Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model

Linear viscoelastic (LVE) measurements as well as non-linear elongation measurements have been performed on stoichiometrically imbalanced polymeric networks to gain insight into the structural influence on the rheological response (Jensen et al., Rheol Acta 49(1):1–13, 2010). In particular, we seek knowledge about the effect of dangling ends and soluble structures. To interpret our recent experimental results, we exploit a molecular model that can predict LVE data and non-linear stress–strain data. The slip-link model has proven to be a robust tool for both LVE and non-linear stress–strain predictions for linear chains (Khaliullin and Schieber, Phys Rev Lett 100(18):188302–188304, 2008, Macromolecules 42(19):7504–7517, 2009; Schieber, J Chem Phys 118(11):5162–5166, 2003), and it is thus used to analyze the experimental results. Initially, we consider a stoichiometrically balanced network, i.e., all strands in the ensemble are attached to the network in both ends. Next we add dangling strands to the network representing the stoichiometric imbalance, or imperfections during curing. By considering monodisperse network strands without dangling ends, we find that the relative low-frequency plateau, G₀/G_N⁰G_0/G_N^0, decreases linearly with the average number of entanglements. The decrease from G_N⁰G_N^0 to G ₀ is a result of monomer fluctuations between entanglements, which is similar to “longitudinal modes” in tube theory. It is found that the slope of G′ is dependent on the fraction of network strands and the structural distribution of the network. The power-law behavior of G ^″ is not yet captured quantitatively by the model, but our results suggest that it is a result of polydisperse dangling and soluble structures.     

Rheological behavior of poly(methyl methacrylate)/polystyrene (PMMA/PS) blends with the addition of PMMA-<Emphasis Type="BoldItalic">ran</Emphasis>-PS

In this work, the dynamic behavior of poly(methyl methacrylate)/polystyrene blend to which P(S_0.5-ran-MMA_0.5) was added was studied. Several blend (ranging from 5 to 20 wt% of dispersed phase) and copolymer (up to 20 wt% with respect to dispersed phase) concentrations were studied. The rheological behavior of the blends was compared to Bousmina’s (Rheol Acta 38:73–83, 1999) and Palierne’s (Rheol Acta 29:204–214, 1990) generalized models. The relaxation spectra of the blends were also inferred, and the results were analyzed in light of the analysis of Jacobs et al. [J Rheol 43:1495–1509, 1999]. The relaxation spectra of the blends with smaller dispersed phase (below 10 wt%) and larger copolymer concentrations (above 0.4 wt%) showed the presence of four relaxation times, two corresponding to the blend phases, τ _F , corresponding to the relaxation of the shape of the dispersed phase of the blend and that can be attributed to the relaxation of Marangoni stresses tangential to the interface between the dispersed phase and matrix. The experimental values of and were used to infer the interfacial tension (Γ) and the interfacial complex shear modulus (β) for the different blends, Γ decreased with increasing copolymer concentration. β decreased with increasing blend dispersed phase concentration and decreasing copolymer concentration. The predictions of Palierne’s generalized model were found to corroborate the experimental data once the values of Γ and β, found analyzing the relaxation spectra, were used in the calculations. Bousmina’s model was found to corroborate the data only for larger dispersed phase concentration. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Relaxation patterns of long,linear, flexible,monodisperse polymers: BSW spectrum revisited

Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation times near τ/τ _max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,” and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit solutions for the dynamic moduli, G ^′(ω), G ^″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations, analysis of linear viscoelastic data allows the direct determination of BSW parameters.     

Conversion of oscillatory shear data from highly filled polymers by direct numerical integration

A procedure was developed to enable the direct numerical integration of the Fourier integral transform equation relating G(t) to G() by considering integration limits that vary as a function of time and which define a range of discrete sub-intervals within the complete frequency domain data set. The method provides results that are in very close agreement to results determined from a relaxation spectrum. However, at low values of time the solution to the variable limit integral transform is sensitive to the absence of a contribution beyond the upper experimental limit of the frequency domain data. G(t) results determined from the conversion of shifted master G() experimental data using the variable limit integral transformation and the relaxation spectrum compared favourably with actual shifted master G(t) experimental data. The former curves were characterised by the same form and trend as the experimental results, confirming that the underlying viscoelastic behaviour is well represented. While the variable limit Fourier integral transform procedure provides a good approximation to relaxation spectrum results, the latter is clearly the more robust method of converting data from the frequency to the time domain.It was observed that the time-temperature superposition procedure used in the construction of shifted master curves can magnify potential differences between the shifted G(t) values determined from the conversion of G() data and the actual experimental G(t) results, when compared to data that has not been shifted to a master curve.     

The development of rheometry for strain-sensitive gelling systems and its application in a study of fibrin–thrombin gel formation

This paper reports simulated sequential frequency sweep data which have been reconstructed from time resolved viscoelastic data obtained by Fourier transform mechanical spectroscopy. Comparisons of the results show that the recording of anomalous values of the stress relaxation power law exponent α at the Gel Point under ‘rapid’ gelling conditions may be due to inappropriate rheological techniques. An appropriate rheometrical criterion is established for the application of sequential frequency sweeps in order to obtain accurate values of α in the formation of strain-sensitive, rapidly formed gels. Furthermore, using appropriate rheometry, we report values of α for fibrin–thrombin gels formed by the addition of thrombin to a physiologically relevant level of human fibrinogen, and relate these values to the microstructure of the fibrin gel network in terms of a fractal dimension. The present study is the first to report a modification of the fractal characteristics of incipient clots in fibrin–thrombin gels due to the availability of thrombin. This work confirms the hypothesis that the self-similar (fractal) stress relaxation behaviour recorded at the Gel Point of samples of coagulating blood (Evans et al. 2010a, b) is associated with the microstructural characteristics of the incipient blood clot’s fibrin network.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Characterizing developing adverse pressure gradient flows subject to surface roughness

An experimental study was conducted to examine the effects of surface roughness and adverse pressure gradient (APG) on the development of a turbulent boundary layer. Hot-wire anemometry measurements were carried out using single and X-wire probes in all regions of a developing APG flow in an open return wind tunnel test section. The same experimental conditions (i.e., T _∞, U _ref, and C _p) were maintained for smooth, k ⁺ = 0, and rough, k ⁺ = 41–60, surfaces with Reynolds number based on momentum thickness, 3,000 < Re _θ < 40,000. The experiment was carefully designed such that the x-dependence in the flow field was known. Despite this fact, only a very small region of the boundary layer showed a balance of the various terms in the integrated boundary layer equation. The skin friction computed from this technique showed up to a 58% increase due to the surface roughness. Various equilibrium parameters were studied and the effect of roughness was investigated. The generated flow was not in equilibrium according to the Clauser (J Aero Sci 21:91–108, 1954) definition due to its developing nature. After a development region, the flow reached the equilibrium condition as defined by Castillo and George (2001), where Λ = const, is the pressure gradient parameter. Moreover, it was found that this equilibrium condition can be used to classify developing APG flows. Furthermore, the Zagarola and Smits (J Fluid Mech 373:33–79, 1998a) scaling of the mean velocity deficit, U _∞δ*/δ, can also be used as a criteria to classify developing APG flows which supports the equilibrium condition of Castillo and George (2001). With this information a ‘full APG region’ was defined.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Development and characterization of a variable turbulence generation system

Experimental turbulent combustion studies require systems that can simulate the turbulence intensities [u′/U ₀ ~ 20–30% (Koutmos and McGuirk in Exp Fluids 7(5):344–354, 1989)] and operating conditions of real systems. Furthermore, it is important to have systems where turbulence intensity can be varied independently of mean flow velocity, as quantities such as turbulent flame speed and turbulent flame brush thickness exhibit complex and not yet fully understood dependencies upon both U ₀ and u′. Finally, high pressure operation in a highly pre-heated environment requires systems that can be sealed, withstand high gas temperatures, and have remotely variable turbulence intensity that does not require system shut down and disassembly. This paper describes the development and characterization of a variable turbulence generation system for turbulent combustion studies. The system is capable of a wide range of turbulence intensities (10–30%) and turbulent Reynolds numbers (140–2,200) over a range of flow velocities. An important aspect of this system is the ability to vary the turbulence intensity remotely, without changing the mean flow velocity. This system is similar to the turbulence generators described by Videto and Santavicca (Combust Sci Technol 76(1):159–164, 1991) and Coppola and Gomez (Exp Therm Fluid Sci 33(7):1037–1048, 2009), where variable blockage ratio slots are located upstream of a contoured nozzle. Vortical structures from the slots impinge on the walls of the contoured nozzle to produce fine-scale turbulence. The flow field was characterized for two nozzle diameters using three-component Laser Doppler velocimetry (LDV) and hotwire anemometry for mean flow velocities from 4 to 50 m/s. This paper describes the key design features of the system, as well as the variation of mean and RMS velocity, integral length scales, and spectra with nozzle diameter, flow velocity, and turbulence generator blockage ratio.     

Statistically Steady Incompressible DNS to Validate a New Correlation for Turbulent Burning Velocity in Turbulent Premixed Combustion

Incompressible 3-D DNS is performed in non-decaying turbulence with single step chemistry to validate a new analytical expression for turbulent burning velocity. The proposed expression is given as a sum of laminar and turbulent contributions, the latter of which is given as a product of turbulent diffusivity in unburned gas and inverse scale of wrinkling at the leading edge. The bending behavior of U _T at higher u′ was successfully reproduced by the proposed expression. It is due to decrease in the inverse scale of wrinkling at the leading edge, which is related with an asymmetric profile of FSD with increasing u′. Good agreement is achieved between the analytical expression and the turbulent burning velocities from DNS throughout the wrinkled, corrugated and thin reaction zone regimes. Results show consistent behavior with most experimental correlations in literature including those by Bradley et al. (Philos Trans R Soc Lond A 338:359–387, 1992), Peters (J Fluid Mech 384:107–132, 1999) and Lipatnikov et al. (Progr Energ Combust Sci 28:1–74, 2002).     

Mass transport effect on the heat transfer coefficient during boiling of multicomponent mixture

In this work a simplified calculation method taking into account the effect of mass transport on the heat transfer coefficient (HTC) during boiling of multicomponent mixture has been elaborated. The calculation results were compared with own experimental data for ternary system methanol–isopropanol–water and Grigoriev data [1] (acetone–methanol–water). The experiments were performed in different hydrodynamic conditions such as: pool boiling and liquid evaporation at the free surface of the falling film. The experimental data covered wide range of heat fluxes from 6 to 30 kW/m² in the case of liquid evaporation from the falling film and from 30 to 240 kW/m² for pool boiling. The analysis of the results indicates that the mass transfer resistance in the liquid phase caused a significant reduction of experimental value HTC in comparison to so-called ideal HTC.     

Relaxation time spectrum molecular weight distribution relationships

Single exponential decay relationships, which define the molecular weight distribution (MWD) of a polymer as a function of the polymer’s relaxation time spectrum (RTS), have been derived by Wu (Polym Eng Sci 28:538–543, 1988) and Thimm et al. (J Rheol 43:1663–1672, 1999). Experimental validation studies with monodisperse polymers, with quite precisely known MWDs, have been used to test their reliability. It has been established that neither formula is always able to accurately recover the MWDs of monodisperse polymers from their experimentally determined RTS. In this paper, different and more general relationships, based on theoretical results of Anderssen and Loy (Bull Aust Math Soc 65:449–460, 2002a) for decays of the form , where the derivative of θ(t) is a completely monotone function, are derived, analyzed, and applied. It is shown how to transform these general relationships to equivalent single exponential decay relationships for which Laplace transform solutions are derived. In order to illustrate the interrelationship between an RTS and its corresponding MWD, an explicit analytic solution is given. The paper concludes with a discussion of the rheological implications for the BSW model.     

A shear rheometer for measuring shear stress and both normal stress differences in polymer melts simultaneously: the MTR 25

The MTR 25 is a multitask rheometer (for shear and squeeze flow) with 25 kg of normal force and a partitioned plate. Torque and normal force are measured at both, the inner disk and the outer ring of the plate. The first and second normal stress differences can be determined from a single test. The axial stiffness is high (10⁷ N/m) by using rigid springs and strain gauges for the load cell. Monodisperse polystyrene (M _w = 206 kg/mol, 180°C) has been sheared in the range from 0.05 to 47 s^− 1. The viscosity and first normal stress difference are highly reproducible. The second normal stress difference scatters and mirrors the instability at the rim. A critical comparison is made between the MTR 25 method and the single transducer evaluation method (RMS 800 method, Schweizer, Rheol Acta 41:337–344, 2002): Both yield excellent and coinciding viscosity and first normal stress difference data. The RMS 800 method gives more stable second normal stress difference data, since the normal force from the outer ring, which is influenced by edge fracture, is not used. Data for the RMS 800 method can be acquired on the MTR 25. The high normal force capacity permits larger samples and higher shear rates than on the RMS 800.

The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer

We show that, in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable. The crucial point here is that the boundary is not flat. In fact (see Beir?o da Veiga et al. in J Math Anal Appl 377:216–227, 2011) if  W = \mathbb R³₊,{\,\Omega = \mathbb R^3_+,} convergence holds in C([0,T]; W^k,p(\mathbb R³₊)){C([0,T]; W^{k,p}(\mathbb R^3_+))}, for arbitrarily large k and p. For this reason, the negative answer given here was not expected.     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

Convergence Rates in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> for Elliptic Homogenization Problems

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems {L_e}{\{\mathcal{L}_\varepsilon\}} with rapidly oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_e}{\{\mathcal{L}_\varepsilon\}} . Most of our results, which rely on the recently established uniform estimates for the L ² Dirichlet and Neumann problems in Kenig and Shen (Math Ann 350:867–917, 2011; Commun Pure Appl Math 64:1–44, 2011) are new even for smooth domains.     

Numerical simulation of fully developed turbulent flow and heat transfer in annular-sector ducts

 The mixing length theory is employed to simulate the fully developed turbulent heat transfer in annular-sector ducts with five apex angles (θ₀=18,20,24,30,40^∘) and four radius ratios (R _o/R _i=2,3,4,5). The Reynolds number range is 10⁴10⁵. The numerical results agree well with an available correlation which was obtained in following parameter range: θ₀=18,20,24,30,40^∘, R _o/R _i=4 and Re=10⁴5×10⁴. The present work demonstrates that the application range of the correlation can be much extended. Apart from the mixing length theory, the kɛ model with wall function and the Reynolds stress model are also employed. None of the friction factor results predicted by the three models agrees well with the test data. For the heat transfer prediction the mixing length theory seems the best for the cases studied. Received on 17 July 2000 / Published online: 29 November 2001     

The relaxation of polymers with linear flexible chains of uniform length

The analysis of dynamic mechanical data indicates that linear flexible polymer chains of uniform length follow a scaling relation during their relaxation, having a linear viscoelastic relaxation spectrum of the formH() = n ₁ G _N ⁰ × (/ _max) ⁿ¹ for _max. Data are well represented with a scaling exponent of about 0.22 for polystyrene and 0.42 for polybutadiene. The plateau modulusG _N ⁰ is a material-specific constant and the longest relaxation time depends on the molecular weight in the expected way. At high frequencies, the scaling behavior is masked by the transition to the glassy response. Surprisingly, this transition seems to follow a Chambon-Winter spectrumH() = C^–n2, which was previously adopted for describing other liquid/solid transitions. The analysis shows that the Rouse spectrum is most suitable for low molecular-weight polymersM M _c , and that the de Gennes-Doi-Edwards spectrum clearly predicts terminal relaxation, but deviates from the observed behavior in the plateau region.Dedicated to Prof. Richard S. Stein on the occasion of his 65th birthday.On sabbatical leave from the University of Linz, Austria.     

Influence of molecular parameters on the stress dependence of viscous and elastic properties of polypropylene melts in shear

The stress dependencies of the steady-state viscosity η and, particularly, that of the steady-state elastic compliance J _e of various linear isotactic polypropylenes (PP) and one long-chain branched PP are investigated using creep-recovery tests. The creep stresses applied range from 2 to 10,000 Pa. In order to discuss the stress-dependent viscosity η and elastic compliance J _e with respect to the influence of the weight average molar mass M _w and the polydispersity factor M _w/M _n the PP are characterized by SEC–MALLS. For the linear PP, linear steady-state elastic compliances J_e⁰J_{\rm e}^0 in the range of 10^− 5–10^− 3 Pa^− 1 are obtained depending on the molar mass distribution. J_e⁰J_{\rm e}^0 of the LCB-PP is distinctly higher and comes to lie at around 10^− 2 Pa^− 1. J_e⁰J_{\rm e}^0 is found to be independent of M _w but strongly dependent on polydispersity. η and J _e decrease with increasing stress. For the linear PP, J _e as a function of the stress τ is temperature independent. The higher M _w/M _n the stronger is the shear thinning of η and the more pronounced is the stress dependence of J _e. For the LCB-PP, the strongest stress dependence of η and J _e is observed. Furthermore, for all PP J _e reacts more sensitively to an increasing stress than η. A qualitative explanation for the stronger stress dependence of J _e compared to η is given by analyzing the contribution of long relaxation times to the viscosity and elasticity.