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.
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.
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.
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.
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