Flow of wormlike micellar solutions around microfluidic cylinders with high aspect ratio and low blockage ratio

Rheological effects on purely-elastic flow asymmetries in the cross-slot geometry

Viscoelastic flows in the cross-slot geometry can undergo a transition from a steady symmetric to a steady asymmetric flow state, ostensibly due to purely-elastic effects arising beyond a critical flow rate, or Weissenberg number Wi. However, some reports suggest that shear thinning of the fluid's viscosity may also play an important role in this transition. We employ a series of polymer solutions of varying rheological properties to investigate in detail how the interplay between fluid elasticity and shear thinning affects the onset and development of asymmetric flows in the cross-slot. Flow velocimetry is performed on each of the polymer solutions, and is used to assess the degree of flow asymmetry I in the cross-slot as a function of both Wi and a dimensionless parameter S quantifying the flow-rate-dependent extent of shear thinning. Typically, the flow field breaks symmetry as Wi is increased beyond a critical value, but the magnitude of I is found to also be dependent on S. For a few specific polymer solutions, the flow field recovers symmetry above a second, higher critical Wi as S becomes small. The experimental results are summarized in a flow state diagram in Wi-S space, showing the relationship between flow asymmetry and fluid rheology. Finally, to gain a deeper understanding of the effects of shear thinning, numerical simulations are performed using the linear simplified Phan-Thien-Tanner model. We demonstrate that the degree of both shear thinning and elasticity of the fluid, and their interplay, are important factors controlling elastic instabilities in the cross-slot geometry.

Using symmetry to control viscoelastic waves in pillar arrays

Solutions of macromolecules exhibit viscoelastic properties and unlike Newtonian fluids, they may break time-reversal symmetry at low Reynolds numbers resulting in elastic turbulence. Furthermore, under some conditions, instead of the chaotic turbulence, the result is large-scale waves in the form of cyclic spatial and temporal concentration variations, as has been shown for macromolecular DNA flowing in microfluidic pillar arrays. We here demonstrate how altering the symmetry of the individual pillars can be used to influence the symmetry of these waves. We control the extent of instabilities in viscoelastic flow by leveraging the effects of the symmetry of the pillars on the waves, demonstrating suppressed viscoelastic fluctuations with relevance for transport and sorting applications, or conversely opening up for enhanced viscoelasticity-mediated mixing. The onset of waves, which changes flow resistance, occurs at different Deborah numbers for flow in different directions through the array of triangular pillars, thus breaking the symmetry of the flow resistance along the device, opening up for using the occurrence of the waves to construct a fluidic diode.

Numerical Study on the Unstable Flow Dynamics of Wormlike Micellar Solutions past a Sphere in the Creeping Flow Regime

The flow dynamics of wormlike micellar solutions around a sphere is a fundamental problem in particle-laden complex fluids but is still understood insufficiently. In this study, the flows of the wormlike micellar solution past a sphere in the creeping flow regime are investigated numerically with the two species, micelles scission/reforming, Vasquez-Cook-McKinley (VCM) and the single-species Giesekus constitutive equations. The two constitutive models both exhibit the shear thinning and the extension hardening rheological properties. There exists a region with a high velocity that exceeds the main stream velocity in the wake of the sphere, forming a stretched wake with a large velocity gradient, when the fluids flow past a sphere at very low Reynolds numbers. We found a quasi-periodic fluctuation of the velocity with the time in the wake of the sphere using the Giesekus model, which shows a qualitative similarity with the results found in present and previous numerical simulations with the VCM model. The results indicate that it is the elasticity of the fluid that causes the flow instability at low Reynolds numbers, and the increase in the elasticity enhances the chaos of the velocity fluctuation. This elastic-induced instability might be the reason for the oscillating falling behaviors of a sphere in wormlike micellar solutions in prior experiments.

Short and long-range cyclic patterns in flows of DNA solutions in microfluidic obstacle arrays

We observe regular patterns emerging across multiple length scales with high-concentration DNA solutions in microfluidic pillar arrays at low Reynolds numbers and high Deborah numbers. Interacting vortices between pillars lead to long-range order in the form of large travelling waves consisting of DNA at high concentration and extension. Waves are formed in quadratic arrays of pillars, while randomizing the position of the pillar in each unit cell of a quadratic array leads to suppression of the long-range patterns. We find that concentrations exceeding the overlap concentration of the DNA enables the waves, and exploring the behavior of the waves as a function of flow rate, buffer composition, concentration and molecular length, we identify elastic effects as central to the origin of the waves. Our work may not only help increase the low throughput that often limits sample processing in microfluidics, it may also provide a platform for further studies of the underlying viscoelastic mechanisms.

Effect of blockage ratio on flow of a viscoelastic wormlike micellar solution past a cylinder in a microchannel

We present experiments on the flow of a viscoelastic wormlike micellar solution around cylinders (radius R) confined in straight microchannels (width W). Thirteen flow geometries are tested where the blockage ratio is varied over a wide range 0.055 ≤ B_R = 2R/W ≤ 0.63. Experiments are performed at negligible Reynolds number, and for Weissenberg numbers Wi = λU/R up to 1000, where U is the average flow speed and λ is the relaxation time of the fluid. Micro-particle image velocimetry is used to characterise the flow state at each B_R and Wi. In all of the geometries, a first critical Weissenberg number marks a transition from symmetric flow to an asymmetric but time-steady flow state, while a second higher critical Weissenberg number marks the onset of time-dependent flows. However, we report a clear shift in behaviour over a narrow intermediate range of 0.33 ≲ B_R ≲ 0.41. Channels with B_R ≤ 0.33 fall in a 'low' B_R regime, with instabilities that originate from the downstream stagnation point, while those with B_R ≥ 0.44 fall in a 'high' BR regime, with instabilities developing at the upstream stagnation point. Behaviour within the newly-identified intermediate B_R regime is complex due to the competing influence of the two stagnation points. We summarise all our results in a flow state diagram covering Wi-B_R parameter space, clearly defining the different regimes of blockage ratio for the first time. Our results contribute to the understanding of the complexities of viscoelastic flow in this benchmark geometry.

Upstream wall vortices in viscoelastic flow past a cylinder

We report a novel inertia-less, elastic flow instability for a viscoelastic, shear-thinning wormlike micellar solution flowing past a microcylinder in a channel with blockage ratio B_R = 2R/W = 0.5 and aspect ratio α = H/W ≈ 5, where R ≈ 100 μm is the cylinder radius, W is the channel width, and H is the channel height. The instability manifests upstream of the cylinder and changes form with increasing Weissenberg number over the range 0.5 ≲ Wi = Uλ/R ≲ 900, where U is the average flow velocity and λ is the terminal relaxation time of the fluid. Beyond a first critical Wi, the instability begins as a bending of the streamlines near the upstream pole of the cylinder that breaks the symmetry of the flow. Beyond a second critical Wi, small, time-steady, and approximately symmetric wall-attached vortices form upstream of the cylinder. Beyond a third critical Wi, the flow becomes time dependent and pulses with a characteristic frequency commensurate with the breakage timescale of the wormlike micelles. This is accompanied by a breaking of the symmetry of the wall-attached vortices, where one vortex becomes considerably larger than the other. Finally, beyond a fourth critical Wi, a vortex forms attached to the upstream pole of the cylinder whose length fluctuates in time. The flow is highly time dependent, and the cylinder-attached vortex and wall-attached vortices compete dynamically for space and time in the channel. Our results add to the rapidly growing understanding of viscoelastic flow instabilities in microfluidic geometries.

Towards Predicting the Onset of Elastic Turbulence in Complex Geometries

Abstract

Flow of complex fluids in porous structures is pertinent in many biological and industrial processes. For these applications, elastic turbulence, a viscoelastic instability occurring at low Re—arising from a non-trivial coupling of fluid rheology and flow geometry—is a common and relevant effect because of significant over-proportional increase in pressure drop and spatio-temporal distortion of the flow field. Therefore, significant efforts have been made to predict the onset of elastic turbulence in flow geometries with constrictions. The onset of flow perturbations to fluid streamlines is not adequately captured by Deborah and Weissenberg numbers. The introduction of more complex dimensionless numbers such as the M-criterion, which was meant as a simple and pragmatic method to predict the onset of elastic instabilities as an order-of-magnitude estimate, has been successful for simpler geometries. However, for more complex geometries which are encountered in many relevant applications, sometimes discrepancies between experimental observation and M-criteria prediction have been encountered. So far these discrepancies have been mainly attributed to the emergence from disorder. In this experimental study, we employ a single channel with multiple constrictions at varying distance and aspect ratios. We show that adjacent constrictions can interact via non-laminar flow field instabilities caused by a combination of individual geometry and viscoelastic rheology depending (besides other factors) explicitly on the distance between adjacent constrictions. This provides intuitive insight on a more conceptual level why the M-criteria predictions are not more precise. Our findings suggest that coupling of rheological effects and fluid geometry is more complex and implicit and controlled by more length scales than are currently employed. For translating bulk fluid, rheology determined by classical rheometry into the effective behaviour in complex porous geometries requires consideration of more than only one repeat element. Our findings open the path towards more accurate prediction of the onset of elastic turbulence, which many applications will benefit.

Article Highlights

Bifurcations in flows of complex fluids around microfluidic cylinders

Flow around a cylinder is a classical problem in fluid dynamics and also one of the benchmarks for testing viscoelastic flows. The problem is of wide relevance to understanding many microscale industrial and biological processes and applications, such as porous media and mucociliary flows. In recent years, we have developed model microfluidic geometries consisting of very slender cylinders fabricated in glass by selective laser-induced etching. The cylinder radius is small compared with the channel width, which allows the effects of the stagnation points in the flow to dominate over the effects of squeezing between the cylinder and the channel walls. Furthermore, the cylinders are contained in high aspect ratio microchannels that render the flow field approximately two-dimensional (2D) and therefore conveniently permit comparison between experiments and 2D numerical simulations. A number of different viscoelastic fluids including wormlike micellar and various polymer solutions have been tested in our devices. Of particular interest to us has been the occurrence of a striking, steady-in-time, flow asymmetry that occurs for certain non-Newtonian fluids when the dimensionless Weissenberg number (quantifying the importance of elastic over viscous forces in the flow) increases above a critical value. In this perspective review, we present a summary of our key findings related to this novel flow instability and present our current understanding of the mechanism for its onset and growth. We believe that the same fundamental mechanism may also underlie some important non-Newtonian phenomena observed in viscoelastic flows around particles, drops, and bubbles, or through geometries composed of multiple bifurcation points such as cylinder arrays and other porous media. Knowledge of the instability we discuss will be important to consider in the design of optimally functional lab-on-a-chip devices in which viscoelastic fluids are to be used.

Constriction length dependent instabilities in the microfluidic entry flow of polymer solutions

Transport phenomena of fluids and particles through contraction and/or expansion geometries have relevance in many applications. Polymer solutions are often the transporter in these processes, giving rise to flow complexities. The separation distance between a contraction and a following expansion in microfluidic entry flow can affect the interplay between the shear and extension force dominated flow regimes, but the process is still little understood. We investigate the rheological responses of such constriction length dependent instabilities with three different polymer solutions and water in planar contraction-expansion microchannels differing only in the constriction length. The viscoelastic polyethylene oxide (PEO) solution is found to exhibit strong constriction length-dependent instabilities in both the contraction and expansion flows. Such a dependence is, however, completely absent from the flow of shear-thinning xanthan gum (XG) solution and Newtonian water. Interestingly, it is only present in the expansion flow of the both shear thinning and viscoelastic polyacrylamide (PAA) solution.

Stagnation points control chaotic fluctuations in viscoelastic porous media flow

Viscoelastic flows through porous media become unstable and chaotic beyond critical flow conditions, impacting widespread industrial and biological processes such as enhanced oil recovery and drug delivery. Understanding the influence of the pore structure or geometry on the onset of flow instability can lead to fundamental insights into these processes and, potentially, to their optimization. Recently, for viscoelastic flows through porous media modeled by arrays of microscopic posts, Walkama et al. [D. M. Walkama, N. Waisbord, J. S. Guasto, Phys. Rev. Lett 124, 164501 (2020)] demonstrated that geometric disorder greatly suppressed the strength of the chaotic fluctuations that arose as the flow rate was increased. However, in that work, disorder was only applied to one originally ordered configuration of posts. Here, we demonstrate experimentally that, given a slightly modified ordered array of posts, introducing disorder can also promote chaotic fluctuations. We provide a unifying explanation for these contrasting results by considering the effect of disorder on the occurrence of stagnation points exposed to the flow field, which depends on the nature of the originally ordered post array. This work provides a general understanding of how pore geometry affects the stability of viscoelastic porous media flows.

Structure-property relationship of a soft colloidal glass in simple and mixed flows

HYPOTHESIS

Under specific conditions, rod-like cellulose nanocrystals (CNC) can assemble into structurally ordered soft glasses (SGs) with anisotropy that can be controlled by applying shear. However, to achieve full structural control of SGs in real industrial processes, their response to mixed shear and extensional kinematics needs to be determined. We hypothesise that by knowing the shear rheology of the CNC-based soft glass and adopting a suitable constitutive model, it is possible to predict the structure-property relationship of the SG under mixed flows.

EXPERIMENTS

We use an aqueous suspension with 2 wt% CNC at 25 mM NaCl to form a structurally ordered SG composed of a CNC network containing nematic domains. We combine rheometry and microfluidic experiments with numerical simulations to study the flow properties of the SG in shear, extension, and mixed flow conditions. Extensional flow is investigated in the Optimised Shape Cross-slot Extensional Rheometer (OSCER), where the SG is exposed to shear-free planar elongation. Mixed flow kinematics are investigated in a benchmark microfluidic cylinder device (MCD) where the SG flows past a confined cylinder in a microchannel.

FINDINGS

The SG in the MCD displays a velocity overshoot (negative wake) and a pronounced CNC alignment downstream of the cylinder. Simulations using the thixotropic elasto-visco-plastic (TEVP) model yield near quantitative agreement of the velocity profiles in simple and mixed flows and capture the structural fingerprint of the material. Our results provide a comprehensive link between the structural behaviour of a CNC-based SG and its mechanistic properties, laying foundations for the development of functional, built-to-order soft materials.

Tristability in Viscoelastic Flow Past Side-by-Side Microcylinders

Viscoelastic flows through microscale porous arrays exhibit complex path selection and switching phenomena. However, understanding this process is limited by a lack of studies linking between a single object and large arrays. Here, we report experiments on viscoelastic flow past side-by-side microcylinders with variable intercylinder gap. With increasing flow rate, a sequence of two imperfect symmetry-breaking bifurcations forces selection of either one or two of the three possible flow paths around the cylinders. Tuning the gap length through the value where the first bifurcation becomes perfect reveals regions of bistability and tristability in a dimensionless flow rate-gap length phase diagram.

Effect of chain scission on flow characteristics of wormlike micellar solutions past a confined microfluidic cylinder: a numerical analysis

Flow past a microfluidic cylinder confined in a channel is considered as one of the benchmark problems for the analysis of transport phenomena of complex fluids. Earlier experiments show the existence of an elastic instability for the flow of a wormlike micellar solution in this model system after a critical value of the Weissenberg number in the creeping flow regime (G. R. Moss and J. P. Rothstein, J. Non-Newtonian Fluid Mech., 2010, 165, 1505-1515; Y. A. Zhao et al., Soft Matter, 2016, 12, 8666-8681; S. J. Haward et al., Soft Matter, 2019, 15, 1927-1941). This study presents a detailed numerical investigation of this elastic instability in this model system using the two-species VCM (Vasquez-Cook-McKinley) constitutive model for the wormlike micellar solution. Inline with the experimental trends, we also observe the existence of a similar elastic instability in this flow once the Weissenberg number exceeds a critical value. However, we additionally find that the elastic instability in this model geometry is greatly influenced by the breakage and reformation dynamics of the wormlike micelles. In particular, the onset of such an elastic instability is delayed or even may be completely suppressed as the micelles become progressively easier to break. Furthermore, this elastic instability is seen to be associated with the elastic wave phenomena which has been recently observed experimentally for polymer solutions. The present study reveals that the speed of such an elastic wave increases non-linearly with the Weissenberg number similar to that seen in polymer solutions.

Purely Elastic Fluid-Structure Interactions in Microfluidics: Implications for Mucociliary Flows

Fluid-structure interactions lie at the heart of the complex, and often highly coordinated, motions of actively driven microscale biological systems (e.g., translating cilia, flagella, and motile cells). Due to the highly viscoelastic nature of most relevant biological fluids and the small length scales involved, the viscous and inertial forces in such flows are dominated by elasticity. However, elastic effects are often overlooked in studies seeking to address phenomena like the synchronization of beating cilia. In this study, unique microfluidic experiments are presented to demonstrate that inertia-free viscoelastic flows can lead to highly regular beating of an immersed (passive) flexible structure, herein named "purely-elastic" fluid-structure interaction. It is also shown how two such flexible structures can achieve an extraordinary degree of synchronization, with a correlation coefficient approaching unity. The synchronization is a result of the generation of localized elastic stresses in the fluid that effectively link the two objects. These purely elastic interactions may be important to consider toward developing a complete understanding of the motions of microscale biological systems.

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.

Oscillations of a cantilevered micro beam driven by a viscoelastic flow instability

The interaction of flexible structures with viscoelastic flows can result in very rich dynamics. In this paper, we present the results of the interactions between the flow of a viscoelastic polymer solution and a cantilevered beam in a confined microfluidic geometry. Cantilevered beams with varying length and flexibility were studied. With increasing flow rate and Weissenberg number, the flow transitioned from a fore-aft symmetric flow to a stable detached vortex upstream of the beam, to a time-dependent unstable vortex shedding. The shedding of the unstable vortex upstream of the beam imposed a time-dependent drag force on the cantilevered beam resulting in flow-induced beam oscillations. The oscillations of the flexible beam were classified into two distinct regimes: a regime with a clear single vortex shedding from upstream of the beam resulting in a sinusoidal beam oscillation pattern with the frequency of oscillation increasing monotonically with Weissenberg number, and a regime at high Weissenberg numbers characterized by 3D viscoelastic instabilities where the frequency of oscillations plateaued. The critical onset of the flow transitions, the mechanism of vortex shedding and the dynamics of the cantilevered beam response are presented in detail here as a function of beam flexibility and flow viscoelasticity.

Flow Patterns of Viscoelastic Fracture Fluids in Porous Media: Influence of Pore-Throat Structures

Viscoelastic surfactant (VES) fluid and hydrolyzed polyacryamide (HPAM) solution are two of the most common fracturing fluids used in the hydraulic fracturing development of unconventional reservoirs. The filtration of fracturing fluids in porous media is mainly determined by the flow patterns in pore-throat structures. In this paper, three different microdevices analogue of porous media allow access to a large range of Deborah number (De) and concomitantly low Reynolds number (Re). Continuous pore-throat structures were applied to study the feedback effect of downstream structure on upstream flow of VES fluid and HPAM solution with Deborah (De) number from 1.11 to 146.4. In the infinite straight channel, flow patterns between VES fluids and HPAM solution were similar. However, as pore length shortened to 800 μm, flow field of VES fluid exhibited the triangle shape with double-peaks velocity patterns. The flow field of HPAM solution presented stable and centralized streamlines when Re was larger than 4.29 × 10^-2. Additionally, when the pore length was further shortened to 400 μm, double-peaks velocity patterns were vanished for VES fluid and the stable convergent flow characteristic of HPAM solution was observed with all flow rates.