Perspectives on viscoelastic flow instabilities and elastic turbulence

Viscoelastic fluids are a common subclass of rheologically complex materials that are encountered in diverse fields from biology to polymer processing. Often the flows of viscoelastic fluids are unstable in situations where ordinary Newtonian fluids are stable, owing to the nonlinear coupling of the elastic and viscous stresses. Perhaps more surprisingly, the instabilities produce flows with many of the hallmarks of turbulence—even though the effective Reynolds numbers may be $O\left(1\right)$ or smaller. We provide perspectives on viscoelastic flow instabilities by integrating the input from speakers at a recent international workshop: historical remarks, characterization of fluids and flows, discussion of experimental and simulation tools, and modern questions and puzzles that motivate further studies of this fascinating subject. The materials here will be useful for researchers and educators alike, especially as the subject continues to evolve in both fundamental understanding and applications in engineering and the sciences.

Figure 1

Flow-induced birefringence of a viscoelastic polymer melt flowing into (a) and exiting (b) a channel die; note that the shapes of the corners are slightly different between the two. The fringe patterns show lines of constant retardance and constant principal stress difference in the flowing melt. As the flow rate into the die increases, the elastic stress differences increase rapidly and fringes become increasingly numerous and closely spaced (c), (d). Beyond a critical imposed pressure drop the flow becomes unsteady and time dependent (e) as reflected in the chaotic fringe pattern, which is an apparent signature of a phenomenon named elastic turbulence. Adapted from [1].

Figure 2

A map of the different viscoelastic flow instabilities that have been documented to date, with references to selected works exemplifying the different kinematics. Please note that the references shown here ([6, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 262]) are numbered with respect to the papers in this Perspective article. Also note that here we have used $U$ to represent the flow velocity scale, instead of $V$ as in the rest of the text.

Figure 3

A three-dimensional representation of the parameter space accessed in processing flows of complex fluids. Taking ratios of each pair of axes results in new dimensionless material parameters that are independent of the kinematics of the flow [75]. Note that here the flow velocity scale is denoted by $U$, instead of $V$ as in the rest of the text, and we have defined the Deborah number De as a ratio of stresses instead of the ratio of timescales presented in the text.

Figure 4

Sketch of a canonical stability diagram for representing the onset of viscoelastic flow instabilities in a complex fluid. The green curves show lines of constant El, while the red and blue curves show the critical Deborah and Reynolds numbers that represent the boundaries of stability.

Figure 5

A stability diagram for viscoelastic flow of dilute solutions of poly(ethylene oxide) (PEO) and hyaluronic acid (HA) through an OSCER (optimized shape cross-slot extensional rheometer) device. The elasticity number of each fluid (shown by the colored lines of constant Wi/Re) is varied by changing the polymer concentration or solvent viscosity. Two distinct modes of instability can be observed at high Weissenberg and Reynolds numbers; the symbols mark the critical conditions for the onset of each mode of instability. Modified from [55].

Figure 6

Order parameter $\mathrm{\Phi }=\overline{S\left(t\right)}$ vs Weissenberg number $\text{Wi}$ for 2D Taylor-Couette flow with a wide gap. The dashed line indicates the fitted scaling law beyond the elastic instability. Upper inset: Secondary flow strength $S\left(t\right)$ plotted vs $t$ for $\text{Wi}=16.3$. Lower inset: Flow resistance quantified by the azimuthal stress on the outer cylinder plotted vs $\text{Wi}$. Adapted from Ref. [154] with permission from EPL, copyright (2018).

Figure 7

Active control of elastic turbulence. Order parameter vs inverse Deborah number, ${\text{De}}^{-1}$, for different Wi; here De is defined as the product of the outer cylinder reversal frequency and the fluid stress relaxation time, while Wi is defined as the product of the outer cylinder maximum angular velocity and the fluid stress relaxation time. Insets: Secondary-flow strength vs time for different $\text{De}$ at $\text{Wi}=21.4$. The time-modulated driving is switched on at $t=250$. Adapted from Ref. [162].

Figure 8

Nonlinear elastic instability in a microfluidic channel flow. (a) Experimental setup showing the initial linear array of cylinders followed by a long parallel shear flow region. (b)–(d) Space-time dye patterns for the case with 15 cylinders for Newtonian and polymeric fluids measured far downstream. (e) Normalized velocity fluctuations as a function of initial perturbation ($n$) and Wi showing the appearance of two branches. (f) Hysteretic behavior, a hallmark of nonlinear subcritical instabilities, is found for polymeric fluids. Modified from [26].

Figure 9

A cycle of coherent structures in ET at Wi = 185 and downstream distances $l/h=36\text{–}41$. The normalized time $t^{*}=t{f}_{\mathrm{el}}$, where $f_{\mathrm{el}}$ is the elastic wave frequency. The quantities reported here are from PIV measurements of velocity fluctuations reported in a reference frame moving with the average fluid velocity. Fluctuating streamwise velocity ($u^{\prime }$), vertical vorticity (${\omega }^{\prime }$), and spanwise gradient of spanwise velocity ($\partial w^{\prime }/\partial z$) are shown in each column with their scales shown on the right as marked in the plot [198].

Figure 10

Dependence of the normalized friction factor $C_f/C_f^{\mathrm{lam}}$ on Wi for three values of nondimensional distance from the third layer of cylinders (at $l/h=0$) $l/h$: (i) $-30$ to 16 (inset), (ii) 26 to 82 (filled circle), (iii) 200 to 250 (open circle), and (iv) Newtonian solvent 26 to 82 (filled black square). Inset shows $C_f/C_f^{\mathrm{lam}}$ vs Wi across three rows of cylinders [198].

Figure 11

(a) Space-time plots at $-0.4<z/W<0.4$ of the streamwise velocity fluctuations, $u^{\prime }(z,t)$ exhibiting elastic wave structures for three values of Wi. The time series are filtered via a band-pass Butterworth filter centered on the spectral peaks to remove background noise. (b) Streamwise velocity, phase averaged at the elastic wave frequency for $\text{Wi}\text{=}\text{407}$ [200].

Figure 12

Transitions to steady asymmetric flow states in various geometries constructed from microscale cylinders as the Weissenberg number is increased beyond a critical value ${\text{Wi}}_c$. (a) Flow past a single cylinder positioned on the flow axis. Reproduced from [222] with permission from the Royal Society of Chemistry. (b) Velocity fields for flow past side-by-side cylinders with different dimensionless intercylinder gap, $G=L_1/(L_1+L_2)$, where $L_1$ and $L_2$ are the cylinder-cylinder and cylinder-wall gaps, respectively. Reproduced from [223] with permission. (c) Velocity fields for flow past two axially aligned cylinders. Reproduced from [218] with permission from John Wiley & Sons, Inc. (d) Retardation fields for flow through a hexagonal array of cylinders (unpublished data, S. Haward). All cases show the flow from left to right of a shear-thinning viscoelastic WLM solution.

Figure 13

Influence of shear thinning and elasticity on the onset and development of asymmetric flow states around a single cylinder. (a) Flow asymmetry occurs only when characteristic shear rates near the cylinder correspond to the shear-thinning region of the flow curve. (Inset) The onset of instability is consistent with the scaling predicted by McKinley et al., indicating that elasticity and curved streamlines in the downstream wake provide the initial perturbation to destabilize the flow [74, 225]. (b) Stability diagram constructed from simulation results with the l-PTT model examining the interplay between shear thinning and strain hardening ($\varepsilon$) [225]; in this literature shear thinning in the l-PTT model is denoted $\beta$ (the label of the horizontal axis). Panels (a) and (b) are reprinted from [225] with the permission of AIP Publishing. (c) Experimental measurements with polymer solutions over a range of concentration also show that the asymmetric flow around a cylinder (characterized by the magnitude of $I$ plotted on the ordinate axis) requires both shear-thinning and elastic effects in the fluid. Reproduced from [91] with permission.

Figure 14

Experimental images of two distinct unstable flow states observed for elastic polymer solution flow through ordered 1D arrays of pore constrictions. Images show fluid pathlines and are adapted from [65] with permission.

Figure 15

(a) Multistability of the unstable flow of polymeric fluid through the pores of a converging-diverging channel. (b) Trace of polymeric stress tensor inside the pores. (c) Probability density function (PDF) of the ratio of eddies to pore area (${\mathrm{A}}_{\mathrm{eddy}}/{\mathrm{A}}_{\mathrm{pore}}$) at different Wi for a channel of 10 closely located pores. ${\mathrm{A}}_{\mathrm{eddy}}$ represents the total area occupied by eddies in an individual pore and ${\mathrm{A}}_{\mathrm{pore}}$ is the total area of the pore. Above a threshold Wi, multistability occurs, and the eddy areas take on a broad range of values. (d) Time-averaged pressure drop ($\langle \mathrm{\Delta }\mathrm{p}\rangle$) across the channels at different Wi. Images are reproduced from [197].

Figure 16

Experiments in which disorder reduces chaotic fluctuations in viscoelastic flows through porous media. (Top row) Normalized, time-averaged speed field in a microfluidic pillar array for a range of geometric disorders ($\text{Wi}\approx 4$). Scale bar, $150\mu \mathrm{m}$. (Bottom row) Local, normalized speed fluctuations as a function of increasing disorder, corresponding to speed fields above. Images are reproduced from [161].

Figure 17

The occurrence of elastic turbulence is spatially heterogeneous throughout a porous medium, reflecting “porous individualism.” Images show the normalized magnitude of root mean square flow fluctuations in different pores and at different flow rates, parameterized by a characteristic Weissenberg number ${\mathrm{Wi}}_I$. Applied flow is from left to right. Pore A becomes unstable at the lowest flow rate, as shown by the red line in the first row. Pore B becomes unstable at the next highest flow rate, shown by the red line in the second row. Pore C becomes unstable only at even higher flow rates. Note that flow velocity magnitude is denoted by $u$ instead of $v$ as in the rest of the text. Modified from [262].

Figure 18

Snapshots of simulations of EIT in (a) channel flow [298] and (b) pipe flow [297]. (a) Color contours indicate polymer stretching, and lines indicate the magnitude $Q$ of the second invariant of the velocity gradient tensor; reproduced with permission from [298]. (b) Isosurfaces indicate $Q$; licensed under a Creative Commons Attribution (CC BY) license.

Figure 19

Snapshot of the finite amplitude Tollmien-Schlichting wave solution at $\mathrm{Re}=3000$, $\mathrm{Wi}=10$ [304]. White contours are wall-normal velocity, colors are deviations of $xx$ polymer stretch from laminar values. Reproduced with permission from [304].

Figure 20

Snapshots of the finite amplitude Tollmien-Schlichting wave solution at $\mathrm{Re}=10000$ and (a) $\mathrm{Wi}=0$, (b) $\mathrm{Wi}=4$, (c) $\mathrm{Wi}=8$, (d) $\mathrm{Wi}=13$ [305]. White contours are wall-normal velocity, and colors are deviations of $xx$ polymer stretch from laminar values.

Figure 21

Schematic representation of various transition scenarios for viscoelastic pipe flow in the Re-Wi plane. The laminar flow is characterized by the Poiseuille velocity profile with rectilinear streamlines. The linearly unstable regions in the interior of the Wi-Re plane, corresponding to the center-mode instability, are marked by a thick black line (solid) for one specific values of $\beta$, while those for other values of $\beta$ are depicted by light gray lines.

Figure 22

Schematic representation of various transition scenarios for viscoelastic plane Poiseuille flow in the Re-Wi plane. The linearly unstable regions in the interior of the Wi-Re plane, corresponding to the center-mode instability, are marked by thick black lines (solid, dashed, or dash-dotted) for specific values of $\beta$, while those for other values of $\beta$ are depicted by light gray lines. Note that the critical value ${\mathrm{Re}}_{\mathrm{c}}\sim 5772$ is defined using the laminar equivalent centerline velocity.

Figure 23

Free surface instabilities in edge fracture. (a) Schematic of a cone and plate device where the interface undergoes an instability. (b) Snapshots from simulations of the Giesekus model between rigid walls. Note that in this figure, $\mathrm{\Gamma }$ is used to denote interfacial tension, while in the corresponding text we use $\sigma$ instead, and $G$ is the elastic modulus; reproduced with permission from [333].

Figure 24

Extensional necking. (a) Experiments of exponential elongation of a filament of a viscoelastic fluid (0.31 wt% polyisobutylene in polybutene) [348]. Modified from [349], with permission from Elsevier. (b) Numerical simulations of the so-called pom-pom model for an imposed strain rate $\dot{\epsilon }$, with the color scale indicative of the tensile stress [368]; see Fig. 9 of the original reference for the time of each image. Modified from [368], with the permission of the Society of Rheology.

Figure 25

The late stages of the blistering process of an aqueous 1000 ppm PEO solution [397]. The first image is at $t$ = 250 ms after the formation of the cylindrical filament; subsequent images are taken every 30 ms. A thin fiber with the small beads is drawn out of the large droplet (red box). The width of the image is about 300 $\mu$m. Image is from R. Sattler, S. Gier, J. Eggers, and C. Wagner, Phys. Fluids 24, 023101 (2012); licensed under a Creative Commons Attribution (CC BY) license.

Figure 26

Dimensionless fluidity and first normal-stress difference $N_1$ in flow past a sphere (sphere-to-tube aspect ratio 1:2) of a WLM solution; solvent fraction $\beta =5\times {10}^{-3}$ and moderate hardening features, $\text{Wi}=10$; see López-Aguilar et al. [434] for further details of the rheology used.