Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

The rheology of powders

This paper studies the slow flow of powders. It is argued that since powders can flow like liquids, there must be equations similar to those of liquids. The phenomenon of a variable density, dilatancy, is described by an analogue of temperature called the compactivity X. Whereas, in thermal physicsT = E/S, powders are controlled byX = V/S. The equations for, v, T of a liquid are replaced by, v, X. An analogy for free energy is described, and the solution to some simple problems of packing and mixing are offered. As an example of rheology, it is shown that the simplest flow equations produce a transition to plug flow in appropriate circumstances.Delivered as a Gold Medal Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Apparent thickening behavior of dilute polystyrene solutions in extensional flows

An experimental investigation was undertaken to study the apparent thickening behavior of dilute polystyrene solutions in extensional flow. Among the parameters investigated were molecular weight, molecular weight distribution, concentration, thermodynamic solvent quality, and solvent viscosity. Apparent relative viscosity was measured as a function of wall shear rate for solutions flowing from a reservoir through a 0.1 mm I.D. tube. As increased, slight shear thinning behavior was observed up until a critical wall shear rate was exceeded, whereupon either a large increase in or small-scale thickening was observed depending on the particular solution under study. As molecular weight or concentration increased, decreased and, the jump in above , increased. increased as thermodynamic solvent quality improved. These results have been interpreted in terms of the polymer chains undergoing a coil-stretch transition at . The observation of a drop-off in at high (above ) was shown to be associated with inertial effects and not with chain fracture due to high extensional rates.     

Neck propagation as a shock wave

Neck propagation in the stretching of elastic solid filaments having a yield point was analyzed using the space one-dimensional thin filament governing equations developed previously by the authors and other researchers. Constitutive model for the filament was assumed to be expressible as engineering tensile stress(X) (tensile force) given as a function of elongational strain with the(X) curve having a yield point maxima followed by a minima and a breaking point greater than the yield point maxima. Also incorporated into the model is the hysteresis of irreversible plastic deformation. When inertia is taken into consideration, the thin filament equations were found to reduce to the nonlinear wave equation ² (X)/ ² =C ₁ ² X/ ² where is Lagrangean space coordinate, is time, andC ₁ is inertia coefficient. The above nonlinear wave equation yields a solutionX(, ) having a stepwise discontinuity inX which propagates along the axis. The zero speed limit of the step wave solution was found to describe the above neck propagation occurring in solid filaments. Furthermore, it was recognized that the nonlinear wave equation was known for many years to also govern the plastic shock wave which propagates axially within a metal rod subjected to a very strong impact on its end. The one-dimensional atmospheric shock wave also was known to be governed by the nonlinear wave equation upon making certain simplifying assumptions. The above and other evidences lead to the conclusion that neck propagation occurring in the extension of solid filament obeying the above(X) function can be formally described as a shock wave.     

Transient stress responses predicted by the internal viscosity model in elongational flow

Predictions are made for the elongational-flow transient rheological properties of the dilute-solution internal viscosity (IV) model developed earlier by Bazua and Williams. Specifically, the elongational viscosity growth function _e ⁺ (t) is presented for abrupt changes in the elongational strain rate . For calculating _e ⁺, a novel treatment of the initial rotation of chain submolecules is required; such rotation occurs in response to the macroscopic step change of at t = 0. Representative are results presented for N = 100 (N = number of submolecules) and = 1000 f and 10000 f (where is the IV coefficient and f is the bead friction coefficient), using h ^* = 0 (as in the original Rouse model) for the hydrodynamic interaction. The major role of IV is to cause the following changes relative to the Rouse model: 1) abrupt stress jump at t = 0 for _e ⁺; 2) general time-retardance of response. There is no qualitative change from the Rouse-model prediction of unbounded il growth when exceeds a critical value ( ), and calculations of submolecule strains at various show that the unbounded- _e ^– behavior arises from unlimited submolecule strains when . However, the time-retardance could delay such growth beyond the timescale of most experiments and spinning processes, so that the instability might not be detected. Finally, _e ⁺ (t) and _e ( ) in the limit are presented for N = 1 and compared with exact predictions for the analogous rigid-rod molecule; close agreement lends support for the new physical approximation introduced for solving the transient dynamics for any N.     

Temperature and concentration dependent viscosity and gelation temperature of ABA triblock copolymer solutions

In solutions of ABA-triblock copolymers in a poor solvent for A thermoreversible gelation can occur. A three-dimensional dynamic network may form and, given the polymer and the solvent, its structure will depend on temperature and polymer mass fraction. The zero-shear rate viscosity of solutions of the triblock-copolymer polystyrene-polyisoprene-polystyrene in n-tetradecane was measured as a function of temperature and polymer mass fraction, and analyzed; the polystyrene blocks contained about 100 monomers, the polyisoprene blocks about 2000 monomers. Empirically, in the viscosity at constant mass fraction plotted versus inverse temperature, two contributions could be discerned; one contribution dominating at high and the other one dominating at low temperatures. In a comparison with theory, the contribution dominating at low temperatures was identified with the Lodge transient network viscosity; some questions remain to be answered, however. An earlier proposal for defining the gelation temperature T _gel is specified for the systems considered, and leads to a gelation curve; T _gel as a function of polymer mass fraction.Mathematical symbols {} functional dependence; e.g., f{x} means f is a function of x - p _log logarithm to the base number p; e.g., ¹⁰log is the common logarithm - exp exponential function with base number e - sin trigonometric sine function - lim limit operation - – in integral sign: Cauchy Principal Value of integral, e.g., - derivative to x - partial derivative to x Latin symbols dimensionless constant - b constant with dimension of absolute temperature - constant with dimension of absolute temperature - B dimensionless constant - c mass fraction - dimensionless constant - constant with dimension of absolute temperature - d ^* dimensionless constant - D{0} constant with dimension of absolute temperature - e base number of natural (or Naperian) logarithm - g distribution function of inverse relaxation times - G relaxation strength relaxation function - h distribution function of relaxation times reaction constant enthalpy of a molecule - H Heaviside unit step function - i complex number defined by i ² = –1 - j{0} constant with dimension of viscosity - j index number - k Boltzmann's constant - k _H Huggins' coefficient - m mass of a molecule - n number - N number - p index number - s entropy of a molecule - t time - T absolute temperature Greek symbols as index: type of polymer molecule - as index: type of polymer molecule - shear as index: type of polymer molecule - shear rate - small variation; e.g. T is a small variation in T relative deviation Dirac delta distribution as index: type of polymer molecule - difference; e.g. is a difference in chemical potential - constant with dimension of absolute temperature - (complex) viscosity - constant with dimension of viscosity - [] intrinsic viscosity number - inverse of relaxation time - chemical potential - number pi; circle circumference divided by its diameter - mass per unit volume - relaxation time shear stress - angular frequency     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

Does N 1 or N 2 control the onset of edge fracture?

The edge fracture instability which occurs at the air/liquid interface in a cone-and-plate rheometer is studied for six elastic polymer solutions. Theoretical models for the onset of edge fracture are evaluated in light of the experimental data. The data are well-described by the Tanner-Keentok model, which predicts that fracture will occur whenever a critical value of the second normal stress difference is exceeded in magnitude.     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

Weissenberg-Rabinowitsch-Auswertung der mit dem Platte-Platte-Rotationsrheometer gemessenen Fließkurve mittels eines Fließgesetzes vom Carreauschen Typ

Zusammenfassung Basierend auf dem das Fließverhalten strukturviskoser Fluide sehr genau beschreibenden sogenannten Carreau-Ansatz wird eine Näherungsformel für die Drehmoment-Schergeschwindigkeit-Charakteristik (scheinbare Fließkurve) des Platte-Platte-Rotationsrheometers (PPR) im stationären Versuch vorgeschlagen. Die gewonnenen Resultate können in die Auswertungseinheit des Platte-Platte-Systems leicht integriert werden und damit das Anwendungsgebiet des PPR-Systems für konzentrierte Polymerlösungen und Polymerschmelzen beträchtlich erweitern.

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A new simplified, but very accurate, formula is proposed for the torque-rate characteristic (apparent flow curve) of the parallel-disk rotational rheometer (PDR) in steady-shear mode, based on the Carreau formula for the viscosity of pseudoplastic fluids. The results can easily be incorporated into the evaluation of the parallel-disk system and therefore the application field of the PDR-system can be extended considerably for concentrated polymer solutions and polymer melts.

     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

Rheological material functions of polymer solutions using a generalized Maxwell model with slip functions Part I: Theory

A generalization of the Maxwell model for polymer systems is derived that replaces the velocity gradient in the Eulerian expression for the upper convected derivative by a tensorial kinematic function. Applying the principle of objectivity this tensorial function is reduced to two scalar slip functions. In shear flows, only one of the two occurs. Material functions are calculated in closed form, and asymptotic conditions are formulated that guarantee isotropic behaviour of the material in sudden strains.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada.     

On the steady simple shear flows of the one-mode Giesekus fluid

Analytical solutions for the plane Couette flow and the plane Poiseuille flow of the one-mode Giesekus fluid without any retardation time have been obtained by considering the domain of definition for each of the two branch solutions which arise due to the presence of the quadratic stress terms in the constitutive equations. For each fixed value of the mobility parametera, the limiting value of the Weissenberg number for the upper branch solution, i.e., the physically realistic solution is determined in terms of the corresponding dimensionless shear stress for the plane Couette flow and in terms of the corresponding dimensionless pressure gradient for the plane Poiseuille flow. In the case of the plane Couette flow, it is shown that fora falling in the range 0a1/2 only the physically realistic solution exists while for 1/2<a 1 a nonphysical solution coexists with the realistic one. In the case of the plane Poiseuille flow, it is shown that the non-physical solution cannot even exist around the center plane of the channel, and the effects of the mobility parameter and the dimensionless pressure gradient on the flow variables are investigated. Possible extensions of the present approach to other steady simple shear flows with and without the introduction of the retardation time are also discussed.     

Analyse und Berechnung der Strömungsvorgänge im zylindrischen Scherspalt von Plastifizieraggregaten

Zusammenfassung Dieser Aufsatz zeigt eine Möglichkeit auf, zylindrische Scherteile einer Plastifiziereinheit, auf der strukturviskose Materialien verarbeitet werden, approximativ zu berechnen. Es ist möglich, den Volumenstrom und Druckabfall, die mittlere Schergeschwindigkeit, Scherdeformation und Schubspannung im Scherspalt zu approximieren. Durch diese Gleichungen wird eine Abschätzung der Verteil- und Zerteilvorgänge im Scherelement möglich.

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A method is described for approximating the flow in cylindrical shearing gaps of plasticating extruder, which is applicable to shear thinning materials. It is possible to calculate the through-put and pressure drop as well as the shear rate, strain and shear stress in the gap. With these equations the distribution and separation process in shearing gaps can be evaluated.

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D Zylinderdurchmesser - d ₁ Schnecken-Kerndurchmesser der Meteringzone - d _s Durchmesser des zylindrischen Scherteils - K Konstante im Potenzfließgesetz - K _0T Koeffizient des Potenzfließgesetzes - L ₁ Länge der Anlaufschräge - L _s Länge des zylindrischen Scherteils - n Fließindex - n ₀ Drehzahl - p Druckabfall über der Scherteillänge - s Scherspalthöhe - T _M Massetemperatur - ₀ Umfangsgeschwindigkeit - _0x Geschwindigkeitskomponente inx-Richtung - _x, _z Geschwindigkeit inx- bzw.z-Richtung als Funktion der Koordinatey - Volumenstrom - x, z Ortskoordinaten - Exponent des Potenzfließgesetzes - Schergeschwindigkeit - mittlere Schergeschwindigkeit - Viskosität - dimensionslose Höhe - Dichte der Schmelze - Schubspannung - _yx, _yz Schubspannungskomponenten - _xx, _zz Normalspannungskomponenten - _ps dimensionsloser Druckgradient - dimensionsloser Volumenstrom - _x, _z dimensionslose Geschwindigkeit inx- bzw.z-Richtung     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Zum Einfluß von Füllstoffen auf das rheologische Verhalten von hochmolekularen Polyethylenschmelzen

Zusammenfassung Zur Sichtbarmachung der Strömungsvorgänge im Bereich der konvergenten Strömung vor einer Kapillaren wurde ein Verfahren entwickelt, das die Eigenschaften bestimmter teilkristalliner Polymere bei der Kristallisation unter Scherung bzw. aus einer vorgescherten Schmelze nutzt. Diese zeigen im erstarrten Zustand sogenannte Fließlinien, die aus in bestimmter Weise auf reihenförmig angeordnete Keime aufgewachsenen Kristallitstrukturen bestehen. Solche linienförmigen Aggregate sind in der Lage, ein Stromlinienbild der Strömung in der Schmelze im erstarrten Polymer wiederzugeben. Für ein LDPE vom Spritzgußtyp konnte die Entwicklung der Einlaufwirbel mit wachsender Schergeschwindigkeit dargestellt werden. Die hochmolekularen Polymertypen erwiesen sich schon bei geringsten Durchsätzen als instabil, die Einlaufwirbel verschwanden dabei vollständig, und die Bilder des erstarrten Polymers legen einen Mechanismus der Instabilitäten nahe, bei dem jeweils unterschiedliche Teilvolumina zeitweise in die Kapillare einströmen, bei Überschreiten einer kritischen Deformation jedoch abreißen und zurückschnellen, während andere Volumenelemente ihren Platz einnehmen. Das Verhalten der Suspensionen unterscheidet sich diesbezüglich nicht von dem der ungefüllten Schmelzen.Ein signifikanter Einfluß der Füllstoffe zeigt sich jedoch beim Einströmen in Kapillaren. Während die ungefüllten Schmelzen nach sehr kurzer Zeit ein laminares Strömungsprofil ausbilden, findet man mit zunehmender Füllung im Bereich stabiler Einströmbedingungen einen sehr hohen Einlaufdruckverlust bei kurzen Kapillaren, nicht jedoch bei Lochdüsen. Dies deutet auf einen Verfestigungsmechanismus bei der Ausbildung des laminaren Strömungsprofils hin, der durch Aggregierungsvorgänge im Bereich der hohen Dehnbeanspruchung in der konvergenten Einlaufströmung bewirkt wird.

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For the visualization of polymer flow in the entry region of a capillary, a method has been developed that uses distinct crystallization phenomena of the partially crystalline polymer itself solidifying from a presheared melt. The solid polymer shows so-called flow lines made up of transcrystalline overgrowth on rows of nuclei capable of showing a picture of the past melt-flow streamlines. The development of circulating flow in the entry region of capillaries could be shown with an injection-molding type LDPE. The higher molecular-weight types appeared to show unstable flow with disappearing circulating flow in the corners even at very low flow rates. The bulk polymer seems to flow into the capillary from certain parts of the reservoir until a critical strain is reached, then snaps back, giving way for the same flow type from another part of the reservoir. The behaviour of the suspensions does not differ in this respect from that of the pure melts.A significant filler influence is, however, shown within the capillary. While unfilled polymer melts reach steady flow conditions only one or two diameters downstream of the capillary entrance, an entry pressure loss increasing with increasing filler content is found in short capillaries (l/d = 5 – 15), but not in no-land dies (l/d 1). This seems to indicate a flow hardening mechanism in the region where the steady laminar flow profile develops and which might result from the very high extensional strain (next to instability) in the converging flow of the capillary entrance.

Zweiter Teil einer vom Fachbereich Chemietechnik der Universität Dortmund genehmigten Dissertation.     

Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit

A hot-film probe has been used to measure slip of high-density polyethylene flowing through a conduit with a rectangular cross section. A transition from no slip to a stick-slip condition has been observed and associated with irregular extrudate shape. Appreciable extrudate roughness was initiated at the same flow rate as that at which the relationship between Nusselt number and Péclet number for heat transfer from the probe departed from the behavior expected for a no-slip condition at the conduit wall. A ₁ constant defined by eq. (A3) - C dimensionless group used in eq. (7) - C _p heat capacity - D constant in eq. (13) - f u _s/u - f _lin defined by eq. (A6) - G storage modulus - G loss modulus - k thermal conductivity - L length of hot film in thex-direction - L _eff effective length of large probe found from eq. (A3) - Nu _L Nusselt number, defined for a lengthL by eq. (2) - (Nu _L)₀ value ofNu _L atPe = 0 (eq. (A 1)) - Pe Péclet number,uL/ - Pe ₀ Péclet number in slip flow, eq. (6) - Pe ₁ Péclet number in shear flow, eq. (4) - q _L average heat flux over hot film of lengthL - R _i resistances defined by figure 8 - R _AB correlation coefficient defined by eq. (14) for signalsA andB - T temperature - T _s temperature of probe surface - T ambient temperature - T T _s –T - u average velocity - u _s slip velocity - V _b voltage indicated in figure 8 - W probe dimension (figure 6) - x distance in flow direction (figure 1) - y distance perpendicular to flow direction (figure 1) - thermal diffusivity,k/C _p - wall shear rate - _5% thickness of lubricating layer during probe calibration for introduction of an error no greater than 5%; see Appendix I - ^* complex viscosity - density - time - _c critical shear stress, eq. (13) - _w wall shear stress - frequency characterizing extrudate distortion (figures 12 and 13), or frequency of oscillation during rheometric characterization (figures 18–20) - * quantity obtained from normalized Nusselt number, eq. (A1), or complex viscosity ^* - A actual (small) probe (see Appendix I) - M model (large) probe (see Appendix I)     

A study of swelling and collapse in polyelectrolyte gels

Swelling and collapse of a polyelectrolyte are shown to be the result of a delicate balance of three thermodynamic forces, viz. osmotic pressure, polymer elasticity, and van der Waals attraction. The behaviour of the polyelectrolyte gel is studied under isotropic pressure and under uniaxial and biaxial loading.