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In the following pages we study the withdrawal of fluid from horizontal channels of finite depth into a vertical slot. The flow caused by a line source in a fluid of finite depth is also considered. Four related problems are considered and each requires the determination of either one or two unknown free surfaces for a range of parameter values. The formulation of each problem leads to a non-linear integral equation, which when solved enables the location of the free surface to be found. In the first problem we study the flow of fluid (under the influence of gravity) through a slot in the lower of two horizontal plane plates. Using a Chebychev series method the problem is reduced to one of solving a system of nonlinear equations which are solved using fixed point iteration. For zero gravity an analytic solution is presented. The results for the numerical and analytical solutions are shown to be in good agreement. Solutions are found for all Fronde numbers. Removing the upper horizontal lid from the first problem to create another free surface leads to the second problem solved in this thesis. Here we focus on the downward flow of fluid from a horizontal uniform channel of finite depth into a vertical slot under the influence of gravity. The nonlinear integral equation is solved for non-zero gravity using collocation and cubic splines together with a Newtonian iteration scheme. The shapes of the top free surface and bottom free surface are IV computed for a range of parameter values. Only supercritical solutions are found where the top free surface attaches smoothly to the vertical wall. In the third problem we study the flow from a vertical slot into a layer of finite depth. Only one free surface is considered and solutions are sought for subcritical flows. A stagnation point exists above the slot. The nonlinear integral equation is solved numerically for non-zero gravity by using collocation, cubic splines and a Newton iteration scheme. An asymptotic solution is also derived for small Fronde numbers. The fourth problem considers the subcritical flow due to a submerged source in a fluid of finite depth with surface tension effects. The nonlinear integral equation is solved numerically by a Newton iteration scheme. The shape of the free surface is computed for a range of parameter values where the effects of surface tension are taken into account. It is shown that surface tension has an effect on the solution behaviour. An asymptotic solution is shown to be in good agreement with the full nonlinear solution for small values of the Froude number

Year: 1995

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oai:researchrepository.murdoch.edu.au:51556

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