Peter H. Gilbert

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.

Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number λω is zero and the Weissenberg number λγ˙0 is above unity), (ii) nonlinear viscoelasticity (where both λω and λγ˙0 exceed unity), and (iii) linear viscoelasticity (where λω exceeds unity and where λγ˙0 approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.

Temperature rise in a verging annular die

Abstract Plastic pipes, tubes or catheters are extruded by pressure-driven flows through annular dies. Whereas die lands are straight, the section connecting the die land to the extruder either converges or diverges, converging when the product is smaller than the extruder barrel, and diverging when larger. In this paper, we carefully consider the converging or diverging connecting flows, in spherical coordinates, for the most common configuration: the Newtonian pressure-driven flow through the annulus between two coapical coaxial cones. We derive the exact analytical solution for the velocity profile, and then use this to arrive at the exact analytical solution for the temperature rise caused by viscous heating. We care about this rise because it often governs maximum throughput, since pipe makers must protect the melt from thermal degradation. We find that both the velocity profile, and the temperature profile, peak over the same conical surface and this surface is nearer the inner die wall. We also provide analytical expressions for the nonlinear pressure profile and the die cooling requirement. We find that this cooling requirement is always higher on the inner cone.

Die Drool and Polymer Degradation

ABSTRACT Die drool is an operational problem associated with polymer extrusion. The extrudate collects outside of the die, necessitating periodic disruptions for cleaning. There is a debate over the mechanism that produces die drool: stress induced fractionation or thermal degradation. This article examines the latter. In cohesive failure, a slip discontinuity develops in the velocity profile, where frictional heating occurs. This slip heating can contribute to resin degradation, resulting in lower molecular weight fragments in the die drool. This article examines resin degradation kinetics and its influence on die drool rates and on the resulting drool layer and bulk polymer concentration profiles. GRAPHICAL ABSTRACT

Die drool and polymer degradation

Die drool is an operational problem associated with polymer extrusion. Extrudate collects outside of the die, necessitating periodic disruptions for cleaning. There exists some debate as to the mechanism that produces die drool: stress induced fractionation or thermal degradation. This paper examines the latter. In cohesive failure, a slip discontinuity develops in the velocity profile, where heat is generated by friction. This slip heating can contribute to resin degradation, resulting in lower molecular weight fragments in the die drool. This paper examines the kinetics of this degradation, its influence on die drool rates, and on the resulting polymer concentration profiles in the drool layer and in the bulk.