Shear-induced structures versus flow instabilities M.A. Fardin,1, 2, ∗ C. Perge,1 N. Taberlet,1 and S. Manneville1 Université de Lyon, Laboratoire de Physique, École Normale Supérieure de Lyon, CNRS UMR 5672, 46 Allée d’Italie, 69364 Lyon cedex 07, France 2 The Academy of Bradylogists (Dated: January 11, 2014) The Taylor-Couette flow of a dilute micellar system known to generate shear-induced structures is investigated through simultaneous rheometry and ultrasonic imaging. We show that flow instabilities must be taken into account since both the Reynolds number and the Weissenberg number may be large. Before nucleation of shear-induced structures, the flow can be inertially unstable, but once shear-induced structures are nucleated the kinematics of the flow become chaotic, in a pattern reminiscent of the inertio-elastic turbulence known in dilute polymer solutions. We outline a general framework for the interplay between flow instabilities and flow-induced structures. PACS numbers: 83.80.Qr, 47.20.-k, 47.27.-i, 47.50.Gj Beside their tremendous industrial importance in applications such as detergence, oil recovery, or drag reduction, micellar systems have long been used as a model system for rheological research [1]. In particular some surfactant systems are known to form rodlike micelles, which can grow to become wormlike and entangled when the concentration in surfactant and/or salt increases, or when the temperature decreases [2, 3]. The dilute regime of rodlike micelles has been studied in the context of highly elastic shear-induced structures (SIS) and associated shear-thickening, whereas the so-called “semidilute” and “concentrated” regimes have been studied in the context of shear-banding, which is an extreme form of shear-thinning where the velocity gradient field becomes inhomogeneous even in simple shear [4]. On the one hand, shear-banding micellar systems have a high viscosity, such that inertial flow instabilities can generally be neglected. Nevertheless semi-dilute and concentrated solutions have high normal stresses, such that the Weissenberg number is large and purely elastic instabilities can develop [5–7], similar to the case of polymeric Boger fluids [8–10]. On the other hand, the possibility for flow instabilities has never been considered thoroughly in dilute, shear-thickening systems even though these systems have both large Reynolds (Re) and Weissenberg (Wi ) numbers [4]. The aim of the present Letter is to fill this gap by showing that both inertial and elastic instabilities are constantly encountered in the Taylor-Couette flow of a well-known dilute micellar system. The sample under study is made of 0.16% wt. hexadecyltrimethylammonium p-toluenesulfonate (CTAT) in water. For this system, which has become a benchmark example of dilute micellar fluids [4], the overlap concentration is estimated at c⋆ ≃ 0.5 wt. % and shearthickening is found over the range 0.05-0.8% wt. [11, 12]. Figure 1 shows the viscosity η of our sample as a function of the applied shear rate γ̇ for three different temperatures T , exhibiting the typical behavior of shearthickening, dilute surfactant systems [4]: a zero-shear vis- 15 η (mPa.s) arXiv:1309.7175v1 [cond-mat.soft] 27 Sep 2013 1 10 5 0 1 2 10 10 γ̇ (1/s) FIG. 1: Viscosity η vs shear rate γ̇ at T = 20, 25, and 30◦ C (filled symbols, from top left to bottom right) for CTAT at 0.16% wt. seeded with 1% wt. polyamide spheres (see text). Open symbols correspond to a CTAT sample free of contrast agents at T = 30◦ C. The addition of polyamide spheres increases the viscosity by about 15% but does not change the overall behavior. Data points correspond to averages over the last 5 s of shear rate steps of total duration 15 s. Vertical dashed lines indicate the onset of shear-thickening γ̇c ≃ 13, 35, and 80 s−1 respectively inferred at T = 20, 25, and 30◦ C from start-up experiments of typical duration 500 s . cosity close to the viscosity of water; then a jump in η at a characteristic shear rate γ̇c that increases with T ; finally, a shear-thinning viscosity branch at high shear rates. This behavior was first explained by postulating the formation of SIS [13]: above γ̇c , micelles grow in length and undergo a transition from rodlike to wormlike aggregates. This microscopic scenario was confirmed through neutron and light scattering experiments [4]. The shear-induced state can then be shear-thinning due to the increasing alignment of the worms. In the present experiments the fluid is sheared in a Taylor-Couette (TC) device adapted to a rheometer (ARG2, TA Instruments) and with dimensions (height H = 60 mm, gap d = 2 mm, radius of the inner rotating cylinder Ri = 23 mm) that ensure the “small gap approximation”, Λ ≡ d/Ri ≃ 0.087 ≪ 1, without any strong end effect at the top and bottom boundaries 2 (d/H ≪ 1), so that the laminar base flow is a simple shear with γ̇ = U/d = ΩRi /d, where U and Ω are the rotor linear and angular velocities respectively. We recall that in a TC device with inner rotation, Taylor showed that a Newtonian fluid becomes unstable and develops a secondary flow made of toroidal counter-rotating vortices [14]. Larson, Shaqfeh and Muller discovered that non-Newtonian fluids can develop a similar vortex flow not driven by inertia, but driven by elasticity [8]. In both cases the instability develops when the Taylor number Ta exceeds a given threshold Ta c . In the purely inertial case Ta = Λ1/2 Re, with Re ≡ τ1 γ̇ = (ρd2 /η)(U/d) = ρdU/η and Ta c ≃ 41 [14, 15], while in the purely elastic case Ta = Λ1/2 Wi , with Wi ≡ τ2 γ̇, where τ2 is a characteristic polymeric relaxation time [8, 16] instead of the viscous dissipation time τ1 , and ρ ≃ 103 kg.m−3 is the fluid density. In the latter case, the value of Ta c depends on the constitutive relation of the non-Newtonian fluid, e.g., linear stability theory predicts Ta c ≃ 6 for the Upper-Convected Maxwell model [8]. More generally the balance between elasticity and inertia can be estimated by the elasticity number E ≡ Wi /Re = τ2 /τ1 . Although predicting the onset of instability for arbitrary elasticity is not yet possible, a range of inertio-elastic instabilities is expected but remains mostly unexplored [17]. Since we expect secondary flows to emerge, our geometry is equipped with a recently developed twodimensional ultrasonic velocimetry technique that allows the simultaneous measurement of 128 velocity profiles over 30 mm along the vertical direction in the TC geometry [18]. We use ultrafast plane wave imaging and cross-correlation of successive images [19] to infer velocity maps from the echoes backscattered by acoustic contrast agents seeding the fluid, namely 1% wt. polyamide spheres (Arkema Orgasol 2002 ES 3 Nat 3, mean diameter 30 µm, relative density 1.03), which do not affect significantly the rheological behavior of our solution (see Fig. 1). This technique yields the component vy (r, z) of the velocity vector, v = (vr , vθ , vz ) in cylindrical coordinates, projected along the acoustic propagation axis y as a function of the radial distance r to the rotor and of the vertical position z with a temporal resolution down to 50 µs [18]. The acoustic axis y is horizontal and makes an angle φ ≃ 10◦ with the normal to the outer cylinder so that vy = cos φ vr + sin φ vθ . Finally we define the v vr measured velocity as v = sinyφ = vθ + tan φ , which coincides with the azimuthal velocity vθ in the case of a purely azimuthal flow v = (0, vθ , 0). More generally v combines contributions from both azimuthal and radial velocity components. Nevertheless, close to instability onset, secondary flows are usually much weaker than the main flow, such that v ≃ vθ , as checked recently on an inertially unstable Newtonian fluid and on an elastically unstable shear-banding fluid [18, 20]. Figure 2 reports the start-up flow of CTAT at T = 25◦ C for γ̇ = 50 s−1 (see also Supp. Movie 1). At very FIG. 2: Spatiotemporal dynamics of CTAT at T = 25◦ C for γ̇ = 50 s−1 (see also Supp. Movie 1). The flow is first inertially unstable and then inertio-elastically unstable due to SIS formation. (a) Global shear stress σ(t) measured by the rheometer (in black) and dimensionless slip velocity vs (t) (in red) [27]. (b) Spatiotemporal diagram (center) of the velocity v(r0 , z, t) at r0 = 0.20 mm from the rotor. The dotted lines show the times t0 = 5 and 22.5 s corresponding to the velocity maps v(r, z, t0 ) shown on the left and right respectively. The color scale is linear and goes from 0 to ΩRi for the velocity maps and from 0.5ΩRi to ΩRi for the diagram. The vertical axis z is oriented downwards with z = 0 being taken at about 6 mm from the top of the TC cell. short times, a laminar boundary layer extends from the inner cylinder to the outer cylinder. A Taylor vortex flow (TVF) then develops for t & 3 s, deforming the main flow which becomes periodic along z: slow moving fluid is brought inward in regions of centripetal radial flow and fast moving fluid is pushed outward in regions of centrifugal radial flow. This initial sequence of events would be exactly similar if the fluid was pure water [18, 21]. Even though such a TVF had never been reported before for dilute micelles, it should not be surprising since, assuming the fluid to be influenced only by inertia in this initial sequence, we have Ta = Λ1/2 Re ≃ 60 > Ta c for γ̇ = 50 s−1 . In computing τ1 , we have used the dynamic viscosity relevant to the short-time behavior, i.e. the zero-shear viscosity η0 ≃ 1 mPa.s. As shown in Fig. 2(a) the onset of TVF at t ≃ 3 s corresponds to a slight increase of the shear stress σ (after an initial spike due to the feedback with the rheometer inertia). This first stress increase is simply due to the formation of vortices breaking the viscometric assumption [18]. In contrast, for t & 10 s, the stress (or alternatively the viscosity) climbs up much more dramatically. Since γ̇ = 50 s−1 falls into the shear-thickening range for T = 25◦ C (see Fig. 1), this next sequence of events can clearly be attributed to slow SIS formation. Meanwhile the structure of the vortex flow is disrupted. The formerly welldefined wavelength and amplitude of the main flow shown in Fig. 2(b,left) are lost and the flow becomes chaoticlike in Fig. 2(b,right). This latter state is reminiscent of 3 the inertio-elastic turbulent state called “disordered oscillations” [22] or “elastically dominated turbulence” [23]. While iso-velocity lines in the initial state can be approximated by harmonic functions, the new secondary flows associated with the SIS deform the main flow intermittently. The state in Fig. 2(b,right) is representative of the asymptotic turbulent flow. It continuously generates large fluctuations in the viscosity and stress, which have been reported before but never accounted for in terms of elastic turbulence [4]. Note also that the turbulent nature of the flow can locally and transiently generate plug flow profiles that may explain some earlier 1D velocity measurements [24, 25]. At lower shear rates, e.g. γ̇ = 20 s−1 < γ̇c and Ta ≃ 24 < 41, the flow is below both thresholds for TVF and SIS formation and remains purely azimuthal as shown in Supp. Fig. 1. As it turned out for T = 25◦ C, the critical shear rate γ̇c for SIS formation and the critical shear rate γ̇TVF ≡ Ta c /(τ1 Λ1/2 ) for the onset of TVF are about the same value γ̇c ≃ γ̇TVF ≃ 35 s−1 in our TC geometry (Λ1/2 ≃ 0.29). In order to separate the inertial TVF and the turbulence associated with SIS more readily, we reproduced similar shear start-up protocols at two other temperatures, T = 20 and 30◦ C, as shown in Fig. 3. Increasing the temperature slightly lowers the zero-shear viscosity (see Fig. 1) so that γ̇TVF only decreases from 45 s−1 at 20◦ C to 33 s−1 at 30◦ C. In contrast, the same temperature change has a much stronger impact on γ̇c . As reported extensively in the literature [4], lowering the temperature leads to easier SIS formation hence shifting γ̇c to lower values. Figure 1 indicates γ̇c ≃ 13 and 80 s−1 at 20◦ C and 30◦ C respectively. Therefore at the highest temperature, we should be able to observe TVF without SIS, whereas SIS without TVF may be expected at the lowest temperature. This scenario is fully confirmed in Fig. 3(a,b) and (c,d) where spatiotemporal dynamics are compared for shear rates such that γ̇TVF < γ̇ < γ̇c and γ̇c < γ̇ < γ̇TVF at T = 30 and 20◦ C respectively. The fact that SIS and TVF can occur separately is an indication that these two phenomena are not consequences of one another. SIS do not need TVF to nucleate, which suggests that the out-of-equilibrium growth of the worms is driven by the base shear flow, as usually postulated [4]. Early velocity measurements at a single height z reported that SIS first form at the inner wall and generate significant slip on this wall [24–26]. The dimensionless slip velocities vs [27] shown in Figs. 2(a) and 3(c) confirm that the onset of wall slip is concomitant with SIS formation, whereas no noticeable wall slip is reported in the presence of TVF alone [see Fig. 3(a)]. Of course TVF does not need SIS since it can occur even in simple molecular fluids. Moreover the spatiotemporal structure of the flow in Figs. 2(b,right) and 3(d,right) are very similar so that TVF appears to have negligible impact on the asymptotic turbulent flow after SIS nucleation. It seems that before the nucleation FIG. 3: Same as Fig. 2 for (a,b) T = 30◦ C and γ̇ = 40 s−1 showing only inertial instability and (c,d) T = 20◦ C and γ̇ = 20 s−1 where the flow slowly develops SIS and the associated inertio-elastic turbulence without initial TVF (see also Supp. Movies 2 and 3 respectively). In (b) r0 = 0.40 mm and t0 = 20 (left) and 175 s (right). In (d) r0 = 0.53 mm and t0 = 50 (left) and 225 s (right). The color scale goes from 0 to ΩRi for all velocity maps and from ΩRi /8 (resp. ΩRi /4) to ΩRi for the diagram in (b) [resp. (d)]. of SIS inertia is dominating the instability of the flow, whereas once SIS are formed elasticity dominates. In a shear-thickening dilute surfactant solution, both τ1 and τ2 depend on γ̇ and t so that Re, Wi , and E also depend on γ̇ and t. To illustrate this, we evaluate a bulk-averaged value of E in the case of Fig. 2 (T = 25◦ C and γ̇ = 50 s−1 ). Assuming the fluid density ρ to remain constant during SIS formation, we first estimate the Reynolds number by Re ≃ γ̇t ρd2 /ηt , where γ̇t (γ̇, t) = γ̇ − vs (t)ΩRi /d is the “true” shear rate corrected for wall slip and ηt (γ̇, t) = σ(t)/γ̇t (γ̇, t) the corresponding “true” viscosity. This yields a decrease from Re ≃ 200 before SIS formation to Re ≃ 7 in the final state. Estimating the characteristic viscoelastic time τ2 of the SIS is more challenging. We take the longest time of the stress relaxation after flow cessation either before or after SIS formation, which yields Wi = γ̇t τ2 ≃ 0 before and Wi ≃ 100 after SIS formation in accordance with the literature [4]. Thus in the early stages of the dynamics 4 30 (2) T ( ◦ C) (1) 25 (3) (4) 20 0 20 40 60 γ̇ (1/s) 80 100 FIG. 4: Stability diagram of CTAT. (1) Laminar flow (◦). (2) Inertial instability (TVF) without SIS (H). (3) Inertioelastic instability of the SIS without initial TVF (N). (4) Inertio-elastic instability of the SIS after initial TVF (). Larger empty symbols correspond to the experiments in Figs. 2 and 3. Stars show the transitions between the various regimes. Solid lines are guides to the eye. E ≃ 0, i.e. inertia dominates, whereas after SIS formation E & 10, validating the dominance of elasticity. In some sense, the dynamics of Fig. 2(b) can be seen as the superposition of the dynamics of Fig. 3(b) and (d) [28]. The stability diagram of Fig. 4 summarizes the interplay between inertial instability (TVF) and elastic instability of the SIS based on an extensive data set. We believe that such a diagram should be systematically sought for in complex fluids subject to flow instabilities in order to sort out the influences of flow-induced microscopic and macroscopic phenomena. Here we find that SIS are always unstable and systematically lead to elastic-like turbulence. Whether or not this is a feature common to all elastic SIS stands out as an open issue. Another fundamental question is the possibility to predict the various boundaries in Fig. 4 using phenomenological approaches or even microscopic models. Deriving an instability criterion based on the elasticity number E and accounting for both Re and Wi thus appears as a first crucial theoretical step. M.-A.F. and C.P. contributed equally to this work. The authors thank S. Lerouge for providing the CTAT sample and for motivating this study. This work was funded by the Institut Universitaire de France and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 258803. ∗ Electronic address: marcantoine.fardin@ens-lyon.fr [1] H. Rehage and H. Hoffmann, Mol. Phys. 74, 933 (1991). [2] J.-F. Berret, in Molecular gels (Springer, 2006), pp. 667– 720. [3] M. E. Cates and S. M. Fielding, Adv. Phys. 55, 799 (2006). [4] S. Lerouge and J.-F. Berret, Adv. Polym. Sci. 230, 1 (2010). [5] S. M. Fielding, Phys. Rev. Lett. 104, 198303 (2010). [6] M.-A. Fardin and S. Lerouge, Eur. Phys. J. E 35, 1 (2012). [7] A. Nicolas and A. Morozov, Phys. Rev. Lett. 108, 088302 (2012). [8] R. G. Larson, E. S. G. Shaqfeh, and S. Muller, J. Fluid Mech. 218, 573 (1990). [9] R. G. Larson, Rheol. Acta 31, 213 (1992). [10] A. N. Morozov and W. van Saarloos, Phys. Rep. 447, 112 (2007). [11] R. Gamez-Corrales, J.-F. Berret, L. Walker, and J. Oberdisse, Langmuir 15, 6755 (1999). [12] M. Truong and L. Walker, Langmuir 16, 7991 (2000). [13] H. Rehage and H. Hoffmann, Rheol. Acta 21, 561 (1982). [14] G. Taylor, Phil. Trans. R. Soc. London. A 223, 289 (1923). [15] R. Donnelly and D. Fultz, Proc. R. Soc. London, Ser. A 258, 101 (1960). [16] G. McKinley, P. Pakdel, and A. Öztekin, J. Non-Newt. Fluid Mech. 67, 19 (1996). [17] S. J. Muller, Korea-Aust. Rheol. J. 20, 117 (2008). [18] T. Gallot, C. Perge, V. Grenard, M.-A. Fardin, N. Taberlet, and S. Manneville, Rev. Sci. Instrum. 84, 045107 (2013). [19] L. Sandrin, S. Manneville, and M. Fink, Appl. Phys. Lett. 78, 1155 (2001). [20] M.-A. Fardin, C. Perge, and S. Manneville, submitted to Eur. Phys. J. E, arXiv:cond-mat/1307.5415 (2013). [21] A. Akonur and R. M. Lueptow, Phys. Fluids 15, 947 (2003). [22] A. Groisman and V. Steinberg, Phys. Rev. Lett. 77, 1480 (1996). [23] C. S. Dutcher and S. J. Muller, J. Rheol. 57, 791 (2013). [24] S. Koch, T. Schneider, and W. Küter, J. Non-Newt. Fluid Mech. 78, 47 (1998). [25] H. Hu, P. Boltenhagen, and D. J. Pine, J. Rheol. 42, 1185 (1998). [26] P. Boltenhagen, Y. Hu, E. Matthys, and D. J. Pine, Europhys. Lett. 38, 389 (1997). [27] Linear fits of the velocity profiles v(r, z) over about 300 µm in the radial direction r are used to estimate the local shear rates close to each wall and extrapolate the fluid velocities v(r = 0, z) and v(r = d, z). The dimensionless total slip velocity is then defined as vs = 1 − [hv(0, z)iz − hv(d, z)iz ]/ΩRi where the average is taken over the 128 simultaneous measurements along the vertical direction z. Error bars in Fig. 2(a) show the standard deviation along z. [28] The SIS formation time in Fig. 3(d) is much longer than in Fig. 2(b). This is due to the fact that in the first case γ̇ & γ̇c and it is well known that the formation time of SIS goes like (γ̇ − γ̇c )−1 [4]. Along the same lines apparent wall slip (≃ 10%) is visible after onset of TVF in Fig. 2(a) while it remains negligible in Fig. 3(a). This can be attributed to a larger distance to γ̇TVF in Fig. 2 and to larger effects of radial velocity components on v and vs . 5 SUPPLEMENTAL MATERIAL Shear-induced structures versus flow instabilities σ (Pa) (a) 0.1 0.6 0.05 0.3 0 vs Sup. Movies 1, 2 and 3 corresponding respectively to Fig. 2(a-b), Fig. 3(a-b) and Fig. 3(c-d) of the paper are available on request from the corresponding author. See the captions of the corresponding figures for details. The movies also display some velocity profiles in the bottom left quadrant. The square symbols correspond to examples of velocity profiles obtained at three different heights z, whereas the circles correspond to the velocity profile obtained by averaging along z. The line represents the purely azimuthal Couette flow profile expected for no-slip boundary conditions. 0 z (mm) (b) 0 10 20 30 0 1 2 0 r (mm) 100 200 300 t (s) 400 0 1 2 r (mm) FIG. 1: Spatiotemporal dynamics of the flow of CTAT for T=25◦ C, at γ̇ = 20 s−1 , below the onsets of both TVF and SIS formation. The flow remains perfectly laminar with no noticeable wall slip. (a) Global shear stress σ(t) measured by the rheometer (in black) and dimensionless slip velocity vs (t) (in red). (b) Spatiotemporal diagram (center) of the velocity v(r0 , z, t) at r0 = 0.3 mm from the rotor. The dotted lines show the times t0 = 50 and 450 s corresponding to the velocity maps v(r, z, t0 ) shown on the left and right respectively. The color scale is linear and goes from 0 to ΩRi .