Anthony Peirce

This thesis considers the problem of semi-infinite hydraulic fractures driven by shear-thinning power-law fluids through a permeable elastic medium. In the recent work by Dontsov and Peirce [Journal of Fluid Mechanics, 784:R1 (2015)], the authors reformulated the governing equations in a way that avoids singular integrals for the case of a Newtonian fluid. Moreover, the authors constructed an approximating ordinary differential equation (ODE) whose solutions accurately describe the fracture opening at little to no computational cost. The present thesis aims to extend their work to the more general case where the fracture is driven by a shear-thinning power-law fluid. In the first two chapters of this thesis we outline the relevant physical modelling and discuss the asymptotic propagation regimes typically encountered in hydraulic fracturing problems. This is followed by Chapters 4 and 5 where we reformulate the governing equations as a non-singular integral equation, and then proceed to construct an approximating ODE. In the final chapter we construct a numerical scheme for solving the non-singular integral equation. Solutions obtained in this way are then used to gauge the accuracy of solution obtained by solving the approximating ODE. The most important results of this thesis center on the accuracy of using the approximating ODE. In the final chapter we find that when the fluid's power-law index is in the range of 0.4 ≤ n ≤ 1, an appropriate method of solving the approximating ODE yields solutions whose relative errors are less than 1%. However, this relative error increases with decreasing values of n so that in the range 0 ≤ n
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Ostwald ripening and chemotaxis are two different mechanisms that describe particle motion throughout a domain. Ostwald ripening describes the redistribution of a solid solution due to energy potentials while chemotaxis is a cellular phenomenon where organisms move based on the presence of chemical gradients in their environment. Despite the two systems coming from disparate fields, they are connected by the late-stage dynamics of interfacial motion.For the Ostwald ripening system we consider the case of N droplets in the asymptotic limit of small radii [formula omitted]. We first derive a system of ODEs that describe the motion of the droplets and then improve this calculation by including higher order terms. Certain properties, such as area preservation and finite time extinction of certain droplets are proved and a numerical example is presented to support the claims.In the chemotaxis model we look at the asymptotic limit of diffusive forces being small compared to that of chemotactic gradients. We use a boundary-fitted coordinate system to derive an equation for the velocity of an arbitrary interface and analyze a few specific examples. The asymptotic results are also explored and confirmed using the finite element and level set methods.Our analysis reveals the mechanism of movement to be motion by curvature in Ostwald ripening and a surface diffusion law in chemotaxis. The governing rules of motion may be different in the two systems but the end result is typically characteristically similar- exchange of mass and smoothing in favor of a larger and more stable configuration of drops.

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This thesis considers the flow of thin fluid film between an elastic sheet and a rigid plane. We derive a mathematical model for the flow from the Navier-Stokes equations using the lubrication approximation and develop numerical and similarity solutions to this problem. An experimental apparatus was developed to investigate this phenomenon, and the results of the mathematical model were compared with experimental data.Chapter 3 examines the evolution of a fixed fluid volume under gravitational forces on a horizontal plane. The evolution of the fluid mass profile and the progression of the fluid front are determined from the numerical solutions, as well as experimentally. The favourable comparison between the numerical solutions and the experimental results establishes the validity of the model.Chapters 4-5 considers the evolution of a thin fluid flow under an elastic on an inclined plane. We establish a traveling wave solution for this flow. A linear stability analysis yields the criterion for the existence of unstable modes and establishes the growth rate and wavelength of the most unstable mode. Instability is promoted by increasing the inclination of the plane. For low angles, the numerical and experimental growth rates were in good agreement, while the wavelengths were experimentally of the same order and numerically computed wavelengths had little variation.The long term behaviour of the fluid front is studied analytically via a similarity solution in Chapter 6.

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