Extensions to the Macroscopic Navier–Stokes Equation

A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcys equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier–Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcys linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix.     

The Solution to the Form of Poisson Equation by the Boundary Element Methods

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On the Static-Limit Solutions to the Navier—Stokes Equations of Compressible Flow

We consider the zero-velocity stationary problem of the Navier--Stokes equations of compressible isentropic flow describing the distribution of the density r \varrho of a fluid in a spatial domain W ì R^N \Omega \subset {\rm R}^N driven by a time-independent potential external force [(f)\vec] = \triangledown F \vec f = \triangledown F . We study the structure of the set of all solutions to the stationary problem having a prescribed mass m > 0 and a prescribed energy. Cardinality of the solution set depends on m and it is either continuum or at most two. Conditions on m for distinguishing these cases have been found. Uniqueness for the stationary system is also studied.     

On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R $ , and $s\ge 1$ , $s\ge r$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.     

On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.     

Developable Surfaces as Generators of the “Isobaric Solutions” to the Euler Equations

The simplest solutions to the Euler equations (1.1) for which the pressure vanishes identically are those representing the motion of lines parallel to a fixed direction moving in the same direction (each line with an independent, given, constant velocity). Are there many other solutions to this problem? If yes, is there a simple characterization of all the initial data (volume occupied by the fluid at time t = 0 and initial velocity that gives rise to the general solutions? In this paper we show that the answer to both questions is positive. We prove, in particular, that there is a natural correspondence between solutions in R² of this problem and (Cartesian pieces of) developable surfaces in R³. See Theorem 3.     

Probabilistic approach to the Lamé equations of linear elastostatics

A probabilistic approach to systems of partial differential equations is developed on the basis of the well-known Feynman–Kac and Bismut formulas providing explicit probabilistic representations of the solutions and of their derivatives of scalar differential equations. Some numerical examples are also included. In particular the Lamé equations of elastostatics are solved and the results are compared with some known exact analytic solutions to demonstrate the efficiency of the approach.     

On the thermo-mechanical coupling in austenite–martensite phase transformation related to the quenching process

The present contribution considers modeling and simulation of the quenching process, presenting an anisothermal model formulated within the framework of continuum mechanics and the thermodynamics of irreversible processes. The energy equation thermo-mechanical coupling terms due to internal and thermal couplings are exploited. In order to analyze the importance of these terms, three different models are considered. The first one is an uncoupled model in the sense that these terms are neglected, corresponding to the rigid body energy equation. In second model, these couplings are represented through the incorporation of a source term in the energy equation associated with the latent heat released during the austenite–martensite phase transformation. Finally, the third model considers all thermo-mechanical coupling terms of the proposed model. Progressive induction hardening of a long cylindrical body is considered as an application of the proposed general formulation. Numerical simulations analyze the effect of the thermo-mechanical coupling terms, comparing the three proposed models.     

From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, II

We consider here the problem of deriving rigorously from renormalized solutions of Boltzmann's equation, globally in time, for general initial conditions and without any additional assumption, solutions of Stokes' equations (together with the strong Boussinesq relation). We also obtain similar results for Euler equations where, however, we need to make an assumption on the high velocities of the solutions of Boltzmann's equation.     

Time Average Geometric Moiré—Back to the Basics

Applicability of time average geometric moiré for elastic oscillating structures is analysed in this paper. Mathematical and numerical models describing the formation of time averaged fringes are carefully constructed without the assumption that dynamic deflections are described by a slowly varying function. Though time average geometric moiré is considered as a classical optical experimental technique, we show that well known relationship between the fringe order, amplitude of oscillation and pitch of the grating in state of equilibrium can be used only when the amplitude is small. Otherwise the inverse problem of fringe interpretation becomes much more complicated and is the object of analysis in this paper. We describe the interpretation of fringes produced by time average geometric moiré in detail and illustrate the complexity of the problem by numerical examples.     

Application of Moiré Interferometry to the Characterization of Orthotropic Materials in the Antisymmetric Configuration using the Virtual Fields Method

Moiré interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moiré patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moiré interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q₁₁, Q₂₂, Q₁₂, and Q₆₆) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moiré interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q₁₁, Q₂₂, Q₁₂ and Q₆₆ values of a unidirectional carbon fiber composite were measured in a real Moiré interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges.     

Applying the Generalized Step-function to Solve the Bending Problems of Nonuniform Beams with Variable Cross Section

The bending problems of nonuniform beams with variable cross-sectioncan be approximated by that of a step beam under sectionally uniform load(including both concentrated forces and couples).In this paper,the conceptof Heaviside function{x-a}~0 will be generalized,and a new function{x-a}~n,n=0,1,2……,will be defined,which may be named as a generalized stepfunction.The rules of operation will also be given to{x-a}~n{x-b}~0.Thereciprocal of the flexural of rigidity1/EJ and the bending moment M(x)can all be expressed in terms of{x-a}~n,and substituted into the differentialequation of the elastic curve of the beam respectively.Thus we mayestablish a set of unified method to solve various types of bending problemsof straight beams.The general solution of the deflection equation will begiven.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L ² norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.     

Extending the Bodner–Partom model to simulate the response of materials with extreme kinematic hardening

This paper demonstrates that, at extreme levels of kinematic hardening, the traditional formulation of the Bodner–Partom model can produce anomalous results. The reasons for this anomalous behaviour are explained, and a reformulated version of the model is presented. This reformulation extends the range of the model to include levels of kinematic hardening that may be problematic in the traditional formulation. The formulation of the model is adjusted so as to retain the rate dependency of the original Bodner–Partom model; and to permit the values of the material parameters used with the traditional formulation to be re-used with the extended model—with the exception only of the hardening coefficients which become dimensionless constants holding different numerical values. This revised formulation also imposes associated flow, thereby ensuring phase consistency between stress and plastic strain during non-proportional loading. In this way, the anomalies are removed, the range and stability of the model is increased, and all the advantages and important features of the Bodner–Partom model are retained.     

Back to the Future? A Re-examination of the Aerodynamics of Flettner-Thom Rotors for Maritime Propulsion

The paper examines, by way of a 3-dimensional, unsteady-RANS analysis, the high-Reynolds-number turbulent flow normal to a cylinder rotating about its axis, thus continuing the re-evaluation of this flow configuration as a potential means of providing low-cost ship propulsion. Comparisons are made between the available experimental and LES data both for the bare cylinder, as employed in Flettner’s notable Atlantic crossing, and the variant advocated by Thom in which close-packed discs are distributed along the cylinder. Our results display close agreement with available experimental and LES data other than the results of Thom (1934). We conclude that the addition of discs by the latter led, at the relatively low Reynolds numbers of his experiments, to the boundary layer on the cylinder being thinned to a point at which the boundary layers became laminar or transitional, thus leading to higher lift coefficients than pertain in turbulent flow.