Three different models, relating to the study of microswimmers immersed in a low Reynolds number fluid, are presented. The underlying, mathematical concepts employed in each are developed using singularity methods of Stokes flow. The first topic concerns the motility of an artificial, three-sphere microswimmer with prescribed, non-reciprocal, internal forces. The swimmer progresses through a low Reynolds number, nonlinear, viscoelastic medium. The model developed illustrates that the presence of the viscoelastic rheology, when compared to a Newtonian environment, increases both the net displacement and swimming efficiency of the microswimmer. The second area concerns biological microswimming, modelling a sperm cell with a hyperactive waveform (vigorous, asymmetric beating), bound to the epithelial walls of the female, reproductive tract. Using resistive-force theory, the model concludes that, for certain regions in parameter space, hyperactivated sperm cells can induce mechanical forces that pull the cell away from the wall binding. This appears to occur via the regulation of the beat amplitude, wavenumber and beat asymmetry. The next topic presents a novel generalisation of slender-body theory that is capable of calculating the approximate flow field around a long, thin, slender body with circular cross sections that vary arbitrarily in radius along a curvilinear centre-line. New, permissible, slender-body shapes include a tapered flagellum and those with ribbed, wave-like structures. Finally, the detailed analytics of the generalised, slender-body theory are exploited to develop a numerical implementation capable of simulating a wider range of slender-body geometries compared to previous studies in the field.

At low Reynolds number flows, the effect of inertia becomes negligible and the fluid motion is dominated by the effect of viscous forces. Understanding of the behaviour of low Reynolds number flows underpins the prediction of the motion of microorganisms and particle sedimentation as well as the development of micro-robots that could potentially swim inside the human body to perform targeted drug/cell delivery and non-invasive microsurgery. The work in this thesis focuses on developing an understanding in the mathematical analysis of objects moving at low Reynolds numbers. A boundary element implementation of the Method of regularized Stokeslets (MRS) is applied to analyse the low Reynolds number flow field around an object of simple shape (sphere and cube). It also showed that the results obtained by a boundary element implementation for an unbounded cube, where singularities are presented in the corners of the cube, agrees with more complex solutions methods such as a GBEM and FEM.A methodology for analysing the effect of walls by locating collocation points on the surface of the walls and the object is presented. First at all, this methodology is validated with a boundary element implementation of the method of images for a sphere at different locations. Then, the method is extended when more than one wall is presented. This methodology is applied to predict the velocity filed of a cube moving in a tow tank at low Reynolds numbers for two different cases with a supporting rod similar to an experimental set-up, and without the supporting rod as in the CFD simulations based on the FVM. The results indicate a good match between CFD and the MRS, and an excellent approximation between the MRS and experimental data from PIV measurements. The drag, thrust and torque generated by helices moving at low Reynolds numbers in an unbounded medium is analysed by the resistive force theory, a slender body theory, and a boundary element method of the MRS. The results show that the resistive force theory predict accurately the drag, thrust and torque of moving helices when the resistive force coefficients are calculated from a slender body theory approximation by calculating independently the resistive force coefficients for translation and rotation, because it is observed that the resistive force coefficients depend also of the nature of motion. Moreover, the thrust generated by helices of different pitch angles is analysed calculated by a CFD numerical simulation based on the FVM and a boundary element implementation, an compared with experimental data. The results also show an excellent prediction between the boundary element implementation, the CFD results and the experimental data. Finally, a boundary element implementation of the MRS is applied to predict swimming of a biomimetic swimmer that mimics the motion of E.coli bacteria in an unbounded medium. The results are compared with the propulsive velocity and induced angular velocity measurement by recording the motion of the biomimetic swimmer in a square tank. It is observed that special care needs to be taken when the biomimetic swimmer is modelled inside the tank, as there is an apparent increment in the calculate thrust propulsion which does not represent a real situation of the biometic swimmer which propels by a power supply. However, this increment does not represent the condition of the biomimetic swimmer and a suggested methodology based on the solution from an unbounded case and when the swimmer is moving inside the tank is presented. In addition, the prediction of the free-swimming velocity for the biomimetic swimmer agrees with the results obtained by the MRS when the resistive force coefficients are calculated from a SBT implementation. The results obtained in this work have showed that a boundary element implementation of the MRS produces results comparable with more complex numerical implementations such as GBEM, FEM, FVM, and also an excellent agreement with results obtained from experimentation. Therefore, it is a suitable and easy to apply methodology to analyse the motion of swimmers at low Reynolds numbers, such as the biomimetic swimmer modelled in this work.

Em geral, uma embarcação de planeio é projetada para atingir elevados níveis de velocidade. Esse atributo de desempenho está diretamente relacionado ao porte da embarcação e à potência instalada em sua planta propulsiva. Tradicionalmente, durante o projeto de uma embarcação, as análises de desempenho são realizadas através de resultados de embarcações já existentes, retirados de séries sistemáticas ou de embarcações já desenvolvidas pelo estaleiro e/ou projetista. Além disso, a determinação dos atributos de desempenho pode ser feita através de métodos empíricos e/ou estatísticos, onde a embarcação é representada através de seus parâmetros geométricos principais; ou a partir de testes em modelos em escala reduzida ou protótipos. No caso específico de embarcações de planeio, o custo dos testes em escala reduzida é muito elevado em relação ao custo de projeto. Isso faz com que a maioria dos projetistas não opte por ensaios experimentais das novas embarcações em desenvolvimento. Ao longo dos últimos anos, o método de Savitsky foi largamente utilizado para se realizar estimativas de potência instalada de uma embarcação de planeio. Esse método utiliza um conjunto de equações semi-empíricas para determinar os esforços atuantes na embarcação, a partir dos quais é possível determinar a posição de equilíbrio de operação e a força propulsora necessária para navegar em uma dada velocidade. O método de Savitsky é muito utilizado nas fases iniciais de projeto, onde a geometria do casco ainda não foi totalmente definida, pois utiliza apenas as características geométricas principais da embarcação para realização das estimativas de esforços. À medida que se avança nas etapas de projeto, aumenta o detalhamento necessário das estimativas de desempenho. Para a realização, por exemplo, do projeto estrutural é necessária uma estimativa do campo de pressão atuante no fundo do casco, o qual não pode ser determinado pelo método de Savitsky. O método computacional implementado nesta dissertação, tem o objetivo de determinar as características do escoamento e o campo de pressão atuante no casco de uma embarcação de planeio navegando em águas calmas. O escoamento é determinado através de um problema de valor de contorno, no qual a superfície molhada no casco é considerada um corpo esbelto. Devido ao uso da teoria de corpo esbelto o problema pode ser tratado, separadamente, em cada seção, onde as condições de contorno são forçadamente respeitadas através de uma distribuição de vórtices. / Generally, a planing craft is designed to achieve high speed levels. This performance attribute is directly related to the boat size and to the propeller plant power. Traditionally, during a boat design, performance analyses are carried out using results taken from systematic series or from others boat previously build by the shipyard and/or designer. Furthermore, performance attributes can be calculated by semi-empirical and/or statistic methods or by tests of reduced scale models. In the specific case of planing boats, the costs of reduced scale tests are too high compared to the design cost itself. Because of this, most designers do not perform experimental tests during the development of new boats. During the last years, the Savitsky method was extensively used to estimate planing craft effective power. The method uses a set of semi-empirical equations to calculate the forces acting on the boat, from which the equilibrium position and the required propeller thrust are determined. During the preliminary phases of planing craft design, the hull geometry hasn\'t been fully defined. Therefore, the Savitsky method is widely used during this phase, because it uses only the main geometrical characteristics to estimate the forces acting on the hull. Advancing toward the final phases of the design process, more detailed information is required. To execute the structural design, for example, the pressure field acting on the hull must be known, which can\'t be estimate using the Savitsky method. The main objective of the present study is to implement a computer method that can be used to estimate the fluid flow and pressure field acting on the hull of a boat moving with forward speed constant in calm water. The fluid flow around the hull is treated as a boundary value problem, in which the wetted hull surface is considered a slender body. The slender body theory enables to solve the problem separately, in each transverse section, where boundary conditions are respected by a sheet of vortices.

Em geral, uma embarcação de planeio é projetada para atingir elevados níveis de velocidade. Esse atributo de desempenho está diretamente relacionado ao porte da embarcação e à potência instalada em sua planta propulsiva. Tradicionalmente, durante o projeto de uma embarcação, as análises de desempenho são realizadas através de resultados de embarcações já existentes, retirados de séries sistemáticas ou de embarcações já desenvolvidas pelo estaleiro e/ou projetista. Além disso, a determinação dos atributos de desempenho pode ser feita através de métodos empíricos e/ou estatísticos, onde a embarcação é representada através de seus parâmetros geométricos principais; ou a partir de testes em modelos em escala reduzida ou protótipos. No caso específico de embarcações de planeio, o custo dos testes em escala reduzida é muito elevado em relação ao custo de projeto. Isso faz com que a maioria dos projetistas não opte por ensaios experimentais das novas embarcações em desenvolvimento. Ao longo dos últimos anos, o método de Savitsky foi largamente utilizado para se realizar estimativas de potência instalada de uma embarcação de planeio. Esse método utiliza um conjunto de equações semi-empíricas para determinar os esforços atuantes na embarcação, a partir dos quais é possível determinar a posição de equilíbrio de operação e a força propulsora necessária para navegar em uma dada velocidade. O método de Savitsky é muito utilizado nas fases iniciais de projeto, onde a geometria do casco ainda não foi totalmente definida, pois utiliza apenas as características geométricas principais da embarcação para realização das estimativas de esforços. À medida que se avança nas etapas de projeto, aumenta o detalhamento necessário das estimativas de desempenho. Para a realização, por exemplo, do projeto estrutural é necessária uma estimativa do campo de pressão atuante no fundo do casco, o qual não pode ser determinado pelo método de Savitsky. O método computacional implementado nesta dissertação, tem o objetivo de determinar as características do escoamento e o campo de pressão atuante no casco de uma embarcação de planeio navegando em águas calmas. O escoamento é determinado através de um problema de valor de contorno, no qual a superfície molhada no casco é considerada um corpo esbelto. Devido ao uso da teoria de corpo esbelto o problema pode ser tratado, separadamente, em cada seção, onde as condições de contorno são forçadamente respeitadas através de uma distribuição de vórtices. / Generally, a planing craft is designed to achieve high speed levels. This performance attribute is directly related to the boat size and to the propeller plant power. Traditionally, during a boat design, performance analyses are carried out using results taken from systematic series or from others boat previously build by the shipyard and/or designer. Furthermore, performance attributes can be calculated by semi-empirical and/or statistic methods or by tests of reduced scale models. In the specific case of planing boats, the costs of reduced scale tests are too high compared to the design cost itself. Because of this, most designers do not perform experimental tests during the development of new boats. During the last years, the Savitsky method was extensively used to estimate planing craft effective power. The method uses a set of semi-empirical equations to calculate the forces acting on the boat, from which the equilibrium position and the required propeller thrust are determined. During the preliminary phases of planing craft design, the hull geometry hasn\'t been fully defined. Therefore, the Savitsky method is widely used during this phase, because it uses only the main geometrical characteristics to estimate the forces acting on the hull. Advancing toward the final phases of the design process, more detailed information is required. To execute the structural design, for example, the pressure field acting on the hull must be known, which can\'t be estimate using the Savitsky method. The main objective of the present study is to implement a computer method that can be used to estimate the fluid flow and pressure field acting on the hull of a boat moving with forward speed constant in calm water. The fluid flow around the hull is treated as a boundary value problem, in which the wetted hull surface is considered a slender body. The slender body theory enables to solve the problem separately, in each transverse section, where boundary conditions are respected by a sheet of vortices.

A planing catamaran is a high-powered, twin-hull water craft that develops the lift which supports its weight, primarily through hydrodynamic water pressure. Presently, there is increasing demand to further develop the catamaran's planing and seakeeping characteristics so that it is more effectively applied in today's modern military and pleasure craft, and offshore industry supply vessels. Over the course of the past ten years, Vorus (1994,1996,1998,2000) has systematically conducted a series of research works on planing craft hydrodynamics. Based on Vorus' planing monohull theory, he has developed and implemented a first order nonlinear model for planing catamarans, embodied in the computer code CatSea. This model is currently applied in planing catamaran design. However, due to the greater complexity of the catamaran flow physics relative to the monohull, Vorus's (first order) catamaran model implemented some important approximations and simplifications which were not considered necessary in the monohull work. The research of this thesis is for relieving the initially implemented approximations in Vorus's first order planing catamaran theory, and further developing and extending the theory and application beyond that currently in use in CatSea. This has been achieved through a detailed theoretical analysis, algorithm development, and careful coding. The research result is a new, complete second order nonlinear hydrodynamic theory for planing catamarans. A detailed numerical comparison of the Vorus's first order nonlinear theory and the second order nonlinear theory developed here is carried out. The second order nonlinear theory and algorithms have been incorporated into a new catamaran design code (NewCat). A detailed mathematical formulation of the base first order CatSea theory, followed by the extended second order theory, is completely documented in this thesis.