Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states.First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.     

On a dynamical system related to fluid mechanics

We introduce a dynamical system strictly related to fluid mechanics and similar to the classical N point vortex system. In the first part we analyze the qualitative behavior of the time evolution and in particular we show the properties of collapse and chaoticity. In the second part of the paper we investigate the relation of the dynamical system with a system of N concentrated large enough smoke rings in an incompressible and inviscid fluid, with axial symmetry and without swirl. We prove the rigorous connection between the two models at time zero for any N. The extension of the same result to any time is obtained only for a smoke ring alone, while for the general case it is just a matter of conjecture. Received June 1998; Revised November 1998     

Exact analytical solution of a generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient

A nonlinear generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient is constructed, in which the total rate of fluid injection into the porous media remains constant. The number of layers in the model can be arbitrary, and thus the generalized model will be very suitable for describing the one-dimensional non-Darcy flow characteristics in low-permeability reservoirs with strong heterogeneity. Through the similarity transformation method, the exact analytical solution of the multiple moving boundary model is obtained, and the formula for the subrate of fluid injection into every layer is provided. Moreover, it is strictly proved that the exact analytical solution can reduce to the solution of Darcy flow as the threshold pressure gradient in different layers simultaneously tends to zero. Through the exact analytical solution, the effects of the layer threshold pressure gradient, the layer permeability ratio, and the layer elastic storage ratio on the moving boundaries, the spatial pressure distributions, the transient pressure, and the layer subrate in low-permeability porous media are discussed. Through comparison of the exact analytical solutions, it is also demonstrated that incorporation of the multiple moving boundary conditions is very necessary in the modeling of non-Darcy flow in heterogeneous multilayered porous media with a threshold pressure gradient, especially when the threshold pressure gradient is large. In particular, an explicit formula is presented for estimating the relative error of the transient pressure introduced by ignoring the moving boundaries in the modeling. All in all, solid theoretical foundations are provided for non-Darcy flow problems in stratified reservoirs with a threshold pressure gradient. They can be very useful for strictly verifying numerical simulation results, and for giving some guidance for project design and optimization of layer production or injection during the development of heterogeneous low-permeability reservoirs and heavy oil reservoirs so as to enhance oil recovery.     

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

The elastic semi-space consolidation under the normal load on a square area

We investigate the elastic saturated semi-space consolidation under the uniform normal load on a square area. The analytical forms for the fluid pressure and the semi-space surface settlement are found by the solutions to the Dirichlet problem for the Laplace equation and the first boundary-value problem for the heat equation.     

A Method for Determining Linear Solutions in Magnetohydrodynamics

Given any two-dimensional and incompressible flow describedby a set of linear partial differential equations, a methodis presented for determining the solution for the dependentvariables (velocity, pressure, etc.). The method is then usedto investigate the following magnetohydrodynamic flows. (a) The flow of an irrotational and inviscid fluid. (b) The flow of a viscous fluid. (c) The flow of an electrically conducting, inviscid fluid inthe presence of a magnetic field aligned with the flow at infinity. (d) The flow of an electrically conducting, viscous fluid inthe presence of a magnetic field having arbitrary direction.     

A regularizing effect of radiation in the equations of fluid dynamics

We investigate the properties of a class of variational solutions to the equations of fluid dynamics when radiation effects are taken into account. The main aim is to prove weak sequential stability of the solution set under certain hypotheses imposed on the pressure, viscosity, and heat conductivity. Copyright © 2004 John Wiley & Sons, Ltd.     

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c₀ is strictly positive and when c₀ is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

Large Time Behavior for a Simplified 1D Model of Fluid–Solid Interaction†

Abstract

In this article we consider a simple model in one space dimension for the interaction between a fluid and a solid represented by a point mass. The fluid is governed by the viscous Burgers equation and the solid mass, which shares the velocity of the fluid, is accelerated by the difference of pressure at both sides of it. We describe the asymptotic behavior of solutions for integrable data using energy estimates and scaling techniques. We prove that the asymptotic profile of the fluid is a self-similar solution of the Burgers equation with an appropriate total mass, and we describe the parabolic trajectory of the point mass. We also prove that, asymptotically, the difference of pressure to both sides of the point mass vanishes.     

Correlation order, merging and diversification

We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses.We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit.We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.     

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.     

Non-isothermal lubrication problem with Tresca fluid–solid interface law. Part II Asymptotic behavior of weak solutions

We study in this paper the asymptotic analysis of an incompressible Newtonian and non-isothermal problem, when one dimension of the fluid domain tends to zero. We prove the strong convergence of the unknowns which are the temperature, the velocity and the pressure of the fluid, we obtain the limit problem with the specific Reynolds equation, and we also prove the uniqueness of the limit temperature velocity and pressure distributions.     

Solution of the boundary value problem for the equations of steady-state flow of a viscous incompressible nonisothermal fluid past a heated rigid spherical particle

We obtain an analytical solution of a boundary value problem for a viscous incompressible nonisothermal fluid assuming an exponential–power law dependence of the fluid viscosity on temperature. A uniqueness theorem for the Navier–Stokes equation linearized with respect to the velocity is proved. We obtain expressions for the mass velocity components and pressure. The solution of the boundary value problem is sought in the form of an expansion in Legendre polynomials.     

On the inverse Laplace transform for C0-semigroups in UMD-spaces

We consider a two dimensional free boundary problem for an inviscid, incompressible and irrotational fluid. Imbedded in the fluid is a gas bubble of prescribed shape. Given some initial configuration we investigate the long time evolution of the potential flow. We prove existence, uniqueness and regularity of a solution.     

An experimental study of fixed points and chaos in the motion of spheres in a Stokes flow

We present the results of an investigation of a novel dynamicalsystem in which one, two or three solid spheres are free tomove in a horizontal rotating cylinder which is completely filledwith a highly viscous fluid. At low rotation rates, steady motionis found where the balls adopt stable equilibrium positionsrotating adjacent to the rising wall at a speed which is inagreement with available theory. At higher cylinder speeds time-dependentmotion sets in via Hopf bifurcations, and when one or two ballsare present the motion is strictly periodic. Perhaps surprisingly,it is found that three balls are required to produce low-dimensionalchaos.     

六维近凯勒流形的Kodaira维数

本文主要研究了六维近凯勒流形的典范丛和Kodaira维数.证明了六维严格近凯勒流形的典范丛是拟全纯平凡的,从而其Kodaira维数为0.特别地,证明了三维复射影空间CP3具有Kodaira维数不为-∞的近复结构.对于齐性的六维严格近凯勒流形,具体构造了它们典范丛的整体生成元.证明了齐性近凯勒流形F3和CP3的Hodge...     

Asymptotic behaviour of the density for one-dimensional Navier-Stokes equations

The existence of a (global in time) solution to the Navier-Stokes equations for barotropic compressible fluids in a bounded interval is already known in the case of vanishing external force field. In this paper we consider these equations for time-independent forces and prove that: (i) there exists a global solution to the usual initial-boundary value problem; (ii) the density of the fluid is bounded and its infimum is greater than zero for infinite time only if the external forces and the pressure satisfy a compatibility condition (which is the same derived in [2] for the existence of a stationary solution having bounded and strictly positive density).     

Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks

Fluid models have recently become an important tool for the study of open multiclass queueing networks. We are interested in a family of such models, which we refer to as head-of-the-line proportional processor sharing (HLPPS) fluid models. Here, the fraction of time spent serving a class present at a station is proportional to the quantity of the class there, with all of the service going into the first customer of each class. To study such models, we employ an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized HLPPS networks with traffic intensity strictly less than 1 are positive Harris recurrent. Related results for FIFO fluid models of Kelly type were obtained in Bramson [4].Partially supported by NSF Grants DMS-93-00612 and DMS-93-04580. The paper was written while the author was in residence at the Institute for Advanced Study.     

On a Time-Dependent Fluid-Solid Coupling in 3D with Nonstandard Boundary Conditions

We study the motion of a Stokes fluid through an elastic half cylinder with thickness. The fluid is driven by a small time-dependent pressure drop between the outflow and the inflow sections. Nonstandard boundary conditions are imposed at the tube’s ends and we consider small displacements of the elastic structure, such that the domains involved are not moving in time. We prove existence and uniqueness of a weak solution for this three dimensional fluid-elastic structure interaction problem. Under appropriate regularity assumptions on the data we obtain strong energy estimates and show the existence of a pressure with improved properties.