No / The advection–dispersion equation (ADE) is one of the most widely used methods for estimating natural stream pollution at different locations and times. In this paper, variational iteration method (VIM) is utilized to obtain a semianalytical solution for 1D ADE in a temporally dependent solute dispersion within uniformsteady flow. Through a computational validation, the effect of different parameters such as uniform flow velocity and dispersion coefficient on the solute concentration values has been investigated. Results show that the change in velocity has a strong effect on fluid density variation. However, when the diffusion coefficient has been increased, the change in flow and velocity behaviors is negligible. To verify the proposed semianalytical solution, the results were compared to analytical solutions and errors were found to be <0.7% in all simulations.

Ce travail concerne l’écoulement incompressible et stationnaire autour d’un cylindre semi-infini en rotation, et ses propriétés de stabilité linéaire. L’effet de la courbure et de la rotation sur la stabilité de cet écoulement est étudié de manière systématique. Avant d’étudier la stabilité, nous calculons d’abord l’écoulement de base. A grand nombre de Reynolds, une couche limite se développe le long du cylindre, ce qui permet d’utiliser l’approximation de couche limite des équations de Navier–Stokes. Ces équations dépendent de deux paramètres de contrôle sans dimension, le nombre de Reynolds (Re) et le taux de rotation (S), et sont résolues numériquement pour obtenir les profils de vitesse et de pression pour une large gamme des paramètres de contrôle. Une couche limite initialement mince s’épaissit avec la distance axiale; ainsi, son épaisseur devient comparable et finalement plus importante que le rayon du cylindre. Au-delà d’un certain taux de rotation, les effets centrifuges conduisent `a un jet de paroi le long d’une portion du cylindre. L’extension axiale de ce jet augmente avec le taux de rotation. L’intensité du jet augmente aussi avec S. Des analyses asymptotiques de l’écoulement à grande distance axiale et à fort taux de rotation sont aussi présentées. L’analyse de stabilité linéaire du précédent écoulement est effectuée dans l’approximation locale. Après une décomposition en modes normaux, les équations des perturbations sont transformées en un problème de valeur propre `a fréquence complexe (ω). Ce problème dépend de cinq paramètres sans dimension: Re, S, la distance axiale normalisée (Z), le nombre d’onde axial (α) et le nombre d’onde azimutal (m). Les équations de stabilité sont résolues numériquement pour étudier les régions instables dans l’espace des paramètres. On observe que de faibles taux de rotation ont un effet important sur la stabilité de l’écoulement. Cette forte déstabilisation est associée à la présence d’un mode quasi-marginal pour le cylindre fixe et qui devient instable pour de petites valeurs de S. Ce phénomène est confirmé par une analyse en perturbation `a petit S. Sans rotation, l’écoulement est stable pour tout Re < 1060, et pour Z > 0.81. Mais, en présence d’une faible rotation, l’instabilité n’est plus limitée par une valeur minimale de Re ou un seuil en Z. Les courbes critiques dans le plan (Z, Re) sont calculées pour une large gamme de S et les conséquences pour la stabilité de l’écoulement discutées. Enfin, un développement asymptotique pour le nombre de Reynolds critique est obtenu, valable aux grandes valeurs de Z. / This work concerns the steady, incompressible flow around a semi-infinite, rotating cylinder and its linear-stability properties. The effect of cylinder curvature and rotation on the stability of this flow is investigated in a systematic manner. Prior to studying its stability, we first compute the basic flow. At large Reynolds numbers, a boundary layer develops along the cylinder. The governing equations are obtained using a boundary-layer approximation to the Navier–Stokes equations. These equations contain two non-dimensional control parameters: the Reynolds number (Re) and the rotation rate (S), and are numerically solved to obtain the velocity and pressure profiles for a wide range of control parameters. The initially thin boundary layer grows in thickness with axial distance, becoming comparable and eventually larger than the cylinder radius. Above a threshold rotation rate, a centrifugal effect leads to the presence of a wall jet for a certain range of streamwise distances. This range widens as the rotation rate increases. Furthermore, the wall jet strengthens as S increases. Asymptotic analyses of the flow at large streamwise distances and at large rotation rates are presented. A linear stability analysis of the above flow is carried out using a local-flow approximation. Upon normal-mode decomposition, the perturbation equations are transformed to an eigenvalue problem in complex frequency (ω). The problem depends on five non-dimensional parameters: Re, S, scaled streamwise direction (Z), streamwise wavenumber (α) and azimuthal wavenumber m. The stability equations are numerically solved to investigate the unstable regions in parameter space. It is found that small amounts of rotation have strong effects on flow stability. Strong destabilization by small rotation is associated with the presence of a nearly neutral mode of the non-rotating cylinder, which becomes unstable at small S. This is further quantified using smallS perturbation theory. In the absence of rotation, the flow is stable for all Re below 1060, and for Z above 0.81. However, in the presence of small rotation, the instability becomes unconstrained by a minimum Re or a threshold in Z. The critical curves in the (Z, Re) plane are computed for a wide range of S and the consequences for stability of the flow described. Finally, a large-Z asymptotic expansion of the critical Reynolds number is obtained.

The inverse nodal problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of the nodal data ( zeros of eigenfunction ). This problem was first defined by McLaughlin[12]. Up till now, the problem on finite intervals has been studied rather thoroughly. Uniqueness, reconstruction and stability problems are all solved. In this thesis, I investigate the inverse nodal problem on semi-infinite intervals q(x) is real and continuous on [0,1) and q(x)!1, as x!1. we have the following proposition. L is in the limit-point case. The spectral function of the differential operator in (1) is a step function which has discontinuities at { k} , k = 0, 1, 2, .... And the corresponding solutions (eigenfunction) k(x) = (x, k) has exactly k zeros on [0,1). Furthermore { k} forms an orthogonal set. Finally we also discuss that density of nodal points and a reconstruction formula on semiinfinite intervals.

We are concerned with forward wave motion simulations in two-dimensional elastic, heterogeneous, semi-infinite media. We use Perfectly Matched Layers (PMLs) to truncate the semi-infinite extent of the physical domain to arrive at a finite computational domain. We use a recently developed hybrid formulation, where the Navier equations for the interior domain are coupled with a mixed formulation for an unsplit-field PML. Here, we implement the hybrid formulation using spectral elements, and report on its performance. The motivation stems from the following considerations: Of concern is the long-time instability that has been reported even in homogeneous and isotropic cases, when the standard complex-stretching function is used in the PML. The onset of the instability is always within the PML zone, and it manifests as error growth in time. It has been suggested that the instability arises when waves impinge at grazing angle on the PML-interior domain interface. Yet, the instability does not always appear. Furthermore, different values of the various PML parameters (mesh density, attenuation strength, order of attenuation function, etc) can either hinder or delay the onset of the instability. It is thus conjectured that the instability is associated with the spectral properties of the discrete operators. In this thesis, we report numerical results based on both Lagrange interpolants, and results based on spectral elements. Spectral elements are explored since they lead to diagonal mass matrices, have improved dispersion error, and, more importantly, have different spectral properties than Lagrangian-based finite elements. Spectral elements are thus used in an attempt to explore whether the reported instability issues could be alleviated. We design numerical experiments involving explosive sources situated at varying depths from the surface, capable of inducing grazing-angle waves. We use the energy decay as the primary metric for reporting the results of comparisons between various spectral element orders and classical Lagrange interpolants. We also report the results of parametric studies. Overall, it is shown that the spectral elements alone are not capable of removing the instability, though, on occasion, they can. Careful parameterization of the PML could also either remove it or alleviate it. The issue remains open. / text

In this dissertation, a new numerical method for solving some problems on the semiinfinite domain is presented. The method is based upon the modified rational Bernoulli functions. These functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then derived and are utilized to reduce the solution of the equations to a system of algebraic equations. This method is used to solve the following problems: Lane-Emden type equations, Volterra’s population model, Blasius equation, and MHD Falkner-Skan equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.

The inhibiting effect of a sharp edge on liquid spreading is well observed during drop interaction with textured surfaces. On groove-textured solid surfaces comprising unidirectional parallel grooves, the edge effect of posts results in the squeezing of drop liquid in the direction perpendicular to the grooves and the stretching of drop liquid along the grooves leading to anisotropy in drop flow, popularly known as wetting anisotropy which has been employed in several engineering applications. A recent study observed that the energy loss incurring at the edges of posts via contact angle hysteresis is primarily responsible for the anisotropic spreading of impacting drops on groove-textured surfaces. The present study aims to elucidate the role of edges on the spreading and receding dynamics of water drops. The experiments of drop impact are carried out on semi-infinite rectangular post comprising a pair of parallel 90-deg edges separated by a distance (post width) comparable to the diameter of impacting drop. The equilibrium shape of drops on the semi-infinite rectangular post is analyzed using open source computational tool Surface Evolver to optimize the ratio of initial droplet diameter to post width. Quantitative measurements of drop impact dynamics on semi-infinite rectangular posts are deduced by analysing high speed videos of impact process captured under three different camera views during experiments. Based on the role of post edges on impacting drops, different regimes of the impacting drops are characterized in terms of drop Weber number and the ratio of diameter of impacting drop to post width. Characteristic features of impact dynamics in each of the regimes are identified and discussed. It is seen that edges play a pivotal role on all stages of impact dynamics regardless of Weber number. Impacts in the regime of completely pinned drops on narrow posts are further analyzed to reveal characteristics of post-spreading oscillations.

The thesis is concerned with the contact mechanics behaviour of non-conforming solids. The geometry of the solids considered gives rise to various contact configurations, from concentrated contacts with circular and elliptical configuration to those of finite line nature, as well as those of less concentrated form such as circular flat punches. The radii of curvature of mating bodies in contact or impact give rise to these various nonconforming contact configurations and affect their contact characteristics, from those considered as semi-infinite solids in accord with the classical Hertzian theory to those that deviate from it. Furthermore, layered solids have been considered, some with higher elastic modulus than that of the substrate material (such as hard protective coatings) and some with low elastic moduli, often employed as tribological coatings (such as solid lubricants). Other bonded layered solids behave in viscoelastic manner, with creep relaxation behaviour under load, and are often used to dampen structural vibration upon impact. Analytic models have been developed for all these solids to predict their contact and impact behaviour and obtain pressure distribution, footprint shape and deformation under both elastostatic and transient dynamic conditions. Only few solutions for thin bonded layered elastic solids have been reported for elastostatic analysis. The analytical model developed in this thesis is in accord with those reported in the literature and is extended to the case of impact of balls, and employed for a number of practical applications. The elastostatic impact of a roller against a semi-infinite elastic half-space is also treated by analytic means, which has not been reported in literature. Two and three-dimensional finite element models have been developed and compared with all the derived analytic methods, and good agreement found in all cases. The finite element approach used has been made into a generic tool for all the contact configurations, elastic and viscoelastic. The physics of the contact mechanical problems is fully explained by analytic, numerical and supporting experimentation and agreement found between all these approaches to a high level of conformance. This level of agreement, the development of various analytical impact models for layered solids and finite line configuration, and the development of a multi-layered viscoelastic transducer with agreed numerical predictions account for the main contributions to knowledge. There are a significant number of findings within the thesis, but the major findings relate to the protective nature of hard coatings and high modulus bonded layered solids, and the verified viscoelastic behaviour of low elastic modulus compressible thin bonded layers. Most importantly, the thesis has created a rational framework for contact/impact of solids of low contact contiguity.

Semi-infinite programming problems is a class of optimization problems in finite dimensional variables which are subject to infinitely many inequality constraints. If the infinite index of inequality constraints depends on the decision variable, then the problem is called generalized semi-infinite programming problem (GSIP). If the infinite index set is fixed, then the problem is called standard semi-infinite programming problem (SIP). In this thesis, convergence of a semismooth Newton method for generalized semi-infinite programming problems with convex lower level problems is investigated. In this method, using nonlinear complementarity problem functions the upper and lower level Karush-Kuhn-Tucker conditions of the optimization problem are reformulated as a semismooth system of equations. A possible violation of strict complementary slackness causes nonsmoothness. In this study, we show that the standard regularity condition for convergence of the semismooth Newton method is satisfied under natural assumptions for semi-infinite programs. In fact, under the Reduction Ansatz in the lower level problem and strong stability in the reduced upper level problem this regularity condition is satisfied. In particular, we do not have to assume strict complementary slackness in the upper level. Furthermore, in this thesis we neither assume strict complementary slackness in the upper nor in the lower level. In the case of violation of strict complementary slackness in the lower level, the auxiliary functions of the locally reduced problem are not necessarily twice continuously differentiable. But still, we can show that a standard regularity condition for quadratic convergence of the semismooth Newton method holds under a natural assumption for semi-infinite programs. Numerical examples from, among others, design centering and robust optimization illustrate the performance of the method.

This study considers the elastostatic contact problem of a semi-infinite cylinder. The cylinder is compressed against a layer lying on a rigid foundation. There is a sharp-edged circular hole in the middle of the foundation. It is assumed that all the contacting surfaces are frictionless and only compressive normal tractions can be transmitted through the interfaces. The contact along interfaces of the elastic layer and the rigid foundation forms a circular area of which outer diameter is unknown. The problem is converted into the singular integral equations of the second kind by means of Hankel and Fourier integral transform techniques. The singular integral equations are then reduced to a system of linear algebraic equations by using Gauss-Lobatto and Gauss-Jacobi integration formulas. This system is then solved numerically. In this study, firstly, the extent of the contact area between the layer and foundation are evaluated. Secondly, contact pressure between the cylinder and layer and contact pressure between the layer and foundation are calculated for various material pairs. Finally, stress intensity factor on the edge of the cylinder and in the end of the sharp-edged hole are calculated.

The presence of notches or cracks causes stresses to amplify in nearby regions. This phenomenon is studied by estimating the Stress Concentration Factor (SCF) for notches, and the Stress Intensity Factor (SIF) for cracks. In the present work, a semi-analytical method under the framework of linear elasticity is developed to give an estimate of these factors, particularly for cracks and notches in finite domains. The solution technique consists of analytically deriving a characteristic equation based on the general solution and homogeneous boundary conditions, and then using the series form of the reduced solution involving the (possibly complex-valued) roots of this characteristic equation to satisfy the remaining non-homogeneous boundary conditions. This last step has to be carried out numerically using, say, a weighted residual method. In contrast to infinite domain problems where a fully analytical solution is often possible, the presence of more boundaries, and a variety in configurations, makes the solution of finite do-main problems much more challenging compared to infinite domain ones, and these challenges are addressed in this work. The method is demonstrated on several classical and new problems including the problems of a semi-circular edge notch in a semi-infinite and finite plate, an elliptical hole in a plate, an edge-crack in a finite plate etc.