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Spindler, Richard

Mathematics

2005

PhD

Here we determine the asymptotic behaviour of small disturbances from boundary conditions in a flat inclined channel. After deriving the shallow water equations and confirming the instability condition for F > 2 for the linearized equations in the boundary value case, we study the non-linear boundary value problem for the weakly unstable region of F slightly larger than 2. We apply multiple scales over long distances to determine the evolution of the solution over space. The evolution equations from multiple scales are found using two different methods: one, the idea of secularity and two, by applying the Fredholm Alternative Theorem. The evolution equations from the two methods agree. In addition, a new averaging method is developed for systems of two hyperbolic partial differential equations and applied to the shallow water equations. The averaging evolution equations differ from multiple scales by one term. Error estimates and range of validity are also derived for the averaging method. For the evolution equations derived from any method, it is found that the solution is dominated by the evolution of the solution along one characteristic. We compare both asymptotic and direct numerical solutions for periodic boundary disturbances. We also determine the shock conditions governing the asymptotic solution and use these conditions to determine the conservative form of the evolution equations. In addition, the shock conditions are utilized to determine the travelling wave solution of the evolution equation. The invariance of the time averaged mass flux for the shallow water equations and the invariance of the time averaged solution to the evolution equations is shown. These results are then used to derive the travelling wave speed of the shallow water wave, based on Dressler's roll waves, and the travelling speed of the evolution equations. Both Dressler's roll waves, first discussed by Dressler in 1949, and the evolution equations travelling wave solution are determined. The resulting graphs compare very favorably. This shows that small boundary disturbances develop into the quasi-steady pattern of roll waves when F, the Froude Number, of the flow exceeds two, and that the asymptotic expansion models this wave development quite well for small boundary disturbances. Finally, we present an application of these results to the problem of transition to slug flow in a channel.