Large Weissenberg Numbers Research Articles

Polymers in turbulence: any better than dumbbells?

Polymer chains in turbulent flows are generally modelled as dumbbells, i.e. two beads joined by a nonlinear spring. The dumbbell only maps a single spatial configuration, described by the polymer end-to-end vector, thus a multi-bead FENE (finitely extensible nonlinear elastic) chain seems a natural improvement for a more accurate characterisation of the polymer spatial conformation. At a large Weissenberg number, a comparison with the more accurate Kuhn chain reveals that the multi-bead FENE chain drastically overestimates the probability of folded configurations. Surprisingly, the dumbbell turns out to be the only meaningful bead-spring model to coarse-grain a polymer macromolecule in turbulent pipe flows.

Effect of nonlinear thermal radiation and homogeneous-heterogeneous reactions on peristaltic propulsion of Johnson-Segalman fluid

The basic aim of this research is to investigate the main features of the peristaltic flow of Johnson-Segalman fluid in a curved flow channel in the presence of a homogeneous-heterogeneous reaction. The fluid is considered electrically conducting with a radial magnetic field effect. The constitutive relation for energy is formulated with the addition of viscous dissipation and nonlinear thermal radiation. The slip and flexible wall boundary conditions are taken into consideration. The lubrication technique is used for the mathematical the modelling of the problem. The solution of the resulting nonlinear system of ODE’s is computed numerically through NDSolve command of Mathematica, and graphical results are prepared for flow variables. It has been clearly noticed that flow is accelerated in the planer medium as compared to the curved channel, and the velocity of the fluid increases with a larger Weissenberg number. Moreover, the thermal profile has an inverse relation to the parameters, i.e., radiation, temperature ratio, and curvature. Furthermore, it is also observed that the magnitude of heat transfer coefficient rises significantly when a comparatively less curved channel is utilized, however, a reduction in concentration profile is witnessed in this case. Alternative impact of homogeneous and heterogeneous reactions on concentration profile are observed through graphical results. Application of the study include the usage of peristaltic pump in chemical industry and in pharmacology industry for the purpose of drug deliveries.

Centrifugal spinning of polymeric solutions: Experiments and modelling

We assess some of the main parameters affecting the performance of Centrifugal Spinning (CS), a method for making nanofibres, using laboratory experiments and modelling. Rheologically characterized polymer solutions of various concentrations are propelled into curved jet trajectories by an in-house CS device and they are visualized via high speed imaging, while final dried fibre morphology diameter distributions are characterized by scanning electron microscopy. Our experiments provide validations of a comprehensive string model, based on the viscoelastic upper convected Maxwell (UCM) constitutive equation for the polymer, allowing us to examine dependence of the CS process on the key dimensionless numbers. We experimentally analyze the effects of the polymer concentration, rotation speed, nozzle diameter, spinneret radius and nozzle angle, and correspondingly use the model to analyze the effects of the Weissenberg number (Wi), Rossby number (Rb), nozzle diameter ratio (ϵa0), spinneret radius ratio (ϵs0), and nozzle angle (α0) on the fibre trajectory and radius. Finally, we use a simple scaling analysis to classify the flow regimes into stable and unstable with respect to jet instabilities and breakup. The results of this analysis agree with the experiments in finding fibres stable at large capillary and Weissenberg numbers.

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.

Effect of Weissenberg number on polymer-laden turbulence

By adopting the direct force formulation in a finitely extensible nonlinear elastic (FENE) model for the feedback force by laden polymers to stationary isotropic turbulence, we clearly identified the energy flow between turbulence and polymers. The stretching motion of turbulence is suppressed by the polymer springs with large Weissenberg number, resulting in suppression of turbulent kinetic energy and energy dissipation.The direct force model seems to be more efficient in providing physical insight than the polymer stress model based on the conformation tensor.

Instability of electroconvection in viscoelastic fluids subjected to unipolar injection

In this paper, a two-dimensional numerical study on the nonlinear behaviors of electrohydrodynamic flows of Oldroyd-B viscoelastic dielectric liquid subjected to unipolar injection is performed via the finite volume method. The entire set of coupled equations, which includes the Navier–Stokes equations, simplified Maxwell’s equations, and conformation transport equations, is solved for the first time. The effects of elasticity on the nonlinear evolution of electroconvection and instability patterns are mainly investigated. Various physical models including free and rigid boundary cases are simulated entirely, and detailed analyses of stability parameters are performed. Convection and fluid motion instability are investigated and explained in detail, with a focus on the onset of motion transitions from a purely conducted state to losing its stability. It is found that the coupling of the electric field with the elasticity field gives rise to new instability and completely new mechanisms. In addition to instabilities such as subcritical bifurcation in electroconvection of Newtonian fluids, supercritical bifurcation and Hopf bifurcation are also possible as the first instability in electroconvection of viscoelastic fluids under free boundary conditions. Under rigid boundary conditions, the system with a large Weissenberg number can also lose its stability via earlier Hopf bifurcation. The stability threshold is not affected by the elastic effect if the Weissenberg number is small enough but decreases when the first instability of the system becomes Hopf bifurcation. Moreover, elasticity promotes the transition from a steady state flow to unsteady convection after the onset of convection. These phenomena are closely related to the elastic parameters.

Three-dimensional viscoelastic instabilities in a four-roll mill geometry at the Stokes limit

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. Both the Oldroyd-B model and FENE-P model of viscoelastic fluids were used. Different temporal behaviors were observed depending on the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal dynamics evolve on long time scales, and simulations were accelerated by using a Graphics Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were prohibitively expensive. For a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. However, for a sufficiently large Weissenberg number, three-dimensional instabilities were observed, creating complex temporal behaviors. For certain Weissenberg values and models, the instability that first emerges is two-dimensional (in the x, y plane), and then the solution develops an instability in the z-direction, whereas for others the z instability comes first. Using a linear perturbation from a steady two-dimensional background solution, extended to three dimensions as constant in the third dimension, it is demonstrated that there is a linear instability for a sufficiently large Weissenberg number, and possible mechanisms for this instability are discussed.

Viscoelastic second normal stress difference dominated multiple-stream particle focusing in microfluidic channels.

Particle focusing in viscoelastic fluid flow is a promising approach for inducing particle separations in microfluidic devices. The results from theoretical studies indicated that multiple stream particle focusing can be realized with a large magnitude of the elastic second normal stress difference (N2). For dilute polymer solutions, theoretical and experimental studies show that the magnitude of N2 is never large, no matter how large the polymer molecular weight nor how high the shear rate. However, for concentrated entangled polymer solutions, the magnitude of N2 becomes large at high shear rates. Therefore, in order to test the hypothesis that N2 can be used to induce multiple particle stream focusing behavior, we perform the systematic study of the effects of increasing carrier fluid polymer concentrations in a microchannel containing fluorescent particles. In a dilute polymer solution, multiple particle stream focusing is not observed, even at high shear rates and large dimensionless Weissenberg number values (Wi ≈ 30) at which the elastic first normal stress difference (N1) and the viscosity shear-thinning should be very large, while in a concentrated entangled polymer solution, we observe that particle streams focused upon the channel centerline bifurcate to form two symmetric off-channel particle streams at higher shear rates. This particle focusing behavior is different from previous multiple-stream focusing phenomena, and that we attribute to the influence of the second normal stress difference N2. This N2 induced multiple stream focusing phenomenon provides a different approach for manipulating the particle trajectory and separation in a microchannel.

Entropy Generation and Consequences of Binary Chemical Reaction on MHD Darcy-Forchheimer Williamson Nanofluid Flow Over Non-Linearly Stretching Surface.

The current article aims to present a numerical analysis of MHD Williamson nanofluid flow maintained to flow through porous medium bounded by a non-linearly stretching flat surface. The second law of thermodynamics was applied to analyze the fluid flow, heat and mass transport as well as the aspects of entropy generation using Buongiorno model. Thermophoresis and Brownian diffusion is considered which appears due to the concentration and random motion of nanoparticles in base fluid, respectively. Uniform magnetic effect is induced but the assumption of tiny magnetic Reynolds number results in zero magnetic induction. The governing equations (PDEs) are transformed into ordinary differential equations (ODEs) using appropriately adjusted transformations. The numerical method is used for solving the so-formulated highly nonlinear problem. The graphical presentation of results highlights that the heat flux receives enhancement for augmented Brownian diffusion. The Bejan number is found to be increasing with a larger Weissenberg number. The tabulated results for skin-friction, Nusselt number and Sherwood number are given. A decent agreement is noted in the results when compared with previously published literature on Williamson nanofluids.

Pore-Scale Flow Characterization of Polymer Solutions in Microfluidic Porous Media.

Polymer solutions are frequently used in enhanced oil recovery and groundwater remediation to improve the recovery of trapped nonaqueous fluids. However, applications are limited by an incomplete understanding of the flow in porous media. The tortuous pore structure imposes both shear and extension, which elongates polymers; moreover, the flow is often at large Weissenberg numbers, Wi, at which polymer elasticity in turn strongly alters the flow. This dynamic elongation can even produce flow instabilities with strong spatial and temporal fluctuations despite the low Reynolds number, Re. Unfortunately, macroscopic approaches are limited in their ability to characterize the pore-scale flow. Thus, understanding how polymer conformations, flow dynamics, and pore geometry together determine these nontrivial flow patterns and impact macroscopic transport remains an outstanding challenge. This review describes how microfluidic tools can shed light on the physics underlying the flow of polymer solutions in porous media at high Wi and low Re. Specifically, microfluidic studies elucidate how steady and unsteady flow behavior depends on pore geometry and solution properties, and how polymer-induced effects impact nonaqueous fluid recovery. This work thus provides new insights for polymer dynamics, non-Newtonian fluid mechanics, and applications such as enhanced oil recovery and groundwater remediation.

Heat transport and entropy optimization in flow of magneto-Williamson nanomaterial with Arrhenius activation energy

BackgroundA newly developed approach in the field of nanotechnology for solving problems and collection of information is the use of nanoparticles. This idea has been further utilized in a better way in pharmaceutical industries. By using nanotechnology, the field of pharmaceutical science has been modernized and redeveloped. The use of nanotechnology in such industries has convinced the scientist to obtain more economical and easier applications. Therefore, with such effectiveness in mind, a theoretical study has been conducted to examine the effects of nonlinear radiative heat flux and magnetohydrodynamics for nanomaterial flow of Williamson fluid over a convectively heated stretchable surface. Brownian diffusion is utilized in mathematical modeling. Furthermore, heat source/sink, viscous dissipation and nonlinear radiative heat flux are examined. Convective boundary condition is implemented. Salient effects of chemical reaction and Arrhenius activation energy in mass transfer are considered. Total entropy rate is obtained through implementation of thermodynamics second law. MethodsThe nonlinear PDEs are reduced into ordinary ones by appropriate similarity transformations. A semi-analytical technique i.e., homotopy method is implemented to obtain the convergent series solutions. ResultsThe obtained results indicate that the velocity of fluid particles increases versus higher fluid parameter. Schmidt number and activation energy variable have opposite effect on concentration. Entropy rate grows up with fluid parameter and Brinkman and Biot numbers while opposite trend is seen for Bejan number. ConclusionsVelocity of the material particles declines through larger estimations of magnetic variable while it upsurges for higher fluid parameter. Thermal distribution shows similar impact for radiative and magnetic variables. Mass concentration decreases against chemical reaction parameter while it increases via activation energy variable. Entropy and Bejan numbers show opposite impacts versus Brinkman number. Skin friction coefficient increases through larger Weissenberg number.

Boundary layer of elastic turbulence

We investigate theoretically the near-wall region in elastic turbulence of a dilute polymer solution in the limit of large Weissenberg number. As has been established experimentally, elastic turbulence possesses a boundary layer where the fluid velocity field can be approximated by a steady shear flow with relatively small fluctuations on the top of it. Assuming that at the bottom of the boundary layer the dissolved polymers can be considered as passive objects, we examine analytically and numerically the statistics of the polymer conformation, which is highly non-uniform in the wall-normal direction. Next, imposing the condition that the passive regime terminates at the border of the boundary layer, we obtain an estimate for the ratio of the mean flow to the magnitude of the flow fluctuations. This ratio is determined by the polymer concentration, the radius of gyration of polymers and their length in the fully extended state. The results of our asymptotic analysis reproduce the qualitative features of elastic turbulence at finite Weissenberg numbers.

A model for the electrical conductivity variation of molten polymer filled with carbon nanotubes under extensional deformation

This work is dedicated to analyzing the variation of conductivity of polymer composites (polystyrene filled with Carbon Nanotubes) under extensional deformation. In a previous work, a conductor-insulator transition has been observed and the predominant role of the polymer dynamics has been brought to light. The evolution of the filler network within a polymer matrix can be described by a kinetic equation that takes into account a structuring mechanism that is controlled by the mobility in the melt matrix and a destruction mechanism that is induced by the extensional deformation. The solution of this equation that describes the filler network at a microscale is used in the percolation law to obtain the macroscopic conductivity of the composite. It turned out that the structuring parameter does not depend on the extensional deformation but only relies on the polymer matrix dynamics. In addition, the breaking parameter only depends on the Hencky strain, whatever the extensional rate. This model has been successfully applied for a large range of filler concentrations and experimental conditions from low to large Weissenberg numbers.

Lubrication in polymer-brush bilayers in the weak interpenetration regime: Molecular dynamics simulations and scaling theories.

We conduct molecular dynamics (MD) simulations and develop scaling laws to quantify the lubrication behavior of weakly interpenetrated polymer brush bilayers in the presence of an external shear force. The weakly interpenetrated regime is characterized by 1<d_{g}/d_{0}<2, where d_{g} is the gap between the opposing surfaces (where the brushes are grafted) and d_{0} is the unperturbed brush height. MD simulations predict that in the shear thinning regime, characterized by a larger shear force or a large Weissenberg number (W), R_{g}^{2}∼W^{0.19} and η∼W^{-0.38}, where R_{g} is the chain extension in the direction of the shear and η is the viscosity. These scaling behaviors, which are distinctly different from that witnessed in strongly compressed regime (for such a regime, characterized by d_{g}/d_{0}<1, R_{g}^{2}∼W^{0.53}, and η∼W^{-0.46}), match excellently with those predicted by our scaling theory.

An Instability Phenomenon in Hele-Shaw Displacements

We study the linear instability of the displacement of an Oldroyd-B fluid by air in a Hele-Shaw cell, motivated by possible applications in Secondary Oil Recovery (SOR) process. Most numerical methodshave usually failed when the Weissenberg numbers $W_i$ (appearing in the constitutive relations) are near 1. We get an approximate formula of the growth rate with a blow-up for some particular $W_i =O(1)$. Therefore the instability at large Weissenberg numbers is due to the model, and not to the computational methods. Our growth rate is quite similar to Saffman-Taylor's formula (obtained when a Newtonian liquid is displaced by air) if $W_1=W_2$.

Formulation-Controlled Positive and Negative First Normal Stress Differences in Waterborne Hydrophobically Modified Ethylene Oxide Urethane (HEUR)-Latex Suspensions.

Hydrophobically modified ethylene oxide urethane (HEUR) associative thickeners are widely used to modify the rheology of waterborne paints. Understanding the normal stress behavior of the HEUR-based paints under high shear is critical for many applications such as brush drag and spreading. We observed that the first normal stress difference, N1, at high shear (large Weissenberg number) can be positive or negative depending on the HEUR hydrophobe strength and concentration. We propose that the algebraic sign of the N1 is primarily controlled by two factors: (a) adsorption of HEURs on the latex surface and (b) the ability of HEURs to form transient molecular bridges between latex particles. Such transient bridges are favored for dispersions with small interparticle distances and dense surface coverages; in these systems; HEUR-bridged latex microstructures flow-align in high shear and exhibit positive N1. In the absence of transient bridges (large interparticle distances, low surface coverage), the dispersion rheology is similar to that of weakly interacting spheres, exhibiting negative N1. The results are summarized in a simplified phase diagram connecting formulation, microstructure, and the N1 behavior.

Concentration fluctuations in polymer solutions under mixed flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under mixed (shear + extensional) flows. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie-Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady state structure factor (Fourier transformed pair correlation function) under general linear flows using a unique method of characteristics solver. Under shear flow, the model predicts butterfly patterns in accord with previous experimental and theoretical work, including a full rotation of peaks past the flow axis. In addition, the magnitude of the structure factor initially grows with the Weissenberg number until reaching a maximum at intermediate shear rates and decaying thereafter. Under mixed flow, the butterfly patterns as well as the location and magnitude of the peak structure factor are strongly tied to both the flow type parameter and the Weissenberg number (the characteristic strain rate). As expected, for flows characterized as strong, the scattering patterns typically appear like a rotated version of pure extension. However, as the flow type approaches the pure shear limit, the influence of shear flow on the butterfly patterns becomes more pronounced. In particular, for large Weissenberg numbers, contrary to expectations, the flow type need not be very near shear flow in order for the scattering patterns to no longer be simply rotated versions of extensional flow.

Large amplitude oscillatory shear of the Giesekus model

An asymptotic solution to the Giesekus constitutive model of polymeric fluids under homogenous, oscillatory simple shear flow at large Weissenberg number, Wi≫1, and large strain amplitude, Wi/De≫1, is constructed. Here, Wi=λ*γ̇0*, where λ* is the polymer relaxation time and γ̇0* is the shear rate amplitude, and De=λ*ω* is a Deborah number, where ω* is the oscillation frequency. Under these conditions, we show that the first normal stress difference is the dominant rheological signal, scaling as G*Wi1/2, where G* is the elastic modulus. The shear stress and second normal stress difference are of order G*. The polymer stress exhibits pronounced nonlinear oscillations, which are partitioned into two temporal regions: (i) A “core region,” on the time scale of λ*, reflecting a balance between anisotropic drag and orientation of polymers in the strong flow; and (ii) a “turning region,” centered around times when the shear flow reverses, whose duration is on the hybrid time scale (λ*)2/3/(γ̇0*)1/3, in which flow-driven orientation, anisotropic drag, and relaxation are all leading order effects. Our asymptotic solution is verified against numerical computations, and a qualitative comparison with relevant experimental observations is presented. Our results can, in principle, be employed to extract the nonlinearity (anisotropic drag) parameter, α, of the Giesekus model from experimental data, without the need to fit the stress waveform over a complete oscillation cycle. Finally, we discuss our findings in relation to recent work on shear banding in oscillatory flows.

Impact of the Peterlin approximation on polymer dynamics in turbulent flows.

We study the impact of the Peterlin approximation on the statistics of the end-to-end separation of polymers in a turbulent flow. The finitely extensible nonlinear elastic (FENE) model and the FENE model with the Peterlin approximation (FENE-P) are numerically integrated along a large number of Lagrangian trajectories resulting from a direct numerical simulation of three-dimensional homogeneous isotropic turbulence. Although the FENE-P model yields results in qualitative agreement with those of the FENE model, quantitative differences emerge. The steady-state probability of large extensions is overestimated by the FENE-P model. The alignment of polymers with the eigenvectors of the rate-of-strain tensor and with the direction of vorticity is weaker when the Peterlin approximation is used. At large Weissenberg numbers, the correlation times of both the extension and of the orientation of polymers are underestimated by the FENE-P model.

Dynamic contact angle measurements of viscoelastic fluids

In this study, the dynamic contact angles of a series of viscoelastic fluids were measured through a modified Wilhelmy plate technique. The advancing and receding contact angles were measured by immersing and withdrawing a PTFE (Teflon) Wilhelmy plate from a reservoir containing a series of different Newtonian and viscoelastic test fluids. The viscoelastic fluids that were tested consisted of either solutions of polyethylene oxide or polyacrylamide in water where the concentration of the polymer was varied to produce solutions with different amounts of fluid elasticity with and without shear thinning. The advancing contact angles of all the viscoelastic fluids tested were found to increase with increasing plate velocity. Conversely, the receding contact angles in each case were found to decrease with increasing contact line velocity. A number of previous measurements have been performed for shear thinning fluids. The measurements presented here are the first to probe the response of the dynamic contact angle at large Weissenberg numbers where the elasticity of the liquid becomes important. For fluids with increased fluid elasticity, the onset of the variation of receding contact angles was found to be delayed to higher contact line velocities and capillary numbers. It was also found that, unlike the case of Newtonian fluids, the cube of advancing and the cube of the receding contact angles were both found to be proportional to the square of the capillary number for highly elastic fluids. Finally, a simple model was proposed to account for the role of elasticity and shear thinning of the viscosity on the dynamic contact angle.