Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes them a very active research area. In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed over the last few years. This approach, is essentially a perturbation technique based on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations and scalar higher-order quasilinear PDEs. As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important, since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition of integrability, based on hydrodynamic reductions, is used throughout, which is a natural analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are given for all classification results obtained.

Thesis (MScEng (Process Engineering))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Mass transfer efficiency in distillation, absorption and stripping depends on both thermodynamic efficiency and hydrodynamic behaviour. Thermodynamic efficiency is dependent on the system kinetics while hydrodynamics is the study of fluid flow behaviour. The focus of this thesis is the hydrodynamic behaviour in tray columns, which affects entrainment. In order to isolate hydrodynamic behaviour from the thermodynamic behaviour that occurs inside sieve tray columns, investigations are conducted under conditions of zero mass transfer. When the gas velocity is sufficiently high to transport liquid droplets to the tray above, entrainment occurs. The onset of entrainment is one of the operating limits that determines the design of the column and thus impacts on the capital cost. By improving the understanding of the parameters that affect entrainment, the design of the tray and column can be improved which will ultimately increase the operability and capacity while reducing capital costs. Existing correlations predicting entrainment in sieve tray columns are based on data generated mainly from an air/water system. Previous publications recommend that more testing should be performed over larger ranges of gas and liquid physical properties. An experimental setup was therefore designed and constructed to test the influence of the following parameters on entrainment: 1. gas and liquid physical properties 2. gas and liquid flow rates 3. tray spacing The experimental setup can also measure weeping rates for a continuation of this project. The hydrodynamic performance of a sieve tray was tested with air and water over a wide range of gas and liquid flow rates and at different downcomer escape areas. It was found that the downcomer escape area should be sized so that the liquid escaping the downcomer always exceeds a velocity of approximately 0.23 m/s in order to create a sufficient liquid seal in the downcomer. For liquid velocities between 0.23 and 0.6 m/s the area of escape did not have an effect on the percentage of liquid entrained. It was also established that entrainment increases with increasing gas velocity. The rate at which entrainment increases as the gas velocity increase depends on the liquid flow rate. As soon as the liquid flow rate exceeded 74 m3/(h.m) a significant increase in entrainment was noted and the gas velocity had to be reduced to maintain a constant entrainment rate. This is because the increased liquid load requires a longer flow path length for the froth to fully develop. The undeveloped froth, caused by the short (455 mm) flow path, then creates a non-uniform froth that is pushed up against the column wall above the downcomer. Consequently, the froth layer is closer to the tray above resulting in most of the droplets ejected from the froth reaching the tray above and increasing entrainment. By reducing the gas velocity, the froth height and ejecting droplet velocity is reduced, resulting in a decrease in entrainment. The results from the experiments followed similar trends to most of the entrainment prediction correlations found in literature, except for the change noted in liquid flow rates above 74 m3/(h.m). There was, however, a significant difference between the experimental results and the correlations developed by Hunt et al. (1955) and Kister and Haas (1988). Although the gas velocities used during the air/water experiments were beyond the suggested range of application developed by Bennett et al. (1995) their air/water correlation followed the results very well. The entrainment prediction correlation developed by Bennett et al. (1995) for non-air/water systems was compared with the experimental air/water results to test for system uniformity. A significant difference was noted between their non-air/water prediction correlation and the air/water results, which motivates the need for a general entrainment prediction correlation over a wider range of gas and liquid physical properties. Based on the shortcomings found in the literature and the observations made during the experiments it is suggested that the influence of liquid flow path length should be investigated so that the effect on entrainment can be quantified. No single correlation was found in the literature, which accurately predicts entrainment for a large range of liquid loads (17 – 112 m3/(h.m)), high superficial gas velocities (3 – 4.6 m/s) and different gas and liquid physical properties. It is therefore recommended that more work be done, as an extension of this project, to investigate the influence of gas and liquid physical properties on entrainment (under zero mass transfer conditions) for a large range of liquid (5 – 74 m3/(h.m)) and gas (2 – 4.6 m/s) flow rates. In order to understand the effect of droplet drag on entrainment, tray spacing should be varied and increased to the extent where droplet ejection velocity is no longer the mechanism for entrainment and droplet drag is responsible for droplet transport to the tray above. Since it is difficult and in most cases impossible to measure exact gas and liquid loads in commercial columns, another method is required to measure or determine entrainment. Since liquid hold-up was found to be directly related to the entrainment rate (Hunt et al. (1955), Payne and Prince (1977) and Van Sinderen et al. (2003) to name but a few), it is suggested that a correlation should be developed between the dynamic pressure drop (liquid hold-up) and entrainment. This will contribute significantly to commercial column operation from a hydrodynamic point of view.

Thesis (MEng)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Tana River, the largest river in Kenya, is an important habit that supports numerous types of life, which creates an attractive environment. Occasionally loss of life and damage to property are experienced during floods. Upstream development of hydropower generation dams without consideration of downstream impacts is well documented in literature and the aftermaths are being well exhibited currently. The aim of the thesis is to investigate and identify factors that affect the flood levels in the Tana River Delta by using a two dimensional model and eventually drawing up a velocity- water depth interaction hazard classification map. Reviews of the literature clearly established that the floods in the delta are not generated by internal rainfall only but also operation of upstream dams accounting for 95% of the flood levels. Consequently, investigations of the impacts of dams have on flow regime were carried out. Probabilistic analysis revealed that post dam seasonal patterns has not been impacted but the magnitude of flood peaks has generally declined due to the attenuation of small peaks. However, large/flash floods (10 year Annual Recurrence Interval (ARI) spill at the dams causing major flooding downstream. Further probabilistic analysis on river discharges and sea water level was carried out to determine various ARI peaks. This incorporated climate change based on the 4th IPCC report. A two-dimensional hydrodynamic model was set up and calibrated with recorded discharges and theoretically derived parameters. Impacts of extreme tidal levels were investigated on the water levels and other factors limiting flood propagation. Finally, the model was used to simulate the 2, 50 and 100 year ARI inclusive of climate change floods and, based on Australian guidelines, flood lines and hazard maps were drawn. The results show that high tides elevate water levels in the delta in combination with the bottleneck effect at the rechanneled canal. The road crossing through the delta has inadequate bridges to convey the floods. The derived flood maps drawn (Figure 6-4) highlight that settlements in the lower delta are located within the 2 year ARI flood lines and that the extent of flooding is similar or less so in 50 and 100 year ARI flood peaks simulated. The model predicted the velocity and water depths with sufficient accuracy and recommendations are made that the study area should be extended upstream, and more field data should be collected to aid in calibration and that land use should be incorporated in flood map classification. In conclusion, the thesis has identified the flood hotspots and factors governing floods. These findings could assist in decision making by various agencies proposing flood mitigation or advocating post dam flooding scenarios. / AFRIKAANSE OPSOMMING: Die Tanarivier is die grootste rivier in Kenia en ’n belangrike habitat vir verskeie tipes diere en plante wat ‘n aantreklike omgewing skep.Verlies aan lewens en skade aan eiendom vind egter somtyds tydens oorstromings plaas. Die bou van damme vir die ontwikkeling van hidrokrag hoër op in die rivier sonder om die impak laer af in ag te neem, asook die gevolge daarvan, word dikwels in die lireratuur beskryf. Die doel met hierdie tesis is om die faktore wat die vloedhoogtes in die Tanarivier beïnvloed met die gebruik van ’n twee-dimensionele hidrodinamiese model te ondersoek en te identifiseer en om dan ’n gevaarsonekaart te teken wat die interaksie tussen waterspoed en -diepte toon. Die oorsig van die literatuur het getoon dat die oorstromings in die delta nie die gevolg is van reën in die binneland nie, maar dat die damme hoër op in die rivier verantwoordelik is vir 95% van die oorstromings. Dus is die impak van die damme op oorstromings ondersoek. Waarskynlikheidsontleding het oor die algemeen getoon dat die na-dam seisoenale vloeipatrone nie beinvloed is nie, maar dat veral die kleiner vloedpieke in die algemeen afgeneem het as gevolg van vloedattenuasie deur die damme. Groter en frats oorstromings (>1:10 ARI) veroorsaak egter steeds dat die damme oorloop en lei tot ernstige oorstromings. Die verdere waarskynlikheidsontleding van riviervloei en die seewatervlak is uitgevoer om die verskillende Jaarlikse Herhaling Periode (JHP) vlakke vas te stel. Dit het klimaatsverandering gegrond op die 4de IPCC verslag ingesluit. ʼn Twee dimensionele hidrodinamiese model is opgestel en gekalibreer teen waargenome vloei en teoreties-afgeleide parameters is gekalibreer. Die impak van uiterste getyvlakke asook faktore wat die oorstromings beperk is ondersoek. Die model is toe gebruik om die 2, 50 en 100 JHP vloedoorstromings te simuleer en vloedlyne en gevaarkaarte is volgens die Australiese riglyne geteken. Die resultate toon dat hooggety die watervlak in die delta laat styg veral in kombinasie met die bottelnek effek van die nuwe kanaal. Daar is te min brûe op die pad wat die delta deurkruis, om die vloede se vloei deur te laat Die kaarte wat geteken is toon dat daar nedersettings in die laer delta binne die 2 JHP jaar vloedarea is en dat die omvang van oorstromings dieselfde of laer is as die 50 en 100 JHP jaar vloedpieke wat gesimuleer is. Die model kan gebruik word om die vloeispoed en waterdieptes redelik akkuraat te voorspel en die volgende aanbevelings word gemaak. Toestande hoër op teen die rivier moet ondersoek word, meer data wat gedurende kalibrasie gebruik kan word moet versamel word en grondgebruik moet in die kaartklassifikasie ingesluit word. Ten slotte is die gevaarpunte vir oorstromings en die faktore wat oorstromings veroorsaak aangetoon. Hierdie bevindinge kan van nut wees as besluite geneem moet word veral wat betref die voorkoming van oorstromings nadat damme gebou is.

A thorough investigation on the existing hydrodynamic lubrication theories and the reverse pumping theories for the conventional lip seal is conducted. On that basis, the algorithms and the methods used in the numerical modeling of the conventional lip seal are modified and applied to the study of the lip seal running against surface textured shafts. For each step of the study, the numerical model is benchmarked against the experimental results. Important physical mechanisms which explain the reverse pumping ability of the triangular surface structures are revealed. Meanwhile, the accuracy of the numerical model is tested. In general, the numerical simulation results match the experimental observation well. However, there are several important discrepancies. For each discrepancy the possible causes are discussed, which benefits the further attempts of the modeling work on the lip seal running against surface textured shafts. The conclusions of this study themselves can be used as a guidance to the design of the surface textured shafts for the lip seal applications. Finally the limitation of the current theories and the modeling methods are discussed and reasonable improvements which can be done are proposed for the future work.

The thermal velocity fluctuations of microscopic particles mediate the transition from microscopic statistical mechanics to macroscopic long-time diffusion. Prior to this work, detection methods lacked the sensitivity necessary to resolve motion at the length and time scales at which thermal velocity fluctuations occur. This dissertation details two experiments which resulted in velocity measurement of the thermal motion of dielectric microspheres suspended by an optical trap in gases and liquids. First, optical tweezers were used to trap glass microspheres in air over a wide range of pressures and a detection system was developed to track the trapped microspheres' trajectories with MHz bandwidth and <100 fm/rt(Hz) position sensitivity. Low-noise trajectory measurements allowed for observation of fluctuations in the instantaneous velocity of a trapped particle with a signal to noise ratio (SNR) of 26 dB, and provided direct verification of the equipartition theorem and of the Maxwell-Boltzmann velocity distribution for a single Brownian particle. Next, the detection technology was further optimized and used to track optically trapped silica and barium titanate glass microspheres in water and acetone with >50 MHz bandwidth and <3 fm/rt(Hz) sensitivity. Brownian motion in a liquid is influenced by hydrodynamic, time-retarded coupling between the particle and the fluid flow its motion generates. Our measurements allowed for instantaneous velocity measurement with an SNR of up to 16 dB and confirmed the Maxwell Boltzmann distribution for Brownian motion in a liquid. The measurements also revealed several unusual features predicted for Brownian motion in the regime of hydrodynamic coupling, including faster-than-exponential decay of the velocity autocorrelation function, correlation of the thermal force and non-zero cross-correlation between the particle's velocity and the thermal force preceding it. / text

The aerodynamic performance of lifting surfaces operating at low Reynolds number conditions is impaired by laminar separation. Understanding of the physical mechanisms and hydrodynamic instabilities that are associated with laminar separation and the formation of laminar separation bubbles (LSBs) is key for the design and development of effective and efficient active flow control (AFC) devices. For the present work, laminar separation and its control were investigated numerically by employing highly-accurate direct numerical simulations (DNS).For a LSB on a curved plate, the primary and secondary instability of the uncontrolled flow were investigated. An inviscid Kelvin-Helmholtz (KH) instability was found to be responsible for the shedding of predominantly two-dimensional (2D) vortices. The onset of transition was caused by temporally-growing three-dimensional (3D) disturbances inside the separated region, which were supported by elliptical and hyperbolic secondary instabilities. The hyperbolic instability was demonstrated to be of absolute/global nature. High-amplitude forcing using pulsed vortex generator jets and 2D time-periodic blowing was found to exploit the KH instability and lead to a significant reduction in bubble size. In addition, the 2D forcing was found to suppress the secondary instabilities such that transition to turbulence was delayed.The role of free-stream turbulence (FST) in the transition process was investigated for a LSB on a flat plate. FST was shown to cause the formation of streamwise-elongated streaks inside the boundary layer. For the uncontrolled LSB, increasing the FST levels led to accelerated transition and a reduction in bubble size. The stage of linear disturbance growth due to the inviscid KH instability was not ``bypassed''. Flow control by means of 2D periodic excitation was found to remain effective, since it could exploit the KH instability and suppress secondary absolute instabilities. Transition was initiated by an interaction of the 2D wave introduced by the forcing and the streamwise boundary-layer streaks. The interaction led to a spanwise modulation of the 2D wave, which was amplified due to a convective elliptical instability.

Among sources of renewable energy, development of tidal energy has traditionally been plagued by relatively high costs and limited availability of sites with sufficiently high tidal amplitudes or flow velocities. However, many recent technology developments and improvements, both in design (e.g. dynamic tidal power, tidal lagoons) and turbine technology (e.g. new axial turbines, crossflow turbines), showed that the economic and environmental costs may be brought down to competitive levels comparing to other conventional energy sources. It has long been identified that the Bay of Fundy is one of the world’s premier locations for the development of tidal power generating systems, since it has some of the world’s largest tidal ranges. Consequently, several proposals have been made in the recent years to find economical ways to harness the power of tides. Presently, there is considerable interest in installing tidal lagoons in the Bay of Fundy. The lagoon concept involves temporarily storing seawater behind an impoundment dike and generating power by gradually releasing the impounded seawater through conventional low-head hydroelectric turbines. A tidal lagoon will inherently modify the tides and tidal currents regime in the vicinity of the lagoon, and possibly induce effects that may be felt throughout the entire Bay of Fundy. The nature of these hydrodynamic impacts will likely depend on the size of the tidal lagoon, its location, and its method of operation. Any changes in the tidal hydrodynamics caused by a tidal lagoon may also impact on the transport of sediments throughout the region and upset ecosystems that are well adapted to existing conditions. The scale and character of the potential hydrodynamic impacts due to tidal lagoons operating in the Bay of Fundy have not been previously investigated. The present study endeavours to investigate these potential impacts to help the development of sustainable, science-based policies for the management and development of clean energy for future generations. After outlining fundamental aspects of tidal power projects taken in consideration in the Bay of Fundy, an analysis of present knowledge on tidal lagoons was conducted in order to provide a focus for subsequent investigations. Hydrodynamic modeling was used to quantify any of the potential hydrodynamic changes induced in the Bay of Fundy due to the presence of tidal lagoons. In the last part of the thesis, new relationships were derived in order to describe the amount of energy removed from tidal lagoons associated with its hydrodynamic impacts.

Tsunamis are among the most terrifying and complex physical phenomena potentially affecting almost all coastal regions of the Earth. Tsunami waves propagate in the ocean over thousands of kilometres away from their generating source at considerable speeds. Among several other tsunamis that occurred during the past decade, the 2004 Indian Ocean Tsunami and the 2011 Tohoku Tsunami in Japan, considered to be the deadliest and costliest natural disasters in the history of mankind, respectively, have hit wide stretches of densely populated coastal areas. During these major events, severe destruction of inland structures resulted from the action of extreme hydrodynamic forces induced by tsunami flooding. Subsequent field surveys in which researchers from the University of Ottawa participated ultimately revealed that, in contrast to seismic forces, such hydrodynamic forces are not taken into proper consideration when designing buildings for tsunami prone areas. In view of these limitations, a novel interdisciplinary hydraulic-structural engineering research program was initiated at the University of Ottawa, in cooperation with the Canadian Hydraulic Centre of the National Research Council, to help develop guidelines for the sound design of nearshore structures located in such areas. The present study aims to simulate the physical laboratory experiments performed within the aforementioned research program using a single-phase three-dimensional weakly compressible Smoothed Particle Hydrodynamics (SPH) numerical model. These experiments consist in the violent impact of rapidly advancing tsunami-like hydraulic bores with individual slender structural elements. Such bores are emulated based on the classic dam-break problem. The quantitatively compared measurements include the time-history of the net base horizontal force and of the pressure distribution acting on columns of square and circular cross-sections, as well as flow characteristics such as bore-front velocity and water surface elevation. Good agreement was obtained. Results show that the magnitude and duration of the impulsive force at initial bore impact depend on the degree of entrapped air in the bore-front. The latter was found to increase considerably if the bed of the experimental flume is covered with a thin water layer of even just a few millimetres. In order to avoid large fluctuations in the pressure field and to obtain accurate simulations of the hydrodynamic forces, a Riemann solver-based formulation of the SPH method is utilized. However, this formulation induces excessive numerical diffusion, as sudden and large water surface deformations, such as splashing at initial bore impact, are less accurately reproduced. To investigate this particular issue, the small-scale physical experiment of Kleefsman et al. (2005) is also considered and modeled. Lastly, taking full advantage of the validated numerical model to better understand the underlying flow dynamics, the influence of the experimental test geometry and of the bed condition (i.e. dry vs. wet) is investigated. Numerical results show that when a bore propagates over a wet bed, its front is both deeper and steeper and it also has a lower velocity compared to when it propagates over a dry bed. These differences significantly affect the pressure distributions and resulting hydrodynamic forces acting on impacted structures.

The discovery of carbon nanotubes by Iijima in 1991 has created a torrent of new research activities. Research on carbon nanotubes ranges from studying their fundamental properties, such as their electron band structure and plasma frequencies, to developing new applications, such as self-assembled nano-circuits and field emission displays. Robust models are now needed to enable a better understanding of the electronic response of carbon nanotubes. We use time-dependent density functional theory to derive a two-fluid two-dimensional (2D) hydrodynamic model describing the collective response of a multiwalled carbon nanotube with dielectric media embedded inside or surrounding the nanotube. We study plasmon hybridization of the nanotube system in the UV range, the stopping force for ion channelling, the dynamical image potential for fast ions, channelled diclusters and point dipoles, and the energy loss for ions with oblique trajectories. Comparisons are made of results obtained from the 2D hydrodynamic model with those obtained from an extension of the 3D Kitagawa model to cylindrical geometries.

The integrability of an m-component system of hydrodynamic type, Ut = v(u)ux, by the generalized hodograph method requires the diagonalizability of the m x m matrix v(u). The diagonalizability is known to be equivalent to the vanishing of the corresponding Haantjes tensor. This idea is applied to hydrodynamic chains - infinite-component systems of hydrodynamic type for which the 00 x 00 matrix v(u) is 'sufficiently sparse'. For such 'sparse' systems the Haantjes tensor is well-defined, and the calculation of its components involves only a finite number of summations. The calculation of the Haantjes tensor is done by using Mathematica to perform symbolic calculations. Certain conservative and Hamiltonian hydrodynamic chains are classified by setting Haantjes tensor equal to zero and solving the resulting system of equations. It is shown that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability of such sysyems. In the cases of the Hamiltonian hydrodynamic chains we were able to first construct one extra conservation law and later a generating function for conservation laws, thus establishing the integrability.