A. Jeffrey Giacomin

Large-amplitude oscillatory shear: comparing parallel-disk with cone-plate flow

We compare the ratio of the amplitudes of the third to the first harmonic of the torque, , measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics of the shear stress, |τ3|/|τ1|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a correction factor with which must be multiplied, to get the quantity |τ3|/|τ1|, where |τ3|/|τ1| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between and τ3/τ1 using the simplest continuum model relevant to large-amplitude oscillatory shear flow: the single relaxation time co-rotational Maxwell model. We focus on the region where the harmonic amplitudes and thus, their ratios, can be fully described with power laws. This gives the expression for , by integrating the explicit analytical solution for the shear stress. In the power law region, we find that, for low Weissenberg numbers, for the third harmonics , and for the fifth harmonics, . We verify these results experimentally. In other words, the heterogeneous flow field of the parallel-disk geometry significantly attenuates the higher harmonics, when compared with the homogeneous, sliding-plate flow. This is because only the outermost part of the sample is subject to the high shear rate amplitude. Furthermore, our expression for the torque in large-amplitude oscillatory parallel-disk flow is also useful for the simplest design of viscous torsional dampers, that is, those incorporating a viscoelastic liquid between two disks.

Padé approximants for large-amplitude oscillatory shear flow

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.

Die drool theory

Abstract When molten plastic is extruded from a die, it sometimes collects on the open face of the die. Known as die drool, this phenomenon costs plastics manufacturers by requiring die cleaning. This has been attributed to many causes, but none of these has led to an equation for the drool rate. In this work, we provide an exact analytical solution for the drool rate, and we base this solution on a postulate of a cohesive slip layer near the die walls. We thus attribute die drool to cohesive failure within the fluid at an internal surface where the fluid slips on itself. We adimensionalize the drool rate with the production rate, and call this the buildup ratio, BR. We provide an exact analytical solution for BR when the cohesive slip layer either slips at the die wall, or when it does not. We examine two important extrusion geometries: slit (which we then extend to pipe) and tube flow. We identify two new experiments: one to measure BR as a function of pressure drop, and another as a function of the die aspect ratio, and we then use our new theory to design droolometers.

Assumed periodicity and dynamic shear stress transduction in rheometry

In viscoelastic property measurements, material is subjected to time unsteady deformations using a rheometer. In step shear strain experiments, for example, the shear strain suddenly jumps to a steady value. In this paper, we develop a method to study the dynamic response of a shear stress transducer in a sliding plate rheometer for any time unsteady rheological test. This general method is developed by first considering a special case of step shear strain for a fluid sample with material ingress in the annular transducer gap. Both the fluid sample and its ingress obey the generalized Maxwell model. Our main mathematical trick is assumed periodicity where a single step in shear strain is treated as the first step in a reciprocating square wave. After solving the problem in the frequency domain, we recover our single step by taking the limit as the period goes to infinity. The transducer eccentricity following step shear is then determined analytically by solving the force balance on the transducer active ...

Viscoelasticity in thermoforming

Abstract This analysis for thermoforming cones from a viscoelastic melt gives the speed of the stresses in the deforming melt. We follow the mechanics of Kershner and Giacomin developed for thermoforming cones from a Newtonian melt, deriving an analytical solution for the corotational Maxwell (CM) model. We have distinguished what happens before and after the melt touches the conical mold (free vs. constrained forming). The CM reduces to the Newtonian solution of Kershner and Giacomin. The upper convected Maxwell (UCM) model is also investigated, but does not yield an analytical solution for conical thermoforming. Our results are confined to forming with a conical mold, the simplest relevant problem in thermoforming, using the simplest relevant constitutive equation, the CM model.

Die Lines in Plastics Extrusion

Die lines in plastics extrusion are annoying surface striations that can destroy extrudate aesthetics and compromise product properties. Here, the available literature on die lines is reviewed to explore how die lines arise and how to minimize them. Many factors influence die line formation, and there are a few ways to suppress them. Future study on die lines should include the non-Newtonian fluid mechanics of the polymer melt.

Exact solutions for oscillatory shear sweep behaviors of complex fluids from the Oldroyd 8-constant framework

In this paper, we provide a new exact framework for analyzing the most commonly measured behaviors in large-amplitude oscillatory shear flow (LAOS), a popular flow 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. By the strain rate sweep, we mean a sequence of LAOS experiments conducted at the same frequency, performed one after another, with increasing shear rate amplitude. 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 coef...

Die drool and die drool theory

When molten plastic is extruded from a die, it sometimes collects on the open face of the die. Known as die drool, this phenomenon costs plastics manufacturers by requiring die cleaning. This has been attributed to many causes, but none of these has led to an equation for the drool rate. In this work we provide an exact analytical solution for the drool rate, and we base this solution on a postulate of a cohesive slip layer near the die walls. We thus attribute die drool to cohesive failure within the fluid at an internal surface where the fluid slips on itself. We adimensionalize the drool rate with the production rate, and call this the build up ratio, BR. We provide an exact analytical solution for BR when the cohesive slip layer either sticks at the wall. We examine the slit geometry corresponding to sheet or film extrusion.

Wire Coating Under Vacuum

A numerical solution is presented for isothermal Newtonian wire coating under vacuum or with an externally applied pressure, Differential equations for the melt cone shape are derived from a force balance on the melt cone. Die design curves are then constructed from the solutions to these equations. These curves are helpful in predicting wire coating operating conditions.

Power Law Model for Tube Coating of Wire

A numerical solution is presented for isothermal power law tube coating of wire under vacuum or with an externally applied pressure. Differential equations for the melt cone shape are derived from a force balance on the melt cone. The effect of shear thinning on the melt cone shape and operating parameters is examined. We show how to construct die design curves from solutions to these equations. These curves help wire coating engineers predict operating conditions.