Cette thèse est consacrée à l'étude de la dynamique des gaz à faible nombre de Mach. Le modèle étudié provient des équations de Navier-Stokes complètes lorsque le nombre de Mach tend vers zéro. On cherche à montrer que le problème de Cauchy correspondant est bien posé. Les cas visqueux et non visqueux sont tous deux considérés. Les coefficients physiques peuvent dépendre de la densité (ou de la température) inconnue. En particulier, nous prenons en compte les effets de conductivité thermique et on autorise de grandes variations d'entropie. Rappelons qu'en absence de diffusion thermique, la limite à faible nombre de Mach implique la condition d'incompressibilité. Dans le cadre étudié ici, en introduisant un nouveau champ de vitesses à divergence nulle, le système devient un couplage non linéaire entre une équation quasi-parabolique pour la densité et un système de type Navier-Stokes (ou Euler) pour la vitesse et la pression. Pour le cas avec viscosité, on établit le résultat classique, à savoir qu'il existe une solution forte existant localement (resp. globalement) en temps pour des données initiales grandes (resp. petites). On considère ici le problème de Cauchy avec données initiales dans des espaces de Besov critiques. Lorsque les coefficients physiques du système vérifient une relation spéciale, le système se simplifie considérablement, et on peut alors établir qu'il existe des solutions faibles globales en temps à énergie finie. Par un argument d'unicité fort-faible, on en déduit que les solutions fortes à énergie finie existent pour tous les temps positifs en dimension deux. Pour le cas sans viscosité, on montre d'abord le caractère bien posé dans des espaces de Besov limites, qui s'injectent dans l'espace des fonctions lipschitziennes. Des critères de prolongement et des estimations du temps de vie sont établis. Si l'on suppose la donnée initiale à énergie finie dans l'espace de Besov limite à exposant de Lebesgue infini, on a également un résultat d'existence locale. En dimension deux, le temps de vie tend vers l'infini quand la densité tend vers une constante positive. Des estimations de produits et de commutateurs, ainsi que des estimations a priori pour les équations paraboliques et pour le système de Stokes (ou d'Euler) à coefficients variables, se trouvent dans l'annexe. Ces estimations reposent sur la théorie de Littlewood-Paley et le calcul paradifférentiel / This thesis is devoted to the study of the dynamics of the gases with small Mach number. The model comes from the complete Navier-Stokes equations when the Mach number goes to zero, and we aim at showing that it is well-posed. The viscous and inviscid cases are both considered. The physical coefficients may depend on the unknown density (or on the unknown temperature).In particular, we consider the effects of the thermal conductivity and hence large variations of entropy are allowed. Recall that if there is no thermal diffusion, then the low Mach number limit just implies the incompressibility condition. In the framework considered here, by introducing a new solenoidal velocity field, the system becomes a nonlinear coupling between a quasi-parabolic equation for the density and an evolutionary Stokes (or Euler) system for the velocity and the pressure. For the case with viscosity, we establish classical results, namely the strong solutions exist locally (resp. globally) in time for big (resp. small) initial data. We consider the Cauchy problem in the critical Besov spaces with the lowest regularity. Under a special relationship between the two physical coefficients, the system recasts in a simpler form and one may prove that there exist weak solutions with finite energy. In dimension two, this implies that strong solutions with finite energy exist for all positive times. In the inviscid case, we first prove the well-posedness result in endpoint Besov spaces, which can be embedded into the set of Lipschitzian functions. Continuation criterions and estimates for the lifespan are both established.If we suppose the initial data to be in the borderline Besov spaces with infinite Lebesgue exponent and to be of finite energy, we also have a local existence result. In dimension two, the lifespan goes to infinity when the density tends to a positive constant. Estimates for products and commutators, together with a priori estimates for the parabolic equations and the Stokes (or Euler) system with variable coefficients, are postponed in the appendix. These estimates are based on the Littlewood-Paley theory and the paradifferential calculus

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2016 / The use of fuels (from crude oil) in vehicles is responsible for one of the biggest environmental challenges; SO2 emission. As a result most countries regulate their sulfur emissions, with the goal of getting to the use of 10 ppm sulfur fuels. These stringent fuel sulfur content requirements have resulted in intensive research being directed at alternative desulfurization technologies which will ensure the treatment of fuels to acceptable sulfur levels. Extractive desulfurization using ionic liquids (IL) may be considered as one of the most promising of these technologies and is the subject of the study presented in this work. This study served two major purposes: (1) to investigate the capacity as well as key parameters which affect the extraction efficiency of the IL; 1-butyl-3-methylimidazolium octylsulfate as a solvent for deep extractive desulfurization of real Fluid Catalytic Cracking Unit (FCCU) diesel fuel samples collected from a typical South African Refinery, (2) to study/find suitable solvents for the regeneration of sulfur-loaded 1-butyl-3-methylimidazolium octylsulfate and the efficiency and effectiveness of the regenerated IL in the desulfurization of diesel fuel. 1-butyl-3-methylimidazolium octylsulfate was selected due to its properties i.e. good extractive ability for S-compounds and insolubility in fuel oils. A 22.1% sulfur removal was achieved in the desulfurization of FCCU feed stream diesel fuel, while 96% sulfur removal was achieved for FCCU product stream diesel fuel. These results show that the IL is more effective in the selective removal of sulfur (S) from FCCU diesel product than from FCCU feed stream, suggesting that fuel sulfur content and stream composition affects the extraction efficiency and effectiveness of the IL. Based on thermodynamic considerations, hexane was selected as the most suitable solvent for the re-extraction of sulfur from spent IL. Regenerated IL was used for desulfurization of diesel and achieved highest sulfur removal of 95% and the IL was regenerated up to four times without appreciable decrease in efficiency. The results obtained herein show that ILs are effective in the desulfurization of real diesel oil samples when the sulfur concentration is not very high. Further studies on the recoverability of ILs as well as their environmental impact need to be done to support findings in this study. / GR2016

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013. / In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is analytically considered using Fourier and Laplace transformations. The main focus of the dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction Implicit method. The one-dimensional parabolic mixed derivative diffusion equation is extended to a two-dimensional analog. In order to do this, the two-dimensional analog is solved using a Crank-Nicholson method and implemented according to the Peaceman- Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour, subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion equation is then implemented with a high-order method to unveil more accurate solutions. An error analysis is executed to show the accuracy differences between the numerical solutions of the general ADI and high-order compact methods. Each of the methods implemented in this dissertation are investigated via the von-Neumann stability analysis to prove stability under certain conditions.

In this project the link between species in solution and the solid state was considered. This is relevant due to the many applications in life where there is this interchange between solid and solution state, for example drug design, environmental metal speciation and the manufacture of materials that are in contact with solution (such as outdoor surface coatings, containers and so on). Complexation of two metal ions, namely cadmium(II) and bismuth(III), was studied. With bismuth showing so much promise in medicinal applications it was pertinent to investigate this interchange since the intake of medication is generally in the solid form which then converts to solution species as it dissolves in the body where it becomes active. For cadmium it is mainly the environmental concerns which we are faced with that call for the examination of speciation of complexes in solution, as well as their disposition upon precipitation or crystallization. A correlation was found between solution species and the complex that was isolated in the crystalline form with regards to pH for a number of metal-ligand species. We show how the results from solution experiments (achieved using direct current polarography) and those of crystalline complexes can complement each other when using species distribution diagrams as the intermediary. The distribution of species can be varied by changing the concentration and ligand-to-metal ratio at which the species distribution diagram is plotted. It is this characteristic which allows the solution and crystalline complexes – which are achieved using differing experimental conditions – to be correlated. The speciation diagram for a metal-ligand system, calculated using formation constants derived from solutions studies, was used in most instances to target specific species for their growth in the solid state. In some cases the solid state structure was used to confirm a suspected solution species, and in others the result was used to identify minor solution species which cannot be detected by the techniques used in determining formation constants. Further, we show that doing solution experiments at a range of temperatures can also aid in elucidating these minor species. The growth of crystalline species at low pH was important for this work because the pH titrations used for solution experiments were conducted from below pH 2 where the diffusion junction potential is large and changing. An in-situ witness ion was incorporated into the experiment to monitor the shifts due to the diffusion junction potential so that they could be compensated for. Additionally, for bismuth-ligand systems, hydrolysis and complexation with nitrates occurs in this same pH region. The formation constants and the species identified below pH 2 therefore do carry some uncertainty, so obtaining crystalline complexes of these species provides further confidence in their prediction in solution.

Neste trabalho, estudamos a teoria de bifurcação para equações diferenciais ordinárias (escrevemos simplesmente EDOs), bem como a existência de ponto de bifurcação para soluções periódicas destas equações. Em seguida, desenvolvemos a teoria, até então inexistente, sobre bifurcação para equações diferenciais ordinárias generalizadas (EDOs generalizadas). Neste desenvolvimento, obtivemos para EDOs generalizadas, um resultado sobre existência de ponto de bifurcação para soluções periódicas. Em seguida, através da correspondência entre EDOs e EDOs generalizadas, obtivemos novos resultados sobre a existência de ponto de bifurcação para soluções periódicas para EDOs clássicas, agora sob a ótica das EDOs generalizadas, quando então, em vez de funções continuamente diferenciáveis, necessitamos, apenas, que as funções envolvidas na EDO sejam integráveis no sentido de Kurzweil-Henstock. Adicionamos, também, um resultado sobre a existência de soluções periódicas de EDOs generalizadas e aplicamos tal resultado para EDOs. A fim de obtermos os resultados que pretendíamos, utilizamos a teoria do grau coincidente. Finalmente, mencionamos que os resultados inéditos deste trabalho estão contidos em [6] / In this work, we study the bifurcation theory for ordinary dierential equations (we write simply ODEs), as well as the existence of a bifurcation point for periodic solutions of these equations. Then we develop the theory of bifurcation for generalized ordinary differential equations (we write generalized ODEs for short). Such theory is new. We obtained an existence result of a bifurcation point for periodic solutions of generalized ODEs. By means of the correspondence of classic ODEs and generalized ODEs, we were able to translate the results to classic ODEs, now in the framework of generalized ODE. This means that instead of the classic hypothesis that the functions involved in the differential equation are continuously differentiable, we only require that they are Kurzweil-Henstock integrable. We also added a result on the existence of a periodic solution of a generalized ODE and we applied such result to classic ODEs. In order to obtain our main results, we employed the coincidence degree theory. Finally, we point out that our results are contained in [6]

Estudou-se o comportamento de íons de cobre (II) em soluções de tiocianato de sódio por via espectrofotométrica e polarográfica. O estudo espectrofotométrico em força iônica 2,0M, a 25°C, levou a determinação da constante de formação da primeira espécie, Cu SCN+, com máximo de absorção a 342 nm, sendo β1 = 56 M-1 e ε1,max = 495?.mol-1 cm-1. Tentativas de estudar a formação sucessiva dos outros três complexos não levaram a resultado satisfatório devido à instabilidade de cor do sistema cobre(II/SCN-. Extenso estudo levou ao esclarecimento do comportamento polarográfico do cobre (II) em tiocianato, onde havia sérias discordâncias na literatura. Comprovou-se que parte do cobre (II) oxida o ligante, passando a cobre (I), havendo formação de SO=4 e HCN. Há três ondas polarográficas: a primeira, catódica, mas sem descontinuidade com o componente anódico de dissolução de mercúrio pelo eletrólito suporte, devido ao cobre (II) remanescente; a segunda trata da redução do cobre (I) ao amálgama de cobre; a terceira é a onda de redução dos íons de hidrogenia liberados pela oxidação do ligante. Fizeram-se estudos de medidas de potencial no sistema (SCN)2/SCN- com tiocianogênio gerado coulometricamente, ou por agentes químicos oxidantes. Novas perspectivas de trabalho são discutidas. Polarogramas no sistema cobre(I)/SCN-, em força iônica 4,0M, a 25°C, levaram ao primeiro estudo completo de equilíbrio do sistema, caracterizado pelas seguintes constantes de equilíbrio: (Ver no arquivo em PDF) Para o cobre (II) foi estimado o seguinte valor de constante: (Ver no arquivo em PDF) / The behavior of copper (II) in sodium thiocyanate solutions was examined by polarography and spectrophotometry. The spectrophotometric study performed in an ionic strength 2.0M at 25°C, lead to the determination of the formation constant β1 = 56 M-1, referred to Cu SCN+ specie. Its absorption maximum is 342 nm and ε1,max = 495?.mol-1 cm-1. Attempts to study the formation of the others successive complexes were unsuccess full due to color instability in the copper(II)/SCN system. Extensive study lead to the elucidation of the polarographic behavior of copper (II) in thiocyanate supporting electrolite. Severe disagreement was found in the literature about this subject. It has been found that copper (II) oxidises partially the ligand to SO=4 and HCN, with partial formation of copper (I). Three waves were well characterized: the first one is cathodic, but with no discontinuity with the anodic wave of mercury dissolution by thiocyanate, resulting from the presente of the remaining copper (II); the second one is referred to the reduction of copper (I) to copper a malgam; the third wave in the reduction of hydrogen ion released during the oxidation of the ligande. Potential measurement were carried out in the (SCN)2/SCN- system, with thiocyanogen generated by coulometry, or by the action of chemical oxidising agents. New prospects and ideas to be worked were discussed. Polarograms from the copper(I)/SCN- system, were obtained in 4,0M ionic strength and 25°C. They lead to the first complete equilibrium study of this system, with the determination of the following equilibrium constants: (See in PDF file) With regard to copper (II) the foliowing formation constant was estimated: (See in PDF file)

Este trabalho é dedicado ao estudo da existência de soluções da equação de Schrödinger \'DELTA\'u + (\'lambda\' a(x) + 1)u = \' u POT. p, u > 0 em \'R POT. N\', onde a \'> ou =\' 0 é uma função contínua e p > 1 é um expoente subcrítico. Métodos Variacionais são empregados para mostrar a existência de uma sequência \' lambda\' IND. n\' \' SETA\' + \'INFINITO\' e da respectiva sequência de soluções \'u IND. lambda IND. n\' convergindo para uma solução de energia mínima do problema de Dirichlet - \'DELTA\' u + u = \'u POT. p\', ; u > 0em \'OMEGA\', u = 0 sobre \'partial\'\' OMEGA\", sendo \"OMEGA\' := int \'a POT. -1\' (0). Além disso, estuda-se o efeito da topologia do conjunto \'OMEGA\' sobre o número de soluções da equação (*) por meio da categoria de Lusternik e Schnirelman / This work is devoted to study the existence of positive solutions of the Schrödinger equation \'DELTA\'u + (\'lambda\'a(x) + 1)u = \' u POT. p\', u > 0 in \'R POT. N\', where a is a nonnegative and continuous function and p > 1 is a subcritical exponent. Variational methods are employed in order to show the existence of a sequence \'lambda\' IND. n\' \"ARROW\' + \'THE INFINITE\' and the respective sequence of solutions converging in \'H POT. 1\' (\'R POT.N\' ) to a least energy solution of the Dirichlet problem - \'DELTA\'u + u = \'u POT. p\' ; u > 0 in \'OMEGA\', u = 0 on \'partial\' \' OMEGA\', where \'OMEGA\' : = int \'a POT. -1 (0) Furthermore, it is studied the effect of the topology of the set \'OMEGA\' on the number of positive solutions of the equation (*) by using the Lusternik and Schnirelman category

Neste trabalho, estudamos a teoria de bifurcação para equações diferenciais ordinárias (escrevemos simplesmente EDOs), bem como a existência de ponto de bifurcação para soluções periódicas destas equações. Em seguida, desenvolvemos a teoria, até então inexistente, sobre bifurcação para equações diferenciais ordinárias generalizadas (EDOs generalizadas). Neste desenvolvimento, obtivemos para EDOs generalizadas, um resultado sobre existência de ponto de bifurcação para soluções periódicas. Em seguida, através da correspondência entre EDOs e EDOs generalizadas, obtivemos novos resultados sobre a existência de ponto de bifurcação para soluções periódicas para EDOs clássicas, agora sob a ótica das EDOs generalizadas, quando então, em vez de funções continuamente diferenciáveis, necessitamos, apenas, que as funções envolvidas na EDO sejam integráveis no sentido de Kurzweil-Henstock. Adicionamos, também, um resultado sobre a existência de soluções periódicas de EDOs generalizadas e aplicamos tal resultado para EDOs. A fim de obtermos os resultados que pretendíamos, utilizamos a teoria do grau coincidente. Finalmente, mencionamos que os resultados inéditos deste trabalho estão contidos em [6] / In this work, we study the bifurcation theory for ordinary dierential equations (we write simply ODEs), as well as the existence of a bifurcation point for periodic solutions of these equations. Then we develop the theory of bifurcation for generalized ordinary differential equations (we write generalized ODEs for short). Such theory is new. We obtained an existence result of a bifurcation point for periodic solutions of generalized ODEs. By means of the correspondence of classic ODEs and generalized ODEs, we were able to translate the results to classic ODEs, now in the framework of generalized ODE. This means that instead of the classic hypothesis that the functions involved in the differential equation are continuously differentiable, we only require that they are Kurzweil-Henstock integrable. We also added a result on the existence of a periodic solution of a generalized ODE and we applied such result to classic ODEs. In order to obtain our main results, we employed the coincidence degree theory. Finally, we point out that our results are contained in [6]

by Niu Aizhen. / "Mar., 2001." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Zheng Yu-kun. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Includes bibliographies.