Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 38–49, July–August, 1998.     

On a supplementary conservation law for the energy equation in a rigid heat conductor

We consider a rigid heat conductor with specified constitutive equations and show that the internal energy equation may be written in the form of a symmetric and conservative hyperbolic system of first order quasi-linear equations for which the Cauchy problem is well-posed. Moreover, such a system is useful to study shocks. Several particular cases are examined.     

A note on the delayed heat equation: Instability with respect to initial data

Working in the context of a simple, one-dimensional, initial-boundary value problem involving homogeneous Dirichlet boundary data, we show that the time delayed heat equation can exhibit a type of instability with respect to the initial condition (IC); specifically, we show that a slight (in the L² sense) change in the IC can change a well-posed problem to an ill-posed one. We also establish that a physically realistic solution is possible only if the IC is of a (very) specific form. The main implication of this study is that the single and dual phase lag models, which have been put forward as possible alternatives to Fourier’s law, are not valid constitutive relations for the thermal flux vector.     

A formal finite element approach for open boundaries in transport and diffusion ground-water problems

In the application of the finite element method to diffusion and convection-dispersion equations over a ground-water domain, the Galerkin technique was used to incorporate Neumann (or second-type) and Cauchy (or third-type) boundary conditions. While mass movement through open boundaries is a priori unknown, these boundaries are usually treated as a zero Neumann condition at some far distance from the domain of interest. Nevertheless, cheaper and better solutions can be obtained if these unknown conditions are adequately incorporated in the weak formulation and in the transient solution schemes (open boundary condition). Theoretical and numerical proofs are given of the equivalences between this approach and a ‘well-posed’ problem in a semi-infinite domain with a zero Neumann condition at a boundary placed at infinity. Transport and diffusion equations were applied in one dimension to show the numerical performances and limitations of this procedure for some linear and non-linear problems. No a priori limitations are foreseen in order to find similar solutions in two or three dimensions. Thus the spatial discretization in the proximity of open boundaries could be drastically reduced to the domain of interest.     

Global solution of the inverse problem for a class of nonlinear evolution equations of dispersive type

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin. Contributed by Chen Yu-shu Foundation item: the National Natural Science Foundation of China (Significance 199990510); the National Key Basic Research Special Foundation of China (G1998020316); Liuhui Center for Applied Mathematics, Nankai University & Tianjin University Biography: Chen Fang-qi (1963-)     

Data completion for the Stokes system

This Note is concerned with the severely ill-posed Cauchy–Stokes problem. This inverse problem is rephrased into an optimization one: An energy-like error functional is introduced. We prove that the optimality condition of the first order is equivalent to solving an interfacial equation which turns out to be a Cauchy–Steklov–Poincaré operator. Numerical trials highlight the efficiency of the present method. To cite this article: A. Ben Abda et al., C. R. Mecanique 337 (2009).     

Mass redistribution method for finite element contact problems in elastodynamics

This paper is devoted to a new method dealing with the semi-discretized finite element unilateral contact problem in elastodynamics. This problem is ill-posed mainly because the nodes on the contact surface have their own inertia. We introduce a method based on an equivalent redistribution of the mass matrix such that there is no inertia on the contact boundary. This leads to a mathematically well-posed and energy conserving problem. Finally, some numerical tests are presented.     

On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.     

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｐｓｅｕｄｏ＿ｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎｗｉｔｈｐｒｉｎｃｉｐａｌｐａｒｔｕｔ -ｕｘｘｔｈａｓｂｅｅｎｓｔｕｄｙｉｎｇｒｅｃｅｎｔｌｙ ,ｂｅｃａｕｓｅｔｈｅｒｅｅｘｉｓｔｓｗｉｄｅｐｈｙｓｉｃａｌｂａｃｋｇｒｏｕｎｄｆｏｒｔｈｉｓｃｌａｓｓｏｆｅｑｕａｔｉｏｎｓ.[1 ]ｓｔｕｄｉｅｄｔｈｅｍｕｌｔｉ＿ｄｉｍｅｎｓｉｏｎｉｎｖｅｒｓｅｐｒｏｂｌｅｍｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｃｌａｓｓｏｆｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕ…     

Impact of Non-uniform Properties on Governing Equations for Fluid Flows in Porous Media

The macroscopic governing equations of a compressible multicomponents flow with non-uniform viscosity and with mass withdrawal (due to heterogeneous reactions) in a porous medium are developed. The method of volume averaging was used to transform local (or microscopic) governing equations into averaged (or macroscopic) governing equations. The impacts of compressibility, non-uniform viscosity, and mass withdrawal on the form of the averaged equations and on the value of the macroscopic transport coefficients were investigated. The results showed that the averaged mass conservation equation is significantly affected by mass withdrawal when a specific criterion on the size of the domain is respected. The results also showed that the form of the averaged momentum equations is not affected by mass withdrawal, by compressibility effects or by non-uniform viscosity, provided that the Reynolds number at the pore level is small. Nonetheless, the velocity field is affected by the heterogeneous reaction via the averaged mass conservation equation, and also by viscosity variations due to the presence of the volume-averaged viscosity (which value changes with position) in the averaged momentum equations. A new closure variable definition was proposed to formulate the closure problem, which avoided the need to solve an integro-differential equation in the closure problem. This formulation was used to show that the permeability tensor only depends on the geometry of the porous medium. In other words, that tensor is independent on whether the fluid is compressible/incompressible, has uniform/non-uniform viscosities, and whether mass withdrawal due to heterogeneous reactions is present/absent.     

The Mean Value Theorem and Converse Theorem of One Class the Fourth-Order Partial Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｍｏｕｓＡｓｇｅｉｒｓｓｏｎｍｅａｎｖａｌｕｅｔｈｅｏｒｅｍｈａｓａｎｓｗｅｒｅｄｔｈａｔｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｓａｒｅｉｌｌ＿ｐｏｓｅｄｔｏｔｈｅｕｌｔｒａ＿ｈｙｐｅｒｂｏｌｉｃｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｆｔｈｅｓｅｃｏｎｄ＿ｏｒｄｅｒ(Δ2 ｘ-Δ2 ｙ)ｕ=0 . ( 1 )Ｔｈｅｒｅｓｕｌｔｃａｎｂｅｕｓｅｄｔｏｐｒｏｖｅｔｈｅｃｏｎｔｉｎｕａｔｉｏｎｏｆｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｉｓｅｑｕａｔｉｏｎ ( 1 ) .Ｓｏｉｔｉｓｉ…     

Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDEs. Spectral eddy-viscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDEs, they actually yield convergence in the graph norm.     

SOLUTIONS OF MAGNETOHYDRODYNAMICS EQUATIONS-THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRAC-PAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS (Ⅳ)

This work is the continuation of the discussion of Refs.[1 3].(A) We turn the Magnetohydrodynamics equaitons of isentropic compresible andnon-dissipative magneto-flow into the form of the ideal hydrodynamics equations in thispaper;we can obtain the general Chaplygin equation from Ref.[3],and the generalsduction of this equation.(B)We apply the theory of functions of a complex rariable under Dirac-paulirepresentation,turn the general Magnetohydrodyamics equations of incompressiblemageto-flow into two nonlinear equaitons for flow function and"magneto-flow function",and obtain the exact stable solution of incompressible magnetohydrodynamics equationsunder the condition of stable magnetic field(i.e.under conditon of equality for kinematicalviscid coefficient or viscid diffusion coefficient with magnetic diffusion coefficient).     

Ill-posedness of the Hydrostatic Euler and Navier–Stokes Equations

We prove that the linearization of the hydrostatic Euler equations at certain parallel shear flows is ill-posed. The result also extends to the hydrostatic Navier–Stokes equations with a small viscosity.     

Local pressure preconditioning method for steady incompressible flows

The convergence and accuracy characteristics of the preconditioned incompressible Euler and Navier–Stokes equations are studied. An object-oriented C++ numerical code has been developed for solving the inviscid and viscous, steady, incompressible flows problems. The code is based on the cell-centred finite volume method. In this scheme, two-dimensional incompressible Euler and Navier–Stokes equations are modified by a robust artificial compressibility (AC) and a local preconditioning matrix of pressure-sensor type. The preconditioned equations are solved with the Jameson's numerical approach, i.e. artificial dissipation and artificial viscosity terms under the form of a fourth- and second-order derivative, respectively. An explicit four-stage Runge–Kutta integration algorithm is applied to obtain the steady-state condition. The computed results include the steady-state solution of flow past the NACA-hydrofoils and a circular cylinder in free stream, for which the numerical results are compared with numerical works of other researchers. Good agreement is observed. The effects of AC parameter, artificial viscosity and dissipation factor, and local preconditioning coefficient on convergence rate and solution accuracy are tested by computing flow over the NACA0012 hydrofoil. In addition, some important design criteria of a preconditioner, such as stiffness reduction, hyperbolicity, symmetrisability, accuracy preservation for M → 0, and M-property have been examined analytically.     

Analytical solution of basic equations set of atmospheric motion

On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typicality and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type of problems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.     

On well-posedness of the dynamic problem for an anisotropic fluid-saturated solid

It is known that a high degree of anisotropy in the constitutive behaviour of a solid may result in the loss of hyperbolicity of the dynamic equations in the form of either complex-conjugate or purely imaginary characteristic wave speeds (flutter ill-posedness and shear band formation, respectively). In the present paper we investigate the characteristic wave speeds in the dynamic problem for a transversely isotropic fluid-saturated porous solid. Three cases are considered: a dry solid and a saturated solid under locally undrained and drained conditions. It is shown that, for given constitutive parameters of the solid skeleton, the dynamic problem for a drained solid may become ill-posed due to the flutter-type loss of hyperbolicity, while the dynamic equations for a dry and an undrained solids remain hyperbolic. For a given solid skeleton, the characteristic wave speeds are strongly influenced by the pore fluid compressibility which, in turn, is extremely sensitive to the presence of a small amount of free gas.     

On the Two-Phase Navier–Stokes Equations with Boussinesq–Scriven Surface Fluid

Two-phase flows with interface modeled as a Boussinesq–Scriven surface fluid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase flows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not well-posed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase flows with surface viscosity.     

A well-posed Euler-Bernoulli beam model incorporating nonlocality and surface energy effect

This study shows that it is possible to develop a well-posed size-dependent model by considering the effect of both nonlocality and surface energy, and the model can provide another effective way of nanomechanics for nanostructures. For a practical but simple problem(an Euler-Bernoulli beam model under bending), the ill-posed issue of the pure nonlocal integral elasticity can be overcome. Therefore, a well-posed governing equation can be developed for the Euler-Bernoulli beams when considering both the pure nonlocal integral elasticity and surface elasticity. Moreover, closed-form solutions are found for the deflections of clamped-clamped(C-C), simply-supported(S-S) and cantilever(C-F) nano-/micro-beams. The effective elastic moduli are obtained in terms of the closed-form solutions since the transfer of physical quantities in the transition region is an important problem for span-scale modeling methods. The nonlocal integral and surface elasticities are adopted to examine the size-dependence of the effective moduli and deflection of Ag beams.     

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID ＊

Introduction Itisreasonabletoconsiderarealfluidasanidealoneinfluidmechanicsundermany conditions.Forinstance,ingeneral,forthedistributionofthefluidfieldaroundtheaerocraft,mostpartofthefluidfieldmayberegardedasidealfluidexceptforasmallpartwhereeffects ofviscousandheatconductioninthethinlayernearthesurfacemustbeconsidered.Evenif thefluidiscompletelysupposedasidealonethroughoutthefluidfield,thequitereasonable resultsarealsogained,thereforestudyingidealfluidhasnotonlytheoreticalsignificancebut also…