A generalized lever rule for shear banding of yield stress fluids

Michela Geri,1, ∗ Brice Saint-Michel,2, 3 Thibaut Divoux,2 Gareth H. McKinley,4 and Sebastien Manneville2, †

1Department of Materials Science and Engineering,Massachusetts Institute of Technology, Cambridge MA 02139, United States

2Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France3Present Address: Univ Gustave Eiffel, CNRS, Ecole des Ponts Paristech,

UMR 8205 Laboratoire Navier, 5 Boulevard Descartes CEDEX 2, 77454 Marne-la-Vallee, France4Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA 02139, United States

(Dated: November 17, 2021)

We study the local dynamics of a yield stress fluid that shows a pronounced non-monotonic flowcurve. Such a mechanically unstable behavior generally leads to the coexistence of a flowing bandwith an arrested region below a critical shear rate γc. Combining ultrasound velocimetry withstandard rheometry, we discover an original shear banding scenario in the decreasing branch of theflow curve, in which the flow profile of the flowing band is set by the applied shear rate γ insteadof γc. As a consequence, the material slips at the walls with a velocity that shows a non-trivialdependence on γ. To capture our experimental observations, we propose a differential version ofthe so-called lever rule that describes the extent of the flowing band and the evolution of wall slipvelocity with shear rate. Our approach therefore provides a generalized lever rule where constraintsimposed in the classical shear banding scenario can be relaxed, while wall slip plays the role of anadditional degree of freedom.

Introduction.- Yield stress fluids (YSF) encompass abroad range of materials, from cosmetics and food prod-ucts to cement pastes and waxy crude oils. All thesematerials have in common a jammed or percolated mi-crostructure that confers upon them solid-like propertiesunder low external stresses. However, for stresses largerthan a critical value, the microstructure yields and thematerial flows like a liquid [1]. The existence of such ashear-induced solid to liquid transition results in a time-dependent mechanical response, referred to as thixotropy,and in complex flow properties, including stick-slip, frac-ture, and more generally heterogeneous flow profiles [2–6]. The latter phenomenon, coined shear banding, hasbeen mainly attributed to the competition between aging(that often results from attractive interactions betweenconstituents) and the rejuvenation imposed by externalshear [7–9]. In practice, such a competition leads to adecreasing branch in the flow curve of shear stress σ vs.shear rate γ, and to the existence of a critical shear rateγc, below which a homogeneous shear flow becomes un-stable [10, 11]. As a result, the local flow profile splitsinto a fluidized region sheared at a critical shear rate γcand an arrested band, whose extent follows the lever rule,i.e., grows proportionally to (γc − γ) as the shear rate γis decreased below γc [5, 10, 12–14].

The above phenomenology was further shown to be af-fected by boundary conditions and by the existence ofslip at the wall. Originally described as a mere artifactthat can be suppressed by well-chosen surface properties,wall slip appears to play a deeper, fundamental role inthe flow properties of YSF [15–17]. For instance, wallslip may affect the yielding transition and modify the

∗ mgeri@mit.edu† sebastien.manneville@ens-lyon.fr

steady-state flow properties of YSF [18, 19]. Nonethe-less, the current consensus remains that wall slip andmore generally boundary conditions can always be tunedindependently of the material rheological response [20].

In this Letter, we rationalize the interplay between wallslip and shear banding in YSF through experiments onmodel paraffin gels, which display a pronounced non-monotonic flow curve under simple steady shear. Ve-locimetry coupled to standard rheometry allows us tounravel an original steady-state shear banding scenario:along the decreasing branch of the flow curve, the shearrate in the fluidized band is controlled by the externallyimposed shear rate γ, rather than being fixed to γc. Asa result, the slip velocity of the fluid at the wall displaysa non-trivial dependence with γ. We propose a simplemodel that describes all the experimental data, in whichwall slip is directly related to the bulk behavior of thematerial based on a differential formulation of the leverrule. This approach remains valid down to very low shearrates, at which cooperative effects come into play as thedimension of the flowing band becomes comparable to theaverage particle size [21–23]. Our results offer a new per-spective on wall slip in YSF, and a generalized approachto the classical lever rule that allows us to relax the con-straints imposed in the classical shear-banding scenario.

Materials and Methods.- Model paraffin gels are ob-tained by dissolving linear n-paraffin chains in heavymineral oil at high temperature (T = 80◦C) and attwo different weight fractions (10 % wt. and 15 % wt.).An example of the gel microstructure after cooling toambient temperature is shown in Fig. 1(a). It con-sists of platelet-like paraffin particles of characteristicsize ` ' 130µm that interact sterically and throughinterparticle crystalline bridges [24, 25]. Each solutionis poured still hot into a homemade Taylor-Couette cellwith a gap H = 1.1 mm mounted on a stress-controlled

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<latexit sha1_base64="Y28Ub1ll6wLnJhXu+OYUvptAv64=">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</latexit>

(c)

<latexit sha1_base64="lMZv5J4wt/T8C+GFXN98sUunITM=">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</latexit>

� = 50 s�1

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� = 0.032 s�1<latexit sha1_base64="HnKZbmD7Yi0EgHIZ2hrrf8KW3YM=">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</latexit>

� = 0.0063 s�1

FIG. 1. (a) Birefringence image of a 10 % wt. paraffin gel sam-

ple cooled under static conditions at T = −0.2◦C/min. Thewidth of the image is 1.25 mm (see also Supplemental Mate-rial). (b) Flow curve of shear stress σ vs. shear rate γ. Veloc-ity profiles u(y) recorded simultaneously allow us to identifythree regimes: (i) homogeneous velocity profile (see inset forγ = 50 s−1); (ii) banded velocity profile with clear wall slip atthe rotor as illustrated in (c) for γ = 0.032 s−1; (iii) curvedvelocity profile as illustrated in (d) for γ = 0.0063 s−1. Thecoordinate y denotes the distance from the fixed wall of theTaylor-Couette cell with gap H = 1.1 mm. Error bars arerepresentative of temporal fluctuations and of experimentaluncertainties on the velocity u. Black lines in (b,c,d) arethe homogeneous velocity profiles expected without wall slip.Colored solid lines in (c,d) are best fits to the velocity pro-files, while the dashed line in (d) shows the linear profile thatwould be expected from the average shear rate in the shearedband (see Supplemental Material for more details).

rheometer (AR-G2, TA Instruments) equipped with two-dimensional ultrasonic velocimetry [26]. The small gapminimizes stress inhomogeneities that give rise to shearlocalization [27]. The cylinder is covered with sandpaperand the outer cup is sandblasted. The gel is obtained bycooling the sample under shear at γ = 50 s−1. A series ofdecreasing shear rate steps is then applied, starting from50 s−1 down to 0.005 s−1, during which velocity maps areacquired (see Supplemental Material for details).

Results.- Figure 1(b) illustrates the steady-state flowbehavior of the 10 % wt. paraffin gel, measured by a de-creasing ramp of shear rate steps. The flow curve shows apronounced decreasing branch, which is delimited by twospecific shear rates: γmin ' 0.09 s−1, at which the stressgoes through a local minimum, and γmax ' 0.016 s−1,

at which the stress goes through a weak local maximum.These two specific shear rates separate three differentflow behaviors as confirmed by velocity profiles recordedsimultaneously to the flow curve. For large shear rates,i.e., γ ≥ γmin (regime i ), the flow curve increases mono-tonically, the corresponding velocity profiles are homo-geneous and wall slip remains negligible as expected forrough walls [see inset of Fig. 1(b)]. Intermediate shearrates, i.e., γmax ≤ γ ≤ γmin (regime ii ), correspond tothe decreasing branch of the flow curve. The base flowis unstable and the velocity profiles separate into two re-gions, a flowing shear band close to the rotor (located aty = 1.1 mm) and an unyielded, solid-like region close tothe stator [see Fig. 1(c) for γ = 0.032 s−1]. The extent ybof the arrested and unyielded band increases as the shearrate decreases, while the average local shear rate in theflowing band γb is always very close to the global imposedshear rate γ. Concomitantly, we measure noticeable slip-page of the flowing band at the rotor with a slip velocityus. Finally, for γ ≤ γmax (regime iii ), the quiescentband takes up about half the gap and the velocity profileof the flowing band shows increasing curvature, while γbremains very close to γ [see Fig. 1(d) for γ = 0.0063 s−1].We emphasize that a 15 % wt. gel shows the same phe-nomenology with larger values of γmin and γmax (see Sup-plemental Fig. S4), and that a steady state is reachedfor all the shear rates investigated, as supported by thestress evolution over time shown in Fig. S1(b) and S4(b)in Supplemental Material.

Figure 2 further analyzes the velocity profiles as a func-tion of γ/γmin for both wax concentrations. Remark-ably, all three observables defined so far and recalledin Fig. 2(d); namely, the normalized extent of the ar-rested band, yb/H [Fig. 2(a)], the normalized average lo-cal shear rate in the flowing band, γb/γ [Fig. 2(b)] and theratio us/uw of the slip velocity at the rotor to the rotorvelocity [Fig. 2(c)], collapse on the same master curves.This hints at a generic underlying physical mechanism,in which both the arrested band and the relative slip ve-locity grow for decreasing shear rates below γmin, whilethe flowing band always experiences the global imposedshear rate γ.

Modeling.- The above observations are in stark con-trast with previous reports on steady-state shear band-ing, in which the flowing band is sheared at the criticalshear rate γc and the “classical” lever rule, yb ∝ (γc− γ),holds for γ < γc. These features have been associatedwith a non-monotonic underlying constitutive behavior,whose experimental signature is a stress plateau in thesteady-state flow curve below γc [10, 11]. In our experi-ments, the extent of the flowing band yb rather displaysa nonlinear, logarithmic dependence on γ [Fig. 2(a)] andthe average local shear rate in the flowing band γb coin-cides (within error bars) with the externally applied shearrate γ [Fig. 2(b)]. Moreover, the fact that strong wall slipgoes along with shear-banded flows suggests that slip-page is key to account for the present steady-state shearbanding. Indeed, as shown in Fig. 3, the slip velocity

3

10-2 10-1 100 101 102 1030

0.5

1

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210-2 10-1 100 101 102 1030

0.5

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0.5

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10% wt.15% wt.

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(b)<latexit sha1_base64="zH98l+LOxXB+M4RZVSGfaQhDazw=">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</latexit>

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FIG. 2. (a) Extent yb of the solid-like band, (b) local shearrate in the flowing band γb and (c) slip velocity us madedimensionless respectively by the gap width H, the imposedshear rate γ and the wall velocity uw = γH. Data for both the10 % wt. (filled circles) and 15 % wt. (open diamonds) gels arepresented as logarithmic functions of the dimensionless shearrate γ/γmin respectively with γmin = 0.09 s−1 and 0.23 s−1 asextracted from the flow curves. Solid lines are fits to Eq. (3)with β = 0.19. Note that, for the curved profiles in regime

iii , γb is defined as the average of the local shear rate in theflowing band. (d) Schematic defining the different quantities.

us displays non-trivial trends as a function of γ for bothparaffin concentrations.

In order to rationalize our observations, we now re-visit the premises of the classical lever rule. First, wenote that the classical lever rule simply derives from thecontinuity of the velocity profile in the absence of wallslip [5, 12]. Second, YSF obeying the lever rule are usu-ally modeled using a structural parameter possessing asingle time scale, which drives the dynamics of both theyield stress at rest and the plastic viscosity under flow[12, 28–30]. Yet, extensive experiments have shown thatmodel paraffin gels as investigated here can build up ayield stress very fast, while their plastic viscosity changesover a much longer timescale [27, 31–35]. Modeling thisclass of thixotropic YSF requires the introduction of twocharacteristic time scales, denoted respectively τy (overwhich the yield stress builds up in the unyielded region)and τv (over which the plastic viscosity changes in theflowing band), with τy � τv. We hypothesize that whenτy � τv, the flowing band cannot adjust its viscosity fastenough to a fixed critical shear rate, here denoted γmin.As a consequence, the local shear rate γb remains closeto the global imposed shear rate γ, while the fluid pref-erentially starts to slip at the boundaries of the shearcell through a thin lubricating layer mainly composed of

10-2 10-1 1000

0.005

0.01

0.015

0.02

0.025

FIG. 3. Slip velocity us as a function of the dimensionless

shear rate γ/γmin in regimes ii and iii for the 10 % wt.

(filled circles, γmin = 0.09 s−1) and 15 % wt. (open diamonds,γmin = 0.23 s−1) gels. The predictions of Eq. (3) for β = 0.19are shown in black solid lines without any additional fittingparameter for both gel concentrations.

solvent.Such a scenario can be supported by the following en-

ergy considerations based on the dissipation per unit vol-

ume averaged over the gap width, P = 1H

∫H0σ(γ)γdy.

In the case of shear banding following the classical leverrule, the stress reaches a plateau value σ0 [5, 12, 36] andP = σ0γchc/H, where hc is the width of the fluidizedband. In contrast, if the velocity profile consists of a flu-idized band with width h = H − yb sheared at γb = γ,and a slip layer close to the rotor as shown by the redline in Fig. 2(d), then P = σ(γb)[γbh + us]/H. Sinceγchc = γbh + us = γH ≡ uw to ensure continuity ofthe velocity profiles, the power dissipated in the classi-cal shear-banding scenario is larger than in the case ofshear banding combined with wall slip, at least whenσ(γ) ≤ σ0. Therefore, it may be energetically favorablefor the system to access at least part of the states withinthe decreasing branch of the flow curve, provided thatwall slip can occur and that there is a clear separation oftime scales τy � τv in the thixotropic YSF.

Within this framework, we can further derive expres-sions for both the extent of the shear band yb and the slipvelocity us as a function of the external imposed shearrate. We start by considering the slip layer as an addi-tional band of finite thickness δs, with the same viscosityµo as the suspending oil, such that us = σδs/µo basedon stress homogeneity within the gap. Perturbing therelation γbh+ us = γH leads to:

hdγb + γbdh+ d

(σδsµo

)= Hdγ. (1)

Since τy � τv, any microstructural change within theflowing band develops much more slowly than in the un-yielded band where the yield stress is building up. Hence,for any given time interval dt, hdγb � γbdh and, up tofirst order, we may neglect the contribution from hdγb.The term related to slip is also much smaller than the two

4

remaining contributions because the slip layer thicknessδs � h,H (typically by two to three orders of magnitude[37–41]) and σ/µo ∼ γµp/µo ∼ 1 s−1, where µp/µo ' 10is the ratio between the plastic viscosity of the gel andthe oil viscosity. Based on these simplifications and sub-stituting γb = γ, the revised lever rule in differential formsimply reads:

γdh ' Hdγ . (2)

Upon integration of Eq. (2) from γmin at which h = Hto γ < γmin at which h = H − yb, we obtain:

ybH

= −β ln

(γ

γmin

)=usuw

, (3)

where β ∼ O(1) is a dimensionless scaling factor thataccounts for all the simplifications leading to Eq. (2),and the last equality results from velocity continuity,which imposes us = γbyb = γyb. Figure 2(a) shows thatEq. (3) provides excellent descriptions of the normalizedextent of the arrested region yb/H and of the normal-ized slip velocity us/uw with β = 0.19 ± 0.05 in both

regimes ii and iii [see black solid lines in Fig. 2(a,c)].Interestingly, the same value of β fits both data setsequally well, which confirms the consistency and robust-ness of the modified lever rule through a priori indepen-dent wall slip and bulk flow measurements. Finally, withuw = γH, Eq. (3) reads us = βHγ ln(γmin/γ), which ac-counts quantitatively very well for our experimental wallslip data [Fig. 3]. In particular, the non-monotonic evo-lution of us with γ appears as a distinctive consequenceof the modified lever rule.

Discussion and conclusion.- The predictions obtainedfrom the generalized differential flow rule hold all alongthe decreasing branch of the flow curve, i.e., in regimeii , as well as in the limit of very low shear rates, i.e.,in regime iii . In the latter regime, the flowing band nolonger results in a uniform shear rate γb = γ, but ratherdisplays a strongly curved velocity profile [see Fig. 1(d)].As detailed in the Supplemental Material, such a cur-vature can be interpreted as a result of cooperative ef-fects when the extent of the flowing band gets smallerthan a few times that of single wax particles ` ' 130µm[21, 22, 42, 43]. In particular, Supplemental Fig. S5shows that a nonlocal model based on a diffusion equa-tion for the fluidity of the YSF correctly fits the velocityprofiles in the flowing band with a cooperativity lengthξ ' 250 µm, very close to the characteristic lateral di-mension of paraffin platelets `. Our results suggest thatin spite of the confinement of the sample microstructure,the generalized lever rule remains valid in regime iii ,provided one defines γb as the average of the local shearrate in the flowing band [see Fig. 2(b)]. Note that thecondition us ≤ uw implies that Eq. (3) should hold downto γ = exp(−1/β)γmin ' 0.0052γmin, which is approxi-mately five times smaller than the minimum shear rateachieved in the present velocimetry experiments. There-fore, future work under even lower rates should assesswhether Eq. (3) remains valid deeper into regime iii .

The approach proposed in this Letter quantitativelydescribes both wall slip and bulk flow heterogeneity inYSF for which the usual assumptions underpinning theclassical lever rule do not hold. Still, our approachdoes not simply constitute an alternative to the stan-dard steady-state shear-banding scenario in the absenceof wall slip. It also encompasses the standard scenariounder appropriate assumptions. Indeed, if τy & τv andno wall slip occurs, the local shear rate can remain fixedat γc and Eq. (2) reduces to γcdh = Hdγ, hence uponintegration we obtain γchc = γH, which boils down tothe classical lever rule.

The possibility of the time scale separation τy � τvand its microscopic origin appear as important yet oftenoverlooked features in many thixotropic YSF, for which itis often assumed that the two time scales coincide. In thecase of the present paraffin gels, the highly anisotropic,platelet-like shape of the interacting microcrystals thatconstitute the gel microstructure [see Fig. 1(a) and mi-croscopy in Supplemental Material] is likely to accountfor a short time scale τy required to rebuild the solid-likeproperties compared to the time scale τv for the evolu-tion of the plastic viscosity. As a result, the decreasingbranch of the flow curve, which is mechanically unstableand usually observed only in transient responses [44–47],becomes increasingly stabilized here.

The above discussion prompts us to also considerthe case of unsteady heterogeneous flows under externalshear. For instance, it is known that shear start-up flowsof YSF and the corresponding yielding transition are ac-companied by wall slip, even in presence of rough bound-ary conditions [18, 48, 49]. During these short periods oftime, YSF, even with isotropic constituents, experienceconditions similar to those reported in the present Let-ter, i.e., τy � τv, which strongly suggests that wall slipalso plays the role of an external degree of freedom dur-ing transient flows. In such framework, Eq. (3) provides afunctional form for the scaling of the slip velocity with theexternally imposed shear rate γ. This expression couldbe coupled to spatially-resolved models, e.g., soft glassyrheology or simpler fluidity models [14], to account forwall slip in transient flows.

Finally, our results at low shear rates, where cooper-ative effects dominate the flow profile, call for incorpo-rating wall slip into spatially-resolved models to furtherinvestigate the interplay between wall slip and nonlocaleffects. To date, spatially-resolved models with coopera-tive effects that successfully account for both steady-stateand complex long-lasting transient flows [50–53], do notinclude wall slip. The present experimental results shouldserve as a benchmark for testing such generalized theo-ries in an effort to fully understand flow processes undervery small imposed shear rates. Above all, they providea more general framework for understanding the com-plexity of heterogeneous flows of YSF of interest in manyengineering, geophysical and biomedical applications.

Acknowledgments.- We are very thankful to Dr. Ba-vand Keshavarz for machining the rotor with great preci-

5

sion. We thank the MIT-France program for supportingthis collaboration between MIT and ENS de Lyon. MG

and GHM are also grateful to Chevron ETC and the MITEnergy Initiative for supporting part of this research.

[1] D. Bonn, M. M. Denn, L. Berthier, T. Divoux, andS. Manneville, Rev. Mod. Phys. 89, 035005 (2017).

[2] J. Persello, A. Magnin, J. Chang, J. M. Piau, and B. Ca-bane, J. Rheol. 38, 1845 (1994).

[3] T. Divoux, D. Tamarii, C. Barentin, and S. Manneville,Phys. Rev. Lett. 104, 208301 (2010).

[4] P. Coussot, J. Non-Newton. Fluid Mech. 211, 31 (2014).[5] T. Divoux, M. A. Fardin, S. Manneville, and S. Lerouge,

Annu. Rev. Fluid Mech. 48, 81 (2016).[6] D. E. V. Andrade and P. Coussot, Soft Matter 15, 8766

(2019).[7] V. Viasnoff and F. Lequeux, Phys. Rev. Lett. 89, 065701

(2002).[8] M. Cloitre, R. Borrega, and L. Leibler, Phys. Rev. Lett.

85, 4819 (2000).[9] D. Bonn, S. Tanase, B. Abou, H. Tanaka, and J. Meu-

nier, Phys. Rev. Lett. 89, 015701 (2002).[10] P. Coussot, J. Raynaud, F. Bertrand, P. Moucheront,

J. Guilbaud, H. Huynh, S. Jarny, and D. Lesueur, Phys.Rev. Lett. 88, 218301 (2002).

[11] S. M. Fielding, Soft Matter 3, 1262 (2007).[12] G. Ovarlez, S. Rodts, X. Chateau, and P. Coussot,

Rheol. Acta 48, 831 (2009).[13] A. Fall, J. Paredes, and D. Bonn, Phys. Rev. Lett. 105,

225502 (2010).[14] S. M. Fielding, Rep. Prog. Phys. 77, 102601 (2014).[15] R. Buscall, J. Rheol. 54, 1177 (2010).[16] M. Cloitre and R. T. Bonnecaze, Rheol. Acta 56, 283

(2017).[17] L. Derzsi, D. Filippi, G. Mistura, M. Pierno, M. Lulli,

M. Sbragaglia, M. Bernaschi, and P. Garstecki, Phys.Rev. E 95, 052602 (2017).

[18] T. Gibaud, C. Barentin, and S. Manneville, Phys. Rev.Lett. 101, 258302 (2008).

[19] T. Gibaud, C. Barentin, N. Taberlet, and S. Manneville,Soft Matter 5, 3026 (2009).

[20] V. Mansard, L. Bocquet, and A. Colin, Soft Matter 10,6984 (2014).

[21] J. Goyon, A. Colin, G. Ovarlez, A. Ajdari, and L. Boc-quet, Nature 454, 84 (2008).

[22] L. Bocquet, A. Colin, and A. Ajdari, Phys. Rev. Lett.103, 036001 (2009).

[23] J. Goyon, A. Colin, and L. Bocquet, Soft Matter 6, 2668(2010).

[24] P. Singh, R. Venkatesan, H. S. Fogler, and N. Nagarajan,AIChE journal 46, 1059 (2000).

[25] Y. Miyazaki and A. G. Marangoni, Mater. Res. Express1, 025101 (2014).

[26] T. Gallot, C. Perge, V. Grenard, M.-A. Fardin, N. Taber-let, and S. Manneville, Rev. Sci. Instrum. 84, 045107(2013).

[27] D. E. Andrade, M. Ferrari, and P. Coussot, J. Non-Newton. Fluid Mech. 279, 104261 (2020).

[28] P. Coussot, Q. Nguyen, H. Huynh, and D. Bonn, Phys.Rev. Lett. 88, 175501 (2002).

[29] A. Ragouilliaux, B. Herzhaft, F. Bertrand, and P. Cous-sot, Rheol. Acta 46, 261 (2006).

[30] I. Cheddadi, P. Saramito, and F. Graner, J. Rheol. 56,213 (2012).

[31] C. J. Dimitriou, R. H. Ewoldt, and G. H. McKinley, J.Rheol. 57, 27 (2013).

[32] R. Mendes, G. Vinay, G. Ovarlez, and P. Coussot, J.Non-Newton. Fluid Mech. 220, 77 (2015).

[33] R. Mendes, G. Vinay, G. Ovarlez, and P. Coussot, J.Rheol. 59, 703 (2015).

[34] M. Geri, R. Venkatesan, K. Sambath, and G. H. Mckin-ley, J. Rheol. 61, 427 (2017).

[35] For a similar chemical composition, fits to an elastovis-coplastic model of thixotropic yield stress fluids lead toτv ' 460 s and τy ' 10 s [34]. Note that the longer timescale τv is much smaller than the one-hour duration ofeach step at a given shear rate, which allows us to con-sider that a steady state is reached for all γ in regimeii .

[36] P. Schall and M. van Hecke, Annu. Rev. Fluid Mech. 42,67 (2010).

[37] J.-B. Salmon, S. Manneville, and A. Colin, Phys. Rev.E 68, 051503 (2003).

[38] J.-B. Salmon, L. Becu, S. Manneville, and A. Colin, Eur.Phys. J. E 10, 209 (2003).

[39] S. P. Meeker, R. T. Bonnecaze, and M. Cloitre, Phys.Rev. Lett. 92, 198302 (2004).

[40] X. Zhang, E. Lorenceau, P. Basset, T. Bourouina,F. Rouyer, J. Goyon, and P. Coussot, Phys. Rev. Lett.119, 208004 (2017).

[41] J. Pemeja, B. Geraud, C. Barentin, and M. Le Merrer,Phys. Rev. Fluids 4, 033301 (2019).

[42] J. R. Seth, C. Locatelli-Champagne, F. Monti, R. T. Bon-necaze, and M. Cloitre, Soft Matter 8, 140 (2012).

[43] M. R. Serial, D. Bonn, T. Huppertz, J. A. Dijksman,J. van Der Gucht, J. van Duynhoven, and C. Terenzi,ArXiv e-prints (2021).

[44] R. Mas and A. Magnin, J. Rheol. 38, 889 (1994).[45] F. Pignon, A. Magnin, and J.-M. Piau, J. Rheol. 40, 573

(1996).[46] P. Grondin, S. Manneville, J.-L. Pozzo, and A. Colin,

Phys. Rev. E 77, 011401 (2008).[47] P. Møller, A. Fall, V. Chikkadi, D. Derks, and D. Bonn,

Philos. Trans. R. Soc. A 367, 5139 (2009).[48] T. Divoux, C. Barentin, and S. Manneville, Soft Matter

7, 9335 (2011).[49] V. Grenard, T. Divoux, N. Taberlet, and S. Manneville,

Soft Matter 10, 1555 (2014).[50] R. Benzi, M. Sbragaglia, M. Bernaschi, S. Succi, and

F. Toschi, Soft Matter 12, 514 (2016).[51] A. Nicolas, E. E. Ferrero, K. Martens, and J.-L. Barrat,

Rev. Mod. Phys. 90, 045006 (2018).[52] R. Benzi, T. Divoux, C. Barentin, S. Manneville,

M. Sbragaglia, and F. Toschi, Phys. Rev. Lett. 123,248001 (2019).

[53] R. Benzi, T. Divoux, C. Barentin, S. Manneville,M. Sbragaglia, and F. Toschi, Phys. Rev. Lett. 127,148003 (2021).

[54] A. E. Smith, J. Chem. Phys. 21, 2229 (1953).

1

A generalized lever rule for shear banding of yield stress fluids

Supplemental Material

I. COMPOSITION AND PREPARATION OF PARAFFIN GELS

Model paraffin gels are obtained by dissolving a single commercial paraffin wax with melting point between 58◦Cand 62◦C (Sigma Aldrich 327212, ASTM D 87) in heavy mineral oil (Sigma Aldrich, 330760). Solid wax is weighed atroom temperature, then added to the liquid mineral oil and the mixture is continuously stirred at high temperature(T = 80◦C) overnight. We prepare two large batches containing 10 % wt. and 15 % wt. in paraffin wax, which areused for all subsequent experiments. Before each experiment, the whole batch is heated back to T = 80◦C on a hotstirring plate and the desired amount of sample is poured inside the Taylor-Couette geometry right before starting thecooling ramp (see Section II below for the detailed protocol). Under these conditions, well above the wax appearancetemperature for both the 10 % wt. and 15 % wt. concentrations, the samples are purely Newtonian at the beginningof each experiment. Therefore, they do not bear any memory of their previous thermal or shear history. Upon cooling,the paraffin chains precipitate out of solution and crystallize, forming discotic platelets with lateral dimensions thatstrongly depend on the cooling rate and a typical thickness . 1µm that corresponds roughly to the length of theparaffin chains [24, 54] (see Section III below for details on the microstructure). Crystalline bridges as well as thesteric interaction among these discotic particles give rise to an elastic network and to a complex nonlinear responseunder shear [24, 25].

II. EXPERIMENTAL SETUP AND PROTOCOL

Our rheo-velocimetry setup is described at length in Ref. [26] and specific details about the analysis of velocity dataare given in Section IV below. In the present work, the Taylor-Couette cell (or concentric-cylinder shear cell) consistsof an outer fixed cylinder (stator) made of sandblasted PMMA and of an inner rotating cylinder (rotor) made of Delrincovered with sandpaper of surface roughness Ra = 35µm. The diameter of the stator is 50 mm and the working gapis H = 1.1 mm, small enough to guarantee a quasi-homogeneous stress field across the gap. We deliberately choose alarge roughness, which is usually supposed to minimize wall slip, to highlight the fact that wall slip sets in during theshear-banding regime in spite of rough boundary conditions. The fact that no wall slip is detected in regime i evenvery close to γmin confirms the effectiveness of the sandpaper in suppressing slip whenever the velocity profile is stable(see Fig. 1 in the main text). We checked that the shear-banding phenomenology reported here is robust and alsoobserved when both surfaces are sandblasted or smooth, although experiments performed without sandpaper clearlyshow an additional significant slip velocity even in regime i .

Supplemental Figure S1(a) depicts the protocol followed in the experiments presented in the main text. Due tothe thermal sensitivity of the ultrasound probe, the water bath surrounding the Taylor-Couette cell and the probemust be kept at temperatures lower than 40◦C. Therefore, the water bath was first stabilized at Ti = 38◦C beforepouring the paraffin–oil mixture heated at T = 80◦C in the Taylor-Couette cell. The sample is subsequently shearedat γcool = 50 s−1 during the whole cooling process. Once the sample temperature equilibrates at Ti, the chiller of thewater bath is set to cool the sample with a cooling rate T = −0.20±0.05◦C/min until a final temperature Tf = 26◦C.After reaching Tf , the mixture is sheared for an additional 20 min. Finally, the steady-state flow curve is measured byprogressively decreasing the shear rate. From γ = 50 s−1 down to 0.1 s−1, the shear rate is swept down logarithmicallywith 10 points per decade between 50 s−1 and 2 s−1 then with 20 points per decade between 2 s−1 and 0.1 s−1. Foreach applied shear rate, the rheometer waits for the stress to vary by less than 3% over 90 s (with a time limit perpoint set to 5 min) before decreasing the shear rate. From γ = 0.1 s−1 down to 0.005 s−1, a constant shear rate isimposed over ∆trheo = 60 min. As seen in Supplemental Fig. S1(b) for the same shear rates as in Fig. 1 in the maintext, this protocol allows us to ensure that all initial transients have died out even at the lowest shear rates whileaccommodating for the time ∆tUSV needed for acquiring ultrasound velocimetry maps at steady state, which rangesfrom much less than 1 min at the highest shear rate to about 30 min at γ = 0.005 s−1.

2

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10 % wt.

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(b)<latexit sha1_base64="UAjg4yq4J/sAgF1HPkfrTSj+Oos=">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</latexit>

(a)<latexit sha1_base64="W7y5RkYGuR3XpxPjtNZ64FUY9Ls=">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</latexit>

� = 0.032 s�1

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� = 0.0063 s�1<latexit sha1_base64="Cw/8EZ41H1KYA/M2S7yeJEJpoSI=">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</latexit>

10 % wt.

FIG. S1. (a) Experimental protocol used for rheo-velocimetry experiments. From top to bottom: temperature T imposed usingan external chiller connected to the water tank surrounding the Taylor-Couette cell, shear rate γ imposed by the rheometerand corresponding stress response σ together with a schematic of the acquisition times ∆trheo and ∆tUSV used respectively forrheological measurements and for ultrasonic velocimetry in different ranges of shear rates. (b) Stress response σ(t) recorded inthe 10 % wt. gel for shear rates γ = 0.032 s−1 (top) and 0.0063 s−1 (bottom), applied during the flow curve of Fig. 1 in themain text (same shear rate values for which the velocity field is shown in Fig. 1 of the main text).

3

III. POLARIZED LIGHT MICROSCOPY OF PARAFFIN GELS

Since paraffin crystallites are birefringent, we use polarized light microscopy to visualize the gel microstructure.A series of images is recorded at room temperature using an ABRIO (CRi, Inc.) camera and software on a NikonTE-2000U inverted microscope using a 20× objective. Two types of preparation protocols are employed in order tocompare the morphology of single wax crystals in the case of static cooling or dynamic cooling (i.e. cooling while alsounder shear).

For the static case, a small visualization chamber is made by laying three stripes (approximately 3 × 20 mm) ofParafilm on a glass slide (75× 25 mm). A cover glass (22× 22 mm) is then sealed on top of the Parafilm by heatingthe slide on a hot plate at about 80◦C, which makes the Parafilm adhesive. Next, the liquid paraffin–oil mixture ispipetted in between the bottom slide and the cover glass at the same temperature of 80◦C. Finally, the cell is quicklyplaced on a Peltier stage to impose the same cooling rate as that used in the rheo-velocimetry experiments, namelyT = −0.2◦C/min, until room temperature is reached.

For the dynamic case, the melted gel sample is spread on a glass slide (75×25 mm) in contact with the Peltier stageof a stress-controlled rheometer (DHR-3, TA Instruments) heated at 80◦C. Shear is applied thanks to a parallel plategeometry of diameter 20 mm covered with sandpaper (same roughness as in the velocimetry experiments) and within

a gap of 1 mm. The sample is then cooled under a constant shear rate γ = 50 s−1 at a cooling rate T = −0.2◦C/minuntil room temperature is reached. At the end of the process, we raise the head of the rheometer and transfer theglass slide directly onto the microscope for visualization.

Images representative of both cooling processes are reported in Supplemental Fig. S2(a) for the static case andSupplemental Fig. S2(c) for the dynamic case. Although the crystallites are arranged very differently in the twocases, most probably due to the flow generated when raising the shearing tool in the last step of the dynamic case,a quantitative analysis of a series of images show very similar distributions of the crystallite characteristic size ` [seeSupplemental Fig. S2(b,d)] and an average size ` ' 130µm in both cases.

(d)

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(b)

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(a)

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(c)

<latexit sha1_base64="WnPzhwMz7k5ZMu1XTHwscuV4wpg=">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</latexit>

FIG. S2. (a) Birefringence image of a 10 % wt. paraffin gel cooled under static conditions at the cooling rate used in the

rheo-velocimetry experiments T = −0.2◦C/min. The yellow dotted frame corresponds to the image shown in Fig. 1(a) in themain text. (b) Probability density function (PDF) of the length ` of the paraffin crystallites obtained from image analysis ofseveral birefringence images. (c) Birefringence image of a 10 % wt. paraffin gel cooled under dynamic conditions at the cooling

rate (T = −0.2◦C/min) and sheared at γ = 50 s−1 used in the rheo-velocimetry experiments. (d) Corresponding probabilitydensity function (PDF) of the crystallite length `. Scale bars in (a) and (b) are both 300µm. The average size of the crystallitesis largely unaffected by shear and we measure 〈`〉 ' 130µm in both cases.

4

IV. ANALYSIS OF VELOCIMETRY DATA

As explained in full details in Ref. [26], our ultrafast ultrasound imaging technique outputs maps of the tangentialvelocity component uθ(y, z, t) as a function of the distance y to the stator, of the vertical position z along theultrasonic probe, and of time t. Such velocity maps result from the cross-correlation of successive ultrasound imagesof the material under shear recorded from the emission and back-scattering of plane pulses sent with a repetitionfrequency that is inversely proportional to the applied shear rate. In the case of the present paraffin gels with waxconcentrations 10 % wt. and 15 % wt., the wax crystallites scatter ultrasound efficiently enough that ultrasoundimages can be obtained directly, without requiring any seeding of the samples by acoustic contrast agents.

Supplemental Figure S3 shows an example of velocity data recorded on the 10 % wt. paraffin gel within thedecreasing branch of the flow curve (γ = 0.025 s−1). As seen in Supplemental Fig. S3(a), the velocity field 〈uθ(y, z, t)〉taveraged over t for the whole acquisition window does not show any significant variation along the vertical directionz. Similarly, the velocity 〈uθ(y, z, t)〉z averaged over z and presented as a function of y and t in the spatiotemporaldiagram of Supplemental Fig. S3(b) fluctuates around a mean without any systematic trend over 103 s. Analogousmaps are produced for all other shear rates under study, which confirms the absence of more complex flow behaviourbesides shear banding and wall slip. This also allows us to focus only on the velocity profiles u(y) = 〈uθ(y, z, t)〉z,taveraged both in time and along the vorticity direction.

To quantify the average local shear rate in the flowing band γb, the extent of the unyielded band yb and the slipvelocity us, each velocity profile (averaged over z) is fitted to a linear profile in the two bands (with zero slope for theunyielded band). The slope of each fitted flowing band provides an estimate for γb and the intercept between the twolinear profiles an estimate for yb, while the slip velocity is extracted by taking the difference between the wall velocityand the velocity calculated based on the fitted linear profile extrapolated at the wall. Error bars in all velocity profilesand calculated quantities are characteristic of the fluctuations in time, but fluctuations in the vorticity direction arevery similar. For shear rates in regime iii , instead of using a linear fit, the curved velocity profile in the flowing band

is fitted for y ∈ [yb, H] by a polynomial of order 3, u(y) =∑3n=0 any

n with ai the polynomial coefficients. From these,

γb is obtained as the average local shear rate, i.e. γb = 1h

∫Hyb

dudy (y)dy = 1

h

∑3n=1 nan

∫Hybyn−1dy, with h = H − yb.

0.5 10

5

10

15

20

25

30

0

0.005

0.01

0.015

0.02

0.025

0.03

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u✓(y, z, t)

FIG. S3. Time-resolved velocity field uθ(y, z, t) recorded in the 10 % wt. paraffin gel under an imposed shear rate γ = 0.025 s−1.(a) Time-averaged velocity map 〈uθ(y, z, t)〉t taken over the whole duration of the ultrasound acquisition. (b) Space-timediagram of the velocity data 〈uθ(y, z, t)〉z averaged over the vertical direction z.

5

V. RHEOLOGY OF A 15 % wt. PARAFFIN GEL

Figure S4 shows some of the results used to extract the data for the 15 % wt. paraffin gel shown with emptydiamonds in Figs. 2 and 3 in the main text. It is clear that the rheology of this 15 % wt. gel is very similar to thatof the 10 % wt. gel: we observe the same regimes in the flow curve [Fig. S4(a)], which correspond to the same typesof velocity profiles [Fig. S4(c,d)] as in Fig. 1 in the main text. As seen from stress evolution over time in Fig. S4(b),a steady state is reached even at very low shear rates.

10-2 10-1 100 101 10210-1

100

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y[mm]

(c)<latexit sha1_base64="jpoQDH9J/yrHUipz2n1onyM1fGE=">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</latexit>

(d)<latexit sha1_base64="ZLfqAydedrsOU1WCtY9G90WCg3o=">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</latexit>

⇥10�3<latexit sha1_base64="VK282tv+012lwcRCf1S51PlTv1M=">AAACJXicbVDLSgMxFE3qq46vVpdugqXgxjLTCroRim5cVrAP6LQlk95pQzMPkkyhDP0Pt7rxa9yJ4MpfMX0sbOuFwMk59+bcHC8WXGnb/saZre2d3b3svnVweHR8ksufNlSUSAZ1FolItjyqQPAQ6pprAa1YAg08AU1v9DDTm2OQikfhs57E0AnoIOQ+Z1QbqutqHoAijt1NryrTXq5gl+x5kU3gLEEBLavWy2Ps9iOWBBBqJqhSbceOdSelUnMmYGq5iYKYshEdQNvAkBq3Tjpfe0qKhukTP5LmhJrM2b8TKQ2UmgSe6QyoHqp1bUb+p7UT7d92Uh7GiYaQLYz8RBAdkVkGpM8lMC0mBlAmudmVsCGVlGmT1MpLQxBjMB9S/tzMsooumDgns9td6ip/apnQnPWINkGjXHIqpfLTdaF6v4wvi87RBbpEDrpBVfSIaqiOGJLoBb2iN/yOP/An/lq0ZvBy5gytFP75BZlio+4=</latexit>

0 600 1200 1800 2400 3000 36000

0.2

0.4

0.6

0.8

1

0 600 1200 1800 2400 3000 36000

0.2

0.4

0.6

0.8

1

(b)<latexit sha1_base64="5QGg4MmKeOWa6N71UsNQEsmSPsw=">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</latexit>

<latexit sha1_base64="RhOdvmCLUWglBQXhWmZSGayWEgI=">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</latexit> �[P

a]

<latexit sha1_base64="/vQUBznnSgICTHf/5nGt8YtuQXc=">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</latexit>

t [s]

<latexit sha1_base64="RhOdvmCLUWglBQXhWmZSGayWEgI=">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</latexit> �[P

a]

<latexit sha1_base64="pDXj30Qdd4EbiLZ1yr5yCt8TQsc=">AAAFvHicddTdTtswFABgw+jGuj/YLndjrdrEJFYlpazsYhOliA0JJGD0RyKlchynRDg/it1CZWVPsKfZ7fYie5s5ocjtqWap0tH5jmP7yLWb8EBIy/q7tPxgpfTw0erj8pOnz56/WFt/2RHxKKWsTWMepz2XCMaDiLVlIDnrJSkjoctZ171u5d4ds1QEcXQuJwnrh2QYBX5AidSpwdo7x4ulcoYkDEmGP2Orau1gZxM7IZFXaajEpfpgZ9lgrWJVG/XGlr2dlxRDB/b21ifbxvY0U0HTcTJYX/mhv0xHIYsk5USIC9tKZF+RVAaUs6zsjARLCL0mQ3ahw4iETPRVcaAMv9UZD/txqn+RxEV2doYioRCT0NWV+T4FtDx5b3NLqeJYwhdgA9Lf6asgSkaSRfRufX/EsYxx3jTsBSmjkk90QGga6CNgekVSQqVu7dz33TArl52I3dBYdzTylOOxTDlJfm7Cs3nqack35PqqlwG7NXYLbWJsAo0ZY9DGjBsdQz00dgjNMmZBE8a+Q+sGU9RXScfOJnB9+Yqr18zU7EUcNGEXj80ix3CRA2MH0M6MnUFrG2tD6xjrQGsZa0HbM7YH7ejI4BHEfWP7C1001oV2aux0wYazOLiLKeH4K6xsmsImNNc16EI8N3YOTQYG5UKj7lHvp2jVPF8R7usCX//HlJ2pGpweR/qib9jv4ZI3sU7XdFq/WPfPEv5/0KlV7Y/V+mm9slufvl2r6DV6gzaQjRpoF31DJ6iNKPqJfqHf6E/pS8krXZfCu9LlpemcV2hulMb/AOckEYY=</latexit>

� = 0.08 s�1<latexit sha1_base64="hR6UYmjGm63F44a0sQ46yc+H/VU=">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</latexit>

� = 0.008 s�1

<latexit sha1_base64="pDXj30Qdd4EbiLZ1yr5yCt8TQsc=">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</latexit>

� = 0.08 s�1

<latexit sha1_base64="hR6UYmjGm63F44a0sQ46yc+H/VU=">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</latexit>

� = 0.008 s�1

FIG. S4. Rheological response of the 15 % wt. paraffin gel: (a) Flow curve σ vs. γ. (b) Stress response σ(t) recorded for shearrates γ = 0.08 s−1 (top) and 0.008 s−1 (bottom) applied during the flow curve (corresponding to highlighted markers indicatedby the arrows). Banded velocity profiles for the same highlighted shear rates: (c) γ = 0.08 s−1 and (d) γ = 0.008 s−1. Blacklines in (c) and (d) are the homogeneous velocity profiles expected without wall slip. Colored solid lines are best fits to thevelocity profiles, while the dashed line in (d) shows the linear profile that would be expected from the average shear rate in thesheared band.

6

VI. FLOW COOPERATIVITY AT VERY LOW SHEAR RATES

Supplemental Figure S5 gathers the normalized velocity profiles u(y)/uw recorded in regime iii for both the10 % wt. and 15 % wt. paraffin gels. When compared to velocity profiles in regime ii [see Fig. 1(c) in the main text

and Supplemental Fig. S4], it is clear that the flowing region shows a much more pronounced curvature in regime iii ,recalling the profiles observed in confined suspensions when spatial cooperativity is present [21, 22, 42]. Inspired bythese results, we solve a nonlocal equation for the local plastic fluidity f(y) = γ(y)/σ to derive an expression for thevelocity profile in the flowing band, i.e. for yb ≤ y ≤ H. This nonlocal equation reads

ξ2∂2f

∂y2= f − fb , (S1)

where ξ is the flow cooperativity length and fb is the bulk fluidity. In our case, since the material outside the flowingband is arrested, we impose fb = f(yb) ≡ 0 and define f(H) ≡ fw > 0 at the moving wall, which leads to:

f(y) = fwsinh[(y − yb)/ξ]

sinh(h/ξ), (S2)

with h = H − yb. Further integrating du(y)dy = γ(y) = f(y)σ and imposing u(yb) = 0 yields:

u(y) = u(H)cosh[(y − yb)/ξ]− 1

cosh(h/ξ)− 1, (S3)

where u(H) = ξfwσcosh(h/ξ)−1sinh(h/ξ) . In Supplemental Fig. S5, we fit the experimental data to Eq. (S3) in order to

determine an estimate of the cooperativity length ξ. In view of the large experimental uncertainty resulting from thevery low velocity levels, we focus on the velocity profiles averaged over all shear rates available in regime iii . Sincefw is not known a priori, we also have to treat u(H) as a fitting parameter. We find ξ ' 200µm, which is comparableto the dimension of the wax crystallites, while u(H)/uw = 0.55 and yb/H = 0.4 for both gel concentrations. Theagreement between theory (blue solid lines) and experiments suggests that nonlocal effects can indeed explain the

curvature observed on the velocity profiles in regime iii .

0 0.5 10

0.5

1

0 0.5 10

0.5

1(b)

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y/H

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)/u

w

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y/H

FIG. S5. Normalized velocity profiles measured over regime iii (a) in the 10 % wt. paraffin gel and (b) in the 15 % wt. paraffingel. Standard deviations are reflective of the temporal fluctuations and of experimental uncertainty on our ultrasound velocitymeasurements at very low shear rates. Dashed lines show the theoretical profile for a Newtonian fluid in the absence of wallslip. Solid lines are fits to the nonlocal profile in Eq. (S3).