with the US and World markets and that it receives and exports past innovations as well as shocks to and from the US and World market.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.

The succession dynamics of a macroalgal community after a flood disturbance in a tropical stream from São Paulo State, southeastern Brazil

The succession dynamics of a macroalgal community in a tropical stream (20º58' S and 49º25' W) was investigated after disturbance by a sequence of intensive rains. High precipitation levels caused almost complete loss of the macroalgal community attached to the substratum and provided a strong pressure against its immediate re-establishment. After this disturbance, a weekly sampling program from May 1999 to January 2000 was established to investigate macroalgal recolonization. The community changed greatly throughout the succession process. The number of species varied from one to seven per sampling. Global abundance of macroalgal community did not reveal a consistent temporal pattern of variation. In early succession stages, the morphological form of tufts dominated, followed by unbranched filaments. Latter succession stages showed the almost exclusive occurrence of gelatinous forms, including filaments and colonies. The succession trajectory was mediated by phosphorus availability in which community composition followed a scheme of changes in growth forms. However, we believe that deterministic and stochastic processes occur in lotic ecosystems, but they are dependent on the length of time considered in the succession analyses.

Tree population and community dynamics in a Brazilian tropical semideciduous forest

The dynamics of the tree community and 30 tree populations were examined in an area of tropical semideciduous forest located on the margin of the Rio Grande, SE Brazil, based on surveys done in 1990 and 1997 in three 0.18 ha plots. The main purpose was to assess whether variations in dynamics were related to topography and the effects of a catastrophic flood in 1992. Rates of mortality and recruitment of trees and gain and loss of basal area in two topographic sites, lower (flooded) and upper (non-flooded), were obtained. Projected trajectories of mean and accelerated growth in diameter were obtained for each species. In both topographic sites, mortality rates surpassed recruitment rates, gain rates of basal area surpassed loss rates, and size distributions changed, with declining proportions of smaller trees. These overall changes were possibly related to increased underground water supply after the 1992 flood as well as to a c. 250-year-old process of primary succession on abandoned gold mines. Possible effects of the 1992 flood showed up in the higher proportions of dead trees in the flooded sites and faster growth rates in the flood-free sites. Species of different regeneration guilds showed particular trends with respect to their demographic changes and diameter growth patterns. Nevertheless, patterns of population dynamics differed between topographic sites for only two species.

Images of the future of privacy: A privacy dynamics framework and a causal layered

The future of privacy in the information age is a highly debated topic. In particular, new and emerging technologies such as ICTs and cognitive technologies are seen as threats to privacy. This thesis explores images of the future of privacy among non-experts within the time frame from the present until the year 2050. The aims of the study are to conceptualise privacy as a social and dynamic phenomenon, to understand how privacy is conceptualised among citizens and to analyse ideal-typical images of the future of privacy using the causal layered analysis method. The theoretical background of the thesis combines critical futures studies and critical realism, and the empirical material is drawn from three focus group sessions held in spring 2012 as part of the PRACTIS project. From a critical realist perspective, privacy is conceptualised as a social institution which creates and maintains boundaries between normative circles and preserves the social freedom of individuals. Privacy changes when actors with particular interests engage in technology-enabled practices which challenge current privacy norms. The thesis adopts a position of technological realism as opposed to determinism or neutralism. In the empirical part, the focus group participants are divided into four clusters based on differences in privacy conceptions and perceived threats and solutions. The clusters are fundamentalists, pragmatists, individualists and collectivists. Correspondingly, four ideal-typical images of the future are composed: ‘drift to low privacy’, ‘continuity and benign evolution’, ‘privatised privacy and an uncertain future’, and ‘responsible future or moral decline’. The images are analysed using the four layers of causal layered analysis: litany, system, worldview and myth. Each image has its strengths and weaknesses. The individualistic images tend to be fatalistic in character while the collectivistic images are somewhat utopian. In addition, the images have two common weaknesses: lack of recognition of ongoing developments and simplistic conceptions of privacy based on a dichotomy between the individual and society. The thesis argues for a dialectical understanding of futures as present images of the future and as outcomes of real processes and mechanisms. The first steps in promoting desirable futures are the awareness of privacy as a social institution, the awareness of current images of the future, including their assumptions and weaknesses, and an attitude of responsibility where futures are seen as the consequences of present choices.

Modelling of ultrasound assisted mixing in Newtonian and non-Newtonian liquids

The purpose of this work is to obtain a better understanding of behaviour of possible ultrasound appliance on fluid media mixing. The research is done in the regard to Newtonian and non-Newtonian fluids. The process of ultrasound appliance on liquids is modelled in COMSOL Multiphysics software. The influence of ultrasound using is introduced as waveform equation. Turbulence modelling is fulfilled by the k-ε model in Newtonian fluid. The modeling of ultrasound assisted mixing in non-Newtonian fluids is based on the power law. To verify modelling results two practical methods are used: Particle Image Velocimetry and measurements of mixing time. Particle Image Velocimetry allows capturing of velocity flow field continuously and presents detailed depiction of liquid dynamics. The second way of verification is the comparison of mixing time of homogeneity. Experimentally achievement of mixing time is done by conductivity measurements. In modelling part mixing time is achieved by special module of COMSOL Multiphysics – the transport of diluted species. Both practical and modelling parts show similar radial mechanism of fluid flow under ultrasound appliance – from the horn tip fluid moves to the bottom and along the walls goes back. Velocity profiles are similar in modelling and experimental part in the case of Newtonian fluid. In the case of non-Newtonian fluid velocity profiles do not agree. The development track of ultrasound-assisted mixing modelling is presented in the thesis.