Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 1991"

Monodromy in the champagne bottle

Abstract: We show that the Hamiltonian description of a particle moving in a potential field shaped like the punt of a champagne bottle (more properly anS 1 symmetric double well) has monodromy, which is a global obstruction to the construction of action-angle variables.

Fluid flow over a thin deformable porous layer

Abstract: The flow of a viscous fluid over a thin, deformable porous layer fixed to the solid wall of a channel is considered. The coupled equations for the fluid velocity and the infinitesimal deformation of the solid matrix within the porous layer are developed using binary mixture theory, Darcy's law and the assumption of linear elasticity. The case of pure shear is solved analytically for the displacement of the solid matrix, the fluid velocity both in the porous medium and the fluid above it. For a thin porous layer the boundary condition for the fluid velocity at the fluid-matrix interface is derived. This condition replaces the usual no slip condition and can be applied without solving for the flow in the porous layer.

On steady state Euler-Poisson models for semiconductors

Abstract: This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity. Also, in the irrotational two- and three-dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift-term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the drift-diffusion model as this parameter tends to zero.

A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

Abstract: LetD⊂RN be a region with smooth boundary∂D. Letp·q>1,p, q≥1. We consider the system:ut=Δu+vp,vt=Δu+uq inD×[0, ∞) withu=v=0 in∂D×[0, ∞) andu0,v0 nonnegative. Letγ=max(p, q). We show that ifD isRN, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (γ+1)/(pq−1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (γ+1)/(pq−1)

On the hydraulic lock exchange problem

Abstract: This paper discusses the fundamental aspects of the hydraulic lock-exchange problem. In a first part the limit of small density ratios of the two fluids involved is discussed by considering the problem of emptying a horizontal water-filled channel by means of pressurized air.

Bounds for eigenvalues of second-order elliptic differential operators

Abstract: We derive bounds for the firstN eigenvalues of a linear second-order elliptic differential operator on a bounded domain, subject to mixed boundary conditions. The results are achieved by a combination of (a generalized version of) Kato's estimates and a homotopy algorithm.

Three-dimensional absolute and convective instabilities, and spatially amplifying the waves in parallel shear flows

Abstract: A formalism for absolute and convective instabilities in parallel shear flows is extended to the three-dimensional case. Assuming that the dispersion relation function is given byD(k, l, ω), wherek andl are wave numbers, andω is a frequency, the analytic criterion is formulated by which a point (k 0,l 0,ω 0) with Imω 0>0 contributes to the absolute instability if and only if one of the two equivalent conditions is satisfied: (i) At least two roots inl of the systemD(k, l, ω)=0,D k (k, l, ω)=0, originating on opposite sides of the reall-axis, collide on thel-plane for the parameter valuesk 0,l 0,ω 0, asω is brought down toω 0. Every point on thek-plane, that corresponds to a point on the collision paths on thel-plane, is itself a coalescence point ofk-roots for a fixedl ofD(k, l, ω)=0, that originate on opposite sides of the realk-axis. (ii) At least two roots ink of the systemD(k, l, ω)=0,D l ,(k, l, ω)=0, originating on opposite sides of the realk-axis, collide on thek-plane for the parameter valuesk 0,l 0,ω 0, asω is brought down toω 0. Every point on thel-plane, that corresponds to a point on the collision paths on thek-plane, is itself a coalescence point ofl-roots for a fixedk ofD(k, l, ω)=0, that originate on opposite sides of the reall-axis.

A demonstration of consistency of an entropy balance with balance of energy

Abstract: It is shown in this note that a general balance of entropy postulated previously with only a limited motivation (based on the form of the energy equation for an inviscid fluid) is consistent with, and can be derived from, a general balance of energy. In this derivation, an early form of entropy balance does not make use of invariance conditions under superposed rigid body motions. However, with the help of the latter invariance conditions, additional results are also derived which provide some insight on the structure of the basic equations in thermomechanics.

Density conservation for a coagulation equation

Abstract: A Theorem is presented which proves that solutions to the coagulation equation are density conserving whenever certain growth assumptions are made on the coagulation kernel.

On the thermomechanical laws for the motion of a phase interface

Abstract: This paper discusses the formulation of balance laws for mass, force, and energy in conjunction with a law of entropy growth for the motion of a sharp evolving phase interface within a continuum framework.

Mass-orthogonal formulation of equations of motion for multibody systems

Abstract: A projection method based on the classical equations of mechanics for the description of multibody systems is presented. The system is described by arbitrary coordinates and velocity parameters. From the constraint equations two complementary projections splitting the space of velocities into the space of admissible and inadmissible velocities respectively are constructed. They are uniquely determined by a condition of mass-orthogonality. A consistent description of the dynamics of the constrained system results. The constraint reactions are given as functions of position and velocities and an explicit system of differential equations for the motion of the constrained system is derived.

Some overdetermined boundary value problems for harmonic functions

Abstract: In this paper we study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunctionu2 of the Stekloff eigenvalue problem in a bounded simply connected plane domain Ω has a constant value of ∥∂u2∥ on ∇Ω, then Ω is a disk

An effective boundary element method of inhomogeneous partial differential equations

Abstract: A method for removing the domain or volume integral arising in boundary integral formulations for linear inhomogeneous partial differential equations is presented. The technique removes the integral by considering a particular solution to the homogeneous partial differential equation which approximates the inhomogeneity in terms of radial basis functions. The remainder of the solution will then satisfy a homogeneous partial differential equation and hence lead to an integral equation with only boundary contributions. Some results for the inhomogeneous Poisson equation and for linear elastostatics with known body forces are presented.

Center manifolds of infinite dimensions I: Main results and applications

Abstract: In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.

Gas flow in pipeline following a rupture computed by a spectral method

Abstract: Breaks in high-pressure pipelines or pipeline systems containing large amounts of flammable or poisonous gas represent a potential hazard. The flow following a sudden rupture is analysed and computed. The governing equations are highly nonlinear and also singular as choking occurs. The spectral method with Legendre-polynomials has been used to compute the blowdown process of a long pipeline. Accurate results have been obtained for short computing time with only few collocation points. The effect of different treatments of the boundary conditions has been studied. For the two limiting cases, adiabatic and isothermal flow, the difference in the mass-flow rate at the broken end has been found to be very small.

Hearing the shape of general doubly-connected domain in R 3 with mixed boundary conditions

Abstract: The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR3 with mixed boundary conditions, from the complete knowledge of the eigenvalues {λn}n=1∞ for the three-dimensional Laplacian, using the asymptotic expansion of the spectral function\(\Theta (t) = \Sigma _{n = 1}^\infty \exp ( - t\lambda _n )\) ast→0.

Existence uniqueness and construction of the solution of the energy transfer problem in a rigid and nonconvex black body

Abstract: This paper is concerned with an important class of heat transfer problems, which arises when the radiation heat transfer is the mechanism for energy transfer from/to a rigid and nonconvex black body. In such situations there will exist a direct energy interchange among points of the body boundary that do not belong to a given “small neighborhood”. Such phenomena are mathematically described by a partial differential equation subjected to nonlinear boundary conditions. It is demonstrated that the problem always admits a solution, which is unique. In addition, an algorithm for solving such problems is presented.

Locally forced critical surface waves in channels of arbitrary cross section

Abstract: Critical long surface waves forced by locally distributed external pressure applied on the free surface in channels of arbitrary cross section are studied in this paper. The fluid under consideration is inviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumed to be near critical, i.e.,u 0=u c +eλ+0(e2), where 0<ɛ≪1 andu c is the critical velocity determined by the geometry of the channel. The external pressure applied on the free surface as the forcing isɛ 2 Pδ(x). Then the first order perturbation of the free surface elevation satisfies a forced Korteweg-de Vries equation (fK-dV). It is shown in this paper that: (i) If $$\lambda \geqslant \lambda _c = (3b^2 P^2 \alpha ^2 /( - 4\beta m_1^2 ))^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern- ulldelimiterspace} 3}} > 0$$ (supercritical), the stationaryfK-dV has two cusped solitary wave solutions; (ii) if $$\lambda< \lambda _L = (3b^2 P^2 \alpha _2 /(\beta m_1^2 ))^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern- ulldelimiterspace} 3}}< 0$$ (subcritical), the stationaryfK-dV has a downstream cnoidal wave solution; (iii) whenλ=λ L , the unique stationary solution of thefK-dV is a wave free hydraulic fall; (iv) ifλ=λ d =−λ L , thefK-dV has a jump solution; and (v) ifλ L <λ<λ c , thefK-dV does not have stationary solutions. Some free surface profiles and bifurcation diagrams are presented.

On bounding the effective conductivity of isotropic composite materials

Abstract: In this paper inequalities for the effective conductivity of isotropic composite materials are derived. These inequalities depend on several coefficients characterizing the microstructure of composites. The obtained coefficients can be exactly calculated for models of a two-component aggregate of multisized, coated ellipsoidal inclusions, packed to fill all space. As a result, new bounds for effective conductivity, considerably narrower than those of Hashin-Shtrikman, are established for such models of composite materials.

Elastic field perturbation by an elliptic inhomogeneity with a sliding interface

Abstract: Muskhelishvili complex potentials are used to solve the problem of an infinite elastic plane containing an elliptic inhomogeneity with a sliding interface but no eigenstrain. The boundary conditions considered are (a) continuity of normal tractions and displacements and vanishing shear tractions at the interface, and (b) vanishing stresses at infinity. After a conformal mapping of the elastic plane, the solution is obtained in terms of a set of infinite algebraic equations yielding the Laurent's expansion coefficients of the complex potentials. Distinct sets of formulae must be written for a circular inhomogeneity (degenerate ellipse) and an elliptic inhomogeneity (no degeneracy), and in both cases no closed-form solution is obtainable. For an elliptic inhomogeneity the solution requires iteration and recursion, and implies vanishing stresses in the homogeneity when the system is loaded with a remote uniform shear parallel to the axes of the ellipse.

Study of the establishment of a laminar liquid jet emerging from a semi-infinite vertical cylindrical pipe under the influence of gravity

Abstract: On etudie l'ecoulement libre d'un jet liquide laminaire axisymetrique emergeant d'une conduite cylindrique verticale semi-infinie, a grands nombres de Reynolds et grands nombres de Froude. Dans de telles conditions, si l'on exclut le cas des tubes capillaires, le nombre de Weber de l'ecoulement est suffisamment grand pour que les effets de tension superficielle s'averent negligeables, ainsi que le confirme cette etude. Les equations sont simplifiees par une analyse de type couche limite. L'utilisation de la methode des developpements asymptotiques raccordes conduit a des solutions approchees semianalytiques faisant apparaitre les parametres de similitude de l'ecoulement. Ces solutions prevoient la distribution des vitesses et la forme du jet sous l'influence des forces de pesanteur et de viscosite. Des resultats experimentaux obtenus par photographie du jet et visualisation holographique du champ des vitesses confirment les previsions theoriques.

Steady two-dimensional flow past a normal flat plate

Abstract: A vorticity/stream function formulation is used to obtain a numerical simulation of steady two-dimensional flow of a viscous incompressible fluid past a normal flat plate for a range of Reynolds numbers. A method of Fornberg [J. Fluid Mech. 98, 819 (1980)] is used to determine upstream and downstream boundary conditions on the stream function. Special care is taken in the neighbourhood of the singularities in vorticity at the plate edges and this is very important because any errors introduced are swept downstream and severely affect such quantities as the length and width of the attached eddies. The computed results are compared with those of a laboratory experiment in which a plane strip is drawn through water and ethylene glycol for the range of Reynolds numbers for which the experimental flow is stable.

The Wigner function for thermal equilibrium

Abstract: We provide a simple mathematical framework which is useful for a precise description of the thermal equilibrium Wigner function and for obtaining approximations to it in both cases of Boltzmann and Fermi Dirac statistics Regular and singular asymptotic expansions of this Wigner function are derived and their validity shown for respectively smooth and discontinuous potentials We also generalize some continuation methods, previously suggested for Boltzmann statistics, to the case of Fermi Dirac statistics

Mixed boundary value problem of potential theory in toroidal coordinates

Abstract: A new approach is presented for solving the title problems. The method presents further extension of previously obtained results to the case of toroidal coordinates. The method allows us to solve non-axisymmetric problems exactly and in closed form, with no integral transform or special function expansions involved. Some integrals of fundamental value, involving distances between several points, are established.

Pulsating entry-flow in a plane channel

Abstract: To obtain results for the title problem, the time-dependent Navier-Stokes equations have been solved numerically. Axial-velocity profiles at various distances from the entrance of the channel are shown for a number of points in time during one period of oscillation. Further some results for the time-dependent inlet length are presented.

The effects of streamline curvature and spanwise rotation on near-surface, turbulent boundary layers

Abstract: A second moment closure model is used to study the mean fields and turbulence structure of spanwise rotating flows and flows with streamline curvature. The effects of flow stabilization and destabilization by rotation and/or curvature and their interpretation in terms of a Rayleigh instability mechanism are discussed in the context of the present model. When applied to the constant flux layer adjacent to a bounding surface, the model provides a similarity theory for flows with spanwise rotation and streamline curvature like that of Monin-Obukhov in the case of density stratified flows. In particular, it is shown that Bradshaw's empirical length scale correction can be derived in terms of the basic constants of the model determined in the absence of rotation and curvature. Also, direct comparisons with experimental data confirm the model predictions. The definitions of strong and mild curvature are discussed and a distinguishing criterion derived.

Wrinkle-free solutions in the theory of annular elastic membranes

Abstract: Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Foppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.

A kinetic method for strictly nonlinear conservation laws

Abstract: Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu 1+f(u)x=0. We choose a class of distribution functionsG=G(t, x, ξ), which are compactly supported with respect to the artificial velocityξ. This can be done in an optimal way, i.e. so that theξ-integral of the solution of the linear kinetic equationG t+ξGx=0 solves the nonlinear conservation law exactly. Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.

Center manifolds of infinite dimensions II: Proofs of the main result

Abstract: In this paper, the part II of a bipartite work, we present the proof of Theorem 1 which was stated in part I, i.e. the precursor of the present paper. This proof establishes the existence of a smooth center manifold, i.e. of a smooth decoupling function for the abstract evolution system considered in part I. In an appendix, the proofs of some auxiliary lemmas are presented, some of which were stated in part I, while the others are related to the present proof of Theorem 1.

Numerical integration of stiff ODE's of singular perturbation type

Abstract: We present a numerical approach for solving in an efficient way stiff systems of singular perturbation type. This approach makes direct use of the theory of singularly perturbed IVP's. We determine numerically an expansion of an invariant manifold with respect to the small (stiffness) parameterɛ and we solve a reduced system of lower dimension no longer stiff on this manifold. The global error of the numerical approximation obtained contains anɛ-dependent term which decreases withɛ. This implies that the error does not increase as the stiffness of the system increases. We also suggest an appropriate implementation of the method and we give an application to some non-trivial example.