Elastic Dumbbell Research Articles

Lopsided elastic dumbbell suspension

Lopsided elastic dumbbell suspension

Perspective on the description of viscoelastic flows via continuum elastic dumbbell models

Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics.

Lubrication approximation of pressure-driven viscoelastic flow in a hyperbolic channel

We investigate theoretically the steady incompressible viscoelastic flow in a hyperbolic contracting channel. The fluid viscoelasticity is modeled using the upper convected Maxwell (UCM), Oldroyd-B, Phan-Thien and Tanner (PTT), Giesekus, and the finite elasticity non-linear elastic dumbbell with the Peterlin approximation (FENE-P) models. We first develop the general governing equations for flow within a non-deformable channel whose cross section varies with the distance from the inlet. We then exploit the classic lubrication approximation, assuming a small aspect ratio of the channel to simplify the original governing equations. The final equations, which we formulate in terms of the stream unction, are then solved analytically using a high-order asymptotic scheme in terms of the Deborah number, De, and the formulas for the average pressure drop are derived up to eight orders in De. The accuracy of the original perturbation solution is enhanced and extended over a wide range of parameters by implementing a convergence acceleration method for truncated series. Furthermore, convergence of the transformed solutions for the average pressure drop is demonstrated. The validity and accuracy of the theoretical results is independently confirmed through comparison with numerical results from simulations performed using high-order finite differences and pseudospectral methods. The results reveal the decrease in the average pressure drop with increasing the Deborah number, the polymer viscosity ratio, and the ratio of the inlet to the outlet height. We also show that the fundamental UCM and Oldroyd-B models can predict the major viscoelastic phenomena for this type of internal and confined lubrication flows, while the effect of the rheological parameters of the PTT, Giesekus, and FENE-P models on the results is minor.

Effect of six weeks of resistance training on bone preservation in older adults: a randomized control trial.

It has been established that chronic resistance exercise contributes to positive changes to bone in older adults. This study evaluated the effect of 6weeks of resistance exercise with either elastic bands or dumbbells vs. a control period on bone morphology of older adults. Fifty-seven adults (mean ± SD; age = 66.5 ± 7.09 yrs; height = 165.2 ± 10.6cm; body mass = 74.5 ± 14.6kg) were randomized into three groups (dumbbell, elastic, or control). Participants underwent a total body dual-energy X-ray absorptiometry (DXA) scan for total body and segmental bone mineral content (BMC) and bone mineral density (BMD) before and following 6-week intervention. Age-matched Z-scores for BMD and BMC were recorded. Data were analyzed using two-way repeated measures ANOVAs and 0.05 significance level. BMCarm improved for the dumbbell group (p = 0.016) after the training, with no change in BMD for any group (p > 0.05). Additionally, significant (time x treatment group) interaction (p = 0.024) of age-matched Z-scores indicated an improvement in only the dumbbell group after 6weeks (p = 0.015), with no change in the elastic group despite them having greater Z-scores than the control group. This study is the first to demonstrate acute normative adaptations as dumbbell-based programs may promote positive maintenance of bone metrics over 6weeks, despite the lack of significant change in absolute BMC or BMD. Adults did not lose relative bone mass with acute exercise using dumbbells as the external load applied and this may lead to positive changes following chronic training. There was no bone-related impact from elastic bands, suggesting a weighted load or force produced relative to gravity is beneficial.

Solving the constitutive equation of dilute polymeric flows: A general Fokker–Planck approach for linear elastic dumbbell models

We present a mesoscopic numerical solver for the constitutive equation of dilute polymer solutions, as described by the Fokker–Planck equation of bead-spring dumbbell models with linear spring forces, such as Oldroyd-B, FENE-P (finitely extensible nonlinear elastic with Peterlin closure), and C-FENE-P (charged FENE-P). The configuration density function of the Fokker–Planck equation is expanded into a series of Hermite polynomials, and we show that only a second order in the expansion is needed to describe the evolution of the polymer-induced stresses. The polymer-induced stress tensor is given exactly through a discrete representation of the configuration distribution function, which is obtained from a finite set of Gauss–Hermite quadrature points. In addition, we derive the conformation tensor equation, which serves as a mathematical quality check for the method. We solve the time evolution of the extra stress tensor for homogeneous flows of dilute polyelectrolyte solutions, conceptualized by the C-FENE-P model, which is a generalization of the FENE-P dumbbell model. The results are shown to be in excellent agreement with analytical and semi-analytical reference results for simple shear and extensional flows. While in this paper the focus is on linear connector forces, we believe that the principles of the derivation are extendable to other force laws.

Extensional rheometry of mobile fluids. Part II: Comparison between the uniaxial, planar, and biaxial extensional rheology of dilute polymer solutions using numerically optimized stagnation point microfluidic devices

Part I of this paper [Haward et al., J. Rheol. 67, 995–1009 (2023)] presents a three-dimensional microfluidic device (the optimized uniaxial and biaxial extensional rheometer, OUBER) for generating near-homogeneous uniaxial and biaxial elongational flows. Here, in Part II, the OUBER device is employed to examine the uniaxial and biaxial extensional rheology of model dilute polymer solutions, compared with measurements made under planar extension in the optimized-shape cross-slot extensional rheometer [OSCER, Haward et al. Phys. Rev. Lett. 109, 128301 (2012)]. In each case, micro-particle image velocimetry is used to measure the extension rate as a function of the imposed flow conditions, and excess pressure drop measurements enable estimation of the tensile stress difference generated in the fluid via a new analysis based on the macroscopic power balance for flow through each device. Based on this analysis, for the most dilute polymer sample tested, which is “ultradilute”, the extensional viscosity is well described by Peterlin’s finitely extensible nonlinear elastic dumbbell model. In this limit, the biaxial extensional viscosity at high Weissenberg numbers (Wi) is half that of the uniaxial and planar extensional viscosities. At higher polymer concentrations, although the fluids remain dilute, the experimental measurements deviate from the model predictions, which is attributed to the onset of intermolecular interactions as the polymer chains unravel in the extensional flows. Of practical significance (and fundamental interest), elastic instability occurs at a significantly lower Wi in uniaxial extensional flow than in either biaxial or planar extensional flow, thereby limiting the utility of this flow type for extensional viscosity measurement.

Master curves for FENE-P fluids in steady shear flow

The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number ( Wi ), i.e. , the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter ( L ), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio, commonly denoted β . An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available (Sureshkumar et al., 1997), but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi / L and it is only in strong shear flows exceeding Wi / L ≳ 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi / L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. • Asymptotic solutions for FENE-P model in steady shear flow are calculated. • Master curves exist for material functions of FENE-P fluids in steady shear flow. • Governing dimensionless parameter is (Weissenberg number)/(finite extensibility). • Reporting this dimensionless parameter clarifies the magnitude of nonlinear effects.

English translation of Giesekus's famous article on the elasticity of liquids

Starting from a suspension of elastic dumbbells or chain-like particle aggregates as a simple model of a viscoelastic liquid, the rheological equation of state is inductively derived into which the concept of an “elastic strain component” (recoverable strain) is incorporated and can be understood as a nonlinear extension of the so-called “generalized Maxwell model.” Here, the liquid is characterized by a sequence of nested continua, each associated with an elasticity law. The consideration of the directional dependence of the mobility of the structural elements requires that the behavior of the liquids represented in this way generally deviates from a Weissenberg liquid.

Drag Reduction in Polymer-Laden Turbulent Pipe Flow

The turbulence of a realistic dilute solution of DNA macromolecules is investigated through a hybrid Eulerian–Lagrangian approach that directly solves the incompressible Navier–Stokes equation alongside the evolution of 108 polymers, modelled as finitely extensible nonlinear elastic (FENE) dumbbells. At a friction Reynolds number of 320 and a Weissenberg number of 2×104, the drag reduction is equal to 26%, which is similar to the one obtained at the lower Reynolds number of 180. The polymers induce an increase in the flow rate and the turbulent kinetic energy, whose axial contribution is predominantly augmented. The stress balance is analysed to investigate the causes of the drag reduction and eventually the effect of the friction Reynolds number on the probability distribution of the polymer configuration. Near the wall, the majority of the polymers are fully stretched and aligned along the streamwise direction, inducing an increase in the turbulence anisotropy.

Drag Reduction in Turbulent Wall-Bounded Flows of Realistic Polymer Solutions.

Suspensions of DNA macromolecules (0.8wppm, 60kbp), modeled as finitely extensible nonlinear elastic dumbbells coupled to the Newtonian fluid, show drag reduction up to 27% at friction Reynolds number 180, saturating at the previously unachieved Weissenberg number ≃10^{4}. At a large Weissenberg number, the drag reduction is entirely induced by the fully stretched polymers, as confirmed by the extensional viscosity field. The polymer extension is strongly non-Gaussian, in contrast to the assumptions of classical viscoelastic models.

A theory for the flow of chemically responsive polymer solutions: Equilibrium and shear-induced phase separation

Chemically responsive polymers are macromolecules that respond to local variations of the chemical composition of the solution by changing their conformation, with notable examples including polyelectrolytes, proteins, and DNA. The polymer conformation changes can occur in response to changes in the pH, the ionic strength, or the concentration of a generic solute that interacts with the polymer. These chemical stimuli can lead to drastic variations of the polymer flexibility and even trigger a transition from a coil to a globule polymer conformation. In many situations, the spatial distribution of the chemical stimuli can be highly inhomogeneous, which can lead to large spatial variations of polymer conformation and of the rheological properties of the mixture. In this paper, we develop a theory for the flow of a mixture of solute and chemically responsive polymers. The approach is valid for generic flows and inhomogeneous distributions of polymers and solutes. To model the polymer conformation changes introduced by the interactions with the solute, we consider the polymers as linear elastic dumbbells whose spring stiffness depends on the solute concentration. We use Onsager’s variational formalism to derive the equations governing the evolution of the variables, which unveils novel couplings between the distribution of dumbbells and that of the solute. Finally, we use a linear stability analysis to show that the governing equations predict an equilibrium phase separation and a distinct shear-induced phase separation whereby a homogeneous distribution of solute and dumbbells spontaneously demix. Similar phase transitions have been observed in previous experiments using stimuli-responsive polymers and may play an important role in living systems.

Prediction of polymer extension, drag reduction, and vortex interaction in direct numerical simulation of turbulent channel flows

Hydrodynamic and viscoelastic interactions between the turbulent fluid within a channel at Reτ=180 and a polymeric phase are investigated numerically using a multiscale hybrid approach. Direct numerical simulations are performed to predict the continuous phase and Brownian dynamics simulations using the finitely extensible nonlinear elastic dumbbell approach are carried out to model the trajectories of polymer extension vectors within the flow, using parallel computations to achieve reasonable computation timeframes on large-scale flows. Upon validating the polymeric configuration solver against theoretical predictions in equilibrium conditions, with excellent agreement observed, the distributions of velocity gradient tensor components are analyzed throughout the channel flow wall-normal regions. Impact on polymer stretching is discussed, with streamwise extension dominant close to the wall, and wall-normal extension driven by high streamwise gradients of wall-normal velocity. In this case, it is shown that chains already possessing high wall-normal extensions may attempt to orientate more in the streamwise direction, causing a curling effect. These effects are observed in instantaneous snapshots of polymer extension, and the effects of the bulk Weissenberg number show that increased WeB leads to more stretched configurations and more streamwise orientated conformities close to the wall, whereas, in the bulk flow and log-law regions, the polymers tend to trace fluid turbulence structures. Chain orientation angles are also considered, with WeB demonstrating little influence on the isotropic distributions in the log-law and bulk flow regions. Polymer–fluid coupling is implemented through a polymer contribution to the viscoelastic stress tensor. The effect of the polymer relaxation time on the turbulent drag reduction is discussed, with greater Weissenberg numbers leading to more impactful reduction. Finally, the velocity gradient tensor invariants are calculated for the drag-reduced flows, with polymers having a significant impact on the Q–R phase diagrams, with the presence of polymers narrowing the range of R values in the wall regions and causing flow structures to become more two-dimensional.

Polymer-turbulence interactions in a complex flow and implications for the drag reduction phenomenon

We present direct numerical simulation data for turbulent duct flow of a finite-extensibility non-linear elastic dumbbell model with the Peterlin approximation (FENE-P) fluid in the high drag reduction regime. While the secondary flow pattern is qualitatively similar to that in a Newtonian fluid, its magnitude is significantly reduced, resulting in a less uniformly distributed velocity profile and hence smaller gradients at the wall. The Reynolds stress tensor in the polymer-laden flow was found to be increasingly anisotropic with most of the turbulent kinetic energy retained in the streamwise component, u′u′¯. We introduce a novel approach for investigating polymer stretching using the anisotropy invariant map of the polymer stress tensor and observe the persistence of both uniaxial and biaxial extension. Analysis of the transport equation for the mean kinetic energy indicates that polymer stretching and relaxation is a highly dissipative process; hence, the introduction of an additional channel for dissipation in a flow is key to drag reduction.

Large Scale Hydrodynamically Coupled Brownian Dynamics Simulations of Polymer Solutions Flowing through Porous Media.

Large scale simulations of polymer flow through porous media provide an important tool for solving problems in enhanced oil recovery, polymer processing and biological applications. In order to include the effects of a wide range of velocity and density fluctuations, we base our work on a coarse-grain particle-based model consisting of polymers following Brownian dynamics coupled to a background fluid flow through momentum conserving interactions. The polymers are represented as Finitely Extensible Non-Linear Elastic (FENE) dumbbells with interactions including slowly decaying transient forces to properly describe dynamic effects of the eliminated degrees of freedom. Model porous media are constructed from arrays of parallel solid beams with circular or square cross-sections, arranged periodically in the plane perpendicular to their axis. No-slip boundary conditions at the solid–fluid interfaces are imposed through interactions with artificial particles embedded within the solid part of the system. We compare the results of our simulations with those of standard Smoothed Particle Hydrodynamics simulations for Newtonian flow through the same porous media. We observe that in all cases the concentration of polymers at steady state is not uniform even though we start the simulations with a uniform polymer concentration, which is indicative of shear-induced cross-flow migration. Furthermore, we see the characteristic flattening of the velocity profile experimentally observed for shear-thinning polymer solutions flowing through channels as opposed to the parabolic Poiseuille flow profile for Newtonian fluids.

On the similarities between the simplified Phan-Thien–Tanner model and the finitely extensible nonlinear elastic dumbbell (Peterlin closure) model in simple and complex flows

For many commonly used viscoelastic constitutive equations, it is well known that the limiting behavior is that of the Oldroyd-B model. Here, we compare the response of the simplified linear form of the Phan-Thien–Tanner model (“sPTT”) [Phan-Thien and Tanner, “A new constitutive equation derived from network theory,” J. Non-Newtonian Fluid Mech. 2, 353–365 (1977)] and the finitely extensible nonlinear elastic (“FENE”) dumbbell model that follows the Peterlin approximation (“FENE-P”) [Bird et al., “Polymer solution rheology based on a finitely extensible bead—Spring chain model,” J. Non-Newtonian Fluid Mech. 7, 213–235 (1980)]. We show that for steady homogeneous flows such as steady simple shear flow or pure extension, the response of both models is identical under precise conditions (ε=1/L2). The similarity of the “spring” functions between the two models is shown to help understand this equivalence despite a different molecular origin of the two models. We then use a numerical approach to investigate the response of the two models when the flow is “complex” in a number of different definitions: first, when the applied deformation field is homogeneous in space but transient in time (so-called “start-up” shear and planar extensional flow), then, as an intermediate step, the start-up of the planar channel flow; and finally, “complex” flows (through a range of geometries), which, although being Eulerian steady, are unsteady in a Lagrangian sense. Although there can be significant differences in transient conditions, especially if the extensibility parameter is small L2>100,ε<0.01, under the limit that the flows remain Eulerian steady, we once again observe very close agreement between the FENE-P dumbbell and sPTT models in complex geometries.

Vanishing viscosity limit to the FENE dumbbell model of polymeric flows

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also.

Non-integrability of a model of elastic dumbbell satellite

We study the integrability of a model of elastic satellite whose centre of mass moves in a circular Keplerian orbit around a gravity centre. The satellite is modelled by two point masses connected by an extensible massless spring that obeys Hooke’s law. It is assumed that the distance between point masses is much smaller than the radius of the orbit, so the orbital motion of the satellite is not perturbed by its rotational motion. The gravity potential of the satellite is expanded into a series with respect to its size up to quadratic terms which describe the gravity gradient torque acting on the satellite. Two cases are considered with Hooke’s centre localised in the centre of mass of the dumbbell and at an arbitrary point along a line connecting both masses. It is shown that the first case appears to be integrable and super-integrable for selected values of the parameter of the system. In the second case, model depends effectively only on one parameter and is non-integrable. In the proof, differential Galois integrability obstructions are used. For the considered sysem, these obstructions are deduced thanks to the recently developed symplectic Kovacic’s algorithm in dimension 4. According to our knowledge, this is the first application of this tool to a physical model.

Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation

Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of a mean-field term to account for intermolecular forces in dilute polymer solutions for background profiles of weak shear and elongation. These investigations involved the solution of the Fokker–Planck equation incorporating a constitutive equation for the second moment. In this paper, we extend this analysis to include the effects of large background shear and elongation beyond the Hookean regime. Further, the constitutive equation is solved for the probability density function which permits the computation of any macroscopic variable, allowing direct comparison of the model predictions with molecular dynamics simulations. It was found that if the concentration effects at equilibrium are taken into account, the FENE-P model gives qualitatively the correct predictions, although the over-shoot in extension in comparison to the infinitely dilute case is significantly underpredicted.

A single bout of hybrid intradialytic exercise did not affect left-ventricular function in exercise-naïve dialysis patients: a randomized, cross-over trial.

Cardiovascular diseases are the leading cause of mortality in end-stage renal disease (ESRD) patients, especially those receiving hemodialysis (HD) therapy. HD has many side effects that are related to patients' hearts, such as recurrent myocardial ischemia and global or segmental left-ventricular dysfunction, which is associated with intradialytic hypotension, long-term loss of systolic function, and high incidence of cardiovascular events and death. Systematic exercise training has a beneficial effect on measures of cardiovascular fitness and reducing cardiovascular risk factors in ESRD. Whether there is an acute benefit of exercise during HD on left-ventricular function is not well known. The current study aimed to investigate whether a single bout of hybrid (aerobic and resistance) intradialytic exercise could affect left-ventricular function during HD sessions. Twenty-one exercise naïve and clinically stable HD patients participated in the study. All participants completed two different HD trials on two different days, separated by 1 week: (1) standard HD and (2) HD including a single bout of hybrid intradialytic exercise. Hybrid intradialytic training included the usual intradialytic cycling followed by resistance training using elastic bands and dumbbells. Echocardiographic assessment of left-ventricular function was completed before HD, half an hour before the end of HD, and 30min after the end of HD. Cohort data for left-ventricular function indices were not different between trials and did not change across time in either the standard HD or HD plus exercise trial. Cohort data for the change in ejection fraction from baseline to during HD did mask considerable inter-individual variability (HD -0 ± 15; HD plus exercise (-2 ± 20). Despite this, the variability was not mediated by the addition of intradialytic hybrid exercise. A single bout of hybrid intradialytic exercise did not affect left-ventricular function during the HD therapy. It is important to determine whether chronic exercise training could beneficially affect left-ventricular function abnormalities often observed during the HD therapy. The study is registered at ClinicalTrials.gov (NCT01721551) as a clinical trial.

Dynamics of electrospun hydrogel filaments in oscillatory microchannel flows: A theoretical and experimental approach

The dynamics of highly flexible micro- and nano-filaments are important to a variety of biological, medical, and industrial problems. The filament configuration variation and cross-stream migration in a microchannel are affected by thermal fluctuations in addition to elastic and viscous forces. Here, hydrogel nano-filaments with small bending Young’s moduli are utilized to elucidate the transitional behavior of elastic Brownian filaments in an oscillatory microchannel flow. A numerical model based on chain elastic dumbbells similar to the Rouse–Zimm model accounting for elastic, viscous, and random Brownian forces is proposed and implemented. In addition, a theoretical model to describe the average orientation–deformation tensor evolution for an ensemble of filaments in an oscillatory flow is proposed. The results are compared with the evolution observed in the experiments.