Finite element analysis of steady-state natural convection problems in fluid-saturated porous media heated from below

In this paper, a progressive asymptotic approach procedure is presented for solving the steady-state Horton-Rogers-Lapwood problem in a fluid-saturated porous medium. The Horton-Rogers-Lapwood problem possesses a bifurcation and, therefore, makes the direct use of conventional finite element methods difficult. Even if the Rayleigh number is high enough to drive the occurrence of natural convection in a fluid-saturated porous medium, the conventional methods will often produce a trivial non-convective solution. This difficulty can be overcome using the progressive asymptotic approach procedure associated with the finite element method. The method considers a series of modified Horton-Rogers-Lapwood problems in which gravity is assumed to tilt a small angle away from vertical. The main idea behind the progressive asymptotic approach procedure is that through solving a sequence of such modified problems with decreasing tilt, an accurate non-zero velocity solution to the Horton-Rogers-Lapwood problem can be obtained. This solution provides a very good initial prediction for the solution to the original Horton-Rogers-Lapwood problem so that the non-zero velocity solution can be successfully obtained when the tilted angle is set to zero. Comparison of numerical solutions with analytical ones to a benchmark problem of any rectangular geometry has demonstrated the usefulness of the present progressive asymptotic approach procedure. Finally, the procedure has been used to investigate the effect of basin shapes on natural convection of pore-fluid in a porous medium. (C) 1997 by John Wiley & Sons, Ltd.

Solution of adsorption problems involving steep moving profiles

The moving finite element collocation method proposed by Kill et al. (1995) Chem. Engng Sci. 51 (4), 2793-2799 for solution of problems with steep gradients is further developed to solve transient problems arising in the field of adsorption. The technique is applied to a model of adsorption in solids with bidisperse pore structures. Numerical solutions were found to match the analytical solution when it exists (i.e. when the adsorption isotherm is linear). The method is simple yet sufficiently accurate for use in adsorption problems, where global collocation methods fail. (C) 1998 Elsevier Science Ltd. All rights reserved.

Theory of multidimensional parametric band-gap simultons

Multidimensional spatiotemporal parametric simultons (simultaneous solitary waves) are possible in a nonlinear chi((2)) medium with a Bragg grating structure, where large effective dispersion occurs near two resonant band gaps for the carrier and second-harmonic field, respectively. The enhanced dispersion allows much reduced interaction lengths, as compared to bulk medium parametric simultons. The nonlinear parametric band-gap medium permits higher-dimensional stationary waves to form. In addition, solitons can occur with lower input powers than conventional nonlinear Schrodinger equation gap solitons. In this paper, the equations for electromagnetic propagation in a grating structure with a parametric nonlinearity are derived from Maxwell's equation using a coupled mode Hamiltonian analysis in one, two, and three spatial dimensions. Simultaneous solitary wave solutions are proved to exist by reducing the equations to the coupled equations describing a nonlinear parametric waveguide, using the effective-mass approximation (EMA). Exact one-dimensional numerical solutions in agreement with the EMA solutions are also given. Direct numerical simulations show that the solutions have similar types of stability properties to the bulk case, providing the carrier waves are tuned to the two Bragg resonances, and the pulses have a width in frequency space less than the band gap. In summary, these equations describe a physically accessible localized nonlinear wave that is stable in up to 3 + 1 dimensions. Possible applications include photonic logic and switching devices. [S1063-651X(98)06109-1].

Finite element modelling of dissipative structures for nonequilibrium chemical reactions in fluid-saturated porous media

We use the finite element method to model and predict the dissipative structures of chemical species for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. In particular, we explore the conditions under which dissipative structures of the species may exist in the Brusselator type of nonequilibrium chemical reaction. Since this is the first time the finite element method and related strategies have been used to study the chemical instability problems in a fluid-saturated porous medium, it is essential to validate the method and strategies before they are put into application. For this purpose, we have rigorously derived the analytical solutions for dissipative structures of chemical species in a benchmark problem, which geometrically is a square. Comparison of the numerical solutions with the analytical ones demonstrates that the proposed numerical method and strategy are robust enough to solve chemical instability problems in a fluid-saturated porous medium. Finally, the related numerical results from two application examples indicate that both the regime and the magnitude of pore-fluid flow have significant effects on the nature of the dissipative structures that developed for a nonequilibrium chemical reaction system in a fluid-saturated porous medium. The motivation for this study is that self-organization under conditions of pore-fluid flow in a porous medium is a potential mechanism of the orebody formation and mineralization in the upper crust of the Earth. (C) 2000 Elsevier Science S.A. All rights reserved.

A two-dimensional analytical solution of groundwater responses to tidal loading in an estuary and ocean

Previous studies on tidal dynamics of coastal aquifers have focussed on the inland propagation of oceanic tides in the cross-shore direction, a configuration that is essentially one-dimensional. Aquifers at natural coasts can also be influenced by tidal waves in nearby estuaries, resulting in a more complex behaviour of head fluctuations in the aquifers. We present an analytical solution to the two-dimensional depth-averaged groundwater flow equation for a semi-infinite aquifer subject to oscillating head conditions at the boundaries. The solution describes the tidal dynamics of a coastal aquifer that is adjacent to a cross-shore estuary. Both the effects of oceanic and estuarine tides on the aquifer are included in the solution. The analytical prediction of the head fluctuations is verified by comparison with numerical solutions computed using a standard finite-difference method. An essential feature of the present analytical solution is the interaction between the cross- and along-shore tidal waves in the aquifer area near the estuary's entry. As the distance from the estuary or coastline increases, the wave interaction is weakened and the aquifer response is reduced, respectively, to the one-dimensional solution for oceanic tides or the solution of Sun (Sun H. A two-dimensional analytical solution of groundwater response to tidal loading in an estuary, Water Resour Res 1997;33:1429-35) for two-dimensional non-interacting tidal waves. (C) 2000 Elsevier Science Ltd. All rights reserved.

Sudden drawdown and drainage of a horizontal aquifer

Drainage of a saturated horizontal aquifer following a sudden drawdown is reanalyzed using the Boussinesq equation. The effect of the finite length of the aquifer is considered in detail. An analytical approximation based on a superposition principle yields a very good estimate of the outflow when compared to accurate numerical solutions. An illustration of the new analytical approach to analyze basin-scale field data is used to demonstrate possible field applications of the new solution.

A director theory for visco-elastic folding instabilities in multilayered rock

A model for finely layered visco-elastic rock proposed by us in previous papers is revisited and generalized to include couple stresses. We begin with an outline of the governing equations for the standard continuum case and apply a computational simulation scheme suitable for problems involving very large deformations. We then consider buckling instabilities in a finite, rectangular domain. Embedded within this domain, parallel to the longer dimension we consider a stiff, layered beam under compression. We analyse folding up to 40% shortening. The standard continuum solution becomes unstable for extreme values of the shear/normal viscosity ratio. The instability is a consequence of the neglect of the bending stiffness/viscosity in the standard continuum model. We suggest considering these effects within the framework of a couple stress theory. Couple stress theories involve second order spatial derivatives of the velocities/displacements in the virtual work principle. To avoid C-1 continuity in the finite element formulation we introduce the spin of the cross sections of the individual layers as an independent variable and enforce equality to the spin of the unit normal vector to the layers (-the director of the layer system-) by means of a penalty method. We illustrate the convergence of the penalty method by means of numerical solutions of simple shears of an infinite layer for increasing values of the penalty parameter. For the shear problem we present solutions assuming that the internal layering is oriented orthogonal to the surfaces of the shear layer initially. For high values of the ratio of the normal-to the shear viscosity the deformation concentrates in thin bands around to the layer surfaces. The effect of couple stresses on the evolution of folds in layered structures is also investigated. (C) 2002 Elsevier Science Ltd. All rights reserved.

Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity

We consider solutions to the second-harmonic generation equations in two-and three-dimensional dispersive media in the form of solitons localized in space and time. As is known, collapse does not take place in these models, which is why the solitons may be stable. The general solution is obtained in an approximate analytical form by means of a variational approach, which also allows the stability of the solutions to be predicted. Then, we directly simulate the two-dimensional case, taking the initial configuration as suggested by the variational approximation. We thus demonstrate that spatiotemporal solitons indeed exist and are stable. Furthermore, they are not, in the general case, equivalent to the previously known cylindrical spatial solitons. Direct simulations generate solitons with some internal oscillations. However, these oscillations neither grow nor do they exhibit any significant radiative damping. Numerical solutions of the stationary version of the equations produce the same solitons in their unperturbed form, i.e., without internal oscillations. Strictly stable solitons exist only if the system has anomalous dispersion at both the fundamental harmonic and second harmonic (SH), including the case of zero dispersion at SH. Quasistationary solitons, decaying extremely slowly into radiation, are found in the presence of weak normal dispersion at the second-harmonic frequency.

Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwell's equation numerical solutions.

A conceptual model for sheet-flow drawn from rapid granular flow theories

33rd IAHR Congress: Water Engineering for a Sustainable Environment

Mathematical modelling of shallow flows: closure models drawn from grain-scale mechanics of sediment transport and flow hydrodynamics

Canadian Journal of Civil Engineering 36(10) 1605–16

Application of standard and refined heat balance integral methods to one-dimensional Stefan problems

The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions to enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions.

Essays in equilibrium asset pricing

This thesis consists of four essays in equilibrium asset pricing. The main topic is investors' heterogeneity: I investigates the equilibrium implications for the financial markets when investors have different attitudes toward risk. The first chapter studies why expected risk and remuneration on the aggregate market are negatively related, even if intuition and standard theory suggest a positive relation. I show that the negative trade-off can obtain in equilibrium if investors' beliefs about economic fundamentals are procyclically biased and the market Sharpe ratio is countercyclical. I verify that such conditions hold in the real markets and I find empirical support for the risk-return dynamics predicted by the model. The second chapter consists of two essays. The first essay studies how het¬erogeneity in risk preferences interacts with other sources of heterogeneity and how this affects asset prices in equilibrium. Using perceived macroeconomic un¬certainty as source of heterogeneity, the model helps to explain some patterns of financial returns, even if heterogeneity is small as suggested by survey data. The second essay determines conditions such that equilibrium prices have analytical solutions when investors have heterogeneous risk attitudes and macroeconomic fundamentals feature latent uncertainty. This approach provides additional in-sights to the previous literature where models require numerical solutions. The third chapter studies why equity claims (i.e. assets paying a single future dividend) feature premia and risk decreasing with the horizon, even if standard models imply the opposite shape. I show that labor relations helps to explain the puzzle. When workers have bargaining power to exploit partial income insurance within the firm, wages are smoother and dividends are riskier than in a standard economy. Distributional risk among workers and shareholders provides a rationale to the equity short-term risk, which leads to downward sloping term structures of premia and risk for equity claim. Résumé Cette thèse se compose de quatre essais dans l'évaluation des actifs d'équilibre. Le sujet principal est l'hétérogénéité des investisseurs: J'étudie les implications d'équilibre pour les marchés financiers où les investisseurs ont des attitudes différentes face au risque. Le première chapitre étudie pourquoi attendus risque et la rémunération sur le marché global sont liées négativement, même si l'intuition et la théorie standard suggèrent une relation positive. Je montre que le compromis négatif peut obtenir en équilibre si les croyances des investisseurs sur les fondamentaux économiques sont procyclique biaisées et le ratio de Sharpe du marché est anticyclique. Je vérifier que ces conditions sont réalisées dans les marchés réels et je trouve un appui empirique à la dynamique risque-rendement prédites par le modèle. Le deuxième chapitre se compose de deux essais. Le première essai étudie com¬ment hétérogénéité dans les préférences de risque inter agit avec d'autres sources d'hétérogénéité et comment cela affecte les prix des actifs en équilibre. Utili¬sation de l'incertitude macroéconomique perù comme source d'hétérogénéité, le modèle permet d'expliquer certaines tendances de rendements financiers, même si l'hétérogénéité est faible comme suggéré par les données d'enquête. Le deuxième essai détermine des conditions telles que les prix d'équilibre disposer de solutions analytiques lorsque les investisseurs ont des attitudes des risques hétérogènes et les fondamentaux macroéconomiques disposent d'incertitude latente. Cette approche fournit un éclairage supplémentaire à la littérature antérieure où les modèles nécessitent des solutions numériques. Le troisième chapitre étudie pourquoi les equity-claims (actifs que paient un seul dividende futur) ont les primes et le risque décroissante avec l'horizon, mme si les modèles standards impliquent la forme opposée. Je montre que les relations de travail contribue à expliquer l'énigme. Lorsque les travailleurs ont le pouvoir de négociation d'exploiter assurance revenu partiel dans l'entreprise, les salaires sont plus lisses et les dividendes sont plus risqués que dans une économie standard. Risque de répartition entre les travailleurs et les actionnaires fournit une justification à le risque à court terme, ce qui conduit à des term-structures en pente descendante des primes et des risques pour les equity-claims.

Devolatilization-induced pressure build-up: Implications for reaction front movement and breccia pipe formation

Generation of fluids during metamorphism can significantly influence the fluid overpressure, and thus the fluid flow in metamorphic terrains. There is currently a large focus on developing numerical reactive transport models, and with it follows the need for analytical solutions to ensure correct numerical implementation. In this study, we derive both analytical and numerical solutions to reaction-induced fluid overpressure, coupled to temperature and fluid flow out of the reacting front. All equations are derived from basic principles of conservation of mass, energy and momentum. We focus on contact metamorphism, where devolatilization reactions are particularly important owing to high thermal fluxes allowing large volumes of fluids to be rapidly generated. The analytical solutions reveal three key factors involved in the pressure build-up: (i) The efficiency of the devolatilizing reaction front (pressure build-up) relative to fluid flow (pressure relaxation), (ii) the reaction temperature relative to the available heat in the system and (iii) the feedback of overpressure on the reaction temperature as a function of the Clapeyron slope. Finally, we apply the model to two geological case scenarios. In the first case, we investigate the influence of fluid overpressure on the movement of the reaction front and show that it can slow down significantly and may even be terminated owing to increased effective reaction temperature. In the second case, the model is applied to constrain the conditions for fracturing and inferred breccia pipe formation in organic-rich shales owing to methane generation in the contact aureole.

One-dimensional solidification of supercooled melts

In this paper a one-phase supercooled Stefan problem, with a nonlinear relation between the phase change temperature and front velocity, is analysed. The model with the standard linear approximation, valid for small supercooling, is first examined asymptotically. The nonlinear case is more difficult to analyse and only two simple asymptotic results are found. Then, we apply an accurate heat balance integral method to make further progress. Finally, we compare the results found against numerical solutions. The results show that for large supercooling the linear model may be highly inaccurate and even qualitatively incorrect. Similarly as the Stefan number β → 1&sup&+&/sup& the classic Neumann solution which exists down to β =1 is far from the linear and nonlinear supercooled solutions and can significantly overpredict the solidification rate.