The buckling of a material subject to stress is a very common phenomenon observed in mechanics. However, the observed buckling of a surface confined hydrogel due to swelling is a unique manifestation of the buckling problem. The reason for buckling is the same in all cases; there is a certain magnitude of force that once exceeded, causes the material to deform itself into a buckling mode. Exactly what that buckling mode is as well as how much force is necessary to cause buckling depends on the material properties. Taking both a finite difference and analytical approach to the problem, it is desired to obtain relationships between the material properties and the predicted buckling modes. These relationships will make it possible for a hydrogel to be designed so that the predicted amount of buckling will occur.

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

This study aims to investigate hydraulics and the flow characteristics of drilling fluids inside annulus and to understand the mechanism of cuttings transport in horizontal and deviated wellbores. For this purpose, initially, extensive experimental studies have been conducted at Middle East Technical University, Petroleum &amp / Natural Gas Engineering Flow Loop using water and numerous drilling fluids for hole inclinations from horizontal to 60 degrees, flow velocities from 0.64 m/s to 3.05 m/s, rate of penetrations from 0.00127 to 0.0038 m/s, and pipe rotations from 0 to 120 rpm. Pressure loss within the test section and stationary and/or moving bed thickness are recorded. New friction factor charts and correlations as a function of Reynolds number and cuttings bed thickness with the presence of pipe rotation for water and drilling fluids in horizontal and deviated wellbores are developed by using experimental data. Meanwhile empirical correlations that can be used easily at the field are proposed for predicting stationary bed thickness and frictional pressure loss using dimensional analysis and the effect of the drilling parameters on hole cleaning is discussed. It has been observed that, the major variable influencing cuttings transport is fluid velocity. Moreover, pipe rotation drastically decreases the critical fluid velocity that is required to prevent the stationary cuttings bed development, especially if the pipe is making an orbital motion. A decrease in the pressure loss is observed due to the bed erosion while rotating the pipe. Cuttings transport in horizontal annulus is modeled using a CFD software for different fluid velocities, pipe rotation speeds and rate of penetrations. The CFD model is verified by using cuttings transport experiments. A mathematical model is also proposed to predict the flow characteristics of Newtonian fluids in concentric horizontal annulus with drillpipe rotation. The Navier-Stokes equations of turbulent flow are numerically solved using finite differences technique. A computer code is developed in Matlab 2007b for the proposed model. The performance of the proposed model is compared with the experimental data which were available in the literature and gathered at METU-PETE Flow Loop as well as Computational Fluids Dynamics (CFD) software. The results showed that the mechanistic model accurately predicts the frictional pressure loss and the velocity profile inside the annuli. The model&rsquo / s frictional pressure loss estimations are within an error range of &plusmn / 10%.

<p>Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. </p><p>We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit.</p><p>As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated. </p>