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Closed-form Analytical Solution (closed-form + analytical_solution)
Selected AbstractsAn Analytical Solution for Ground Water Transit Time through Unconfined AquifersGROUND WATER, Issue 4 2005R. Chesnaux An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions. [source] Analytical solution of the harmonic waves diffraction by a cylindrical lined cavity in poroelastic saturated mediumINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 5 2007Y. S. Karinski Abstract This paper presents a model for the analysis of plane waves diffraction at a cavity in an infinite homogeneous poroelastic saturated medium, lined by a lining composed of four equal segments. An elastic boundary layer is placed between the cavity lining and the infinite porous medium. The boundary layer is simulated by ,elastic boundary conditions' in which the bulk matrix stress is proportional to the relative displacement between the lining and the surrounding medium matrix boundary. In addition, fluid impermeability through the intermediate layer is assumed. For the frequencies, that differ from the pseudoresonanse frequencies, the problem was reduced to the problem of an ideal elastic medium. A closed-form analytical solution of the problem was obtained using Fourier,Bessel series, the convergence of which was proven. It was shown that the number of series terms required to obtain a desired level of accuracy can be determined in advance. The influence of the medium porosity on the medium dynamic stress concentration was studied. Copyright © 2006 John Wiley & Sons, Ltd. [source] Dynamics of jet streaks in a stratified quasi-geostrophic atmosphere: Steady-state representationsTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 600 2004Philip Cunningham Abstract The structure and dynamics of jet streaks in the extratropical upper troposphere are examined in the context of a continuously stratified quasi-geostrophic (QG) framework. It is hypothesized that jet streaks may result from the superposition of monopolar or dipolar vortices of mesoscale dimensions with the enhanced potential-vorticity gradients constituting the tropopause. Based on this hypothesis, steady-state monopolar and dipolar vortices in a uniform zonal background flow on an f -plane are investigated for their applicability as idealized dynamical representations of jet streaks. The representations of jet streaks satisfy the nonlinear governing equations of the continuously stratified QG framework: the monopolar vortex is specified in terms of axisymmetric distributions of QG potential vorticity in the interior of the domain and perturbation potential temperature on upper (tropopause) and lower (surface) boundaries, whereas the dipolar vortex is adapted from a closed-form analytical solution for the geostrophic stream function. Through the incorporation of vertical structure and divergent circulations, these representations of jet streaks extend those presented previously by the authors using a non-divergent barotropic model. It is shown that these vortex representations display characteristic signatures similar to those observed in atmospheric jet streaks. In particular, both the monopole and the dipole exhibit an ageostrophic wind directed towards lower geopotential height in the entrance region of the streak and towards higher geopotential height in the exit region. For the monopole, this ageostrophic wind is entirely rotational and there is no vertical motion. For the dipole, the rotational part of the ageostrophic wind dominates the divergent part; the latter is associated with a four-cell pattern of vertical velocity similar to that described in conceptual models of straight jet streaks. For both the monopole and the dipole, the jet streak is induced by the vortex structure such that the wind speed maximum translates at the same speed as the individual vortices; this translation speed is slower than the maximum wind speed in the core of the speed maximum, consistent with observations of jet streaks. It is proposed that the above representations provide a formal theoretical foundation for the conceptual models of jet streaks prevalent in the literature; these conceptual models typically are based on heuristic kinematic or parcel arguments and not on consistent solutions to a physically plausible set of equations. The representations also provide a foundation upon which to explore the unsteady behaviour of jet streaks in terms of the superposition of monopolar and dipolar vortices with non-uniform zonal background flows. Copyright © 2004 Royal Meteorological Society. [source] Interaction of exchange and differential relaxation in the saturation recovery behavior of the binary spin-bath model for magnetization transferCONCEPTS IN MAGNETIC RESONANCE, Issue 4 2006Gunther Helms Abstract Most closed-form analytical solutions of the binary spin-bath are difficult to interpret in terms of underlying physics. The key notions are the presence of a kinetic and a thermal equilibrium and that the time course of saturation recovery under conditions of fast exchange can be understood as conjoint relaxation and lossless transfer. By introducing a suitable parameter, it is shown how exchange and differential relaxation counteract each other: the amount of transferred saturation (transfer term) is altered and the kinetic equilibrium appears slightly disturbed (difference term). Although the factorization formally represents the general solution of saturation recovery in the binary spin-bath, this interpretation applies only to the case of fast exchange and slow relaxation. By calculating the set of parameters for a wide range of hypothetical relaxation rates, it was shown that the difference term is crucial to describe the transition to the slow-exchange limit. The transfer term vanishes as the two pools appear decoupled in this approximation. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 291,298, 2006. [source] Error estimation of closed-form solution for annual rate of structural collapseEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 15 2008Brendon A. Bradley Abstract With the increasing emphasis of performance-based earthquake engineering in the engineering community, several investigations have been presented outlining simplified approaches suitable for performance-based seismic design (PBSD). Central to most of these PBSD approaches is the use of closed-form analytical solutions to the probabilistic integral equations representing the rate of exceedance of key performance measures. Situations where such closed-form solutions are not appropriate primarily relate to the problem of extrapolation outside of the region in which parameters of the closed-form solution are fit. This study presents a critical review of the closed-form solution for the annual rate of structural collapse. The closed-form solution requires the assumptions of lognormality of the collapse fragility and power model form of the ground motion hazard, of which the latter is more significant regarding the error of the closed-form solution. Via a parametric study, the key variables contributing to the error between the closed-form solution and solution via numerical integration are illustrated. As these key variables cannot be easily measured, it casts doubt on the use of such closed-form solutions in future PBSD, especially considering the simple and efficient nature of using direct numerical integration to obtain the solution. Copyright © 2008 John Wiley & Sons, Ltd. [source] | ||||||||||