Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, b o [(h)\tilde]_s/([(h)\tilde]_s +[(h)\tilde]_p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where [(h)\tilde]_p\widetilde{\eta}_p and [(h)\tilde]_s\widetilde{\eta}_s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, [(R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, [(H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=[(l)\tilde][(U)\tilde]/[(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=[(r)\tilde][(g)\tilde][(H)\tilde]²/([(h)\tilde]_p +[(h)\tilde]_s )[(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on [(H)\tilde]\tilde{H} and the average film velocity, [(U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain [(H)\tilde]\tilde{H} a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ.     

In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Shear and elongational flow behavior of acrylic thickener solutions

We investigate the effect of hydrophobic aggregation in alkali-swellable acrylic thickener solutions on shear and extensional flow properties at technically relevant polymer concentrations using the commercial thickener Sterocoll FD as model system. Apparent molecular weight of aggregates in water is M _w  ≈ 10⁸ g/mol and decreases by more than an order of magnitude in ethanol. Zero shear viscosity η ₀ is low and shear thinning is weak compared to the high molecular weight of the aggregates. Linear viscoelastic relaxation is described by the Zimm theory up to frequencies of 10⁴ rad/s, demonstrating that no entanglements are present in these solutions. This is further supported by the concentration dependence of η ₀ and is attributed to strong association within the aggregates. Extensional flow behavior is characterized using the capillary break-up extensional rheometry technique including high-speed imaging. Solutions with ϕ ≥ 1% undergo uniform deformation and show pronounced strain hardening up to large Hencky strains. Elongational relaxation times are more than one order of magnitude lower than the longest shear relaxation times, suggesting that aggregates cannot withstand strong flows and do not contribute to the elongational viscosity.

Group classification of one-dimensional nonisentropic equations of fluids with internal inertia

A systematic application of the group analysis method for modeling fluids with internal inertia is presented. The equations studied include models such as the nonlinear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Iordanski Zhurnal Prikladnoj Mekhaniki i Tekhnitheskoj Fiziki 3, 102–111, 1960; Kogarko Dokl. AS USSR 137, 1331–1333, 1961; Wijngaarden J. Fluid Mech. 33, 465–474, 1968), and the dispersive shallow water model (Green and Naghdi J. Fluid Mech. 78, 237–246, 1976; Salmon 1988). These models are obtained for special types of the potential function W(r,[(r)\dot],S){W(\rho,\dot \rho,S)} (Gavrilyuk and Teshukov Continuum Mech. Thermodyn. 13, 365–382, 2001). The main feature of the present paper is the study of the potential functions with W _S  ≠ 0. The group classification separates these models into 73 different classes.     

A procedure to determine the viscosity function from experimental data of capillary flow

The accurate calculation of the viscosity η as function of the shear rate &γdot; from capillary viscometry is still a matter of debate in the literature. In fact, this problem involves the inversion of an integral equation, which leads to multiple solutions due to the unavoidable noise present in the experimental data. The purpose of this work is to develop an efficient procedure to determine the viscosity function from experimental data of capillary flow without presenting the difficulties inherent in other methods discussed previously in the literature. The system identification procedure is used here to estimate the parameters of a viscosity model, which is appropriately selected for the fluid under study through preliminary calculations involving the apparent shear rate – shear stress data. Once the model is chosen by satisfying criteria for the fit goodness and its parameters are evaluated, a smooth and continuous function η(γdot;) is obtained in the range of experimental shear rates. The procedure proposed is also applicable to fluids in shear flow that present two Newtonian plateaus, as it is typically found in macromolecular dilute solutions. The mean value theorem of continuous functions is used to reduce significantly the computational time. Received: 15 November 1999 Accepted: 7 November 2000     

The role of the mathematical mean value theorem (MVT) in rheometry: an easy way to convert apparent flow curves into correct ones

The mean value theorem of integral calculus guarantees that the apparent viscosity η _a can easily be converted into the correct viscosity η. For ordinary liquids there is a direct identity between η _a and η but the apparent shear rate (or apparent shear stress) has to be shifted to the representative shear rate γ˙^ (or representative shear stress τ^). A model free approximation scheme is introduced which implies a constant shift factor. The corresponding approximation for η is acceptable for liquids most commonly encountered. For plastic fluids the relation between η and η _a is more complex since it involves a function depending upon α; the yield stress relative to the maximum stress within the viscometer. Using the same approximation scheme as before the shift factor will involve α as well. The corresponding approximation of η is shown to be acceptable for the whole range of α. Received: 7 February 2000/Accepted: 15 February 2000     

Planar elongation of soft polymeric networks

A new test fixture for the filament stretch rheometer (FSR) has been developed to measure planar elongation of soft polymeric networks with application towards pressure-sensitive adhesives (PSAs). The concept of this new geometry is to elongate a tube-like sample by keeping the perimeter constant. To validate this new technique, soft polymeric networks of poly(propylene oxide) (PPO) were investigated during deformation. Particle tracking and video recording were used to detect to what extent the imposed strain rate and the sample perimeter remained constant. It was observed that, by using an appropriate choice of initial sample height, perimeter, and thickness, the planar stretch ratio will follow l(t) = h(t)/h₀ = exp([(e)\dot] t)\lambda(t) = h(t)/h_0= \exp({\dot{\varepsilon}} t), with h(t) being the height at time t and [(e)\dot]{\dot{\varepsilon}} the imposed constant strain rate. The perimeter would decrease by a few percent only, which is found to be negligible. The ideal planar extension in this new fixture was confirmed by finite element simulations. Analysis of the stress difference, σ _zz  − σ _xx , showed a network response similar to that of the classical neo-Hookean model. As the Deborah number was increased, the stress difference deviated more from the classical prediction due to the dynamic structures in the material. A modified Lodge model using characteristic parameters from linear viscoelastic measurements gave very good stress predictions at all Deborah numbers used in the quasi-linear regime.     

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.     

Molecular orientation in sheared molten thermotropic random copolyester

The macromolecular alignment and texture orientation in sheared thermotropic copolyester were investigated using in situ wide-angle X-ray scattering (WAXS) and polarizing optical microscopy (POM). The molecular behavior was correlated with viscoelastic properties. The polymer is a random copolyester based on 60 mol% 1,4-hydroxybenzoic acid (B) and 40 mol% ethylene terephthalate (ET) units. X-ray scattering showed that the molecular chains were aligned along the flow direction. The degree of molecular orientation, , is an increasing function of the applied shear rate. However, rheo-optics showed that shear flow could not orient the polydomain texture, i.e., neither defect stretching nor elimination of defects was observed. Instead, shear compressed the microdomains and gave rise to long-range orientation correlations. Rheology showed that the nematic melt is viscoelastic, the loss modulus G″ dominates the elastic modulus G′, and the dynamic viscosity η* is shear thinning. Moreover, the steady shear viscosity, η, also behaved shear thinning, while the first normal stress difference N ₁ remained positive. The empirical Cox–Merz rule did not hold, , within the shear rate range studied. The microscopic and rheological properties suggest that B–ET is a flow-aligning nematic polymer.     

Electrorheological determination of the Leslie viscosity coefficient (α2) of a main-chain liquid crystal polymer in a nematic solvent

Analysis of the electrorheological response of thermotropic solutions of a main-chain liquid crystal polymer (MCLCP) in a low molar mass nematic solvent is performed at a fixed shear rate as a function of the applied field strength. The Leslie viscosity coefficient α₂ can be obtained by least squares fits to an equation describing the balance between the viscous and electric torques, formulated via the two dimensional Leslie-Ericksen-Parodi theory. We find that the increment Δα₂ on dissolving the MCLCP increases linearly with molecular weight, consistent with earlier light scattering measurements of the increment in the twist viscosity, Δγ₁, and also with previous electrorheological measurements of the increment in the Miesowicz viscosity Δη_c. Received: 1 December 1998 Accepted: 28 April 1999     

Uniaxial elongational behavior of poly(vinyl chloride) physical gel

A poly(vinyl chloride) (PVC,  M_w = 102×10³)(\mbox{PVC,}\;{\rm M}_{\rm w} =102\times 10^3) di-octyl phthalate (DOP) gel with PVC content of 20 wt.% was prepared by a solvent evaporation method. The dynamic viscoelsticity and elongational viscosity of the PVC/DOP gel were measured at various temperatures. The gel exhibited a typical sol–gel transition behavior with elevating temperature. The critical gel temperature (T_gel) characterized with a power–law relationship between the storage and loss moduli, G^′ and G^″, and frequency ω, G￠=G^￠￠/tan  ( np/2 ) μ wⁿ{G}^\prime={G}^{\prime\prime}{\rm /tan}\;\left( {{n}\pi {\rm /2}} \right)\propto \omega ^{n}, was observed to be 152°C. The elongational viscosity of the gel was measured below the T_gel. The gel exhibited strong strain hardening. Elongational viscosity against strain plot was independent of strain rate. This finding is different from the elongational viscosity behavior of linear polymer solutions and melts. The stress–strain relations were expressed by the neo-Hookean model at high temperature (135°C) near the T_gel. However, the stress–strain curves were deviated from the neo-Hookean model at smaller strain with decreasing temperature. These results indicated that this physical gel behaves as the neo-Hookean model at low cross-linking point, and is deviated from the neo-Hookean model with increasing of the PVC crystallites worked as the cross-linking junctions.     

Recoverable strains and retardation times of monodisperse suspensions of silicon dioxide spheres in poly(dimethylsiloxane)

Steady-state viscosities η, steady-state recoverable strains γ _rs and characteristic retardation time τ _1/2 were measured for suspensions of monodisperse silicon dioxide (SiO₂) spheres in poly(dimethylsiloxane) (PDMS) with various volume fractions Φ of the suspended spheres at various creep stresses σ ₀. Two different regions are found in plots of η/η _m vs γ _rs, where η/η _m denotes the relative viscosity of the suspensions. In one region, η/η _m is proportional to γ _rs, while γ _rs is independent of η/η _m in the other region. In both regions, τ _1/2 is the functions of the shear strain rate in the steady-state of creep test independently of Φ. The origin of the elasticity is related to the ‘maximally distorted’ cages recovered owing to the repulsive interaction between the SiO₂ spheres and recovery of the cages in the shear-induced clusters of the suspended spheres.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

Droplet deformation of a strongly shear thinning dense suspension of polymeric micelles

We investigated the deformation of a strong shear thinning droplet undergoing simple shear flow in a Newtonian liquid. The droplet was an aqueous solution of poly(ethylene oxide) end capped with an alkyl group that forms spherical micelles in aqueous solution. At high concentrations and below a critical temperature, the jammed micelles showed strong shear thinning behaviour, and neither a yield stress nor a Newtonian viscosity was observed. At small shear rates, the droplet rotated and aligned in the flow, but did not deform or only very weakly. At high shear rates, the droplet deformation increased with increasing shear rate. The deformed droplet did not relax after the shear was stopped except for a modest rounding of the edges. For each shear rate, an apparent viscosity, η _ad, of the equivalent Newtonian droplet was calculated assuming affine deformation. η _ad showed a power law dependence on the capillary number Ca with an exponent of − 1.8 and was larger than the shear viscosity of the micelle suspension at the same shear rates. The results were explained by the existence of a strong gradient of the viscosity inside the droplet leading to a very low viscosity fluid layer near the droplet/matrix interface.     

Viscosity of concentrated noncolloidal bidisperse suspensions

At the same solid volume fraction (Φ) the relative viscosity (η _r ) of a concentrated noncolloidal bidisperse suspension of hard spherical particles is lower than that of a monodisperse suspension. In this paper a semi-analytical viscosity model of noncolloidal bidisperse suspensions is derived using an integration method. In this model the random loose packing density obtained by computer simulation is taken as the limit of solid volume fraction Φ _m which depends upon both the diameter ratio (λ) of large to small particles and the volume fraction of large particles (ξ=Φ _l /Φ). This model shows that at high solid volume fraction, Φ > 0.40, both λ and ξ significantly influence η _r . For example, at Φ=0.5, it predicts that for monodisperse suspensions η _r =70, while for bidisperse suspensions (λ=2 and ξ=0.7) η _r =40. Comparison shows that, at high solid volume fraction (0.4–0.5), the relative viscosity predicted by this model is in good agreement with that predicted by the work of Shapiro and Probstein (1992) and of Patlazhan (1993), but is higher than that predicted by the work of others. Received: 27 February 2001 Accepted: 25 April 2001     

A scaling law near the primary resonance of the quasiperiodic Mathieu equation

We analyze the quasiperiodic damped Mathieu equation

Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts

In high shear rate capillary rheometry the combined effect of pressure dependent viscosity and dissipative heating becomes significant. Analytical expressions are derived to treat curved Bagley plots and throttle experiments. End effects are taken into account by using an effective length over radius ratio. The non-adiabatic case is described using a lump heat transfer coefficient ? following Hay et al. (1999). The latter enters into the dissipative heating coefficient % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfKttLearuavTnhis1MBaeXatLxBI9gBam % XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB % Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf % euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9 % q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba % WaaqaafaaakeaacqaH1oqzdaWgaaWcbaGaemiCaahabeaakiabg2da % 9iabeg8aYnaaCaaaleqabaGaeyOeI0IaeGymaedaaOWaaeWaaeaacq % WGJbWydaWgaaWcbaGaemiCaahabeaakiabgUcaRmaalyaabaGaeu4M % dWeabaGafmyBa0MbaiaaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacq % GHsislcqaIXaqmaaaaaa!4D6C! e_p = r ^{- 1} ( c_p + L \mathord

Extensional-flow-induced crystallization of isotactic polypropylene

A filament stretching extensional rheometer with a custom-built oven was used to investigate the effect of uniaxial flow on the crystallization of polypropylene. Prior to stretching, samples were heated to a temperature well above the melt temperature to erase their thermal and mechanical histories and the Janeschitz-Kriegl protocol was applied. The samples were stretched at extension rates in the range of 0.01 s^-1 ￡ [(e)\dot] ￡ 0.75 s^-10.01\,\mbox{s}^{-1}\le \dot{{\varepsilon }}\le 0.75\,{\rm s}^{-1} to a final strain of ε = 3.0. After stretching, the samples were allowed to crystallize isothermally. Differential scanning calorimetry was applied to the crystallized samples to measure the degree of crystallinity. The results showed that a minimum extension rate is required for an increase in percent crystallization to occur and that there is an extension rate for which percent crystallization is maximized. No increase in crystallization was observed for extension rates below a critical extension rate corresponding to a Weissenberg number of approximately Wi = 1. Below this Weissenberg number, the flow is not strong enough to align the contour path of the polymer chains within the melt and as a result there is no change in the final percent crystallization from the quiescent state. Beyond this critical extension rate, the percent crystallization was observed to increase to a maximum, which was 18% greater than the quiescent case, before decaying again at higher extension rates. The increase in crystallinity is likely due to flow-induced orientation and alignment of contour path of the polymer chains in the flow direction. Polarized light microscopy verified an increase in number of spherulites and a decrease in spherulite size with increasing extension rate. In addition, small angle X-ray scattering showed a 7% decrease in inter-lamellar spacing at the transition to flow-induced crystallization. Although an increase in strain resulted in a slight increase in percent crystallization, no significant trends were observed. Crystallization kinetics were examined as a function of extension rate by observing the time required for molten samples to crystallize under uniaxial flow. The crystallization time was defined as the time at which a sudden increase in the transient force measurement was observed. The crystallization time was found to decrease as one over the extension rate, even for extension rates where no increase in percent crystallization was observed. As a result, the onset of extensional-flow-induced crystallization was found to occur at a constant value of strain equal to ε _c  = 5.8.     

Influence of NaClO3 on the rheological behaviour of a micellar solution of CPCl

It is now well know that a small addition of salt to a micellar solution often increases the zero-shear viscosity η₀ of the solution, the understanding of the behaviour at high salt content is more questionable. In this situation, addition of more salt induces a decrease of η₀. In this experimental work we investigate the linear and non-linear rheological behaviour of a new micellar system: CPCl (surfactant)/NaClO₃ (salt). Studies of the evolution of η₀ as well as G₀ (the elastic modulus) or τ_R (the relaxation time) are in agreement with the hypothesis of a diminution of the mean micellar length when, after the maximum η₀, the salt content increases. In the non-linear behaviour (non-Newtonian viscosity) the evolution of γ˙ _c, (which defines the occurrence of the shear thinning) with salt concentration C_S is also in agreement with such a hypothesis. Received: 29 March 1999/Accepted: 20 March 2000     

The Regularity of Weak Solutions of the 3D Navier–Stokes Equations in {B^{-1}_{\infty,\infty}}

We show that if a Leray–Hopf solution u of the three-dimensional Navier–Stokes equation belongs to C((0,T]; B^-1_￥,￥){C((0,T]; B^{-1}_{\infty,\infty})} or its jumps in the B^-1_￥,￥{B^{-1}_{\infty,\infty}}-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya–Prodi–Serrin criterion.