Three-dimensional Partial Differential Equations Research Articles

Nonlinear stability criteria for the motion of three-dimensional quasigeostrophic flow on a beta -plane

The authors study the nonlinear stability of three-dimensional quasigeostrophic motion on a beta -plane. The criteria obtained are applicable to perturbations of both initial data and parameters in models. For verifying criteria parallel to Arnold's second theorem given by other authors, one has to calculate the lowest eigenvalue of a three-dimensional elliptic partial differential equation with variable coefficients and, for the given criteria, one needs only do the same for the two-dimensional Laplacian.

An appropriate mesh representation for tetrahedral finite volume computations

The amounts of information used for the solution of three-dimensional partial differential equations in engineering applications is formidable. Part of the problem is that in unstructured meshes (adequate for advanced numerical techniques) the information is local in nature. An alternate representation of an unstructured tetrahedral mesh is proposed. Besides having some storage advantages over representations commonly used for finite volume computations, we show that preprocessing complexity is lower than in the conventional representation. The resulting simplified structure may result in significant gains for applications that go through a number of mesh refinement cycles.

Three-dimensional transient mass transfer model for laser chemical vapor deposition of titanium on stationary finite slabs

A mathematical model for mass transfer inside a laser chemical vapor deposition (LCVD) chamber is presented for pure titanium deposition from titanium tetrabromide on a stainless-steel (SS 304) substrate. A set of five three-dimensional, transient, and nonlinear partial differential equations is solved to account for the diffusion of various species inside the LCVD chamber and the thermal decomposition of titanium tetrabromide, titanium dibromide, and titanium monobromide at the surface of the substrate in order to determine the thickness of the titanium film deposited on the substrate. The model is used for studying the effects of various parameters on the deposition and shape of titanium films. The parameters which were varied are as follows: laser power, activation energy, total pressure inside the LCVD chamber, and partial pressure of titanium tetrabromide. Under certain conditions, the deposited titanium film is found to have a volcanic profile.

Analytical models for growth by metal organic vapour phase epitaxy: I. Isothermal models

For a proper description of growth by metal organic vapour phase epitaxy the three-dimensional Navier-Stokes partial differential equations need to be solved which govern the following series of processes: (i) transport by diffusion and flow through the gas phase, (ii) reactions which take place in this gas phase, (iii) reactions which take place at the surface. The authors are at first interested in the medium- and higher-temperature regions, which cover the growth determined by diffusion through the gas phase (medium temperature) and the growth that is determined by the desorption of growth species (higher temperature). Using a number of well justified assumptions one can reduce the problem to a two-dimensional one. For the diffusion-limited region (i.e. medium-temperature region) the effect of different flow profiles (plug flow, parabolic flow, linear increasing velocity and combination of plug and linear profile) on the growth rate has been studied under isothermal conditions. It was found that all profiles yield the same growth rate within a few per cent, so that it suffices to use the simple plug flow profile in growth rate calculations. It is also shown that axial diffusion is an important effect only at the end of long reactors. Finally a model is derived in which surface reaction kinetics is combined with the diffusion-limited model for the isothermal case.

Optical Analog Solutions Of Partial Differential And Integral Equations

Confocal feedback systems exist in a variety of forms and can solve a wide range of partial differential equations (PDEs) and integral equations (lEs). In this paper we describe several of these feedback systems and how they can be applied to provide optical analog solutions to PDEs of constant coefficients (e.g., diffusion, Poisson's, and wave equations), PDEs of variable coefficients (e.g., modified Helmholtz equations), three-dimensional PDEs, four-dimen-sional PDEs, and IEs (e.g., Fredholm and Volterra equations). The important advantage of obtaining the solutions by optical analog methods rather than digital methods is speed. The disadvantage is solution accuracy, although the accuracy obtainable with optical feedback is better than without feedback. To further improve solution accuracy, we suggest the replacement of simple spherical mirrors by Mangin mirrors and the incorporation of coherent image amplification by photorefractive crystals (e.g., BaTiO3 or BSO) in the confocal systems.

A Fourier method for three-dimensional partial differential equations in periodic geometry. Application: HELIAC

A numerical method based on Fourier expansions and finite differences is presented. The method is demonstrated by solving a scalar, three-dimensional elliptic equation arising in MFE research, but has applicability to a wider class of problems. The scheme solves equations whose solutions are expected to be periodic in one or more of the independent variables.

An advanced integral model for cooling tower plume dispersion

An advanced integral model is developed for predicting cooling tower plume rise from single natural draft cooling towers. The theoretical formulation of the model is aimed at avoiding many of the pitfalls and unnecessary assumptions of existing models. The model is based on a careful integration of the three-dimensional partial differential equations of conservation across the plume cross-section; radial profiles of temperature, velocity, and total water are assumed to be Gaussian in shape. The model includes a treatment of plume thermodynamics and tower downwash effects. The model has been calibrated with a wide range of laboratory data. Verification of the model with single-tower field data from Chalk Point, Paradise, Lünen, Gardanne and Philippsburg reveals good results.

Prediction of air pollutant concentrations in Tokushima Prefecture, Japan

This paper deals with the prediction problem of air pollutant concentrations over the industrial area of Tokushima Prefecture, Japan. The mathematical model used for the prediction of air pollution, which describes the transport and diffusion of pollutants from stack emissions into the atmosphere, is expressed by the three-dimensional partial differential equation known as the advection diffusion model with initial and boundary conditions. Diffusion coefficients characterizing this model are determined effectively from the statistical processing of the available measured data obtained at two monitoring stations. The measured data of sulpher dioxide concentrations are classified by season, weather and wind speed, and then the identification of these coefficients for each class is carried out by using a line search method because of its numerical stability. The estimation theory is extensively applied to this model for obtaining the useful estimates of the spatial and temporal concentration distributions on the basis of actual measured data, in which the analytical solution of this model is given by using the Green's function under some reasonable assumptions and the suitable transformation. The effectiveness of the proposed approach for the prediction of air pollutant is indicated in simulation results.

A global model of thermospheric dynamics—I. Wind and density fields derived from a phenomenological temperature

This paper describes the mathematical formulation and implementation of a numerical model of the neutral upper atmosphere based on the equations of conservation of mass, momentum and energy in the altitude range 120–500 km. The model is three-dimensional and includes the effects of viscosity, ion drag, the Coriolis force and the nonlinear terms in the equations of motion. The Galerkin method is used as an efficient alternative to finite-difference approaches for the solution of three-dimensional partial differential equations. In the preliminary computations discussed here, the upper atmosphere is represented by a single fluid with molecular weight and temperature fields taken from the phenomenological model of Jacchia (1971). The horizontal and vertical wind fields and the density of the thermosphere are discussed, with particular attention to the diurnally averaged wind fields. Comparisions with previous theoretical and observational investigations are made.

Theory of isotachophoresis development of concentration boundaries

Three-dimensional partial differential equations describing the migration of ions through a stationary medium under the influence of an electric field are developed and then applied to the analysis of steady-state and unsteady-state properties of isotachophoresis in one dimension. The shape of the front region between two ion species in the steady state is discussed, and the time development of the system from an arbitrary initial distribution of ions to the steady state is analyzed and illustrated by a numerical example.

Dynamic estimation of air pollution

The advection-diffusion model of air pollution over an urban area is developed. The region is subdivided into a grid and the three-dimensional partial differential equation of the pollution concentration is reduced to a linear vector difference equation. Along with this discrete equation, a stochastic model of air pollution is considered and pollution concentrations over the area are estimated from observed data generated by a few monitor points. High dimensionality of the resulting Kalman filter equation is avoided by using a discrete form of Chandrasekar-type equations. Diffusion coefficients in the advection-diffusion model, which are difficult to specify accurately, are also identified.

Anisotropic waves in elastoplastic soils

The Prandtl-Reuss theory extended to consider a medium which is compressible in the plastic region is utilized to study waves in elastoplastic soils. The von Mises yield condition is applied with the yield stress a function of the pressure, which is typical of soils. An arbitrary pressure-density relationship with noncoincident loading and unloading paths is also allowed. The full nonlinear three-dimensional unsteady partial differential equations are analyzed by determining their characteristic or discontinuity surfaces. The characteristic analysis results in equations for the wave speed of infinitesimal disturbances in the medium as a function of the orientation of the wave normal vector, the stress, the shear modulus, the bulk modulus and the density. These equations have been coded for digital computer solution and automatic plotting of the three-dimensional normal wave speed diagrams and their corresponding wave surfaces or fronts. Results in the form of normal wave speed diagrams and wave surfaces reveal the nature of the anisotropic waves which are found in soils in the nonlinear elastic and plastic regimes. The variation of the waves with variation in the parameters is also found. Most of the phenomena of cusped wave surfaces and intersecting normal wave speed diagrams found in anisotropic elastic waves, magnetofluid dynamics and crystal optics are also observed in the present results. In addition, it is found that certain waves do not propagate (their wave speeds become imaginary) in directions normal to the axis of the principal deviatoric stress whose absolute value is maximum.

THE APPLICATION OF RELAXATION METHODS TO SATISFY NORMAL-GRADIENT BOUNDARY CONDITIONS ASSOCIATED WITH THREE-DIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS

THE APPLICATION OF RELAXATION METHODS TO SATISFY NORMAL-GRADIENT BOUNDARY CONDITIONS ASSOCIATED WITH THREE-DIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS

FINITE-AMPLITUDE THREE-DIMENSIONAL HARMONIC WAVES ON THE SPHERICAL EARTH

Abstract By making use of the quasi-nondivergent approximation, the potential vorticity equation is reduced to an equation in the stream function ψ. Assuming that the motion is of permanent wave type, a first integral of this nonlinear vorticity equation is obtained, which itself is a linear three-dimensional partial differential equation in ψ. This equation has been solved as a boundary-value problem by the method of separation of variables. It is found that the latitudinal-amplitude functions of these waves satisfy a spheroidal-wave equation while the vertical-amplitude functions are given by Bessel and Hankel functions of the argument lp/p0, where l is a parameter depending on both the static stability and the nodal number r. The eigenvalues µmr of these wave solutions are connected with the parameter l2 by a transcendental relation. We have expanded µmr into a power series of l2 and obtained the various coefficients, up to that of the fourth power of l2. The latitudinal- and vertical-amplitude functio...