Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Step and steady shear responses of nearly monodisperse highly entangled 1,4-polybutadiene solutions

Nonlinear relaxation dynamics of highly entangled solutions of high molecular weight 1,4-polybutadiene (PB) in a PB oligomer are studied in steady shear and step shear flows. Polymer entanglement densities vary in the range 14hN/N_e(J)⣴, allowing systematic investigation of entanglement effects on nonlinear rheological response. In agreement with previous steady shear studies using well entangled polystyrene solutions, a flow regime is found where both the steady-state shear stress and first normal stress difference remain constant or increase quite slowly with shear rate, leading to a plateau in the steady-state orientation angle. The magnitude of the average orientation angle in the plateau range is in accordance with predictions of a recent theory by Islam and Archer (2001). In step shear, the nonlinear relaxation modulus G(t,%) is approximately factorable into time-dependent G(t) and strain-dependent h(%) functions only at long times, t>5_k, where 5_k,O(F_d0). This finding is consistent with earlier observations for entangled polystyrene solutions; however the complex crossing pattern in G(t,%)h^-1(%) that precede factorability in the latter materials is not observed. For all but the most entangled sample, apparent shear damping functions h (%,t)=(G(t,%))/(G(t)) immediately following imposition of shear are in nearly quantitative accord with the damping function h_DEIA predicted by Doi-Edwards theory.     

Elongational flow and rheology of monodisperse polymers in solution

We describe the utilization of idealized stagnation point extensional flows, produced by opposed jets, for birefringence visualization of induced molecular strain and flow resistance measurements. We identify rheological changes associated with the coil---stretch transition which occurs beyond a critical strain-rate in elongational flow-fields. In dilute solutions of monodisperse atactic polystyrene, increases in extensional viscosity are observed as isolated molecules become stretched. The largest increases in extensional viscosity, however, are found only beyond a critical concentration and strain rate, and are associated with the stretching of transient networks of interacting molecules. These results parallel similar effects seen in porous media flow and capillary entrance experiments. We determine the molecular weight dependence of the critical concentration which scales as M^−0.55 in agreement with pairwise interaction of coils, but is much lower than conventional values of the critical polymer concentration, c^*. We believed that polydispersity may play an important role in the development of such transient networks, and in controlling the degradation behaviour during flow.     

Prediction of the interfacial shear-stress in vertical annular flow

Many improvements of the Wallis correlation for the interfacial friction in annular flow have been proposed in the literature. These improvements give in general a better fit to data, however, their physical basis is not always justified. In this work, we present a physical approach to predict the interfacial shear-stress, based on the theory on roughness in single-phase turbulent pipe flows. Using measured interfacial shear-stress data and measured data on roll waves, which provide most of the contribution to the liquid film roughness, we show that the interfacial shear-stress in vertical annular flow is in very close agreement with the theory. We show that the sand-grain roughness of the liquid film is not equal to four times the mean film thickness, as it is assumed in the Wallis correlation. Instead, the sand-grain roughness is proportional to the wave height, and the proportionality constant can be predicted accurately using the roughness density (or solidity). Furthermore, we show that our annular flow, which is in similar conditions to others in the literature, is fully rough. Hence, the bulk Reynolds number should not appear in the prediction of the interfacial friction coefficient, as is often done in the improvements of the Wallis correlation proposed in the literature.     

Interfacial instability in turbulent flow over a liquid film in a channel

We revisit the stability of a deformable interface that separates a fully-developed turbulent gas flow from a thin layer of laminar liquid. Although this problem has received considerable attention previously, a model that requires no fitting parameters and that uses a base-state profile that has been validated against experiments is, as yet, unavailable. Furthermore, the significance of wave-induced perturbations in turbulent stresses remains unclear. To address these outstanding issues, we investigate this problem and introduce a turbulent base-state velocity that requires specification of a flow rate or a pressure drop only; no adjustable parameters are necessary. This base state is validated extensively against available experimental data as well as the results of direct numerical simulations. In addition, the effect of perturbations in the turbulent stress distributions is investigated, and demonstrated to be small for cases wherein the liquid layer is thin. The detailed modelling of the liquid layer also elicits two unstable modes, ‘interfacial’ and ‘internal’, with the former being the more dominant of the two. We show that it is possible for interfacial roughness to reduce the growth rate of the interfacial mode in relation to that of the internal one, promoting the latter, to the status of most dangerous mode. Additionally, we introduce an approximate measure to distinguish between ‘slow’ and ‘fast’ waves, the latter being the case for ‘critical-layer’-induced instabilities; we demonstrate that for the parameter ranges studied, the large majority of the waves are ‘slow’. Finally, comparisons of our linear stability predictions are made with experimental data in terms of critical parameters for onset of wave-formation, wave speeds and wavelengths; these yield agreement within the bounds of experimental error.     

Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts

The interfacial deformation and stability of two-(A-B) as well as three-layer symmetric (A-B-A) and asymmetric (A-B-C) pressure-driven flow of viscoelastic fluids has been investigated. Flow visualization in conjunction with digital image processing has been used to observe and measure the rate of encapsulation and interfacial stability/instability of the flow. Specifically, the encapsulation behavior as well as stability/instability of the interface and the corresponding growth or decay rate of disturbances as a function of various important parameters, namely, number of layers and their arrangement, layer depth ratio, viscosity and elasticity ratio as well as disturbance frequency, have been investigated. Based on these experiments, we have shown that the encapsulation phenomena occurs irrespective of the stability/instability of the interface and in cases when both encapsulation and instability occur simultaneously their coupling leads to highly complex and three-dimensional interfacial wave patterns. Moreover, it has been shown that the simple notion that less viscous fluids encapsulate more viscous fluids is incorrect and depending on the wetting properties of the fluid as well as their first and second normal stresses the reverse could occur. Additionally, in two- and three-layer flows it has been shown that by placing a thin, less viscous layer adjacent to the wall longwave disturbances can be stabilized while short and intermediate wavelength disturbances are stabilized when the more elastic fluid is the majority component. Furthermore, in three-layer flows it has been demonstrated that in the linear instability regime no dynamic interaction between the two interfaces is possible for short and intermediate wavenumber disturbances. However, in the nonlinear stability regime dynamic interactions between interfaces have been observed in this range of disturbance wavenumbers leading to highly chaotic flows. Finally, in the parameter space of this study no subcritical bifurcations were observed while supercritical bifurcations resulting in waves with a pointed front and a gradual tail were observed.     

A linear model of the stick-slip phenomena in polymer flow in rheometers

A linear approach was employed for the qualitative theoretical study of the stick-slip phenomena in polymer flows in rheometers. For this purpose, the familiar three-constant rheological equation and a linear wall friction law (i.e. wall stress — slip velocity dependence) were used with some additional hypotheses about the onset of the stick-slip behaviour. The friction law used was derived from a crude molecular approach. On the basis of these equations the inertialess stickslip behaviour of a viscoelastic liquid flowing through a capillary at a constant flow rate was considered. To be able to describe some transient phenomena in this problem, inertial effects (as an example) were taken into account. Furthermore, the distortions on the extrudate surface due to the slip phenomena inside the capillary were described theoretically within the framework of a linear approximation. In the final part of the study the possibility of rapid stochastization was discussed for rotational stick-slip flow of polymers in a cone-plate rheometer.     

Determination of flow sub-regimes in stratified air–water channel flow using LDV spectra

In gas–liquid stratified flows, pressure drop and transport across the interface are strongly influenced by the interfacial wave structure, making the determination of interfacial topography in this kind of flows very important. An objective way of characterizing the wave pattern present in the interface is proposed here. The method consists in analysing the spectra of the signal obtained from Laser Doppler Velocimetry (LDV) measurements of fluctuations occurring close to the air-sheared interface. Transitions are defined by the appearance and disappearance of peaks in the frequency spectra. The method was applied to study the transition regimes of a stratified air–water flow in a square-cross section channel. A flow pattern map for air–water channel flow is presented and compared with the maps available from the literature.     

Theory of interfacial dynamics of nematic polymers

 The interfacial momentum and torque balance equations for deforming interfaces between nematic polymers and isotropic viscous fluids are derived and analyzed with respect to shape selection and interfacial nematic ordering. It is found that the interfacial momentum balance equation for nematic interfaces involves bending forces that act normal to the interface, and that interfacial pressure jumps may exist even for planar surfaces. In addition tangential forces on nematic interfaces arise in the presence of surface gradients of the tensor order parameter. The torque balance equation shows that couple stress jumps are balanced by the surface molecular field. The interfacial balance equations are shown to be coupled such that nematic ordering depends on shape and vice versa. The governing dimensionless numbers for deforming nematic polymer interfaces are identified and the limiting regimes are discussed in reference to related experimental data. It is found that the ratio of Frank elasticity to surface anchoring controls whether the surface tensor order parameter deviates from its preferred equilibrium value. Whether the shape is affected, depends on the relative magnitudes of the isotropic surface tension, Frank bulk elasticity, and anchoring energy, and capillary number. Received: 16 April 1999/Accepted: 19 August 1999     

Shear and elongational flow behavior of acrylic thickener solutions. Part II: effect of gel content

We have investigated the effect of crosslink density on shear and elongational flow properties of alkali-swellable acrylic thickener solutions using a mixing series of the two commercial thickeners Sterocoll FD and Sterocoll D as model system. Linear viscoelastic moduli show a smooth transition from weakly elastic to gel-like behavior. Steady shear data are very well described by a single mode Giesekus model at all mixing ratios. Extensional flow behavior has been characterized using the CaBER technique. Corresponding decay of filament diameter is also well fitted by the Giesekus model, except for the highest crosslink densities, when filament deformation is highly non-uniform, but the non-linearity parameter α, which is independent of the mixing ratio, is two orders of magnitude higher in shear compared to elongational flow. Shear relaxation times increase by orders of magnitude, but the characteristic elongational relaxation time decreases weakly, as gel content increases. Accordingly, variation of gel content is a valuable tool to adjust the low shear viscosity in a wide range while keeping extensional flow resistance essentially constant.     

Purely elastic interfacial instabilities in superposed flow of polymeric fluids

Purely elastic interfacial stability of superposed plane Poiseuille flow of polymeric liquids has been investigated utilizing both asymptotic and numerical techniques. It is shown that these instabilities are caused by an unfavorable jump in the first normal stress difference across the fluid interface. To determine the significance of these instabilities in finite experimental geometries, a comparison between the maximum growth rates of purely elastic instabilities with instabilities driven primarily by a viscosity or a combined viscosity and elasticity difference is made. Based on this comparison, it is shown that purely elastic interfacial instabilities can play a major role in superposed flow of polymeric liquids in finite experimental geometries.     

Prediction of the slug-to-churn flow transition in vertical two-phase flow

An assessment is made of the various viewpoints on the slug-to-churn flow transition in vertical upward flow in the light of recent experimental results obtained at Harwell Laboratory. It is found that the flooding model of McQuillan & Whalley and the bubble entrainment model of Barnea & Brauner give satisfactory results at low and high liquid flow rates, respectively. An improved model for flooding, which takes account of the effect of the falling film, has been proposed. It is shown that this new model is in good agreement with experimental results at both low and high liquid flow rates.     

Suspensions of monodisperse spheres in polymer melts: particle size effects in extensional flow

Suspensions in polymeric, viscoelastic liquids have been studied in uniaxial extensional flow. The fibre wind-up technique has been used for this purpose. The effects of particle size and particle volume fraction have been investigated, using monodisperse, spherical particles. The results have been compared with shear flow data on the same materials. The values of the relative extensional viscosities at low stretching rates are in agreement with the relative shear viscosities and relative moduli. This indicates that hydrodynamic forces are stronger than the particle interaction forces. At larger strain rates strain hardening occurs; it is suppressed when particles are added. Small aggregating particles reduce the strain hardening more strongly than larger particles; strain hardening can even be totally eliminated. When further increasing the stretching rate, hydrodynamic effects dominate again and the effect of particle size effect on strain hardening disappears.     

Stokes flow before plane boundary with mixed stick-slip boundary conditions

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the Stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.     

Interfacial drag of two-phase flow in porous media

In this paper, the influence of the interfacial drag on the pressure loss of combined liquid and vapour flow through particulate porous media is investigated. Motivation for this is the coolability of fragmented corium which may be expected during a severe accident in a nuclear power plant. Cooling water is evaporated due to the particles decay heat. To reach coolability, the outflowing steam has to be replaced by inflowing water.     

An effective temperature theory for the nonequilibrium behavior of amorphous polymers

Amorphous polymers lack an organized microstructure, yet they exhibit structural evolution, where physical properties change with time, temperature, and inelastic deformation. To describe the influence of structural evolution on the mechanical behavior of amorphous polymers, we developed a thermomechanical theory that introduces the effective temperature as a thermodynamic state variable representing the nonequilibrium configurational structure. The theory couples the evolution of the effective temperature and internal state variables to describe the temperature-dependent and rate-dependent inelastic response through the glass transition. We applied the theory to model the effect of temperature, strain rate, aging time, and plastic pre-deformation on the uniaxial compression response and enthalpy change with temperature of an acrylate network. The results showed excellent agreement with experiments and demonstrate the ability of the effective temperature theory to explain the complex thermomechanical behavior of amorphous polymers.     

The axisymmetric long-wave interfacial stability of core-annular flow of power-law fluid with surfactant

The long wave stability of core-annular flow of power-law fluids with an axial pressure gradient is investigated at low Reynolds number. The interface between the two fluids is populated with an insoluble surfactant. The analytic solution for the growth rate of perturbation is obtained with long wave approximation. We are mainly concerned with the effects of shear-thinning/thickening property and interfacial surfactant on the flow stability. The results show that the influence of shear-thinning/thickening property accounts to the change of the capillary number. For a clean interface, the shear-thinning property enhances the capillary instability when the interface is close to the pipe wall. The converse is true when the interface is close to the pipe centerline. For shear-thickening fluids, the situation is reversed. When the interface is close to the pipe centerline, the capillary instability can be restrained due to the influence of surfactant. A parameter set can be found under which the flow is linearly stable.     

Modeling spurt and stress oscillations in flows of molten polymers

The paper presents an approach for modeling polymer flows with non-slip, slip and changing non-slip — slip boundary conditions at the wall. The model consists of a viscoelastic constitutive equation for polymer flows in the bulk, prediction of the transition from non-slip to sliding boundary conditions, a wall slip model, and a model for the compressibility effects in capillary polymer flows. The bulk viscoelastic constitutive equation contains a hardening parameter which is solely determined by the polymer molecular characteristics. It delimits the conditions for the onset of solid, rubber-like behavior. The non-monotone wall slip model introduced for polymer melts, modifies a slip model derived from a simple stochastic model of interface molecular dynamics for cross-linked elastomers. The predictions for the onset of spurt, as well as the numerical simulations of hysteresis, spurt, and stress oscillations are demonstrated. They are also compared with available data for a high molecular weight, narrow distributed polyisoprene. By using this model beyond the critical conditions, many of the qualitative features of the spurt and oscillations observed in capillary and Couette flows of molten polymers, are described.Notations upper convected derivative of elastic strain tensor - f, f_m, f_min dimensionless (sliding) shear friction characteristics, and its maximum and minimum - G Hookean elastic modulus - G_p plateau modulus - G, G storage and loss moduli - I₁, I₂ first and second invariant of strain tensor - I₁, I₀ capillary and barrel lengths - M non-dimensional mass flow rate - M_C critical molecular weight - M^*, M_e molecular weights of a statistical segment, and of polymer chain between entanglements - M_n, M_W number average and weight average molecular weights - m, k two fitting parameters of slip model - _s , _s ^o nominal and characteristic sliding velocities - u non-dimensional sliding velocity - u _sc initial (infinitesimal) slip velocity - u ₁ upper limit of u on the lower branch - u ₂ lower limit of u on the upper branch - u _max value of u corresponding to f_min - u _min value of u corresponding to f_max - U piston speed - Q nominal volumetric flow rate - q non-dimensional volumetric flow rate - R, R_o capillary and barrel radii - M non-dimensional mass flow rate     

Stick-slip transition in capillary flow of linear polyethylene: 3. Surface conditions

Effects of surface topology and energy on the stick-slip transition were studied in capillary flow of highly entangled polyethylene (PE) melts. Surface roughness was shown to increase the critical stress of the stick-slip transition because of the increased resistance to interfacial disentanglement. Lowering the surface energy of a smooth die wall by treatment with a fluorocarbon completely eliminates the stick-slip transition and produces massive interfacial slip at PE/wall boundary down to a stress level of 0.05 MPa. On the other hand, considerable roughness on the same low energy surface can produce a stick hydrodynamic boundary condition and restore the stick-slip transition despite the weak PE/wall interfacial interactions. Additionally, a slip-slip transition was found in the die with a nearly non-adsorbing wall that appears to involve a secondary chain-debonding process.