This report is circulated to persons believed to have an active interest in the subject matter; it is intended to furnish rapid communication and to stimulate comment, including corrections of possible errors.

When polymeric liquids undergo large‐amplitude shearing oscillations, the shear stress responds as a Fourier series, the higher harmonics of which are caused by fluid nonlinearity. Previous work on large‐amplitude oscillatory shear flow has produced analytical solutions for the first few harmonics of a Fourier series for the shear stress response (none beyond the fifth) or for the normal stress difference responses (none beyond the fourth) [JNNFM, 166, 1081 (2011)], but this growing subdiscipline of macromolecular physics has yet to produce an exact solution. Here, we derive what we believe to be the first exact analytical solution for the response of the extra stress tensor in large‐amplitude oscillatory shear flow. Our solution, unique and in closed form, includes both the normal stress differences and the shear stress for both startup and alternance. We solve the corotational Maxwell model as a pair of nonlinear‐coupled ordinary differential equations, simultaneously. We choose the corotational Maxwell model because this two‐parameter model (η0 and λ) is the simplest constitutive model relevant to large‐amplitude oscillatory shear flow, and because it has previously been found to be accurate for molten plastics (when multiple relaxation times are used). By relevant we mean that the model predicts higher harmonics. We find good agreement between the first few harmonics of our exact solution, and of our previous approximate expressions (obtained using the Goddard integral transform). Our exact solution agrees closely with the measured behavior for molten plastics, not only at alternance, but also in startup.

Large-amplitude oscillatory shear flow (LAOS) is a popular for studying the nonlinear physics of complex fluids. Specifically, the strain rate sweep (also called the strain sweep) is used routinely to identify the onset of nonlinearity. In this paper, we give exact expressions for the nonlinear complex viscosity and the corresponding nonlinear complex normal stress coefficients for the Oldroyd 8-constant framework for oscillatory shear sweeps. We choose the Oldroyd 8-constant framework for its rich diversity of popular special cases (we list 18 of these). We evaluate the Fourier integrals of our previous exact solution to get exact expressions for the real and imaginary parts of the complex viscosity, and for the complex normal stress coefficients, as functions of both test frequency and shear rate amplitude. For our comparisons with data, we use the Spriggs relations to generalize the Oldroyd 8-constant framework to multimode. We explore the role of infinite shear rate viscosity on strain rate sweep responses for the special case of the corotational Jeffreys fluid. We find that raising η ∞ , raises the real part of the complex viscosity, and lowers the imaginary. In our worked examples, we thus first use the corotational Jeffreys fluid, and then, for greater accuracy, we use the Johnson-Segalman fluid, to describe the strain rate sweep response of molten atactic polystyrene. Our generalization yields unequivocally, a longest fluid relaxation time, used to assign Weissenberg and Deborah numbers to each oscillatory shear flow experiment. We then locate each experiment in Pipkin space. () are the real and minus [10] the imaginary parts of the nonlinear complex viscosities, η n * , and first and second normal stress coefficients, Ψ 1,n * and Ψ 2,n * . Some prefer to write Eq. (6) as (see Eq.(3) of [11]):

This report is circulated to persons believed to have an active interest in the subject matter; it is intended to furnish rapid communication and to stimulate comment, including corrections of possible errors.

No abstract

To solidify plastic pipe, the pipe is transported through a long cooling chamber. Inside this chamber, inside the pipe, the plastic remains molten, and this inner surface solidifies last. The flow due to the self-weight of the molten plastic then causes the product to thicken on bottom (and to thin on top). This is why plastic pipe is normally extruded from an eccentric die, and specifically, from a die where the mandrel (also called centerpiece [62] or core [12]) is decentered downward. This paper focuses on the consequences of this decentering in eccentric cylindrical coordinates. Specifically, when the molten polymer is viscoelastic, as is normally the case, an downward lateral force, F x , is exerted on the mandrel. The die eccentricity also affects the positive axial force on the mandrel, F z . These forces govern how rigidly the mandrel must be attached (normally, on a spider die, by eight bolts). We use the method of Jones (1964), called polymer process partitioning, designed for the Oldroyd 8-constant constitutive model. We produce a method for estimating F x and F z . We also obtain an expression for the shape of the extruded pipe, whose thickness scales with average velocity at each angular positions ⌣ v z θ , by integrating the velocity profile, ⌣ v z ξ ,θ ( ) through the thickness of the eccentric annulus (with respect to ξ ). We further include expressions for the stresses in the extruded polymer melt. These expressions can be used to estimate the upper-bound for the stress that is frozen into the outermost layer of the plastic pipe (since this layer is quenched first, shortly after extrusion). We include detailed dimensional worked examples to help process engineers with their pipe die designs.

General rigid bead-rod theory [O. Hassager, “Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60(10), 4001–4008 (1974)] explains polymer viscoelasticity from macromolecular orientation. By means of general rigid bead-rod theory, we relate the complex viscosity of polymeric liquids to the architecture of axisymmetric macromolecules. In this work, we explore the zero-shear and complex viscosities of 24 different axisymmetric polymer configurations. When nondimensionalized with the zero-shear viscosity, the complex viscosity depends on the dimensionless frequency and the sole dimensionless architectural parameter, the macromolecular lopsidedness. In this work, in this way, we compare and contrast the elastic and viscous components of the complex viscosities of macromolecular chains that are straight, branched, ringed, or star-branched. We explore the effects of branch position along a straight chain, branched-chain backbone length, branched-chain branch-functionality, branch spacing along a straight chain (including pom-poms), the number of branches along a straight chain, ringed polymer perimeter, branch-functionality in planar stars, and branch dimensionality.

Analytical solutions for either the shear stress or the normal stress differences in large-amplitude oscillatory shear flow, both for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. Here, we explore improving the accuracy of these truncated series by replacing them with ratios of polynomials. Specifically, we examine replacing the truncated series solution for the corotational Maxwell model with its Padé approximants for the shear stress response and for the normal stress differences. We find these Padé approximants to agree closely with the corresponding exact solution, and we learn that with the right approximants, one can nearly eliminate the inaccuracies of the truncated expansions.