Analytic derivation of the Cox�CMerz rule using the MLD ��toy�� model for polydisperse linear polymers

A general constitutive formalism, the ??na?ve?? polydisperse MLD model, has been developed by Mead et al. (Macromolecules 31:7895?C7914, 1998) and Mead (Rheol Acta 46:369?C395, 2007) at both the tube coordinate level and the mathematically simplified ??toy?? level independent of the tube coordinate. The model includes constraint release generated by convection-driven chain retraction (which is equivalent to ??convective constraint release?? (CCR)), reptation, and tube contour length fluctuations. The properties of the mathematically simplified na?ve polydisperse ??toy?? MLD model are explored in linear and nonlinear steady shear flows where we analytically derive the Cox?CMerz rule relating the steady shear viscosity to the modulus of the linear viscoelastic dynamic viscosity. The Cox?CMerz rule relating the linear viscoelastic material properties and the nonlinear material properties is shown to be a direct consequence of convective constraint release. The specific feature of CCR that leads to this result is that the relaxation rate due to convective constraint release is proportional to the shear rate, $\dot{{\gamma }}$ , independent of molecular weight. The viability of this well-known empirical relationship is a direct consequence of a coincidence in the mathematical structure of the linear viscoelastic material properties and convective constraint release. There is no physical analogy or relationship between the molecular relaxation mechanisms operative in linear (diffusive relaxation) and nonlinear (convective relaxation) flow regimes. The polydisperse MLD model predictions of the individual molecular weight component contributions to the flow curve, and interpretations thereof, are effectively identical to those first postulated by Bersted (J Appl Polym Sci 19:2167?C2177, 1975, J Appl Polym Sci 20:2705?C2714, 1976). Following the theoretical developments, a limited experimental study is executed with a commercial polydisperse polystyrene melt. Nearly quantitative agreement between the polydisperse MLD theory and experimental measurements of steady-shear viscosity and dynamic moduli is achieved over a wide range of shear rates.     

On the Motive Power of Chemical Transformations in?Open Systems

The classical basic concepts of cyclic processes and the efficiency of heat engines are used here to conjecture about the laws of thermodynamics for open systems that can exchange matter with a surrounding environment. An ideal chemomechanical elastic bar is envisioned that changes its stiffness while undergoing a chemical transformation which is, in turn, influenced by the axial strain of the bar. Stable equilibrium states are identified as minimizers of the total energy, which is assumed to be nonconvex in type. If the bar is loaded and then alternatively placed in environments at chemical potentials either ?? _i or ?? _s >?? _i , a reversible cycle analogous to the classical Carnot cycle may be traced. In this case, the environmental ??chemical potential?? plays the role of the temperature and the ??chemical work?? the role of heat. For the system, the main form of interaction with the exterior, other than mechanical work, is the exchange of mass of a component at different environmental chemical potentials. It is then possible to obtain an elementary theory of chemical engines in which efficiency estimates (in terms of environmental chemical potentials) and related pertinent issues can be discussed. This model may serve as a basis for analyzing coupled chemo-mechanical processes occurring in materials such as ionized gels for possible applications as actuators, and to interpret complex phenomena in biological systems, such as the chemical kinetics of smooth muscles.     

On the selective local stabilization of the mixed Q1–P0 element

In this paper we study the stability and performance of the quadrilateral finite element ??₁–??₀ (bili‐ near/constant) for the Stokes equations. We set up a framework to show the stability of the element for a wide range of meshes with macroelement patches. We apply the new theory to show the stability of ??₁–??₀ elements on some previously studied meshes and on some newly suggested meshes. Nevertheless such earlier and newly suggested meshes are not effective in practice, compared to the traditional unstable meshes for the ??₁–??₀ element. The new theory leads naturally to a general idea in treating instability of square ??₁–??₀ elements by the local stabilization on macroelement patches of larger, but fixed sizes. The good performance of the traditional ??₁–??₀ square elements with filtering can be kept in some cases after the local stabilization. Some numerical tests are provided to support the theory and to show the performance of stabilized ??₁–??₀ elements. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability and approximability of the 𝒫1–𝒫0 element for Stokes equations

In this paper we study the stability and approximability of the ??₁–??₀ element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the ??₁–??₀ element (bilinear/constant on quadrilaterals) and the ??₁–??₀ element. We apply many stability results on the ??₁–??₀ element to the analysis of the ??₁–??₀ element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families. As a byproduct, we also obtain a basis of divergence‐free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence‐free, ??₁ finite element spaces in computation. Copyright © 2006 John Wiley & Sons, Ltd.     

The Mesoscopic Eulerian Approach for Evaporating Droplets Interacting with Turbulent Flows

This paper presents a statistical approach, known as mesoscopic Eulerian formalism (Février et al., J Fluid Mech 533:1?C46, 2005), which is extended in order to model a cloud of inertial evaporating droplets interacting with a turbulent carrier flow. This approach is checked in a non-isothermal droplet-laden turbulent planar jet by means of a priori tests. The ??measurement?? of the mesoscopic particle-velocity and particle-temperature moments is accomplished by using the Eulerian particle fields computed from a Direct Numerical Simulation (DNS) coupled with a Lagrangian approach for the droplets. The results of this work show the ability of such an approach to describe the evaporating dispersed phase interacting with turbulent flows.     

L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

We study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t ?? ??, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy??s law. In this paper, we prove that any L ^?? weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural L ¹ topology with decay rates to the Barenblatt profile of the porous medium equation. The density function tends to the Barenblatt solution of the porous medium equation while the momentum is described by Darcy??s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.     

Stress-Dependent Elastic Behaviour of a Titanium Alloy at Elevated Temperatures

In this study the influence of stress and temperature on the elastic modulus during fully-reversed low cycle fatigue of the titanium alloy Ti6242 is examined. The change of the elastic properties with stress manifests itself in a crescent-like shaped hysteresis loop of stress vs. plastic strain at very low amplitudes, i.e. below the technical yield stress. A quadratic extension of Hooke??s law with a second constant ??k?? is applied. The parameters are determined all along the unloading curve in tension and compression. The approach results in the alignment of the hysteresis loop so that they become vertical, i.e. the elastic strain is accurately described. The value and sign of ??k?? depend on whether the deformation occurs in tension or compression. Like the Young??s modulus E ₀, ??k?? also depends on temperature. At temperatures up to 550°C the values of ??k?? in tension and compression do not change during fatigue life. However, at 650°C thermally activated slip processes lead to changes of both, E ₀ and ??k??.     

Characterization of Anisotropic Polymeric Foam Under Static and Dynamic Loading

An orthotropic polymeric foam with transverse isotropy (Divinycell H250) used in composite sandwich structures was characterized at various strain rates. Uniaxial experiments were conducted along principal material axes as well as along off-axis directions under tension, compression, and shear to determine engineering constants, such as Young??s and shear moduli. Uniaxial strain experiments were conducted to determine mathematical stiffness constants, i. e., C _ij . An optimum specimen aspect ratio for these tests was selected by means of finite element analysis. Quasi-static and intermediate strain rate tests were conducted in a servo-hydraulic testing machine. High strain rate tests were conducted using a split Hopkinson Pressure Bar system built for the purpose using polymeric (polycarbonate) bars. The polycarbonate material has an impedance that is closer to that of foam than metals and results in lower noise to signal ratios and longer loading pulses. It was determined by analysis and verified experimentally that the loading pulses applied, propagated along the polycarbonate rods at nearly constant phase velocity with very low attenuation and dispersion. Material properties of the foam were obtained at three strain rates, quasi-static (10^?4 s^?1), intermediate (1 s^?1), and high (10³ s^?1) strain rates. A simple model proposed for the Young??s modulus of the foam was in very good agreement with the present and published experimental results.     

Turbulence Models and Large Eddy Simulations Applied to Ascending Mixed Convection Flows

Coolant flows in the cores of current gas-cooled nuclear reactors consist of ascending vertical flows in a large number of parallel passages. Under post-trip conditions such heated turbulent flows may be significantly modified from the forced convection condition by the action of buoyancy, and the thermal-hydraulic regime is no longer one of pure forced convection. These modifications are associated primarily with changes to the turbulence structure. Flow laminarization may occur, and in that event heat transfer rates may be as low as 40% of those in the corresponding forced convection case. The present work is concerned with the modelling of such ??mixed?? convection flows in a vertical heated pipe. All fluid properties are assumed to be constant and buoyancy is accounted for within the Boussinesq approximation. Six different Eddy Viscosity Models (EVMs) are examined against experimental measurements and the direct numerical simulation (DNS) data of You et?al. (Int J Heat Mass Transfer 46:1613?C1627, 2003). The EVMs selected for study embody distinct physical refinements with respect to the parent high-Reynolds-number k-?? model. Large Eddy Simulations employing the classical Smagorinsky sub-grid-scale model are also presented. It is found that the early EVM scheme of Launder and Sharma (1974) is the turbulence model in the closest agreement with direct simulation results for the ratio of mixed-to-forced convection Nusselt number, Nu/Nu ₀; the model in the poorest accord with the DNS data is a k-??-SST formulation. However, in relation to comparisons with both numerical and experimental data for forced convection Nusselt number, Nu ₀, the present work reveals that some of the more recent models perform better than the Launder-Sharma scheme. No single scheme is in consistently close agreement with the numerical simulation flow profiles.     

The Alternative Model of Water Vapour Sorption in Porous Building Materials

Mathematical modelling of water sorption in porous building materials is considered. The explanations of inadequacies of both Brunauer?CDeming?CDeming?CTeller model for the high relative humidities range and Frenkel?CHalsey?CHill model for the low relative humidities range reported by Pavlík et?al. (Transp. Porous Med. 91:939?C954, 2012) are proposed. The generalized D??Arcy and Watt (GDW) model is proposed as a simpler alternative for a procedure of experimental isotherms fitting proposed by Pavlík et?al. The suitability of the GDW equation to describe the water sorption isotherms in the building materials for the whole range of relative humidities is confirmed.     

Spacetime Interpretation of Torsion in Prismatic Bodies

A non-linear theory for the plastic deformation of prismatic bodies is constructed which interpolates between Prandtl??s linear soap-film approximation and Nádai??s sand-pile model. Geometrically Prandtl??s soap film and Nádai??s wavefront are unified into a single smooth surface of constant mean curvature in three-dimensional Minkowski spacetime.     

Variational Convergences of Dual Energy Functionals for Elastic Materials with a ε-Thin Strong Inclusion

We give a new derivation, based on the complementary energy formulation, of a simplified model for a multi-structure made up of two anisotropic hyper-elastic bodies connected by a thin strong material layer. The model is obtained by identifying the Mosco-limit of the stored complementary energy functional when the thickness is of order ?? and the stiffness of order 1/?? where ?? is a positive real adimensional parameter. In order to prove the existence of the displacement associated with the stress we use a suitable weak version of the Saint-Venant compatibility condition also known as Donati??s theorem.     

Large Eddy Simulation of Turbulent Convective Cavity Flow

The large eddy simulation model with Smagorinsky subgrid-scale model was applied to two-dimensional turbulent convective cavity flow. The Reynolds number is lying from 1×10⁴ to 4×10⁵ and Archimedes number from 0 to 0.4. The simulation results were compared with the k?? model results and experimental results wherever possible. Flow results were in good agreement with experimental data across the mid-planes. Effects of Smagorinsky constant and grid resolution were investigated.     

Nonlinear model identification for Artemia population motion

In this paper, two different nonlinear models for Artemia swarming are derived. In order to generate the data suitable for identification, a robot driving the Artemia population has been built. The obtained data have been then used to identify the parameters of a model based on Newton??s equations and a black-box NARX model implemented by neural networks. The performances obtained validate the physical hypotheses underlying the gray-box model.     

Nonlinear Weakly Curved Rod by Γ-Convergence

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use an approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity. We do not impose any constitutional behavior of the material and work in a general framework. To derive the model, by means of ??-convergence, we need to set the order of strain energy (i.e., its relation to the thickness of the body h). We analyze the situation when the strain energy (divided by the order of volume) is of the order h ⁴. This is the same approach as the one used in F?ppl-von Kármán model for plates and the analogous model for rods. The obtained model is analogous to Marguerre-von Kármán for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.     

Normal stress differences in large-amplitude oscillatory shear flow for the corotational ��ANSR�� model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant ?? ₀ controlling the ratio ??₂/??₁ for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.     

Linear and non-linear rheology of linear polydisperse polyethylene

Rheological techniques, size-exclusion chromatography, and molecular spectroscopy are the most widely used tools for describing polymer molecular structure in polyolefins. The detection of long-chain branching, and to some extent, its quantification, have been based on quantifying the deviation of polyethylene??s (PE) rheological behavior from that of a linear reference. Although metallocene-based PE has been extensively studied, linear polydisperse originating from Ziegler or Chromium-based catalysts are not often thoroughly considered, despite their high industrial importance. Within this work, we study the linear and non-linear rheology of a set of polydisperse PEs, for which the topological linearity is confirmed by GPC-MALLS measurements. Thus, we can safely quantify the effect of broad molecular weight distribution, high and ultra-high molecular weight fractions on rheological quantities and model parameters. Specifically, the zero-shear viscosity, ?? ₀ vs. M _w, relaxation spectra, phase lag vs. the complex modulus plot (van Gurp?CPalmen method) were applied and significant deviations from the ??rheologically linear?? behavior were observed, attributed only to M _w, M _z and polydispersity. Since the elongational viscosity was typical of linear PE, large-amplitude oscillatory shear and FT-Rheology were applied to quantify the non-linear rheological behavior. The latter was described by a single parameter, $Q=I_{3/1}/\gamma_0^2$ , which for linear polydisperse PE was correlated to the high molecular weight fraction and was constant over a broad range of applied Deborah numbers for the respective excitation frequencies. Since we need to correlate structural features such as broad MWD and HMW to polymer performance under processing conditions, we have to extend the analysis of linear rheological parameters, such as zero-shear viscosity, to non-linear parameters, e.g., the Q parameter quantified and used here.     

Renormalization of Planar Analytic Saddles

Let ${X = \lambda_{1} x_{1}{\frac{\partial}{\partial {x_1}}} + \lambda_2 x_2 \frac{\partial}{\partial {x_2}} + O(|x|^2)}$ be an analytic vector field near x = 0. We suppose that the linear part of this vector field has real eigenvalues ??₁, ??₂ and that the ratio ${\eta = -\frac{\lambda_1}{\lambda_2}}$ is a positive irrational number. In a previous paper of the first author and P. De Maesschalck, it was shown that any analytic saddle can be conjugated analytically to a form ??as close as desired?? to the formal normal form. In this paper we will iterate and renormalize these conjugacies. The iteration of this procedure will be strongly connected to the diophantine properties of ?? and we will establish the convergence of this process. A consequence of this convergence will be the two dimensional version of the by now classical linearization theorem of Bruno.     

Selection of NARX models estimated using weighted least squares method via GIC-based method and l 1-norm regularization methods

We investigate the model selection problem for nonlinear autoregressive with exogenous variables models estimated using the weighted least squares (WLS) method. Because WLS changes the statistical property of data under study and violates the assumptions imposed on the well-developed model evaluation and selection methods (e.g. Akaike??s information criterion, Schwarz??s Bayesian information criterion, and the error reduction ratio based methods), therefore, new approaches should be investigated. In this research, an information criterion based method and two l ₁-norm regularization methods are taken into consideration: (a) in the former method, for models estimated using WLS, we first derive an information criterion in terms of the generalized information criterion (GIC, proposed by Konishi and Kitagawa in Biometrica 83(4):875?C890, 1996), which is a theoretical framework for the analysis and extension of information criteria via a statistical functional approach. Then we develop a robust selection procedure by combining the GIC-based forward stepwise method with Subsampling; (b)?in the latter two methods, we employ the l ₁-norm regularization methods, including Lasso and adaptive Lasso, to select models estimated with WLS. Finally, a numerical example is given to test and compare the performance of the three methods.