Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

Maximal Regularity of the Time - Periodic Linearized Navier–Stokes System

Time-periodic solutions to the linearized Navier–Stokes system in the n-dimensional whole-space are investigated. For time-periodic data in L ^q -spaces, maximal regularity and corresponding a priori estimates for the associated time-periodic solutions are established. More specifically, a Banach space of time-periodic vector fields is identified with the property that the linearized Navier–Stokes operator maps this space homeomorphically onto the L ^q -space of time-periodic data.     

Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.     

Crack-tip dynamic isochromatics in the presence of small-scale yielding

A plasticity correction factor for the dynamic stress-intensity factor,K _I ^dyn , associated with a propagating crack tip in the presence of small-scale yielding, is derived from Kanninen's solution for a constant-velocity Yoffe crack with a Dugdale-strip yield zone. Distortions in the otherwise elastic isochromatics surrounding the constant-velocity crack tip are also studied by the use of this model. This plasticity correction factor is then used to evaluateK _I ^dyn from the dynamic isochromatics of a propagating crack in a 3.2-mm-thick polycarbonate wedge-loaded rectangular double-cantilever-beam specimen. The correctedK _I ^dyn is in good agreement with the corresponding values computed by a dynamic, elastic-plastic finite-element code executed in its generation mode.     

Experiments on sedimentation beneath downward-facing inclined walls

It has been known since Boycott (1920) that sedimentation rates can be several times higher within inclined walls than in vertical vessels. Beneath the downward-facing inclined wall a thin boundary-layer of clear liquid is formed with an upward velocity. Mass continuity brings about the observed effect. Ponder (1925), and Nakamura & Kuroda (1937) proposed a global kinematic model without any detail of the flow field to predict the sedimentation rate. Recently, two asymptotic theories for describing the flow field became available. The investigation by Acrivos & Herbolzheimer (1979) requires both Re²/Gr and Re⁴/Gr to be small, with Re being a sedimentation Reynolds number and Gr a sedimentation Grashof number. The analysis due to Schneider (1982) is valid for small values of Re²/Gr and large values of Re⁴/Gr. This paper presents experimental sedimentation data in a²/Gr and large values of symmetrical, roof-shaped vessel. The experiments with spherical glass beads in a variable mixture of glycerine and water cover the whole parameter range from very small to very large values of Re⁴/Gr. Therefore, both theories could be verified. In the case of small values of Re⁴/Gr, strong waves beneath the downward-facing wall were observed and graphically evaluated in order to find a basis for further theoretical approach. Furthermore, an upward flow of particles in a sublayer of the boundary-layer leads te protruding “borns” and an oscillation of the originally horizontal discontinuity separating the suspension from the clear liquid. The measured particle distribution and deviations from a monodispersed suspension seems to be a possible explanation for this effect, which is investigated theoretically in a separate paper (Schaflinger 1984). A further experimental shortcoming, the shallow depth of the settling vessel, that was necessary for an optoelectronic measuring of the actual volume fraction solids within the bulk, may affect the convective motion when the sedimentation Reynols number is small. However, in this case the experimental data are in conformity with the theoretical predictions which do not consider the influence of wall friction on the sedimentation behaviour.     

Optimal convergence rates for three-dimensional turbulent flow equations

In this paper,the convergence rates of solutions to the three-dimensional turbulent flow equations are considered.By combining the Lp-Lq estimate for the linearized equations and an elaborate energy method,the convergence rates are obtained in various norms for the solution to the equilibrium state in the whole space when the initial perturbation of the equilibrium state is small in the H3-framework.More precisely,the optimal convergence rates of the solutions and their first-order derivatives in the L2-norm are obtained when the Lp-norm of the perturbation is bounded for some p ∈[1,6/5).     

Design of analog variable fractional order differentiator and integrator

This paper deals with the design of analog variable fractional order differentiator s ^m and integrator s ^?m , for 0<m<1, for a given frequency band, a subject that has not been yet investigated. The main feature of this analog variable fractional order integrator or differentiator is that its frequency characteristics can be changed without redesigning a new one. First, analog rational function approximation of the fractional order differentiator s ^m and integrator s ^?m are derived with the new idea to keep all its poles to be independent of the fractional orders?m. Next, we have used the polynomial interpolation method to design the variable fractional order analog integrator and differentiator that can be implemented by an analog structure like the digital Farrow structure. Finally, some examples are presented to illustrate the efficiency and the effectiveness of the proposed design method.     

Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ?⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N? 2)/N,0}, in which the limits of classical solutions are also continuous in ? ^N as extended functions with values in ?₊∪{∞}. We introduce a precise class of extended continuous solutions ? _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ? _c has an initial trace in ?⁺, and (iii) that the solutions in ? _c are limits of classical solutions. Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈??, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ? ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N? 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense. As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

On the Transformation Property of the Deformation Gradient under a Change of Frame

If the deformation gradients are denoted by F and F ^* respectively before and after a change of frame, they are related by the transformation formula, F ^*=QF, where Q is the orthogonal transformation associated with the change of frame. Although it has been pointed out that this relation is valid “provided that the reference configuration be unaffected by the change of frame” (see p. 308 of [1]), this formula is found in most textbook of Continuum Mechanics, and is used, without further justification, in deriving the condition of material frame-indifference, ?(QF)=Q?(F)Q ^T for the constitutive function ? of the stress tensor of an elastic body. In this note, we shall analyze the effect of change of frame on the transformation property of the deformation gradient, and show that the above transformation formula is not valid in general. However, we shall confirm the validity of the above well-known condition of material frame-indifference without the assumption that the reference configuration be unaffected by the change of frame.     

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

The equilibrium range cascades of wind-generated waves

It was established theoretically by Zakharov and Filonenko (1966) that the direct cascade region of the equilibrium range of the spectrum should follow an f⁻⁴ variation. This has since been verified from experimental data, by Toba (1973), Donclan et al (1985) and others. In this study we present a numerical verification of this f⁻⁴ variation, assuming physically realistic parameterizations for nonlinear wave-wave interactions, S_nl, for energy input to waves by the wind, S_in and removed by wave - breaking dissipation, S_ds.     

Characterization of the flow for a single fluid in an excavation damaged zoneCaractérisation de l'écoulement d'un fluide monophasique en zone endommagée

The aim of this Note is to quantify the change of characteristics of the media of an Excavated Damaged Zone (EDZ) affected by several fractures. For this, we consider Darcy flow through matrix blocks and fractures with permeability of order ε²δ^θ and 1 respectively. ε is the size of a typical porous block, δ representing the relative size of the fracture and θ is a parameter characterising the permeability ratio. We derive the global behavior from the limit as ε and δ tend to zero. The resulting homogenized equation is of dual-porosity type for θ=2, but it is a simple-porosity model with effective coefficients for θ>2, and there is no flow at the macroscopic level when 0<θ<2. To cite this article: B. Amaziane et al., C. R. Mecanique 332 (2004).     

Strong Traces for Solutions of Multidimensional Scalar Conservation Laws

In this paper we consider multidimensional scalar conservation laws without BV estimates defined in a subset Ω??⁺×? ^d . We show that, with a non-degeneracy hypothesis on the flux, we can define a strong notion of trace at the boundary of Ω reached by L ¹ convergence.     

Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum

Formation of sharp vorticity gradients in two-dimensional (2D) hydrodynamic turbulence and their influence on the turbulent spectra are considered. The analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines is developed and compressibility of this mapping appears as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as k ^?3 at large k, which appear to take the same form as the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the k dependence of the spectrum due to the sharp gradients coincides with the Saffman spectrum: E(k) ~ k ^?4. Numerical investigations of decaying turbulence reveal exponential growth of di-vorticity with a spatial distributed along straight lines. Thus, indicating strong anisotropy and accordingly the spectrum is close to the k ^?3-spectrum.     

Modeling of vibration-dissociation oxygen kinetics at temperatures of 4,000-11,000 K

The time profiles of vibrational molecular oxygen temperature T _v measured earlier in experiments behind a strong shock wave were used for testing the theoretical and empirical models of thermal nonequilibrium dissociation of molecules. To do this, dissociating gas flows behind the strong shock wave front were calculated with account for these models. If the initial gas temperature behind the wave front T ₀ < 6.5×10³ K, the models well describe changing the temperature with time. However, for T ₀ > 7×10³ K neither of the models tested describes the measured temperature profiles satisfactorily. Using the empirical model proposed in the present study made it possible to satisfactorily describe the vibrational temperature evolution observed in experiments at temperatures up to 11×10³ K.     

The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity

Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R ². If the matrix fields are respectively of class C ² and C ¹, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W ^1,∞ _loc and L ^∞ _loc, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense.     

Effect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus

Measurement apparatus designed and constructed according to conceptions of the authors, enabled a more precise calculation of the heat transfer coefficient with the balance and gradient methods. Construction and use of the apparatus and devices are described below, results of experimental investigations for horizontal and vertical, isothermal, flat plates obtained independently with the balance and gradient methods, are also presented. The following equations were found:Nu=0.612 · (Ra)^1/4 10⁴ ≦Ra ≦ 10⁸ for vertical platesNu=0.766 · (Ra)^1/5 10⁴ ≦Ra ≦ 10⁷ Nu=0.173 · (Ra)^1/3 10⁵ ≦Ra≦ 10⁸ for horizontal plates. On the basis of the results obtained from both these methods, differences of natural convection acting from vertical and horizontal plates are discussed. The usefulness of the balance and gradient methods have been considered for qualitative and quantitative investigations of heat transfer by natural convection.     

Development of empirical correlations to predict the secondary droplet size of impacting droplets onto heated surfaces

The present article reports an experimental analysis of the mechanisms of secondary atomization which occur at the impact of individual droplets onto heated targets. The experiments follow those reported in a previous article (Moreira et al. 2007) and encompass the use of different liquids and impact conditions. An image analysis system is combined with a phase Doppler interferometer to measure extended size distributions, which cover the full range of diameters generated at all heat transfer regimes. The results evidence that disintegration mechanisms depend on the heat transfer regimes; therefore, a universal relation cannot be devised for the outcome of droplet impact. Analysis shows that droplets impacting within the nucleate-boiling regime break-up by a thermal-induced mechanism associated with the vapour pressure at bubble nucleation sites, combined with liquid surface tension. On the other hand, within the film-boiling regime, disintegration is associated with radial disruption of the rim at the early instants after impact, as in non-heated targets, and with the rupture of the ligaments of the cellular structures. Functional relations available at the literature, mostly developed for impacts onto non-heated surfaces, are well fitted to the experimental results obtained within the film-boiling regime, since the break-up mechanisms are qualitatively similar. On the other hand, such relations cannot predict the secondary atomization occurring within the nucleate-boiling regime, as the break-up mechanisms within this regime have significantly different characteristics. In this context, the present article recognizes the relevance of the relations devised for ‘cold impacts’, to fit the size of secondary droplets within the film-boiling regime, as the correlation formulated here has a similar form: SMD/D ₀ = f(We, Re) ~ A ₁ We _N ^?0.6 Re ^?0.23 and proposes a new correlation for impacts within the nucleate-boiling regime: SMD/D ₀ = f(We, Re, Ja) ~ A ₂ We _N ^?0.14 Re ^?011 Ja ^?03. These correlations are observed to hold for impacts onto rough surfaces with dimensionless roughness R _a/D ₀ smaller than 2E-3, but not for larger roughness amplitudes, for which the data are quite scattered.     

Inverse Estimation of the Effective Diffusion of the Filter in the In Situ Diffusion and Retention (DR) Experiment

Porous filters are often used in laboratory and in situ diffusion and retention experiments. The proper interpretation of these experiments requires knowing the effective diffusion, D _e, of the filter which is commonly determined from laboratory diffusion experiments or estimated from the filter porosity. The D _e of the filter in the in situ experiment may differ from the D _e of the filter measured in the laboratory due to pore clogging. Here, we present an inverse method to estimate the D _e of the filter of in situ diffusion experiments. The method has been tested for several sampling schemes, numbers of synthetic data, N, and standard deviations of the noise, ??. It has been applied to the following tracers used in the in situ diffusion and retention (DR) experiment performed in the Opalinus clay at Mont Terri underground research laboratory: HTO/HDO, Br^?,I^?, ²² Na⁺,¹³³ Ba²⁺,⁸⁵ Sr²⁺, Cs⁺/¹³⁷Cs⁺, and ⁶⁰Co²⁺. The estimation error increases with the standard deviation of the noise of the data and decreases with the number of data. It is smallest for sorbing tracers. The D _e of the filter can be properly estimated from 12 data collected within the first 3?days for conservative tracers as long as ????? 0.02 and for sorbing tracers as long as ??????0.05. The estimate of D _e for conservative tracers is poor when data are collected from a 10-day experiment with daily sampling. The convergence of the estimation algorithm for conservative tracers improves by starting with a value of the D _e smaller than the true value. The choice of the initial value of D _e does not affect the convergence of the estimation algorithm for sorbing tracers. Filter clogging and vertical flow though the filter can influence the tracer transport through the filter. The use of the D _e of the filter obtained from a laboratory test for the in situ experiment may result in large errors for strongly sorbing tracers. Such errors can be overcome by estimating the equivalent D _e of the filter with the proposed inverse method which will be useful for the design of in situ diffusion experiments.