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Displacement flow of non-Newtonian fluids in an eccentric annulus Pelipenko, Sviatoslav

Abstract

In this thesis we derive, analyze and devise a method of numerical solution for a Hele-Shaw model of displacement flow of non-Newtonian fluids in an eccentric annulus. The physical problem stems from an industrial process of oil well cementing during the well's construction and successful mathematical modelling and solution of the problem allows for optimization of the process resulting in economic and environmental benefits. Here we outline derivation of the model based on using the long-thin geometry of the annular domain to reduce the flow equations to two spatial dimensions together with the Hershel-Bukley constitutive equations and time evolution equation. Therefrom we obtain analytical solution for the form of the interface in cases of concentric annulus and small annular eccentricity. We proceed to put the problem into its variational and minimization formulation from which we show the existence and uniqueness of a weak solution to the original model. We apply an iterative augmented Lagrangian method to obtain this solution together with a Flux Corrected Transport method for time evolution to arrive at a fully 2-D numerical simulation of the flow. We derive another model using ideas of thin-film and lubrication flows which allows for a quicker prediction of the displacement flow type. We compare the predictions for the flow type based on the lubrication model to those obtained using the 2-D flow simulations. We conclude with a discussion on the significance of results achieved in this work together with relative merits and limitations of the derived models and solutions.

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