The present thesis consists of two parts; both are devoted to two celebrated old problems in fluid dynamics. The first deals with symmetry breaking in a liquid layer flowing down an inclined plane. The second problem concerns the equilibrium and symmetry breaking of interfacial polygonal patterns generated by a system of vortices arranged on a circular ring. The first problem dates back to Nusselt (1916) who obtained the solution for the basic flow. Since then, thin layers of liquid falling down inclined plane continues to be the subject of extensive studies for both their practical applications and theoretical value. In this thesis, the problem is approached analytically. Three new mathematical models are proposed. The first two involve three and four equations respectively. These produce linear stability results that agree fairly with past experimental outcomes and results obtained with similar models. For a deeper and qualitative analysis a lower dimension model that retains the physics is needed. Hence, a two-equation model (involving only two fundamental flow parameters namely the film thickness and flow rate) is derived. The new model taking account of the shear stress at the free surface is shown to be superior to the existing two-equation model of Usha and Uma in Phys Fluid (2004). The influence of electrical and magnetic fields on the stability of falling film of an electrically conductor fluid is also investigated. In comparison with the model of Korsunsky ( Eur.J.F.M.1999 ) for higher Reynolds numbers. The proposed model takes account of the inertia terms, which are of second order with respect to a small parameter namely the long wave parameter. As shown through the chapter four of the part one, the proposed two-equation model improves significantly Korsunsky's model. The second problem dates back to Kelvin (1867) who hypothesized atoms to be point vortices arranged in circular ring forming symmetrical polygonal patterns. Although, the atomic vortex model is long abandoned, the problem of system of point vortices has become of great interest in superfluidity and by analogy in plasma physics. Moreover, polygonal patterns, which are the signature of the presence of vortices, equally distributed in) rings were also observed in several engineering problems and geophysical flows in nature. In fluid dynamics, polygonal patterns become clearly visible in swirling flows where the vortex core is hollow. The empty core can eventually support polygonal shapes (up to hexagonal). The first experimental report on the phenomenon was by Vatistas in 1990. In this thesis the phenomenon is revisited using image-processing technique that allows a deeper and more precise investigation. The dynamics of the patterns is investigated and for the first time the transition from one pattern to another is explored in detail. The stability condition for a system of point vortices on circular ring derived first by J.J Thomson (1897) and generalized later by Havelock (1931) for N point vortices including the influence of circular boundaries surrounding the equilibrium is confirmed. Frequency locking between the pattern and the disk frequencies which are suspected in the previous experiments is established and quantified. Moreover, the transition from the elliptical to the hexagonal pattern is found that it follows a "devil's staircase" scenario. Due to the similarity between the problem under the scope and other fields of physics, the present experimental results are anticipated to go beyond the field of fluid mechanics.
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