Showing papers on "Boundary value problem published in 1976"

Notes on black-hole evaporation

Abstract: This paper examines various aspects of black-hole evaporation. A two-dimensional model is investigated where it is shown that using fermion-boson cancellation on the stress-energy tensor reduces the energy outflow to zero, while other noncovariant techniques give the Hawking result. A technique for replacing the collapse by boundary conditions on the past horizon is developed which retains the essential features of the collapse while eliminating some of the difficulties. This set of boundary conditions is also suggested as the most natural set for a preexistent black hole. The behavior of particle detectors under acceleration is investigated where it is shown that an accelerated detector even in flat spacetime will detect particles in the vacuum. The similarity of this case with the behavior of a detector near the black hole is brought out, and it is shown that a geodesic detector near the horizon will not see the Hawking flux of particles. Finally, the work of Berger, Chitre, Nutku, and Moncrief on scalar geons is corrected, and the spherically symmetric coupled scalar-gravitation Hamiltonian is presented in the hope that someone can apply it to the problem of black-hole evaporation.

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

Abstract: This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.

Frontal solution program for unsymmetric matrices

Abstract: A frontal solution program is presented which may be used for the solution of unsymmetric matrix equations arising in certain applications of the finite element method to boundary value problems. Based on the Gaussian elimination algorithm, it has advantages over band matrix methods in that core requirements and computation times may be considerably reduced; furthermore numbering of the finite element mesh may be completed in an arbitrary manner. The program is written in FORTRAN and a glossary of terms is provided.

Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics

Abstract: Initial-boundary value problems for several systems of partial differential equations from fluid dynamics are discussed. Both rigid wall and open boundary problems are treated. Boundary conditions are formulated and shown to yield well-posed problems for the Eulerian equations for gas dynamics, the shallow-water equations, and linearized constant coefficient versions of the incompressible, anelastic equations. The "primitive" hydrostatic meteorological equations are shown to be ill-posed with any specification of local, pointwise boundary conditions. Analysis of simplified versions of this system illustrates the mechanism responsible for ill-posedness.

Modified Nonlinear Schrödinger Equation for Alfvén Waves Propagating along the Magnetic Field in Cold Plasmas

Abstract: The basic nonlinear equation which describes the Alfven waves, with small but finite amplitude propagating along the magnetic field in cold plasmas, is derived modifying the reductive perturbation method proposed by Taniuti and Wei. Then as a result, the nonlinear dispersion relation is obtained through a procedure which clarifies the physical meaning. Furthermore, the modified nonlinear Schrodinger equation which describes the modulated Alfven wave more correctly than the previous works is proposed. An example of the nonlinear phenomena is shown by the numerical calculations of the initial value problem, using our basic equation for the Alfven waves.

A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer

Abstract: A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity plane on which the real and imaginary parts of the secular determinant are zero. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two-wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards phase velocity equal to unity and the spacing between eigenvalues goes to zero. The original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow.

Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory

Abstract: A method is presented for accurately solving the Schrodinger equation for the reactive collision of an atom with a diatomic molecule in three dimensions on a single Born–Oppenheimer potential energy surface. The Schrodinger equation is first expressed in body‐fixed coordinates. The wavefunction is then expanded in a set of vibration–rotation functions, and the resulting coupled equations are integrated in each of the three arrangement channel regions to generate primitive solutions. Next, these are smoothly matched to each other on three matching surfaces which appropriately separate the arrangement channel regions. The resulting matched solutions are linearly combined to generate wavefunctions which satisfy the reactance and scattering matrix boundary conditions, from which the corresponding R and S matrices are obtained. The scattering amplitudes in the helicity representation are easily calculated from the body fixed S matrices, and from these scattering amplitudes several types of differential and int...

Stokes flow for a stokeslet between two parallel flat plates

Abstract: Velocity and pressure fields for Stokes flow due to a force singularity of arbitrary orientation and arbitrary distance between two parallel plates are found, using the image technique and a Fourier transform. Two alternative expressions for the solution, one in terms of infinite integrals and the other in terms of infinite series, are given. The infinite series solution is especially suitable for computation purposes being an exponentially decreasing series. From the series the “far field” behaviour is extracted. It is found that a force singularity parallel to the two planes has a far field behaviour of source and image for the parallel components (a two dimensional source doublet of height-dependent strength) whereas the normal component, and all fields due to a force singularity normal to the planes, die out exponentially. Velocity fields are compared with those of the one plane case. An estimate of the influence of the second wall and when its effect can be disregarded is obtained.

A moving fluid interface on a rough surface

Abstract: When an interface between two fluids moves in contact with a solid boundary, the Navier-Stokes equations and the no-slip boundary condition provide an unsatisfactory theoretical model, because they predict an undefined velocity at the contact line and a non-integrable stress on the solid boundary. If the surface irregularities are included in the model, the flow on a length scale large compared with their size can be calculated, using a slip coefficient and treating the surface as smooth.A simple type of corrugated surface is examined, and the effective slip coefficient calculated, for grooves of finite and infinite depth. The slip coefficient when the grooves are filled with one fluid and another fluid flows over them is also calculated. It is suggested that, when a fluid displaces another on a rough surface, the displaced fluid remains in the hollows on the surface, thus providing a partly fluid boundary for the displacing fluid and leading to a slip coefficient for the flow.Fluid contained between two vertical plates and rising between them provides a simple example of a flow for which the solution can be found with and without a slip coefficient. With slip present, the force on the plates is finite and its value is calculated.

Boundary conditions and non-equilibrium thermodynamics

Abstract: The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.

Transport equation techniques for the deposition of auroral electrons

Abstract: Auroral electron scattering and energy loss are calculated by using for the first time a multiangle equation of transfer at all energies. The results are compared with those obtained by using a Fokker-Planck equation. Both equations have been solved in terms of their eigensolutions. The equation of transfer has also been solved by numerical integration. Fokker-Planck solutions agree well with equation of transfer solutions above 3 keV but deviate increasingly at lower energies. A comparison is made between the present Fokker-Planck results and those of M. Walt at 10 keV, giving good agreement. Energy deposition rates are also found to agree satisfactorily with those obtained previously. The accuracy of integration of the transfer equation is tested by comparing results obtained by the eigenvalue method and the direct integration method. Differences of less than 5% were found at all altitudes, energies, and pitch angles. The predicted backscatter near the upper energy boundary is sensitive to the boundary condition there. Backscatter results for various boundary conditions in energy show both this and the effects of the propagation of the boundary condition toward lower energies. Solutions to the equation of transfer are given between 10 eV and 20 keV, based on a measured auroral electron spectrum. These solutions are compared with similar results by Banks et al. (1974). The results agree above 3 keV but differ below that energy, a finding which is consistent with our comparisons of solutions of the equation of transfer and the Fokker-Planck equation.

The spatial viscous instability of axisymmetric jets

Abstract: The stability of three axisymmetric jet profiles is reviewed. These profiles represent the development of an incompressible jet from a nearly top-hat profile to a fully developed jet profile. The disturbance equations for arbitrary mode number in a region of zero shear, which provide the boundary conditions for the numerical solution, are solved analytically through use of the disturbance vorticity equations. Numerical solutions for the spatial stability for the axisymmetric (n = 0) disturbance and the asymmetric n = l disturbance are presented. Previously published calculations of least stable modes are shown to be incorrectly interpreted and their actual mode types are given. The critical Reynolds number is found to increase as the profile varies from a top-hat to a fully developed jet form. Closed contours of constant amplification, which are unusual in free shear flows, are shown to exist for the n = 1 disturbance in the fully developed jet region. A fluctuation energy balance is used to justify the occurrence of this destabilizing effect of decreasing Reynolds number.

Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain

Abstract: Summary The aim of this talk is to describe the main results about the regularity (as well as the singularities) of the solutions of an elliptic boundary value problem in domains with a non smooth boundary. Up to now, results concerning general elliptic boundary value problems, are available only if the domain is two-dimensional, mainly with a polygonal boundary. On the contrary few results are known in higher dimension when the domain presents singularities more complicated than conical points. For instance for a domain with edges but with no vertices, only boundary value problems for elliptic operator of the second order have been investigated. I will give a brief survey of recent results on this subject with precise bibliographical references together with some new contributions to the study of a second order elliptic boundary value problem in a three dimensional polyhedron. Except for new results the material is presented without proofs. The topics mentioned above are closely related to the study of mixed boundary value problems for an elliptic equation and of problems with interface conditions (conjugacy problems). For these problems, I will only mention briefly the results which are similar to those concerning our main subject, with bibliographical references.

Integral formalism for surface waves in piezoelectric crystals. Existence considerations

Abstract: An integral formalism for surface waves in piezoelectric half‐infinite solids valid up to the critical velocity is developed. Various boundary conditions are considered and, in particular, the problem of which boundary conditions allow surface‐wave solutions for velocities below the limiting velocity vL is discussed in detail. It is proved that (a) with a mechanically free surface and zero dielectric constant for adjoining medium, at most one solution is possible for v

The Effect of Heater Size, Location, Aspect Ratio, and Boundary Conditions on Two-Dimensional, Laminar, Natural Convection in Rectangular Channels

Abstract: The effect of localized heating in rectangular channels was studied by solving the partial differential equations for the conservation of mass, momentum, and energy numerically using an unsteady state formulation and the alternating-direction-implicit method. The heating element was a long, horizontal, isothermal strip located in one, otherwise-insulated vertical wall. The opposing wall was maintained at a lower uniform temperature and the upper and lower surfaces were insulated or maintained at the lower temperature. Computations were carried out for Pr = 0.7, 0 less than or equal to Ra less than or equal to 10/sup 5/, a complete range of heater widths and locations and a wide range of aspect ratios. Flow visualization studies and comparison with prior computed results for a limiting case confirm the validity of the computed values. The computed rates of heat transfer and circulation provide guidance for locating heaters or coolers.

Electromagnetic fields in the presence of rotating bodies

Abstract: Field calculations in the presence of rotating bodies with symmetry of revolution can be performed in the (inertial) laboratory frame of reference. Specific results are presented for a rotating circular cylinder immersed in a plane wave of the E or H type. Particular emphasis is put on the low-frequency limit, but some numerical data are also given for a typical frequency in the "resonance" region. The analysis becomes more complicated in the absence of symmetry of revolution. It is then necessary to solve the problem in a rotating system of coordinates. Maxwell's equations are written in these coordinates, together with the relevant constitutive equations and boundary conditions. The general formalism is applied to a typical two-dimensional configuration, viz., a cylinder immersed in an incident E wave. Considerable simplification obtains if all material velocities are negligible with respect to c, a condition which is always met in practice. Even simpler results are obtained if the cross-sectional dimensions of the cylinder are small with respect to λ. Some numerical results are presented, at low frequencies, for a dielectric cylinder of rectangular cross section.

The calculation of near-wake flows

Abstract: Calculated flow properties are compared with measurements obtained in twodimensional isothermal wakes with and without recirculation. The equations of continuity and momentum were solved numerically together with equations which formed a turbulence model. Calculations were made using three turbulence models : the first comprised transport equations for turbulence kinetic energy and the rate of turbulence dissipation; the second and third comprised equations for the rate of turbulence dissipation and two forms of Reynolds-stress equations characterized by different redistribution terms. The results show that, for wakes without recirculation, the particular turbulence model is less important than the boundary condition assumed in the plane of the trailing edge of the body; though the Reynolds-stress models do, of course, provide a better representation of the individual normal stresses. In the case of wakes with recirculation, both the length of the recirculation region and the rate of spread of the downstream wake are underestimated. The second discrepancy is particularly evident and appears to stem from the form of the dissipation equation. A suggestion for improving the modelling of this equation is provided together with necessary justification.

A Time-Dependent Lateral Boundary Scheme for Limited-Area Primitive Equation Models

Abstract: Before high-resolution numerical models can be of use operationally, they must be restricted to a limited domain, thus necessitating lateral boundary conditions which allow the changes outside the limited domain to influence the results while not contaminating the forecast with spurious boundary-reflected energy. Such a set of time-dependent lateral boundary conditions are presented in this paper. This boundary condition set is investigated using the linear analytic and finite-difference advection equations, the non-linear finite-difference shallow-water equations, and the hydrostatic primitive equations. The results illustrate how the boundary condition transforms long- and medium-length interior advective and gravity waves into short waves which can then be removed by a low pass filter, thereby giving the appearance that the exiting wave simply passed through the boundary. The results also indicate that large-scale advective and gravity waves enter the forecast domain with little degradation. T...

Analysis of trapped‐energy resonators operating in overtones of coupled thickness‐shear and thickness‐twist

Abstract: The equations of three‐dimensional linear piezoelectricity are applied in the analysis of trapped‐energy resonators with rectangular electrodes vibrating in coupled thickness shear and thickness twist in the vicinity of the fundmantal and odd overtone thickness‐shear frequencies. Closed form asymptotic expressions for the frequency wave‐number dispersion relations for the fundamental and odd overtone coupled thickness‐shear and thickness‐twist waves near cutoff are obtained for both the electroded and unelectroded regions of the trapped‐energy resonator. The influence of piezoelectric stiffening, electrode mass loading, and electrical shorting is included in the analysis. Simple approximate boundary conditions at a junction between an electroded and unelectroded region of the plate are obtained in a manner exhibiting the natural limitations inherent in the approximation. In order that these boundary conditions can be satisfied at each such junction, in the adjacent regions the wave numbers in the direction of the junction line are assumed to be the same. The boundary conditions to be satisfied at the junctions between the unelectroded corner region and the unelectroded regions adjacent to the electroded region are obtained from an extended version of the variational principle of linear piezoelectricity. These latter conditions result in the form of the solution in the corner region. One result of the foregoing analysis is the determination of a two‐dimensional condition which is a generalization of Bechmann’s number in one dimension. The above‐mentioned dispersion relations and edge conditions are applied in the analysis of the steady‐state vibrations of a trapped‐energy resonator and a lumped parameter representation of the admittance, which is valid in the vicinity of a resonance, is obtained.Subject Classification: [43]40.24; [43]85.52, [43]85.32.

Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media

Abstract: Maxwell's equations can be cast into a basic differential operator equation, the curlcurl equation, which lends itself easily to variational treatment. Various forms of this equation are associated with problems of practical importance. The formulation includes the treatment of loss-free anisotropic media. The boundary conditions associated with electromagnetic-field problems are treated in detail and the uniqueness of the solution is discussed. A functional is derived for the curlcurl equation in Cartesian and cylindrical coordinates.