Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

The effect of pre-shear on the extensional rheology of wormlike micelle solutions

The effect of initial microstructural deformation, alignment, and morphology on the response of wormlike micelle solutions in transient uniaxial extensional flows is investigated using a pre-shear device attached to a filament stretching rheometer. In filament stretching experiments, increasing the strength and the duration of the pre-shear just before stretch is found to delay the onset of strain hardening. In these experiments, the wormlike micelle solution filaments fail through a rupture near the axial midplane. The value of the elastic tensile stress at rupture is found to decrease with increasing pre-shear rate and duration. The most dramatic effects are observed at shear rates for which shear banding has been independently observed. The reduction in the strain hardening suggests that pre-shear before filament stretching might break down the wormlike micelles reducing their size before stretch. Strain hardening is also observed in capillary breakup rheometry experiments; however, the pre-sheared wormlike micelle solutions strain harden faster, achieve larger steady-state extensional viscosities and an increase in the extensional relaxation time with increasing shear rate and duration. The difference between the response of the wormlike micelles in filament stretching and capillary breakup experiments demonstrates the sensitivity of these self-assembling micelle networks to pre-conditioning.     

Impact dynamics of drops on thin films of viscoelastic wormlike micelle solutions

The impact dynamics of water drops on thin films of viscoelastic wormlike micelle solutions is experimentally studied using a high-speed digital video camera at frame rates up to 4000 frame/s. The composition and thickness of the thin film is modified to investigate the effect of fluid rheology on the evolution of crown growth, the formation of satellite droplets and the formation of the Worthington jet. The experiments are performed using a series of wormlike micelle solutions composed of a surfactant, cetyltrimethylammonium bromide (CTAB), and a salt, sodium salicylate (NaSal), in deionized water. The linear viscoelastic shear rheology of the wormlike micelle solutions is well described by a Maxwell model with a single relaxation time while the steady shear rheology is found to shear thin quite heavily. In transient homogeneous uniaxial extension, the wormlike micelle solutions demonstrate significant strain hardening. The size and velocity of the impacting drop is varied to study the relative importance of Weber, Ohnesorge, and Deborah numbers on the impact dynamics. The addition of elasticity to the thin film fluid is found to suppress the crown growth and the formation of satellite drops with the largest effects observed at small film thicknesses. A new form of the splashing threshold is postulated which accounts for the effects of viscoelasticity and collapses the satellite droplet data onto a single master curve dependent only on dimensionless film thickness and the underlying surface roughness. Additionally, a plateau is observed in the growth of the maximum height of the Worthington jet height with increasing impact velocity. It is postulated that the complex behavior of the Worthington jet growth is the result of a dissipative mechanism stemming from the scission of wormlike micelles.     

Complex transition to chaotic flow in a periodic array of cylinders

This paper presents a numerical study of the transition to chaos of the flow of a Newtonian fluid in a periodic array of cylinders between two parallel walls. Using tools from dynamical system theory, we identify and characterize the different solutions to the Navier-Stokes equations at different values of the Reynolds number. We show that a very complex transition to chaos occurs for this problem where we first observe two incommensurate frequencies and then a frequency locking followed by a few period doublings following Feigenbaum's route to turbulence.     

Slow flow through a periodic array of spheres

Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction $\text{c}$ of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0($\text{c}{}^{10}$) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0($\text{c}{}^3$) on infinitely long cylinders in square and hexagonal arrays.     

Wave propagation through a periodic array of inclined cracks

In the frame of wave propagation in damaged (elastic) solids, an analytical approach for normal penetration of a plane wave through a periodic array of inclined cracks is developed. The problem is reduced to an integral equation holding over the length of each crack; approximated forms (of one-mode and low-frequency types) are then given to the kernel, so as to derive explicit formulas for the reflection and transmission coefficients. Numerical resolution of the relevant equations finally provides some graphs that are compared.     

Viscoelastic flow of polymer solutions around a periodic,linear array of cylinders: comparisons of predictions for microstructure and flow fields

Numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model; the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbell model of Chilcott–Rallison (CR). In the Giesekus model, intramolecular forces are described by a Hookean spring, whereas a finitely extensible spring whose modulus is given by the Warner approximation is used in both the FENE-P and CR models. Hydro dynamic drag on the beads is taken to be anisotropic for the Giesekus model and isotropic for the other two models. The CR and FENE-P models differ subtly in their approximate treatment of the nonlinear force law. The three models exhibit very similar rheological behavior in viscometric flow and steady elongational flow, with the notable exception that the viscosity for the CR model is shear-rate independent. Finite element simulations are performed by using two different formulations: the elastic-viscous split-stress gradient (EVSS-G) method and a new variant of this formulation, the discrete EVSS-G (DEVSS-G) formulation, in which the elliptic stabilization term is added only to the discrete version of the momentum equation, and the constitutive equation is solved directly in terms of the polymer contribution to the stress tensor. Calculations are performed for all models up to a Weissenberg number We, where the configuration tensor 〈QQ〉 loses positive definiteness. However, by locally refining the mesh in the gap region, the positive definiteness of 〈QQ〉 is recovered. The flow and stress fields predicted by the three constitutive equations are qualitatively similar. A `birefringent strand' of highly stretched polymer molecules, which appears to emanate from the rear stagnation point in the cylinder, strengthens as We is increased. Not surprisingly, the molecular extension computed for the Giesekus model is considerably larger than that of the two FENE spring models. The drag force on the cylinders differs for the FENE-P and CR models, because of the difference in the shear-thinning viscosity resulting from the different approximations used in these models.     

Flow analysis for wormlike micellar solutions in an axisymmetric capillary channel

Flows of wormlike micellar solutions in an axisymmetric capillary channel were studied both numerically and experimentally. In the experiments, an aqueous solution of cetyltrimethylammonium bromide (CTAB) with sodium salicylate (NaSal) was used as a test fluid. The mole concentration of CTAB is 0.03 mol/l, and that of NaSal is 0.06 mol/l. The velocity distribution was measured with a particle tracking velocimetry and flow visualization experiments were performed. The velocity profile showed a plug-like shape and had inflection points where the velocity gradient rapidly changed. High-shear-rate regions near the channel wall spread with increasing the average velocity. Moreover, the flow turned out to be unstable at high average velocities, and when the flow was unstable, white turbidity was observed near the capillary wall. Shear rates showing a white turbidity were included in the range of shear rate where a shear-rate jump in a flow curve occurred. These results suggest that both the characteristic velocity profile and the emergence of white turbidity relate the shear-rate-jump property of wormlike micellar solution. In the numerical analysis, startup flows were considered. A modified Bautista–Manero model was employed as a constitutive equation, and startup flows at a constant average velocity were numerically simulated. The velocity profile at steady state predicted by the numerical simulation adequately agreed with corresponding experimental data. The velocity profile changes from Newton-like to plug-like with time. Inflection points in velocity profile appeared and moved towards the center-side with time. Temporal changes in both velocity gradient and fluidity indicated that the behavior in velocity depended on the shear-rate-jump property of wormlike micellar solution. The velocity gradient rapidly changed around the inflection point and the range of velocity gradient corresponds to that where a white turbidity was observed in the experiments. An erratum to this article can be found at     

Effects of surfactant and salt concentrations on capillary flow and its entry flow for wormlike micelle solutions

Dynamic viscoelasticities and flow properties were measured for aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) to examine the effects of surfactant (C _D) and salt (C _S). The relaxation time λ of a single mode Maxwell model was obtained, and the relationship between λ and free NaSal concentration was discussed. The relation between λ and was applied to the classification of flow curves, which were obtained using a capillary rheometer. In the flow curves, a shear rate jump occurred at low shear rates for the solutions with low , while bending was seen at high shear rates for all the flow curves. On the other hand, vortex growth at the salient corner in the entrance region of the capillary was also investigated. Four different flow patterns were identified: Newtonian-like flow (A), steady vortex flow (B), periodically oscillated flow (C), and perfectly unstable flow (D). In the steady vortex of the flow pattern B, the vortex length increased with increasing shear rate. In the flow patterns C and D, white turbidity was observed. Furthermore, the relation between λ and was also applied to the discussion on the development of the vortex.     

New stress intensity factor solutions of a periodic array of cracks in residual stress field

Many important applications of crack mechanics involve self-equilibrating residual or thermal stress fields. For these types of problems, the traditional fracture mechanics approach based on the superposition principle has ignored the effect of crack surface contact when the crack-tip propagates into the residual compressive region. Contact between the crack faces and the wedging action are responsible for subsequent crack-tip reopening, which often leads to a much larger mode I stress intensity factor. In this study, an analytical approach is used to study the effect of crack face contact for a period array of collinear cracks embedded in several typical residual stress fields. It is found that the nonlinear contact between crack surfaces dominates the cracking behavior in residual/thermal stress fields, which is responsible for crack coalescence.     

Anomalous flow properties of spherical micelle surfactant solutions passing through small-sized slits

The flow properties of complex fluids, such as aqueous solutions of polymers and surfactants, have been investigated in many studies, which revealed interesting and anomalous tendencies for several types of complex fluids in abrupt contraction and expansion flows, such as flows passing through small-sized orifices and slits. In the study, the jet thrust and excess pressure drop (net differential pressure) for experimentally observing their flow properties of water and aqueous solutions of several types of surfactants with spherical micelles in slit flows were measured. Different properties were observed for various surfactant solutions depending on the charge of the solute. The resultant jet thrust and excess pressure drop of cationic and non-ionic surfactant solutions were lower than the experimental values of water. For anionic surfactant solutions, the experimental results were similar to those of water. The types can be arranged as cationic > non-ionic > anionic in order of diminishing the jet thrust and excess pressure drop. Moreover, the effect of strong strain, boundary slip, contraction ratio, size effect, concentration, and interface phenomena was discussed.     

Flow of dilute polyacrylamide solutions through a sudden planar contraction

The flow of partially hydrolyzed polyacrylamide solutions through a 10:1.2 sudden planar contraction was investigated by means of laser-Doppler anemometry. The resultant velocity profiles are compared with those for Newtonian water flow. It is shown that velocity profiles of dilute high molecular weight HPAM solutions of concentrations of 25 ppm and 50 ppm exhibit a velocity maximum upstream of the sudden planar contraction. They first appear near the wall and move towards the channel axis as the flow approaches the contraction. Furthermore, it is found that the centreline velocity profiles of the polymer solution show an earlier response to the downstream flow restriction than water. This is associated with enhanced recirculation regions in front of the sudden channel contraction.Streamlines calculated from the experimentally obtained velocity data reveal all the characteristics of a diverging flow field upstream of the contraction. The experiments reveal that, for the volume flow rate investigated, the flow of dilute polymers remains two-dimensional in the centre throughout the major part of the channel.The addition of small amounts of the divalent salt CaCl₂ reduces the polymer effects to pure Newtonian flow behaviour.     

Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

     

Flow of a viscoelastic fluid between eccentric cylinders

Summary An incompressible non-Newtonian fluid flows in the annular region between two eccentric circular cylinders under a uniform constant longitudinal pressure gradient. The flow is assumed to be sufficiently slow, so that the second-order Rivlin-Ericksen constitutive equation is applicable. The resultant forces exerted by the fluid on the cylinders are calculated.

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Zusammenfassung Eine inkompressible nicht-newtonsche Flüssigkeit fließt unter der Einwirkung eines gleichförmigen konstanten longitudinalen Druckgradienten durch einen von zwei exzentrischen Kreiszylindern begrenzten Ringspalt. Die Strömung wird als so langsam angenommen, daß das Stoffgesetz einer Rivlin-Ericksen-Flüssigkeit zweiter Ordnung angewandt werden kann. Die von der Flüssigkeit auf die beiden Zylinder ausgeübten resultierenden Kräfte werden berechnet.

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With 12 figures and 3 tables     

Flow of a viscoelastic fluid between eccentric cylinders

The velocity field in the annular region between two eccentric cylinders for a second-order fluid is determined. Second-order velocity terms appear only as a result of the interaction between imposed axial and planar motions. When only one of the two motions is imposed by the boundary conditions, the velocity field coincides with that of a Navier-Stokes fluid.The results of the present study are to be used in a future investigation of the stresses, forces, and torques acting on the walls of the cylinders to enable a rheometer based on the present boundary value problem to be suggested.     

High-bandwidth one-and two-particle microrheology in solutions of wormlike micelles

We have developed a large-bandwidth two-particle microrheology technique to measure loss and storage moduli of viscoelastic materials from 0.1 Hz to about 100 kHz using laser trapping and interferometry. We found that quantitative agreement between one- and two-particle microrheology exist in entangled solutions of wormlike micelles chosen as a simple model viscoelastic system. These results validate both experimental method and data interpretation. The consistent results also prove that in a simple system, where the solution length scales are much smaller than the micron probe size, one-particle microrheology can accurately measure bulk viscoelastic parameters.This paper was presented at the AERC 2005     

Pressure-driven flow of wormlike micellar solutions in rectilinear microchannels

In this paper the inhomogeneous response of the (two species) VCM model (Vasquez et al., A network scission model for wormlike micellar solutions. I. Model formulation and homogeneous flow predictions, J. Non-Newtonian Fluid Mech. 144 (2007) 122–139) is examined in steady rectilinear pressure-driven flow through a planar channel. This microstructural network model incorporates elastically active network connections that break and reform mimicking the behavior of concentrated wormlike micellar solutions. The constitutive model, which includes non-local effects arising from Brownian motion and from the coupling between the stress and the microstructure (finite length worms), consists of a set of coupled nonlinear partial differential equations describing the two micellar species (a long species ‘A’ and a shorter species ‘B’) which relax due to reptative and Rouse-like mechanisms as well as rupture of the long micellar chains. In pressure-driven flow, the velocity profile predicted by the VCM model deviates from the regular parabolic profile expected for a Newtonian fluid and exhibits a complex spatial structure. An apparent slip layer develops near the wall as a consequence of the microstructural boundary conditions and the shear-induced diffusion and rupture of the micellar species. Above a critical pressure drop, the flow exhibits shear banding with a high shear rate band located near the channel walls. This pressure-driven shear banding transition or ‘spurt’ has been observed experimentally in macroscopic and microscopic channel flow experiments. The detailed structure of the shear banding profiles and the resulting flow curves predicted by the model depend on the magnitude of the dimensionless diffusion parameter. For small channel dimensions, the solutions exhibit ‘non-local’ effects that are consistent with very recent experiments in microfluidic geometries (Masselon et al., Influence of boundary conditions and confinement on non local effects in flows of wormlike micellar systems, Phys. Rev. E 81 (2010) 021502).