Viscosity enhancement In non-Newtonian flow of dilute polymer solutions through crystallographic porous media

We report results of the flow of dilute mono-disperse solutions of atactic poly(styrene) in di-octyl phthalate through regular crystallographic packed beds of spheres. Pressure drop measurements made as a function of flow rate across simple cubic and body centred cubic arrays of spheres have been used to estimate the specific viscosities of the polymer solutions as a function of the superficial Deborah number. Through both structures the onset Deborah number for the non-Newtonian increase in specific viscosity is found to be low when compared on the basis of well-characterized Zimm relaxation times. Surprisingly it is found that polymer solutions achieve a greater maximum specific viscosity in the simple cubic than in the body centred cubic array, a result contrary to prior expectations due to the absence of trailing stagnation points in the simple cubic structure. It is hypothesised that the trailing stagnation points in the body centred cubic array may be screened from the flow field by strands of oriented polymer and that, as such, the periodic variations in cross-sectional area of the flow (which are more severe in the simple cubic array) may play the most significant role in causing polymer extension and in enhancing the non-Newtonian viscosity.     

Viscosity enhancement in non-Newtonian flow of dilute aqueous polymer solutions through crystallographic and random porous media

We report results on the flow of dilute aqueous solutions of hydrolysed poly(acrylamide) (HPAA) through beds of spheres packed in simple cubic and body-centred cubic crystallographic arrays. Pressure drop measurements made across the arrays as a function of the flow rate have been used to estimate the specific viscosities of the HPAA solutions as a function of the superficial strain rate. It is found that greater non-Newtonian increases in the specific viscosity occur in the body-centred cubic array, which is thought to be due to the presence of trailing stagnation points, which are not present in the simple cubic array. Experiments have been performed using HPAA solutions in the presence of mono- and divalent cations at various concentrations and, for validity, have been compared with results obtained from a traditional randomly packed porous medium. In addition, a study of mechanical degradation of the polymer in flow through the crystallographic arrays has been carried out and reveals a greater rate of degradation in the body-centred array and also a significant increase in degradation with salt concentration.     

Extensional flow of polymer solutions through porous media

The flow behaviour of various polymer solutions of non-hydrolyzed polyacrylamide, hydrolyzed polyacrylamide, polyox and Xanthan was investigated in a plexiglass column having a succession of enlargements and constrictions, and compared with the flow behaviour and mechanical degradation of a solution of non-hydrolyzed polyacrylamide in a packed column of non-consolidated sand. The flow behaviour of this solution was found to be very similar in both the sand pack and plexiglass pore.Apart from the Xanthan solution, all other polymer solutions showed a viscoelastic behaviour in the plexiglass pore. The onset of viscoelastic behaviour, which has previously been defined using the shear rate ( ), stretch rate ( _s ) and Ellis number (E ₁), could be more precisely evaluated using a modified stretch rate (S _G). The pressure losses across the plexiglass pore for different polymer solutions of the same type were found to follow a unique curve provided the suggested group (S _G) was used, a situation which was not achieved with the other rheological parameters.The multipass mechanical degradation of the non-hydrolized polyacrylamide was tested through the sand pack against the suggested group (S _G) and Maerker's group (M _a). It was found that the loss of the solution viscoelasticity due to multipass mechanical degradation was better represented usingS _G thanM _a. A cross-sectional area (cm²) - C ^* critical concentration of polymer (ppm) - d plexiglass pore enlargement diameter - D average sand grain diameter (cm) - e equivalent width for the plexiglass pore - E ₁ Ellis number (a Deborah number) - F _R resistance factor - F _Ri resistance factor at the first pass - h height of the flow path of the plexiglass pore - K power-law constant - K _h,K _w effective permeability to hydrocarbon and water, respectively (10^–8 cm²) - M _a Maerker's group for a given porosity (s^–1) - M _ai value ofM _a at the first pass - N _D Deborah number - n power-law index - Q flow rate (cm³/s) - R capillary radius (cm) - R _g radius of gyration - S _G suggested group of rheological parameters representing a modified maximum stretch rate (s^–1) - S _Gi value ofS _G at the first pass - T _R,t characteristic time for the fluid (s) - t _s residence time (s) - V ₀ superficial velocity (cm/s) - V mean velocity of flow through a porous medium (cm/s) - average axial velocity in the enlargement section of the plexiglass pore (cm/s) - V ₁,V ₂ maximum velocity at a plexiglass enlargement neck and centre - [] intrincis viscosity - viscosity (mPa s) - _r relative viscosity (ratio of the viscosity of the polymer solution to that of the solvent) - shear rate (s^–1) - _s stretch rate (s^–1) - characteristic time for the polymer solution (s)     

Dynamics of dilute polymer solutions

Taking volume effects, hydrodynamic no-flow conditions, and internal viscosity into account in the molecular dynamics has made it possible to formulate an equation of flow for a dilute polymer solution which in the region of comparatively slowly varying motions is described by the available experimental facts [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–146, January–February, 1988.     

Thermal diffusion of dilute polymer solutions

In this paper diffusion of a dilute solution of elastic dumbbell model macromolecules under non-isothermal conditions is studied. Using the center of mass definition for the local polymer concentration, the diffusive flux contains a thermal diffusion dyadic d _T .  To get some idea of thermal diffusion d _T is evaluated for steady state isothermal conditions. Explicit results are presented for some homogeneous flows. It is shown that if the polymeric number density is defined via the beads (of the dumbbell) – termed n _b – then the diffusive flux j contains , where τ ^c is the intramolecular contribution to the bulk stress. Though the form of the diffusion equation for n _b thus differs from the corresponding one for n, it is shown that for essentially unbounded systems differences between n and n _b are small. Since the results involve the translational diffusion coefficient they can readily be taken over for Rouse coils. Received: 23 September 1997 Accepted: 5 June 1998     

Dissipative stresses in dilute polymer solutions

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed.     

Converging radial flow of dilute polymer solutions

A characteristic time of dilute polymer solutions is determined from a converging radial flow experiment. The influence of the intradisk separation and of the polymer concentration on this time is studied. Present results are compared to results obtained with a diaphragm. The main limitations of the experiments are pointed out.     

Normal stress effect in dilute polymer solutions

Rheologica Acta - The non-linear, steady flow behavior of dilute solutions of poly ethylene oxide at high shear rates is experimentally studied. The shear stress-shear rate relations at low and...     

Universal viscometric functions for dilute polymer solutions

The Gaussian approximation has been shown previously to be excellent for the treatment of hydrodynamic effects in dilute polymer solutions. However, the computational time required to find the viscometric functions in simple shear flow is prohibitively long for chains with a large number of beads. Here we introduce a new approximation which retains the accuracy of the Gaussian approximation but is significantly less computationally intensive. Thus the rheological behavior of long chains may be explored. Extrapolation of results obtained numerically for long chains to the infinite chain length limit is shown to lead to predictions independent of model parameters. As a result, within the context of the approximation introduced here, universal viscometric functions for dilute polymer solutions in simple shear flow under theta conditions are obtained.     

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.     

Kinetic theory and rheology of dilute polymer solutions

Summary A summary is given of some recent attempts to relate the results of the kinetic theory of rigid and flexible macromolecules to continuum mechanics results.With 1 table     

Hartree self-consistent field approach towards dilute polymer solutions

Summary The presence of elastic forces in the spring and bead model of polymer solutions suggests a discussion in terms of modes rather than of beads. The transition from a many-mode picture to a one-mode picture is effected by a factorization of the many-mode probability density into one-mode probability densities. These obey a system of coupled integro-differential equations of the Hartree type. Working in the Rouse mode representation and beginning with the longmolecule free-draining approximation good one-mode probability densities were derived which take into account Oseen's hydrodynamic interactions in an average fashion and form the starting basis for the well-known Hartree self-consistent procedure which goes beyond the approximation of the averaged Oseen tensor. This procedure can be used to improve the values of the viscosity and normal stress functions which we have calculated for the case of simple shear flow. The shear stress and the two relevant normal stress differences are non-zero and depend on the velocity gradient in a non-linear manner. The calculation of explicit numerical results requires computer work of moderate extent for which all required formulae are given in the appendix.

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Zusammenfassung Das Auftreten elastischer Kräfte im Feder-Kugel-Modell (spring-bead-model) der Polymerlösungen legt eine Diskussion mit Hilfe der Schwingungs-Moden anstelle der Teilchenbewegung nahe. Der Übergang von einer Viel-Moden-Beschreibung zu einer Ein-Moden-Darstellung wird durch die Faktorisierung der Viel-Moden-Wahrscheinlichkeitsdichte zu Ein-Moden-Dichten erreicht. Diese gehorchen einem System gekoppelter Integrodifferentialgleichungen vom Hartree-Typ. Beginnend mit der Näherung frei durchspülter langer Moleküle werden brauchbare Ein-Moden-Wahrscheinlichkeitsdichten in der Moden-Darstellung vonRouse abgeleitet. Diese berücksichtigen die Oseenschen hydrodynamischen Wechselwirkungen im Mittel und bilden die Ausgangsnäherung für die Hartreesche Selbstkonsistenz-Methode, die über die Näherung des gemittelten Oseentensors hinausführt. Während die allgemeine Darstellung dieser Methode einer späteren Untersuchung vorbehalten bleiben soll, werden hier die Viskosität und die Normalspannungsfunktionen für den Fall der einfachen Scherströmung berechnet. Die Schubspannung und die beiden relevanten Normalspannungs-differenzen verschwinden nicht, sondern hängen in nichtlinearer Weise vom Geschwindigkeitsgradienten ab. Die Berechnung expliziter numerischer Resultate erfordert Computerarbeit von mäßigem Umfang. Alle hierfür benötigten Formeln werden in einem Anhang zusammengestellt.

     

Quantitative description of rheological properties of dilute polymer solutions

The Rouse-Zimm model has been generalized to describe the rheology of dilute polymer solutions at moderate shear rates by allowing for a reduced-shear-rate dependent slippage of the carrier solvent in the neighbourhood of the polymer coil with respect to the macroscopic continuum. A comparison of the theoretical predictions with published viscosity data for monodisperse polymer systems shows good agreement. Thus it appears to be sufficient to measure the intrinsic viscosity in order to estimate all the material functions of interest for the steady and dynamic shear flow of dilute polymer solutions.