Disorder Suppresses Chaos in Viscoelastic Flows

Derek M. Walkama, Nicolas Waisbord, and Jeffrey S. Guasto

Phys. Rev. Lett. 124, 164501 – Published 20 April 2020

Abstract

Viscoelastic flows through microstructured geometries transition from steady to time dependent and chaotic dynamics under critical flow conditions. However, the implications of geometric disorder for flow stability are unknown. We measure the onset of spatiotemporal velocity fluctuations for a viscoelastic flow through microfluidic pillar arrays, having controlled variations of geometric disorder. Introducing a small perturbation into the pillar array ( $\sim 10 %$ of the lattice constant) delays the onset of the instability to higher flow speed, and yet larger disorders ( $\geq 25 %$ ) suppress the transition to chaos. We show that disorder introduces preferential flow paths that promote shear over extensional deformation and enhance flow stability by locally reducing polymer stretching.

Received 11 July 2019
Revised 24 December 2019
Accepted 23 March 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.164501

Physics Subject Headings (PhySH)

Complex fluids Flow instability Flows in porous media Non-Newtonian fluids

Viscoelasticity

Fluid DynamicsPolymers & Soft Matter

Authors & Affiliations

Derek M. Walkama ^1,2, Nicolas Waisbord¹, and Jeffrey S. Guasto ^1,*

¹Department of Mechanical Engineering, Tufts University, 200 College Avenue, Medford, Massachusetts 02155, USA
²Department of Physics and Astronomy, Tufts University, 574 Boston Avenue, Medford, Massachusetts 02155, USA

^*Corresponding author. Jeffrey.Guasto@tufts.edu

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Vol. 124, Iss. 16 — 24 April 2020

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Images

Figure 1
Disorder reduces chaotic fluctuations in viscoelastic flows. (a) Normalized, time-averaged speed field, $\bar{u} (r) / \max [\bar{u} (r)]$ , in a microfluidic pillar array for a range of Weissenberg numbers, Wi, and geometric disorders, $β$ (40% of full field of view shown; see also Supplemental Movies 1–5 [35]). Scale bar, $150 μ m$ . (b) Local, normalized speed field fluctuations, ${\tilde{u}}_{rms} (r)$ , as a function of increasing Wi, corresponding to speed fields for $β = 0$ [green box in (a)]. (c) Local, normalized speed fluctuations as a function of increasing disorder, corresponding to speed fields for $Wi \approx 4$ [magenta box in (a)].
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Figure 2
Pore-scale velocity fluctuations are indicative of elastic instability. (a) Normalized, time-averaged speed field for the ordered lattice ( $β = 0$ , $Wi \approx 4$ ). Streamlines (black) are computed from measured flow fields, and pillar locations (gray circles) are used to discretize throat cross sections (gray lines). Scale bar, $70 μ m$ . (b) Schematic of the throat flow profile $\tilde{u} (λ_{i}, t)$ and spatially averaged speed ${\tilde{u}}_{i} (t)$ with local coordinate $λ_{i}$ . (c) Normalized, time-averaged speed field for a disordered lattice ( $β = 1$ , $Wi \approx 4$ ). (d),(e) Measured instantaneous [ $\tilde{u} (λ_{i}, t)$ ; green] and time-averaged ( $⟨ \tilde{u} (λ_{i}, t) ⟩_{t}$ ; black) throat speed profiles (left) and flow speed fluctuation, ${\tilde{u}}_{i}^{'} = {\tilde{u}}_{i} (t) - ⟨ {\tilde{u}}_{i} (t) ⟩_{t}$ , kymographs (right) for individual throats from (d) ordered and (e) disordered channels [blue throats in (a) and (c), respectively; $Wi \approx 4$ ]. Instantaneous speed profiles (left, green) correspond to indicated kymograph time (right, green). (f) Normalized, instantaneous throat flow speed for disordered ( $β = 0$ ) and ordered ( $β = 1$ ) throats at three different Wi values [blue throats in (a) and (c), respectively].
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Figure 3
Disorder delays the onset of viscoelastic instability. (a) Normalized temporal throat speed fluctuations as a function of Weissenberg number Wi for a range of geometric disorders $β$ . Newtonian (97% glycerol; black circles) and shear-thinning (3000 ppm xanthan gum; black stars) control experiments shown for $β = 0$ (upper horizontal axis). (b) Normalized spatial fluctuations of time-averaged throat flow speed as a function of Wi in various disorders. Dotted lines represent measured Newtonian values. (c) Temporal and spatial fluctuations of both viscoelastic and Newtonian fluids for various $β$ at fixed mean flow speed corresponding to $Wi \approx 4$ .
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Figure 4
Disorder regulates flow-type history and the transition to chaotic flow. (a) Flow-type parameter maps measured from time-averaged velocity fields $\bar{u} (r)$ for ordered and disordered channels at low and high Wi. Particle trajectories (green) shown for one relaxation time, $τ$ (lower right). Scale bar, $100 μ m$ . (b) Flow-type autocorrelation along tracer trajectories for $β = 0$ (left) and $β = 1$ (right). (c) Ensemble-averaged flow type along tracer trajectories over one relaxation time.
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