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Volume 15, Issue 4, December 1971

Oscillatory Shear of Nonlinear Fluids I. Preliminary Investigation
View Description Hide DescriptionExperimental oscillatory shear stress data were compared to the nonlinear behavior predicted from steady‐state viscosity versus shear rate data. Both types of experimental data were obtained using a modified Weiseenberg Rheogoniometer, while finite difference techniques predicted the oscillatory data for nonlinear purely viscous materials. For nonlinear viscous fluids whose relaxation times are short compared to the period of oscillation, the harmonic amplitudes and phases are predictable from the steady‐state viscosity versus shear rate function. Thixotropic and viscoelastic fluids showed large discrepancies between experimental and predicted data, indicating significant time dependence. These results indicate the suitability of oscillatorytesting procedures for investigation of time‐dependent materials.

Fourier Transform Method in Linear Viscoelastic Analysis: The Vibrating Viscoelastic Reed
View Description Hide DescriptionWhen linear viscoelasticity is formulated in a Fourier transform space, it is not necessary to assume any specific mathematical model, such as assemblies of Voigt or Maxwell elements, or differential operator equations, to completely describe the behavior of linear viscoelasticmaterials under any stress‐strain history. Indeed, the Fourier transform method is quite applicable even for complex vibration problems. In this paper we apply the method to the vibrating viscoelastic reed under two modes of oscillation, fixed at one end and subjected to a sinusoidal loading at the other, and subjected to a sinusoidal force at one end with the other end left free to oscillate. Theoretical results were in excellent quantitative agreement with experiments performed on a reed of poly(t‐butyl ethylene oxide) using both modes of excitation. We expressed the results in terms of corrections to the approximate relationship /(resonance frequency).

Time‐Dependent Polymer Rheology under Constant Stress and under Constant Shear Conditions
View Description Hide DescriptionThe kinetic rate theory previously presented for describing non‐Newtonian phenomena has been further modified to predict the flow behavior of viscoelastic materials under constant stress conditions. The thixotropic shear stress or shear rate is predicted by the kinetic theory, and the experimental stress or shear rate is obtained by modifying the thixotropic value by a stress or shear rate retardation term. The retardation term stems from a Maxwellian approach for stress retardation. In order to test the validity of this approach, transient and steady‐state data have been obtained for two solutions of polymethylmethacrylate in diethyl‐phthalate. Both constant stress measurements and constant shear rate data have been taken over a broad range. In a systematic manner as suggested by the theory, the parameters were evaluated from constant stress data, and were in turn used to predict constant shear rate data as well as the constant stress measurements. It should be emphasized that the parameters were not obtained from the best empirical fit to the data but were evaluated in a manner suggested by the theory. The agreement between theory and data was good enough to ascertain that the approach is adequate for correlating polymer rheology data. The overall average absolute mean deviation ranged from 4.2% for the steady‐state measurements to 11.2% for the constant stress transient measurements. It was further observed that stress overshoot at constant shear rate conditions normally occurred when the Deborah number was greater than unity. Gradual stress growth curves were observed when the number was less than unity.

The Effect of Small Amounts of Long and Short Branches on the Steady State Flow Behavior of Polyethylene‐Wax Blends: Application of Concentration Superposition
View Description Hide DescriptionSteady‐state flow curves were obtained on a series of polyethylene‐wax blends over a shear rate range of to Wax concentration extended from 0 to 85%. The wax was a low molecular weight polyethylene ( and ). Seven polyethylenes with melt indices ranging 0.3 to 9 and ranging from 3 to 15 were examined. Included in the samples were homopolymers, copolymers with a small amount of short branches, copolymers with a small amount of short and long branches, and low density polyethylene. Concentration superpossability was demonstrated with these samples. For the high molecular weight polymers, superposition provided a means of estimating the low shear Newtonian viscosity, which could not be directly measured. Values of were compared to obtained by GPC. The effects of branches on the concentration shift factors for superposition were discussed.

Precision Falling Sphere Viscometry
View Description Hide DescriptionTerminal velocity data of spheres falling in a polyisobutylene (PIB) solution were obtained in sealed tubes. The tubes were easily invertible so that the fall of a sphere could be repeated as often as desired. The sealed tubes have the further advantage that degradation of the fluid is greatly reduced when compared to similar experiments in open tubes. Precisely reproducible velocities were obtained by careful temperature control and by measurement of the radial eccentricity. From such data it is possible to calculate the zero‐shear viscosity, provided the range of effective shear rates is sufficiently small. To acquire data in this range it is necessary to use spheres with small effective mass (actual mass less the mass of the displaced fluid). Spheres of various materials (nylon, ruby, steel, and carbide) were used, and their properties were checked by dropping them in a Newtonian fluid of known viscosity. In some cases the sphere properties were found to fall outside the tolerances specified by the manufacturers. A test of the absolute accuracy of the falling sphere method was made with a calibrated oil supplied by the Cannon Instrument Company. The viscositymeasured with the spheres is within half a percent of the value specified. The data have been analyzed with formulae derived from perturbation calculations based on the theory of Rivlin‐Ericksen fluids. These formulae include the effects of walls and fluid inertia. The third order theory predicts the initial departure from Stokes law. Ideally can be obtained by extrapolation of data in the range of the third order theory. However, for the PIB solution this range appears not to exist or else it falls below that of most of the data. Since the above extrapolation was not feasible, data were taken in tubes of four sizes, and was then deduced from the wall effect formulas. The value so obtained was found to be in good agreement with the values obtained from an extrapolation which assumes the apparent viscosity based on Stokes law varies exponentially with the shear stress. This type of limiting behavior contradicts the third order theory but describes the data remarkably well.

Forced Vibration of Damped Circular and Annular Membranes
View Description Hide DescriptionThe response of circular and annular membranes to sinusoidal vibration is described theoretically. In contrast to prior investigations, emphasis is placed here on the forced vibration of membranes with damping, which is accounted for by writing the membrane tension as a complex quantity. The membranes, of outer radius a, are driven symmetrically as follows: by a ring force of arbitrary radius; by a force that is uniformly distributed within a concentric circle, also of arbitrary radius; or by a central “point” force. The point force can be visualized as the limiting case of the ring‐ or distributed‐force excitation when the radius of application being some numerical multiplier ⩽1.0—approaches zero. As this limit is approached, both the force transmitted to the membrane boundary and the transfer impedance become independent of the value of μ; however, the expressions for driving‐point impedance approach a common value that is proportional to Since, in practice, the impressed force cannot be applied at a point of zero radius, μ can realistically be assigned a small value for which the logarithmic term remains finite and for which meaningful calculations of driving‐point impedance can be made. In the example considered, the area over which the point force is applied is times smaller than the membrane area The physical significance of graphical results that show the frequency dependence of impedance and transmissibility is discussed, and the effect of mass loading the membrane is described. Simple expressions are given from which the membrane damping factor can be deduced if the transmissibility across the membrane is measured at resonance.

Developing Velocity Profiles on the Downstream Side of a Contraction For Inelastic Polymer Solutions
View Description Hide DescriptionDeveloping velocity profiles on the downstream side of a 2 to 1 contraction were measured using a technique involving streak photography. The test fluids employed were water and squeous solutions of Methocel. The flow properties of all fluids were measured using an R16 Weissenberg rheogoniometer, with the shear stress‐shear rate data being fitted by a power‐law model. The results indicate that the velocity profile at the entrance of a 2 to 1 contraction is uniform for and and that the analysis presented by Collins and Schowalter can be used to accurately predict the entry length requirements on the downstream side of a 2 to 1 contraction. Photographs of the flow field through the contraction at low Reynolds numbers clearly show the presence of a stationary vortex on the upstream side of the contraction when axial diffusion of momentum is significant. The stationary vortex is absent at higher Reynolds numbers when the effect of axial diffusion of momentum can be neglected.

On the Ultrasonic Dynamic Viscosities in Superposed Oscillatory Shear
View Description Hide DescriptionThe dynamic shear viscosities for both in‐line and transverse oscillations, superposed on steady simple shearing, have been measured experimentally. It has been found that at large values of the frequency of oscillation, these dynamic viscosities approach the zero shear dynamic viscosity,measured according to the linear theory of viscoelasticity. It is the purpose of this note to derive these ultrasonic limits by a theoretical study of the incompressible BKZ fluid. It is also shown by a counterexample that these limits do not hold for all simple fluids.

Hydrodynamic Interaction Effects in Rigid Dumbbell Suspensions. I. Kinetic Theory
View Description Hide DescriptionA macromolecular solution is represented by the simple model of rigid dumbbells suspended in a Newtonian fluid with Brownian motion included; hydrodynamicinteraction is taken into account. The expressions for the “diffusion equation” and the stress tensor are given. Example results using the equations for the rigid dumbbell suspension modified to include the hydrodynamicinteraction effect are obtained for two flows: steady shearing flow and small‐amplitude oscillatory motion.

On the Consideration of Dynamic Effect in Folding Deformation of a Layered Viscoelastic Medium in Compression
View Description Hide DescriptionPrevious works on folding deformation of a layered viscoelastic medium in compression preclude the consideration of dynamic effect as the contribution due to this effect is assumed to be negligibly small because of sufficiently slow deformation rate. In this discussion the dynamic effect is included in finding a modified expression for the dominant wavelength and the analysis is carried through to deduce criteria for the existence of the dominant wavelength under known prestress conditions. A workable estimate to detect the circumstances within the framework of classical linear theory of elasticity under which the results obtained by quasi‐static and dynamic considerations differ significantly is constructed in later part. The presence of such significant deviation of values indicates the deformation rate is not sufficiently slow and therefore the inertia term characterizing the dynamic effect can not be neglected. To elucidate further a graph giving the values of the ratio of the dominant wavelengths in two considerations is provided in the end.

Mill Processing Behavior of Ethylene‐Propylene‐Hexadiene Terpolymers
View Description Hide DescriptionThe mill processing behavior of ethylene/propylene/hexadiene terpolymers, one type of EPDM, has been described. In agreement with the general observations of Tokita and White, the millingcharacteristics of the polymers change from an elastic band (region 2) to a crumbling bag (region 3) and then to a viscoelastic fluid (region 4) as the temperature is increased from 5°C to 175°C. The 2–3 transition can be treated as a failure process which is time and temperature dependent. Hence, a continuous curve relating the transition nip setting; and the temperature exists for each polymer sample. The transition nip setting at a given temperature (or the transition temperature at a given nip width) was found to be wider (or higher) for samples of broader molecular weight distribution.

Hydrorheology of Clay Soils
View Description Hide DescriptionA multiple integral theory is advanced to describe the influence of water content on the mechanical behavior of clay soils. Experimental data from three series of creep tests and one set of relaxation tests over a wide range from a few seconds to beyond 10,000 minutes are reported and analyzed. In the range of experimental observations the theoretical results show good agreement with the experimental data.

Note: An Integral Constitutive Equation and Rate‐Dependent Relaxation Spectrum
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