Steady Extensional Flow Research Articles

Full linear Phan-Thien–Tanner fluid model: Exact analytical solutions for steady, startup, and cessation regimes of shear and extensional flows

Exact, fully explicit, purely real analytical expressions for the material functions describing steady, startup, and cessation regimes of shear flows and of planar, uniaxial, and biaxial extensional flows of full linear Phan-Thien–Tanner fluids are obtained. These expressions, which have no analogs in the literature, are formulated in compact, beautiful forms, partially due to the unique scaling procedure reducing the number of the model parameters from four to one. The properties of the material functions are investigated in detail. For steady extensional flows, the possible shapes of the extensional viscosity curves are described and the conditions for these shapes to occur are determined. For startup flows, it is found when exactly the stress dynamics is oscillatory, and, in this case, a detailed characterization of oscillations is given, which includes expressions for the position and magnitude of stress overshoots and undershoots.

Predictions for flow-induced scission in well-entangled living polymers: The “living Rolie-Poly” model

In polymeric materials subject to both polymerization reactions and flow, there can be a complex interplay between reactions and stress relaxation processes. For example, reversible scission reactions can “shuffle” stresses across the molecular weight distribution, narrowing the stress relaxation spectra and decreasing the typical stress relaxation time. In addition, flow can stretch chains and make them more likely to break apart, leading to changes in the underlying reaction kinetics. Existing strategies for modeling the coupling between reactions and flow in polymer systems are limited in their range of applicability or dubious in their underlying approximations. Here, we develop a more flexible modeling approach with coupled population balance models that move both material and stress across the molecular weight distribution. The full model, which we call the “living Rolie Poly” (LRP) model, reproduces some earlier findings on linear rheology and offers new insights into nonlinear rheology and the role of flow-induced scission. For systems that are not inclined to shear band in the absence of flow-induced scission, we predict that flow-induced scission produces an additional shear thinning effect for steady shear flow, and in steady extensional flow, we predict that flow-induced scission acts like a finite extensibility correction, preventing the divergence of the steady viscosity. The LRP model is too complex to use in spatially resolved calculations or complex flow geometries at this time, but a “simplified” model with no such limitation arises naturally in the “fast--breaking” limit and shows good agreement with the full LRP model predictions.

Dynamics and rheology of ring-linear blend semidilute solutions in extensional flow: Single molecule experiments

Ring polymers exhibit unique flow properties due to their closed chain topology. Despite recent progress, we have not yet achieved a full understanding of the nonequilibrium flow behavior of rings in nondilute solutions where intermolecular interactions greatly influence chain dynamics. In this work, we directly observe the dynamics of DNA rings in semidilute ring-linear polymer blends using single molecule techniques. We systematically investigate ring polymer relaxation dynamics from high extension and transient and steady-state stretching dynamics in a planar extensional flow for a series of ring-linear blends with varying ring fraction. Our results show multiple molecular subpopulations for ring relaxation in ring-linear blends, as well as large conformational fluctuations for rings in a steady extensional flow, even long after the initial transient stretching process has subsided. We further quantify the magnitude and characteristic time scales of ring conformational fluctuations as a function of blend composition. Interestingly, we find that the magnitude of ring conformational fluctuations follows a nonmonotonic response with increasing ring fraction, first increasing at low ring fraction and then substantially decreasing at large ring fraction in ring-linear blends. A unique set of ring polymer conformations are observed during the transient stretching process, which highlights the prevalence of molecular individualism and supports the notion of complex intermolecular interactions in ring-linear polymer blends. In particular, our results suggest that transient intermolecular structures form in ring-linear blends due to a combination of direct forces due to linear chains threading through open rings and indirect forces due to hydrodynamic interactions; these combined effects lead to large conformational fluctuations of rings over distributed time scales. Taken together, our results provide a new molecular understanding of ring polymer dynamics in ring-linear blends in the nonequilibrium flow.

Start-up and cessation of steady shear and extensional flows: Exact analytical solutions for the affine linear Phan-Thien–Tanner fluid model

Exact analytical solutions for start-up and cessation flows are obtained for the affine linear Phan-Thien–Tanner fluid model. They include the results for start-up and cessation of steady shear flows, of steady uniaxial and biaxial extensional flows, and of steady planar extensional flows. The solutions obtained show that at start-up of steady shear flows, the stresses go through quasi-periodic exponentially damped oscillations while approaching their steady-flow values (so that stress overshoots are present); at start-up of steady extensional flows, the stresses grow monotonically, while at cessation of steady shear and extensional flows, the stresses decay quickly and non-exponentially. The steady-flow rheology of the fluid is also reviewed, the exact analytical solutions obtained in this work for steady shear and extensional flows being simpler than the alternative formulas found in the literature. The properties of steady and transient solutions, including their asymptotic behavior at low and high Weissenberg numbers, are investigated in detail. Generalization to the multimode version of the Phan-Thien–Tanner model is also discussed. Thus, this work provides a complete analytical description of the rheology of the affine linear Phan-Thien–Tanner fluid in start-up, cessation, and steady regimes of shear and extensional flows.

A charged finitely extensible dumbbell model: Explaining rheology of dilute polyelectrolyte solutions

A robust non-Newtonian fluid model of dilute polyelectrolyte solutions is derived from kinetic theory arguments. Polyelectrolyte molecules are modeled as finitely elongated nonlinear elastic dumbbells, where effective charges (interacting through a simple Coulomb force) are added to the beads in order to model the repulsion between the charged sections of polyelectrolyte chains. It is shown that the relative strength of this repulsion is regulated by the electric-to-elastic energy ratio, E, which is one of the key parameters of the model. In particular, E accounts for the intrinsic rigidity of polyelectrolyte molecules and can be used to explain the impact of solvent salinity on polyelectrolyte rheology. With two preaveraging approximations, the constitutive equations of the resulting fluid model are formulated in closed form. Material functions predicted by the model for steady shear flow, steady extensional flow, small-amplitude oscillatory shear flow, and start-up and cessation of steady shear flow are obtained and investigated using a combination of analytical and numerical methods. In particular, it is shown how these material functions depend on E. The two limiting cases of the model—uncharged dumbbells (E = 0) and rigid dumbbells (E → ∞)—are included in the analysis. It is found that despite its simplicity, the model predicts most of the experimentally observed rheological features of polyelectrolyte solutions.

An exploration and further study of an enhanced Oldroyd model

The Oldroyd 6-constant constitutive model for polymeric liquids [J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. R. Soc. A 200, 523 (1950)] was shown in the work of Bird [“A modification of the Oldroyd model for rigid dumbbell suspensions with Brownian motion,” Z. Angew. Math. Phys. 23, 157 (1972)] to have insufficient flexibility even to describe all second-order time-dependent behaviors of the simple molecular model of rigid dumbbells in solution. Bird proposed an enhancement of the Oldroyd model that would remove this deficiency. The advantage of such a continuum constitutive model is that it is far easier to use in solving specific flow problems than the more physical, but far more cumbersome, molecular models. Thus, if this enhanced constitutive model is indeed able to replicate the results of the molecular models, it should be extremely useful over the applicable range. Here, we employ the enhanced constitutive model of Bird [Z. Angew. Math. Phys. 23, 157 (1972)] to solve four different flow problems, and we verify that our solutions coincide exactly with those of a molecular model: through third order for rectilinear steady shear flow and steady extensional flow, and for one steady non-rectilinear flow (eccentric disk rheometer flow); and through second order for one unsteady (i.e., oscillatory) shear flow, thus providing strong confirmation of the physical veracity and utility of this continuum constitutive model, at least for small and moderate shear or extension rates.

Dynamics of Star Polymers in Fast Extensional Flow and Stress Relaxation

We confirm the observation from Ianniruberto and Marrucci [Macromolecules 2013, 46, 267−275] that entangled melts of branched polystyrenes behave like linear polystyrenes in the steady state of fast extensional flow, by measuring a linear, an asymmetric star, and a symmetric star polystyrene with the same span molecular weight (180 kg/mol). We show that all three melts reach the same extensional steady-state viscosity in fast extensional flow (faster than the inverse Rouse time). We further measure stress relaxation following steady extensional flow for the three melts. We show that initially they relax in a similar way, most likely via arm retraction, at short time, but behave differently at long time due to both the length of the arm and the branch point. The terminal relaxation is described by a Doi and Edwards based model, i.e., considering pure orientational relaxation.

A theory for non-Newtonian viscoelastic polymeric liquids

In many existing theories for incompressible polymeric liquids the Cauchy stress is decomposed as T=−p1+Sv+Se, where p is an arbitrary pressure, Sv=2μsD a deviatoric viscous stress with μs a viscosity and D the deviatoric stretching tensor, and Se is a deviatoric elastic stress which is introduced to account for stiffening arising from the alignment of long-chain polymer molecules during flow. A constitutive equation for Se needs to be prescribed and there are a large number of different proposals in the literature, with most proposals involving a hypoelastic rate constitutive equation for Se given in terms of a suitable frame-indifferent rate, which is usually taken as the Oldroyd or upper-convected rate. As is well-known, a hypoelastic equation for the stress is not thermodynamically consistent, in the sense that the constitutive equation for Se is not derived form a free energy function.The purpose of this paper is to present an alternative — thermodynamically-consistent and frame-indifferent — continuum theory for incompressible viscoelastic liquids. The theory is based on a Kröner-type multiplicative decomposition of the deformation gradient F of the form F = FeFp. In this theory the elastic stress Se is derived from a free-energy function which is prescribed in terms of a suitable measure based on the unimodular elastic distortion tensor Fe. This relation is supplemented by an evolution equation for the unimodular plastic distortion tensor Fp — the plastic flow rule.We study the response of the constitutive theory in steady simple shearing and steady extensional flows. We show: (i) that the theory qualitatively reproduces the experimentally-observed transient shear-thinning and normal stress effects during shearing flows of a polymer melt; and (ii) that it also reproduces the transient extensional response of a polymer melt.

Lubricated extensional flow of viscoelastic fluids in a convergent microchannel

Some of the fluids routinely used in microfluidic devices have micro and nanostructures that can generate viscoelastic behavior. In this paper, an experimental technique is developed to estimate the extensional flow resistance of dilute polymer solutions using flow in a lubricated, converging, and microfluidic channel. The channel has a two-dimensional hyperbolic profile, and the viscous or viscoelastic core fluid is lubricated by a less viscous Newtonian fluid that is introduced through side channels upstream. Differences in flow patterns are observed between miscible and immiscible lubricants. Stable flows are achieved with an immiscible lubricant and with a suitable interfacial tension between the core fluid and the lubricant. Steady extensional flows are established at low Reynolds numbers (Re ≪ 1 for the core fluid), at moderate Hencky strains (3 < ɛH < 3.9), and over a wide range of Deborah numbers (0 < De < 20). Pressure drop and flow rate measurements are made, and the velocity field is characteri...

The effect of surfactant on bubble and thread dynamics

AbstractTwo examples from the dynamics of surfactant‐laden interfacial flow are considered. In the first example, a bubble is rapidly stretched by an imposed flow to form a dumb‐bell shape, then the imposed flow is relaxed, so that the bubble evolves solely under the action of surfactant‐modified surface tension. In the second example, a surfactant‐coated bubble is continually stretched by a steady extensional flow. At sufficiently small strain rates, steady bubble shapes are found, whereas at larger strain rates a long‐wave model of the dynamics predicts behavior that is reminiscent of experimentally observed tip‐streaming. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Monte Carlo simulation of steady extensional flows

An efficient nonequilibrium Monte Carlo method using the Bond fluctuation model is used to simulate uniaxial and planar extension of dilute polymer solutions. The time scale is obtained from the stress relaxation of a fully stretched chain and can be related to the longest relaxation time of a real molecule, while the length scale is taken to be the statistical Kuhn segment length. The method leads to τ1∼N2.16 and D∼N−1.02 for a freely draining chain with an excluded volume constraint, where τ1, N, and D are the longest relaxation time, chain length, and diffusion coefficient, respectively. The finite extensibility of the bond causes extension thinning following the coil-stretch transition. The Monte Carlo predictions for the transient extension of isolated DNA molecules in a planar extensional flow agree reasonably well with published experimental measurements and Brownian dynamics simulations. “Molecular individualism” is observed in the unraveling of the polymer chain.

Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation

The shape of a two-dimensional viscous drop deforming in several time-dependent flow fields, including that due to a potential vortex, has been studied. Vortex flow was approximated by linearizing the induced velocity field at the drop centre, giving rise to an extensional flow with rotating axes of stretching. A generalization of the potential vortex, a flow we have called rotating extensional flow, occurs when the frequency of revolution of the flow is varied independently of the shear rate. Drops subjected to this forcing flow exhibit an interesting resonance phenomenon. Finally we have studied drop deformation in an oscillatory extensional flow.Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We investigate the effects of interfacial tension, periodicity, viscosity ratio, and Reynolds number on the drop dynamics. The simulation reveals interesting behaviour for steady stretching flows, as well as time-dependent flows. For a steady extensional flow, the drop deformation is found to be non-monotonic with time in its approach to an equilibrium value. At sufficiently high Reynolds numbers, the drop experiences multiple growth–collapse cycles, with possible axes reversal, before reaching a final shape. For a vortex flow, the long-time deformation reaches a steady value, and the drop attains a revolving steady elliptic shape. For rotating extensional flows as well as oscillatory extensional flows, the maximum value of deformation displays resonance with variation in parameters, first increasing and then decreasing with increasing interfacial tension or forcing frequency. A simple ODE model with proper forcing is offered to explain the observed phenomena.

Some asymptotic results for high rates of strain

The predictions of some constitutive equations for rheological behaviour at high rates of strain are investigated. One new result is the formal establishment of limits for shear flow of an integral model which uses the Seth strain measure. The work is related to earlier work on uniaxial and planar extension for rate equation models. Old and new results are tabulated and the behaviour of different models is classified. A recent idea of Harlen is used to `fix up' the upper convected Maxwell model (in the sense of removing the upper limit on rate of strain for steady extensional flow).

Start-up of a strongly extensional flow of a dilute polymer solution

We consider an impulsively started flow of a dilute monodisperse polymer solution for which the Reynolds number is low but the Weissenberg number may be large. The polymers are modelled as linear-locked dumbbells, and the aspect ratio of the flow geometry (a ‘cross-slot’) is taken to be sufficiently large that lubrication methods can be applied. In the limit in which the polymers are highly extended, the flow may be described asymptotically using a birefringent strand approximation (O.G. Harlen, J.M. Rallison and M.D. Chilcott, J. Non-Newtonian Fluid Mech., 34 (1990) 319-349) and we determine how such strands develop from rest. The flow is computed using a novel mixed Lagrangian-Eulerian formulation that is particularly well suited to the asymptotic method. The numerical method is checked by comparing its results with known expressions for steady extensional flows. We find that the time evolution takes place in a number of stages of which the first two or three are sharply delineated and in each of which the strand near the stagnation point suddenly becomes thicker, after which a front of increased strand width propagates downstream at constant velocity. The strain-rate at the stagnation point correspondingly first undershoots and then overshoots at each stage. Successive stages take progressively longer. These theoretical results are found to be in qualitative agreement with measurements of optical birefringence in a two-roll mill made by Geffroy and Leal [J. Non-Newtonian Fluid Mech., 35 (1990) 361-400] on the Boger fluid M1, and on a monodisperse polystyrene solution. [J. Polymer Sci., Part B, 30 (1992) 1329-1348].

Measurement of biaxial extensional viscosity of wheat flour doughs

The biaxial extensional viscosity of wheat flour doughs was measured using the lubricated squeeze film apparatus. Two disks of different diameters, 80 and 57.2 mm, were employed, with an extension rate (0.011 s−1) which remained approximately constant with both disks. Using polydimethylsiloxane a steady state lasted from 40 to 70 s. Obtaining steady extensional flow with wheat doughs, on the other hand, necessitated 10 to 200 s depending on the magnitude of normal stresses which ranged from 5.018 to 0.361 KPa. Doughs with different protein contents (13.2%, 16.0%, and 18.8% based on 14%MB) showed different biaxial extensional viscosities. These increased with increasing protein content. At an extensional rate of 7.3×10−5 s−1 the biaxial extensional viscosity approached 6η; Trouton’s ratio (ratio of ηE to η) increases with extension rate. The differences between biaxial extensional viscosities and shear viscosities become more profound when viscosities are plotted versus stresses.

On energy transfer in resonant bubble oscillations

In this paper, energy transfer mechanism due to resonant interactions of bubble shape and volume deformation modes is studied. In the process, a number of recent investigations that have focused on specific sets of initial conditions are generalized. Using phase-space analysis and a reduction of the mode dynamics equations to an integrable Hamiltonian, two types of resonant interactions are considered. The first is the so-called one–two resonance, when the natural frequency of the breathing mode is twice (or approximately twice) that of a shape mode. In this case, it is found that there is nearly always a continuous, periodic exchange of energy between the shape and volume modes. However, there exists a particular class of initial conditions, that was the subject of most earlier studies, for which the energy transfer is one way, from the shape mode to the volume mode. The second type of resonant interaction occurs when the natural frequency of the breathing mode is approximately equal to that of the shape mode. Here, it is found that periodic energy exchange can take place if equilibrium shape of the bubble is nonspherical due, for example, to a surface pressure distribution or a steady extensional flow. The analysis of the one–two resonance case is used to explain the stability of an oscillating spherical bubble to small perturbations of shape. It is found that instability of small volume oscillations can lead to large shape oscillation.

Single polymer chain in an extensional flow: n-->0 model.

By formulating the problem of a single polymer chain in a steady extensional flow as a zero-component field theory, the scaling exponent for the flow rate is obtained exactly in terms of known static and linear dynamic exponents.

Studies on Droplet Deformation and Breakup. I. Droplet Deformation in Extensional Flow

The extensional deformation of a viscoelastic droplet suspended in a viscoelastic medium was investigated, both theoretically and experimentally. A theoretical analysis was carried out on the deformation of a droplet suspended in a steady extensional flow field, where both fluids may be represented by the second‐order fluid model. The study took into account the effects of the elasticity, viscosity, and interfacial tension of the fluids concerned. The shape of a deformed droplet is determined by solving the equations of motion for both fluids (droplet phase and medium). A perturbation technique was employed and the iteration method was used to determine the shape of the droplet undergoing extensional deformation. Series solutions of stream function inside and outside the droplet, pressure distribution around the droplet, and the deformation of the droplet were obtained. For the experimental study, a transparent flow channel consisting of a conical section and a straight cylindrical tube was constructed. Along its central axis, the conical section provides extensional flow. Droplets of known volume were injected in the conical section at the centerline of the flow channel, through which a viscoelastic fluid was flowing at a constant flow rate. The deformation patterns of droplets were recorded on both movie and still films as they were traveling along the centerline of the conical section. Tracer particles were used to determine the axial velocity profiles. The suspending liquids were aqueous solutions of polyacrylamide at various concentrations (viscoelastic fluids) and corn syrup (Newtonian fluid). For viscoelastic droplets, polyisobutylene dissolved in decalin at various concentrations was used, and for Newtonian droplets, Indopols and benzene were used. It was observed that the droplets, initially spherical, were slightly deformed in the upper section of the cone and greatly elongated at the entrance region of the cylindrical tube. The viscoelastic droplets were less deformable than the Newtonian droplets, and highly viscoelastic media gave rise to large deformations of droplets. The deformability of droplets was analyzed based on the flow conditions and the rheological properties of the fluids concerned. A comparison is made between the theoretically predicted and experimentally observed shapes of droplets in the conical section, only where extension rate is constant.

Orthogonal stagnation flow, a framework for steady extensional flow experiments

A new framework for viewing steady extensional flow is presented: Steady orthogonal stagnation flow is an ideal which one should strive to reach in an actual experiment. The kinematics and the stress boundary conditions are developed for stagnation flow in general and in two special cases: the impingement of two circular streams and of two planar sheets. Contemplating this ideal clarifies the advantages and disadvantages of current experiments, thereby pointing the way towards new experiments; a number are suggested. Axisymmetric (with\(\dot \varepsilon _x \) > 0) and planar stagnation flow within a lubricated die look particularly promising. Some preliminary experimental results are given for uniaxial extension of a polyacrylamide solution in a water lubricated die.

Propagation of acoustic waves in steady extensional flows on non-Newtonian fluids

An analysis for the calculation of the attenuation and sound speed of a plane acoustic wave propagating through a non-Newtonian fluid that is undergoing a simple steady extensional motion is presented. The constitutive representation for the stress tensor in terms of the deformation gradient is developed for the model of a simple fluid, An explicit form is obtained for the superposed acoustic motion on the basic flow by using a functional Taylor's series having linear forms that are represented by integrals whose kernels are relaxation functions that depend on the extensional rates. An alternative representation is given as rate-dependent complex viscosities that depend on the relative orientation of the acoustic vector to the axes of extension. Several examples are described that are experimentally realizable and that can provide information on the rheological nature of the non-Newtonian fluid. [Work supported by NSF.]