Volume 16, Issue 2, July 1972
Index of content:

Nearly Viscometric Flow in the Disk and Cylinder System. I: Theoretical
View Description Hide DescriptionFor the steady flow of viscoelastic liquids a perturbation theory is developed for a weak secondary flow superimposed on an arbitrary primary flow. The dynamic and kinematic equations are used in a simplified form appropriate to slow flow, but the constitutive equations are not so approximated, being allowed to retain general nonlinear viscoelastic effects. The equations are used to study flow driven by a disk rotating atop a cylindrical tank. The governing equations are solved using Fourier‐Bessel series. Comparison with experimental data is given in part II.

Nearly Viscometric Flow of Viscoelastic Fluids in the Disk and Cylinder System. II: Experimental
View Description Hide DescriptionRadial and tangential velocity distributions are measured for laminar flow of two Newtonian and one viscoelastic fluids in the disk and cylinder system. Velocities are determined from time lapse photographs of small particles suspended in the fluid which is illuminated in thin sections by a collimated beam of light. Measurements are made at Reynolds numbers of 0.0616, 24.4, and 97.6 for the Newtonian fluids and at 0.0371 for the viscoelastic fluid. The distributions are compared with the analyses of Kramer and Johnson in Part I. A toroidal secondary flow is observed which circulates in opposite directions for Newtonian and viscoelastic fluids, and a study of the dependence of the secondary flow direction on fluid properties and disk speed is reported. Complex flow patterns which appear in the region in which the direction of secondary flow changes are described. For Newtonian fluids tangential velocity computations agree to ±5% with the experimental measurements up to a Reynolds number of 24.4, and predicted radial velocities agree with experimental data within 20% at the Reynolds number of 24.4. At a Reynolds number of 97.6 predicted tangential velocities differ from experimental measurements by as much as ±50% while radial velocities differ from experimental values by a factor of two to three. The second order fluid predicts tangential velocities which are twice as large as experimental values and radial velocities which are in error by as much as a factor of six. The WJFLMB model predicts tangential and radial velocities which agree with experiment to within ±20% and ±40%, respectively. A Weissenberg Rheogoniometer is used to measure non‐Newtonian viscosity, complex viscosity, and primary normal stress differences of the viscoelastic fluids, and to measureviscosities of the Newtonian fluids.

Flow of a Nonlinear Heat‐Conducting Viscoelastic Fluid
View Description Hide DescriptionThe Koh‐Eringen formulation of nonlinear thermoviscoelasticity is used to study the behavior of a heat‐conducting Rivlin‐Ericksen fluid characterized by a set of constitutive equations involving the coupled dependence of stress and heat flux on temperature gradient and the first two Rivlin‐Ericksen tensors. Slow steady‐state flow of the fluid under a small temperature gradient is considered. With the use of a perturbation technique, the basic equations and boundary conditions are derived for each increasing order of approximation. The first order equations are precisely those of a Newtonian fluid. Succeeding higher order corrections involve linear equations also although the results of preceding lower order approximations are used. The given boundary conditions are satisfied in the first order analysis. Homogeneous boundary conditions are then applicable for higher order corrections. Two specific problems are considered: (1) pressure flow of the fluid between two parallel plates in relative motion and at different constant temperatures; and (2) Couette‐helical flow in the annular region between two coaxial circular cylinders in relative translational velocity parallel to the axis and at two different constant temperatures. Numerical results are plotted for several Eckert numbers and other thermomechanical parameters.

Prediction of the Bagley End Correction Factor and Die Swell for Capillary Flow of an Oldroyd Rate Type Fluid
View Description Hide DescriptionThe significance of the Bagley capillary end correction factor and capillary die swell are analytically investigated for a fluid obeying the quasilinear Oldroyd rate type constitutive equation which incorporates the Jaumann differential operator. The Goddard and Miller integration technique is used, under the assumption that the fluid enters the capillary with a fully developed velocity profile, to invert the constitutive equation and to produce an explicit relation for the pressure drop in the capillary. The inlet stress field decay in the capillary is shown to produce both nonlinear die swell behavior and nonlinear pressure drop behavior with increasing The theory is substantiated, in part, with experimental results.

Linear Stability of Plane Poiseuille Flow of Viscoelastic Liquids
View Description Hide DescriptionThe stability of plane Poiseuille flow to infinitesimal perturbations was studied for the second order and Maxwell fluid rheological models. When the Deborah number based disturbance propagation is small the second order fluid is a consistent constitutive equation and the two models give identical results. This occurs for elasticity number (E) less than At higher values of E the second order fluid cannot be used. The critical Reynolds number is a slowly decreasing function of E up to and a rapidly decreasing function subsequently in the region where fluid relaxation effects become important. For a sufficiently elastic fluid the flow transition is governed by a new mode of the Orr‐Sommerfeld equation and differs qualitatively from that for a Newtonian liquid. The results suggest the likelihood of low Reynolds numberinstability in highly elastic liquids.

Nonlinear Stability of Plane Poiseuille Flow of Viscoelastic Liquids
View Description Hide DescriptionThe stability of plane Poiseuille flow to finite disturbances was studied for the second order fluid using the cascade (Stuart‐Watson‐Reynolds‐Potter) method. The analysis was carried out for elasticity numbers up to In this range the Deborah number based upon disturbance propagation is small and the second order fluid is a consistent constitutive approximation. In contrast to linear theory in this region, elasticity has a stabilizing effect on finite disturbances relative to the Newtonian liquid and the difference between stability behavior of Newtonian and slightly elastic liquids tends to disappear.

The Free Energy Constitutive Equation for Polymer Solutions from the Dumbbell Model
View Description Hide DescriptionThe simplest model of flexible macromolecules in a dilute solution is the elastic dumbbell (or bead‐spring) model. This has been widely used for purely mechanical theories of the stress. In this work the model is used to determine the constitutive equation for the free energy and for the stress under nonisothermal conditions. The results are shown to be consistent with the general thermodynamic theory of simple fluids with fading memory.

End Corrections in Capillary Flow of Polymer Melts with Nonzero Exit Pressures
View Description Hide DescriptionCapillary flow data for a polypropylene melt exhibiting a positive exit pressure are analyzed using the normal Bagley correction procedure. The results obtained by this method are shown to be the same as those derived by the Hagenbach‐Couette procedure. The agreement is exact because the Bagley procedure implicitly involves a linear extrapolation to zero exit pressure. Thus it is not necessary to subtract the exit pressure from total pressure drop as suggested by other workers. The Bagley procedure correctly adjusts data obtained for polymer melts in capillary flow experiments even if a positive exit pressure exists.

Microstructure of Melt Blended Polymer Systems
View Description Hide DescriptionTwo thermodynamically incompatible polymers were melt blended. The resulting microstructure and, consequently, the physical properties were dependent on the blending conditions and the rheological properties of the components. In this study the blending conditions were held constant while the rheological properties were varied and correlated with dispersioncharacteristics of the phases and the physical properties of the composite. A series of five atactic polystyrenes (weight average molecular weight from 75,000 to 400,000) and five linear polyethylenes (weight average molecular weight from 43,000 to 347,000) were used. The blending was done over a range of composition ratios in a normal stress extruder. Viscosity and normal stress of each pure component were measured in torsional flow. The mixed extrudates were examined by light interference microscopy,scanning electron microscopy, and solvent leaching. The nature and the fineness of the microstructure were interpreted in terms of the composition ratios, mixing history, and the melt rheological properties of the components. It was concluded that the fineness of the minor phase in a shear mixing process is a function of the difference between both the viscous and elastic behavior of the components.

Velocity Profiles in Tube Flow
View Description Hide DescriptionIt is proposed to show that a paper presented by Gerrard, which asserts that the velocity profile of a fluid flowing through a tube can be uniquely determined by measuring the thrust and the corresponding flow rate, is incorrectly formulated. An example is discussed that demonstrates the nonuniqueness of the velocity profile at a fixed ratio of thrust to flow rate.

Note: Predictions of the Spriggs Viscoelastic Model for the Biaxial Extensional Viscosity
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