On supercavitating flow past a symmetric wedge

Résumé On considère le problème non-linéaire de l'écoulement plan d'un fluide en présence d'un coin symétrique avec cavitation, à l'aide d'un modèle à cavité ouverte. On en donne la solution analytique par l'application de la théorie des transformations conformes et des techniques de Riemann-Hilbert.     

The fast multipole method for the symmetric boundary integral formulation

^** Email: of{at}mathematik.uni-stuttgart.de^*** Email: o.steinbach{at}tugraz.at^**** Email: wendland{at}mathematik.uni-stuttgart.de A symmetric Galerkin boundary-element method is used for thesolution of boundary-value problems with mixed boundary conditionsof Dirichlet and Neumann type. As a model problem we considerthe Laplace equation. When an iterative scheme is employed forsolving the resulting linear system, the discrete boundary integraloperators are realized by the fast multipole method. While thesingle-layer potential can be implemented straightforwardlyas in the original algorithm for particle simulation, the double-layerpotential and its adjoint operator are approximated by the applicationof normal derivatives to the multipole series for the kernelof the single-layer potential. The Galerkin discretization ofthe hypersingular integral operator is reduced to the single-layerpotential via integration by parts. We finally present a correspondingstability and error analysis for these approximations by thefast multipole method of the boundary integral operators. Itis shown that the use of the fast multipole method does notharm the optimal asymptotic convergence. The resulting linearsystem is solved by a GMRES scheme which is preconditioned bythe use of hierarchical strategies as already employed in thefast multipole method. Our numerical examples are in agreementwith the theoretical results.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.     

Axially symmetric flow with free boundary

We prove the solvability of a boundary-value problem with the Bernoulli condition in the form of an inequality on a free boundary. By using the Rietz method, we construct an approximate solution that converges to an exact solution in the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 477–487, April, 1995.     

Similarity solutions of radially symmetric two-phase flow

When both the diffusivityD and fractional flow functionf have a power law dependence on the water content , i.e.D=D _o andf=⁺¹, the nonlinear transport equation for radially symmetric two phase flow can, in certain circumstances, be reduced to a weakly coupled system of two first order nonlinear ordinary differential equations. Numerical solutions of these equations for a constant flux boundary conditionV _wo and comparison with experimental data are given. In particular, when the fluxV _wo and a are related byV _wo( + 1)/D _o=2, a new fully explicit analytical solution is found as (r, t)=(1 – r ²/4D _ot)^1/ forr ² < 4D _ot/ and (r, t)=0 forr ² 4D _ot/ We show that the existence of this exact soution is due to the presence of a Lagrangian symmetry.     

Transonic flow on an axially symmetric torus

On a Riemannian manifold the existence (and uniqueness) of subsonic gas flows with prescribed circulation has been previously established (Acta Math.125 1970, 57–73). If the manifold is a torus of revolution then the gas dynamics equation reduces to a nonlinear ordinary differential equation and the flow can be described explicitly. We show that, as the circulations are increased, one obtains a complete family of solutions: smooth subsonic, smooth transonic, transonic with shocks, and smooth supersonic flows.     

New formulation and relaxation to solve a concave-cost network flow problem

In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

A note on the nonlinear stability of a spatially symmetric vlasov-possion flow

We give sufficient conditions for the nonlinear stability of a, possibly non-smooth, spatially homogeneous Vlasov-Poisson flow. Research partially supported by Italian CNR and Ministero della Pubblica Istruzione.     

Stabilization of horizontal symmetric flow of a liquid film by a heat flux controller

Linearized equations of motion are used to analyze the stability of flow of a liquid film on the walls of a plane-parallel horizontal channel which melt in a stream of hot viscous gas. For materials with a high specific heat of fusion, this flow is shown to be unstable relative to small-amplitude long-wave perturbations.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 74–78, 1986.     

Harmonic map heat flow for axially symmetric data

We examine the harmonic map heat flow problem for maps between the three-dimensional ball and the two-sphere. We give blow-up results for certain initial data. We establish convergence results for suitable axially symmetric initial data, and discuss generalizations to higher dimensions.     

Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.     

Covering a symmetric poset by symmetric chains

We prove a min-max result on special partially ordered sets, a conjecture of András Frank. As corollaries we deduce Dilworth's theorem and the well-known min-max formula for the minimum size edge cover of a graph.Research supported by the Netherlands Organization for Scientific Research (NWO)     

A Lagrangian formulation for steady three-dimensional Newton-Busemann flow

A new Lagrangian formulation for steady three dimensional inviscid flow over rigid bodies is developed. First, the continuity equation is eliminated by the use of two stream functions. This is followed by a transformation to new independent variables, two of which are these stream functions and the third one is a Lagrangian time distinct from the Eulerian time. This Lagrangian formulation with the use of Lagrangian time requires only three independent variables and allows the free boundary problem of flow with shock wave to be rendered a fixed boundary one thereby making it easier to solve. In the Newtonian limit the governing equations reduce to the geodesic equations of the body surface. For flow past two-dimensional bodies, bodies of revolution, and conical bodies and wings, the problems are solved to within quadrature. All known Newtonian flow solutions are found to be special cases of the present theory.     

A convergent finite element formulation for transonic flow

Summary A finite element formulation for the full potential equation in the case of two-dimensional transonic flow is presented. The formulation is based on an optimal control approach developed by Glowinski and Pironneau. The solution of the full potential equation is obtained by a minimization problem. Using a new compactness result it is possible to prove convergence for the solutions of the minimization problem. The a priori assumption of existence and uniqueness of a weak solution of the full potential equation satisfying an entropy condition implies that the limit function must be the solution. It is possible to extend the convergence result to the case of three-dimensional transonic potential flow.The research reported here was supported by a grant from the Stiftung Volkswagenwerk, Federal Republic of Germany. It is a part of the doctoral thesis of the above author, Universität Stuttgart 1989     

Graphical solution of axially symmetric problems of plastic flow

Résumé Une méthode graphique de solution des problèmes dans le cas de symétrie axiale a été proposée pour des corps rigides, parfaitement plastiques, obéissants au critère d'écoulement de Coulomb-Tresca et à l'hypothèse du potentiel plastique. Deux cas ont été considérés: d'une part des régimes de Haar-Kármán pour lesquels la contrainte circonférencielle est égale à l'une des contraintes principales contenues dans le plan axial, et d'autre part des régimes duor lesquels l'une des vitesses de déformation principales dans le plan axial est nulle.