Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

Effect of a controlled pre-deformation history on extensional viscosity of dilute polymer solutions

We use a modified filament stretching rheometer to quantify the influence of a known controlled pre-shear history on the transient extensional viscosity of a dilute polymer solution. Two different types of pre-deformation are explored; both influence the subsequent stretching significantly, albeit in opposite ways. Small-amplitude oscillatory straining parallel to the direction of stretching enhances strain hardening and accelerates the tensile stress growth toward the steady-state value. Conversely, steady torsional shearing orthogonal to the direction of stretching retards strain hardening and results in a delayed approach to steady-state elongational flow. In both cases, the final steady-state extensional viscosity is the same as that observed with no pre-shearing. Calculations using a finitely extensible nonlinear elastic Peterlin dumbbell model qualitatively capture the trends observed in experiments, enabling interpretation of these observations in terms of the degree of polymer chain stretching imposed by the flow before extensional stretching.     

Numerical modelling of step–strain for stretched filaments

This article addresses the modelling of filament-stretching/step–strain deformation under viscoelastic capillary break-up configurations of the CaBER-type. Start-up, prior to step–strain, is conducted under constant stretch-rate synchronous plate retraction with impulsive sessation of plate motion. The study encompasses variation in material rheology, appealing to Oldroyd, Geisekus and Phan-Thien/Tanner-type models, which display differences in shear and extensional viscosity properties (shear thinning/extension hardening). Two different viscosity ratio settings are considered to reflect high- and low-solvent viscosity constituent components; the former representing typical Boger fluids, the latter high-polymer concentration fluids. We compare and contrast results for three alternative filament aspect ratios at the onset of step–strain. Throughout the step–strain period, we have been able to successfully capture such physical features as drainage to the filament feet, necking at the filament centre, and periods with travelling waves through the axial filament length. In addition, we have identified the suppressive influence that larger capillary forces have upon radial fluctuations, and the minor impact that gravitational forces have upon the ensuing deformation. From this study, estimates for rheometrical data have been derived in terms of characteristic material relaxation time and apparent extensional viscosity. The computational techniques employed include a compressed-mesh (CM) procedure, an Arbitrary Lagrangian–Eulerian scheme (ALE) and a free-surface particle tracking technique. Spatial discretisation of the problem is accomplished through a hybrid finite element/finite volume algorithm implemented in the form of a time-stepping incremental pressure-correction formulation.     

Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation

Primary atomization of liquid injected at high speed into still air is investigated to elucidate physical processes by direct numerical simulation. With sufficient grid resolution, ligament and droplet formation can be captured in a physically sound way. Ligament formation is triggered by the liquid jet tip roll-up, and later ligaments are also produced from the disturbed liquid core surface in the upstream. Ligament production direction is affected by gas vortices. Disturbances are fed from the liquid jet tip toward upstream through vortices and droplet re-collision. When the local gas Weber number is O(1), ligaments are created, thus the ligament or droplet scale becomes smaller as the bulk Weber number increases. Observation of droplet formation from a ligament provides insights into the relevance between the actual droplet formation and pinch-off from a slow liquid jet in laboratory experiments. In the spray, the dominant mode is the short-wave mode driven by propagative capillary wave from the ligament tip. An injection nozzle that is necessary for a slow jet is absent for a ligament, thus the long-wave (Rayleigh) mode is basically not seen without the effect of stretch. By the present simulation, a series of physical processes have been revealed. The present result will be extended to LES modeling in the future.     

Ligament formation in sheared liquid–gas layers

We perform numerical simulations of two-phase liquid–gas sheared layers, with the objective of studying atomization. The Navier–Stokes equations for two-dimensional incompressible flow are solved in a periodic domain. A volume-of-fluid method is used to track the interface. The density ratio is kept around 10. The calculations show good agreement with a fully viscous Orr–Sommerfeld linear theory over several orders of magnitude of interface growth. The nonlinear development shows the growth of finger-like structures, or ligaments, and the detachment of droplets. The effect of the Weber and Reynolds numbers, the boundary layer width and the initial perturbation amplitude are discussed through a number of typical cases. Inversion of the liquid boundary layer is shown to yield more readily ligaments bending upwards and is thus more likely to produce droplets.     

Using filament stretching rheometry to predict strand formation and “processability” in adhesives and other non-Newtonian fluids

The spinning of polymeric fibers, the processing of numerous foodstuffs and the peel and tack characteristics of adhesives are all associated with the formation, stability and, ultimately, the longevity of thin fluid `strands'. This tendency to form strands is usually described in terms of the tackiness of the fluid or by heuristic concepts such as `stringiness' (Lakrout et al. J Adhesion 1999). The dynamics of such processes are complicated due to spatially and temporally non-homogeneous growth of extensional stresses, the action of capillary forces and the evaporation of volatile solvents. We describe the development and application of a simple instrument referred to as a microfilament rheometer (MFR) that can be used to readily differentiate between the dynamical response of different pressure-sensitive adhesive fluid formulations. The device relies on a quantitative observation of the rate of extensional thinning or `necking' of a thin viscous or viscoelastic fluid filament in which the solvent is free to evaporate across the free surface. This high-resolution measurement of the radial profile provides a direct indication of the ultimate time to break up of the fluid filament. This critical time is a sensitive function of the rheological properties of the fluid and the mass transfer characteristics of the solvent, and can be conveniently reported in terms of a new dimensionless quantity we refer to as a processability parameter P. We demonstrate the usefulness of this technique by presenting our results in the form of a case study in which we measure the visco-elasto-capillary thinning of slender liquid filaments for a number of different commercial polymer/solvent formulations and relate this to the reported processing performance of the materials. We also compare the MFR observations with the prediction of a simple 1D theory derived from the governing equations that model the capillary thinning of an adhesive filament. Received: 22 December 1999/Accepted: 4 January 2000     

Evaluation by means of stress relaxation (after a step strain) experiments of the viscoelastic behavior of polymer melts in uniaxial extension

 As is widely acknowledged, morphology in most materials is far more sensitive to extensional than to shear deformations but, unfortunately, due to the experimental difficulties involved, there are no non-destructive, morphology probing techniques in such flows, i.e., the equivalent of stress relaxation and oscillatory experiments in shear flows. This paper tries to overcome some of those drawbacks by proposing an experimental technique that allows stress relaxation experiments after a step strain in uniaxial extension to be performed. The benefits of this technique are twofold: (a) while the deformation is small enough for the response to be in the linear viscoelastic regime it constitutes a probe of the microstructure of the material and (b) it allows the departure to the non-linear regime to be studied, useful, for example, for the definition of the damping function in uniaxial extensional flow or for the study of the response of materials to fast transient flows with a strong extensional component, such as contraction flows. In this work the proposed technique, which requires a correction to the apparent (theoretical) strain rate in order to allow the calculation of the true Hencky strains attained during the strain step, is tested and validated for two polyisobutylene melts. Received: 9 April 2001 Accepted: 26 July 2001     

Consideration of the yarn–yarn interactions in meso/macro discrete model of fabric. Part I: Single yarn behaviour

A mesoscopic discrete model of dry fabric has been developed, based on the yarn–yarn interactions occurring at the yarns crossing points. The fabric yarns, described initially by a Fourier series development, are discretized into elastic straight bars represented by stretching springs and connected at frictionless hinges by rotational springs. The motion of each node is described by a lateral displacement and a rotation. The expression of the reaction force exerted by the transverse yarns at the contact points is assessed, from which the work of the reaction forces is established. The equilibrium shape of the yarn is obtained as the minimum of its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. Simulations of a traction curve of a single yarn are performed, that evidence the effect of the yarn interactions. The two principal deformation mechanisms, the variation of undulation and the yarn stretching, are separately analysed.