A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

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Vorticity and Regularity for Viscous Incompressible Flows under the Dirichlet Boundary Condition. Results and Related Open Problems

In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R ³ must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary condition in the half-space R ₊³. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition. The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand, by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.      

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

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Further Comments on “Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work”

This note is concerned with the assertion of Barletta and Nield (2009a) that “a fluid with a thermal expansion coefficient greater than that of a perfect gas (β > β _perfect gas) is of marginal or no interest in the framework of convection in porous media”, and that for a remark of Magyari (Transp. Porous Media, 2009) about the forced convection eigenflow solutions, the circumstance β > β _perfect gas does not represent “a sound physical basis”. Here, it is shown, however, that these assertions are in contradiction with the experimentally measured values of β for important technical fluids as e.g., air, nitrogen, carbon dioxide, and ammonia where, in the temperature range between −20 and +100°C, just the inequality β > β _perfect gas holds.     

The “Butterfly Effect” in a Porous Slab

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution [(T)\tilde]_w ( x ){\tilde {T}_w \left( x \right)}, when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

Rheological analysis of highly concentrated w/o emulsions

A series of highly concentrated lipophilic cosmetic emulsions were analysed, in order to determine their rheological and textural properties, as a function of their microstructure. The originality of this study lies in the methodology used, especially the shear-stress scanning analysis. The results of a very powerful and comprehensive dynamic rheological analysis suggest the existence of two critical volume fraction values: besides the “close-packed” value φ _c , a “slack-packed” value φ₀, close to 0.60 could be demonstrated. It has been shown that the close-packed structure is stable under shear; in constrast, the slack-packed configuration, defined as φ₀<φ<φ _c is unstable under shear. A comparison with theoretical models, especially that of Princen, showed good agreement and allowed the close-packed value φ _c to be defined more precisely as 0.67. The gap between 0.67 and 0.74 is probably indicative of a highly polydisperse distribution, as confirmed by microscopic analysis. Flow experiments confirmed the validity of Princen‘s model. Received: 20 February 1997 Accepted: 20 January 1998     

On the Domain of Validity of Brinkman’s Equation

An increasing number of articles are adopting Brinkman’s equation in place of Darcy’s law for describing flow in porous media. That poses the question of the respective domains of validity of both laws, as well as the question of the value of the effective viscosity μ ^e which is present in Brinkman’s equation. These two topics are addressed in this article, mainly by a priori estimates and by recalling existing analyses. Three main classes of porous media can be distinguished: “classical” porous media with a connected solid structure where the pore surface S _p is a function of the characteristic pore size l _p (such as for cylindrical pores), swarms of low concentration fixed particles where the pore surface is a function of the characteristic particle size l _s , and fiber-made porous media at low solid concentration where the pore surface is a function of the fiber diameter. If Brinkman’s 3D flow equation is valid to describe the flow of a Newtonian fluid through a swarm of fixed particles or fibrous media at low concentration under very precise conditions (Lévy 1983), then we show that it cannot apply to the flow of such a fluid through classical porous media.     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Turbulence in the Stratified and Rotating World Ocean

This is an overview of knowledge, derived mainly from observations, of turbulence in the stratified and rotating World Ocean from the 1960s, when mesoscale motions with scales of 30–150 km and 100 days were discovered by neutrally buoyant floats, to the present decade and the use of SF₆“purposeful tracer” release study in the North Atlantic. Most of the ocean is stably stratified, but it contains a rotational turbulent “continua” and isolated rotating eddies, as well as Rossby waves, and many features similar to those of, say, planetary atmospheres. It differs however because (a) the presence of lateral boundaries, the continental land masses, islands, and seamounts, provides constraints to the circulation and to the propagation of eddies, and possibly substantial sources and sinks of eddy motion; (b) channels connecting the oceans to land-locked seas (e.g., the Mediterranean; the formation of water with anomalous properties, “natural tracers”, e.g., temperature or salinity, allows “interthermocline eddies” to be readily detected); and (c) convection and differential seasonal latitudinal forcing introduce upper-ocean variability and intrathermocline eddies. The focus of research interest has moved from “turbulence”per se towards study of its consequences, for example towards dispersion of material particles and dissolved solutes, and the meridional and inter-basin transfers of heat. Received 2 December 1996 and accepted 18 September 1997     

Detection and quantification of industrial polyethylene branching topologies via Fourier-transform rheology,NMR and simulation using the Pom-pom model

The significance of sparse long-chain branching in polyolefines towards mechanical properties is well-known. Topology is a very important structural property of polyethylene, as is molecular weight distribution. The method of Fourier-transform rheology (FTR) and melt state nuclear magnetic resonance (NMR) is applied for the detection and quantification of branching topology (number of branches per molecule), for industrial polyethylenes of various molecular weight and molecular weight distributions. FT rheology consists of studying the development of higher harmonics contribution of the stress response to a large amplitude oscillatory shear deformation. In particular, when applying large-amplitude oscillatory shear (LAOS), one observes the development of mechanical higher harmonic contributions at 3ω ₁, 5ω ₁,..., in the shear stress response. We correlate the relative intensity, I _3/1, and phase Φ ₃ of these harmonics with structural properties of industrial polyethylene, i.e. polymer topology and molecular weight distribution. Experiments are complemented by numerical simulations, using a multimode differential Pom-pom constitutive model (DCPP formulation), by fitting to the experimental linear and nonlinear viscoelastic behaviours. Simulation results in the nonlinear regime are correlated with molecular properties of the “pom-pom” macromolecular architecture. Qualitative agreement is found between predicted and experimental FT rheology results.     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.     

Local mass/heat transfer from a wall-mounted block in rectangular channel flow

This investigation explores the mass/heat transfer from a wall-mounted block in a rectangular fully developed channel flow. The naphthalene sublimation scheme was used to measure the level of local mass transfer from the block’s surfaces. The heat transfer coefficient can be obtained by analogy between heat and mass transfer. The effects of the Reynolds number on the local mass transfer from the block’s surfaces have been widely discussed. Results showed that, owing to the flow complexity induced by vortices around the block, the block’s surfaces appeared four different spatial Sherwood number distributions, termed “Wave type”, “U type”, “Slant type”, and “Pit type”. A change in the Reynolds number significantly altered the spatial Sherwood number distributions on the block’s surfaces. Besides, four correlations between the Reynolds number and the surface-averaged Sherwood number were presented for the front, top, side, and rear surfaces of the block at a given block’s height, for the purpose of practical applications.     

Uniformly valid asymptotic solutions of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate of four clamped edges with variable thickness

By using “the method of modified two-variable”,“the method of mixing perturbation ”and introducing four small parameters, the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied. And the uniformly valid asymptotic solution of Nth- order for ε1 and Mth- order for ε2 of the deflection functions and stress function are obtained.     

Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization

In this chapter of the special issue of the journal “Transport in Porous Media,” on the topic “Flow and transport above permeable domains,” we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has a solid mathematical background. The rigorous mathematical justification of the presented results is given in the corresponding articles of the authors. The plan is as follows: We start with the section “Introduction” where we give an overview and comparison with interface conditions obtained using other approaches. In Sect. 2, we give a very short derivation of the Darcy law by homogenization, using the two-scale expansion in the typical pore size parameter ε. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Sect. 3, we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly and show that it is valid at order O(ε ²). Next, in Sect. 4 we give a derivation of the Beavers–Joseph–Saffman interface condition and of the pressure jump condition, using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface law and being based on the rigorous mathematical result. In addition, we show the invariance of the law with respect to the small variations in the choice of the interface position. Finally, there is a short concluding section. The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN).     

Dynamic flight stability of hovering insects

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the “rigid body” assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the “rigid body” assumption is tested and how differences in size and wing kinematics influence the applicability of the “rigid body” assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the “rigid body” assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the “rigid body” assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M _u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z _w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode. The project supported by the National Natural Science Foundation of China (10232010 and 10472008).     

Free volume dependence of polymer viscosity

The Simha–Somcynsky (S–S) equation of state (eos) was used to compute the free volume parameter, h, from the pressure–volume–temperature (PVT) dependencies of eight molten polymers. The predicted by eos variation of h with T and P was confirmed by the positron annihilation lifetime spectroscopy; good agreement was found for h(P = constant) = h(T) as well as for h(T = constant) = h(P). Capillary shear viscosity (η) data of the same polymers (measured at three temperatures and six pressures up to 700 bars), were plotted as logη vs 1/h, the latter computed for T and P at which η was measured. In previous works, such a plot for solvents and silicone oils resulted in a “master curve” for the liquid, in a wide range of T and P. However, for molten polymers, no superposition of data onto a “master curve” could be found. The superposition could be obtained allowing the characteristic pressure reducing parameter, P*, to vary. The necessity for using a “rheological” characteristic pressure reducing parameter, P*_R = κP*, with κ = 1 to 2.1 indicates that the free volume parameter extracted from the thermodynamic equilibrium data may not fully describe the dynamic behavior. After eliminating possibility of other sources for the deviation, the most likely culprit seems to be the presence of structures in polymer melts at temperatures above the glass transition, T _g. For example, it was observed that for amorphous polymers at T ≅ 1.52T _g the factor κ = 1, and the deviation vanish.     

Isotherme und nicht-isotherme Spannungsrelaxation nach stationärer Scherströmung

Summary A review of the different possibilities for formulating viscoelastic models or theories is given. In steady shear flow such theories allow one to interrelate the various viscometric parameters for a given polymer.The relaxation time model proposed byBogue was chosen because of its relative simplicity. With this choice no independent parameter is introduced into the theory.In the original model an effective relaxation time, based on an integration of the strain rate history, was used. In the present work, a generalized averaging mode for the relaxation time is proposed to allow nonsteady deformation histories and non-isothermal temperature histories to be analysed. The advantage of the new mode becomes clear when either isothermal or non-isothermal stress relaxation following isothermal steady state flow is considered. The effect of the steady shear persists into the relaxation period even though no shear is being imposed then.The relaxation times and moduli for a high density polyethylene were determined and used to calculate the isothermal shear stress relaxation following cessation of steady state shear flow. The calculated results are in good agreement with the experimental data ofMenges and coworkers (50, 51).

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Zusammenfassung Es wird ein Überblick über die Struktur der viskoelastischen Modelltheorien vom Integral-Typ gegeben. Anhand der stationären einfachen Scherströmung wird gezeigt, daß sich alle viskosimetrischen Größen eines Kunststoffes im Rahmen einer solchen Theorie miteinander verknüpfen lassen.Zur Ausarbeitung eines Modells wählen wir wegen ihrer Einfachheit die Boguesche Relaxationszeit. Im Rahmen unserer Untersuchungen stellen wir fest, daß mit der Wahl dieser Relaxationszeit kein unabhängiger Parameter in die Modelltheorie eingeführt wird.Um auch instationäre Strömungsvorgänge analysieren zu können, wird eine neue mittlere Relaxationszeit definiert. Für isotherme Strömungen führt diese Mittelwertbildung zum selben Resultat wie mit einer gemittelten zweiten Invarianten des Deformationsgeschwindigkeitstensors. Der Vorteil dieser Mittelwertbildung zeigt sich deutlich bei der nicht-isothermen Spannungsrelaxation nach stationärer isothermer Scherströmung. In den dafür abgeleiteten Gleichungen ist auch weiterhin ein Einfluß der Deformationsgeschwindigkeit bzw. des Schergradienten enthalten.Schließlich werden noch Relaxationszeiten und -moduln eines Polyäthylens hoher Dichte bestimmt und daraus anschließend die isotherme Relaxation der Schubspannung nach einer stationären Scherströmung berechnet. Die erzielte Übereinstimmung mit den experimentellen Ergebnissen vonMenges und Mitarbeitern (50, 51) ist gut.

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Nomenclature a ₁,a ₂ Faktoren - a _T WLF-Faktor - b Parameter in der Bogueschen Relaxationszeit - c Cauchyscher Deformationstensor - c ^–1 Fingerscher Deformationstensor - d [s^–1] Deformationsgeschwindigkeitstensor - g _ij metrischer Tensor - i, l Summationsparameter - m ₁,m ₂ [Pa s^–1] Relaxationsfunktionen - t [s] Zeit - t _w [s] gewichtete Zeit - t,t,s [s] Integrationsparameter - C ₁ Konstante des WLF-Faktors - C ₂ [°C] Konstante des WLF-Faktors - E dimensionslose Schubspannung - G _i [Pa] Relaxationsmodul - H() [Pa] Relaxationsspektrum - N ₁,N ₂ [Pa] erste bzw. zweite Normalspannungsdifferenz - P [Pa] hydrostatischer Druck - T [°C] Temperatur - T ₀ [°C] Bezugstemperatur des WLF-Faktors - U [Pa s^–1] skalare Funktion - W [Pa] Verformungsenergie - Schergeschwindigkeit - [Pa s] Scherviskosität - ₁₂ [Pa] Schubspannung - ₁₁, ₂₂, ₃₃ [Pa] Normalspannungen in Richtung der drei RaumkoordinatenX ₁,X ₂,X ₃ - _i [s] Relaxationszeit - _i ⁰ [s] Relaxationszeit bei Bezugstemperatur - _ieff [s] effektive Relaxationszeit - mittlere Relaxationszeit - I,II,III Invarianten des Fingerschen Deformationstensors - I _d [s^–1] Invarianten des Deformationsgeschwindigkeitstensors - II _d [s^–2] Invarianten des Deformationsgeschwindigkeitstensors - III _d [s^–3] Invarianten des Deformationsgeschwindigkeitstensors Mit 6 Abbildungen und 1 Tabelle     

Influence of elastic material compressibility on parameters of an expanding spherical stress wave

We investigated the influence of elastic material compressibility on parameters of an expanding spherical stress wave. The material compressibility is represented by Poisson’s ratio, ν, in this paper. The stress wave is generated by a pressure produced inside a spherical cavity surrounded by the isotropic elastic material. The analytical closed form formulae determining the dynamic state of the mechanical parameters (displacement, particle velocity, strains, stresses, and material density) in the material have been derived. These formulae were obtained for surge pressure p(t) = p ₀ = const inside the cavity. From analysis of these formulae, it is shown that the Poisson’s ratio substantially influences the course of material parameters in space and time. All parameters intensively decrease in space together with an increase of the Lagrangian coordinate, r. On the contrary, these parameters oscillate versus time around their static values. These oscillations decay in the course of time. We can mark out two ranges of parameter ν values in which vibrations of the parameters are “damped” at a different rate. Thus, Poisson’s ratio in the range below about 0.4 causes intense decay of parameter oscillations. On the other hand in the range 0.4 < ν < 0.5, i.e. in quasi-incompressible materials, the “damping” of parameter vibrations is very low. In the limiting case when ν = 0.5, i.e. in the incompressible material, “damping” vanishes, and the parameters harmonically oscillate around their static values. The abnormal behaviour of the material occurs in the range 0.4 < ν < 0.5. In this case, an insignificant increase of Poisson’s ratio causes a considerable increase of the parameter vibration amplitude and decrease of vibration “damping”.      

Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

A Near-Asymptotics analysis of the turbulence energy spectrum is presented that accounts for the effects of finite Reynolds number recently reported by Mydlarski and Warhaft [21]. From dimensional and physical considerations (following Kolmogorov and von Karman), proper scalings are defined for both low and high wavenumbers, but with functions describing the entire range of the spectrum. The scaling for low wavenumbers uses the kinetic energy and the integral scale, L, based on the integral of the correlation function. The fact that the two scaled profiles describe the entire spectrum for finite values of Reynolds number, but reduce to different profiles in the limit, is used to determine their functional forms in the “overlap” region that both retain in the limit. The spectra in the overlap follow a power law, E(k) =Ck _{−5/3 + μ}, where μ and C are Reynolds number dependent. In the limit of infinite Reynolds number, μ → 0 and C → constant, so the Kolmogorov/Obukhov theory is recovered in the limit. Explicit expressions for μ and the other parameters are obtained, and these are compared to the Mydlarski/Warhaft data. To get a better estimate of the exponent from the experimental data, existing models for low and high wavenumbers are modified to account for the Reynolds number dependence. They are then used to build a spectral model covering all the range of wavenumbers at every Reynolds number. Experimental data from grid-generated turbulence are examined and found to be in good agreement with the theory and the model. Finally, from the theory and data, an explicit form for the Reynolds number dependence of φ = ɛL/u ₃ is obtained. This revised version was published online in July 2006 with corrections to the Cover Date.