| Dissertation |
Thesis (Ph.D.) --NUI, 2008 at Department of Applied Mathematics, UCC. |
| Summary |
In this thesis a theoretical framework for investigating the motion of a deformable ellipsoid subject to a surrounding flow of different viscosity is developed via an extension of Jeffery's (1922) rigid solution. The approach of Mulchrone and Walsh (2006) is taken whereby the internal deformation is determined by equating the internal and external stresses at the ellipsoid boundary. The stability properties of equilibria and orbits of the system are then examined. Equilibria are found to display a variety of stability properties depending, not only on the initial conditions, but also on the viscosity contrast between inclusion and matrix and the bulk flow deformation. Periodic orbits and their stability are studied for the passive case where it is possible to analytically determine the trajectory and its period. For particular initial conditions, numerical results suggest that periodic motions exist when the viscosities are different. A study of rigid ellipsoid motion under simple shear is then undertaken. It is found that chaotic behaviour is exhibited under certain initial conditions. Poincaré maps and Lyapunov exponents show that this behaviour is most likely to occur when the ellipsoid is sufficiently prolate and results agree well with previous studies of this phenomenon. The equations describing the ellipsoidal motion are than adapted to determine the evolution of a deformable layer. Finite strain and vorticity of layers are examined under pure and simple shear and it is found that non-rigid layers rotate passively regardless of viscosity contrast. Next, a system is developed to account for the rotational disturbances of rigid ellipsoids caused by interactions between neighbours. Rather than adjusting the perturbed velocity field, an interaction term is appended to the equations that determine the orientation. This term depends on the neighbouring particles' degree of coupling, proximity and relative position. Simulations reveal a preferred orientation at an angle to the shear direction that increases with the degree of coupling. Finally, digital images obtained from X-ray computed tomography scans are analysed for the estimation of inclusion shape and orientation. Three-dimensional computer imagery and segmentation algorithms are used to visualise and isolate the regions of interest. Theses regions are approximated by best-fit ellipsoids and the mean best-fit ellipsoid is used as a measure of preferred inclusion orientation. A Windows program is developed to implement these procedures and results found from both manufactured and natural data are presented. These results show that the radiodensity contrast plays a major role in the ability of the software to isolate inclusions from their matrix and hence determine rock fabric. |
| Subject |
Ellipsoids.
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Geology -- Mathematics.
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| Collection |
Theses Ph.D.
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Theses Applied Mathematics Department
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| Description |
289 p. : ill. ; 30 cm. |
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