^{1,a)}, Christophe Perge

^{1}, Vincent Grenard

^{1}, Marc-Antoine Fardin

^{1}, Nicolas Taberlet

^{1}and Sébastien Manneville

^{1,b)}

### Abstract

We describe a technique coupling standard rheology and ultrasonic imaging with promising applications to characterization of soft materials under shear. Plane wave imaging using an ultrafast scanner allows to follow the local dynamics of fluids sheared between two concentric cylinders with frame rates as high as 10 000 images per second, while simultaneously monitoring the shear rate, shear stress, and viscosity as a function of time. The capacities of this “rheo-ultrasound” instrument are illustrated on two examples: (i) the classical case of the Taylor-Couette instability in a simple viscous fluid and (ii) the unstable shear-banded flow of a non-Newtonian wormlike micellar solution.

We acknowledge invaluable technical help from D. Israel at TA Instruments and J. Lachèvre at Lecoeur Electronique. T. Divoux and S. Lerouge are thanked for fruitful discussions. We acknowledge funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 258803.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

A. Rheological measurements

B. Ultrasonic scanner and probe

C. Ultrasonic data processing

1. Plane wave imaging

2. From raw speckle data to ultrasonicimages

3. Displacement field

III. CALIBRATION AND TEST IN A NEWTONIAN FLUID

A. Calibration procedure

B. Shear start-up in a Newtonian fluid

C. Taylor-Couetteinstability in a Newtonian fluid

IV. ILLUSTRATION IN A WORMLIKE MICELLAR SOLUTION

V. CONCLUSION AND PERSPECTIVES

### Key Topics

- Rotating flows
- 32.0
- Ultrasonics
- 29.0
- Flow instabilities
- 27.0
- Ultrasonography
- 26.0
- Taylor Couette flows
- 22.0

## Figures

Sketch and picture of the experimental setup. (a) Three-dimensional general view. (b) Top view of the gap of the Couette cell together with the path of the acoustic beam and the various axes and angles defined in the text. (c) Picture showing the water tank with the ultrasonic transducer facing a smooth, transparent Taylor-Couette geometry with dimensions (*R* _{1} = 23 mm and *R* _{2} = 25 mm) smaller than that used in the text.

Sketch and picture of the experimental setup. (a) Three-dimensional general view. (b) Top view of the gap of the Couette cell together with the path of the acoustic beam and the various axes and angles defined in the text. (c) Picture showing the water tank with the ultrasonic transducer facing a smooth, transparent Taylor-Couette geometry with dimensions (*R* _{1} = 23 mm and *R* _{2} = 25 mm) smaller than that used in the text.

(a) Raw speckle signal *s* _{ i }(*t*, *z*) recorded after a single plane wave emission as a function of time *t* and transducer position *z* along the array. (b) Corrected speckle signal after removal of the fixed echoes in *s* _{ i }(*t*, *z*) (see text). The dashed lines at *t* = 38.7 and 41.5 μs indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . The signals are coded using linear color levels. Experiment performed in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} (see the supplementary material for movie 1). ^{ 32 }

(a) Raw speckle signal *s* _{ i }(*t*, *z*) recorded after a single plane wave emission as a function of time *t* and transducer position *z* along the array. (b) Corrected speckle signal after removal of the fixed echoes in *s* _{ i }(*t*, *z*) (see text). The dashed lines at *t* = 38.7 and 41.5 μs indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . The signals are coded using linear color levels. Experiment performed in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} (see the supplementary material for movie 1). ^{ 32 }

(a) Beam-formed image *S* _{ i }(*y*, *z*) computed from the corrected speckle signal of Fig. 2(b) and shown as a function of *y*, the distance to the transducer array, and *z*, the position along the array. *S* _{ i } is normalized by its maximum value and coded in linear color levels. (b) Two successive beam-formed speckle signals *S* _{ i }(*y*, *z*) (in black) and *S* _{ i+1}(*y*, *z*) (in red) for a given position *z* = 14.9 mm along the transducer array. *S* _{ i } and *S* _{ i+1} correspond to two different plane wave emissions separated by δ*t* = 2 ms. (c) Enlargement of *S* _{ i }(*y*, *z*) (in black) and *S* _{ i+1}(*y*, *z*) (in red) over a window of width Δ*y* = 2λ close to the inner bob [indicated as a dashed box in (b)]. This evidences a noticeable displacement of the speckle to the right when going from *S* _{ i } to *S* _{ i+1}. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 (see also the supplementary material for movie 2). ^{ 32 }

(a) Beam-formed image *S* _{ i }(*y*, *z*) computed from the corrected speckle signal of Fig. 2(b) and shown as a function of *y*, the distance to the transducer array, and *z*, the position along the array. *S* _{ i } is normalized by its maximum value and coded in linear color levels. (b) Two successive beam-formed speckle signals *S* _{ i }(*y*, *z*) (in black) and *S* _{ i+1}(*y*, *z*) (in red) for a given position *z* = 14.9 mm along the transducer array. *S* _{ i } and *S* _{ i+1} correspond to two different plane wave emissions separated by δ*t* = 2 ms. (c) Enlargement of *S* _{ i }(*y*, *z*) (in black) and *S* _{ i+1}(*y*, *z*) (in red) over a window of width Δ*y* = 2λ close to the inner bob [indicated as a dashed box in (b)]. This evidences a noticeable displacement of the speckle to the right when going from *S* _{ i } to *S* _{ i+1}. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 (see also the supplementary material for movie 2). ^{ 32 }

(a) Displacement map δ*y* _{ i }(*y*, *z*) computed from two successive beam-formed images *S* _{ i } and *S* _{ i+1} separated by δ*t* = 2 ms. (b) Displacement map ⟨δ*y* _{ i }(*y*, *z*)⟩_{ i } averaged over 199 correlations between 200 successive images. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 .

(a) Displacement map δ*y* _{ i }(*y*, *z*) computed from two successive beam-formed images *S* _{ i } and *S* _{ i+1} separated by δ*t* = 2 ms. (b) Displacement map ⟨δ*y* _{ i }(*y*, *z*)⟩_{ i } averaged over 199 correlations between 200 successive images. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 .

(a) Axial velocity profiles *v* _{ y, i }(*y*, *z*) computed from the two successive beam-formed images *S* _{ i } and *S* _{ i+1} used in Fig. 4(a) and shown for *z* = 7.4 (⧫), 15.1 (•), and 22.4 mm (■). (b) Axial velocity profiles *v* _{ y }(*y*, *z*) = ⟨*v* _{ i }(*y*, *z*)⟩_{ i } averaged over 199 correlations between 200 successive images. The grey line shows the best linear fit of the full data set averaged over *z*. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 .

(a) Axial velocity profiles *v* _{ y, i }(*y*, *z*) computed from the two successive beam-formed images *S* _{ i } and *S* _{ i+1} used in Fig. 4(a) and shown for *z* = 7.4 (⧫), 15.1 (•), and 22.4 mm (■). (b) Axial velocity profiles *v* _{ y }(*y*, *z*) = ⟨*v* _{ i }(*y*, *z*)⟩_{ i } averaged over 199 correlations between 200 successive images. The grey line shows the best linear fit of the full data set averaged over *z*. The dashed lines at *y* = 28.7 and 30.7 mm indicate the limits of the gap as inferred from the calibration procedure described in Sec. III A . Same experiment as in Fig. 2 .

Tangential velocity profiles *v*(*r*) deduced from the calibration procedure [Eqs. (5) and (6) ] with *y* _{0} = 28.66 mm and ϕ = 5.2° for different applied shear rates: (•), 10 (□), 15 (•), and 20 s^{−1} (▼). Data are averaged over 199 successive correlations and over the vertical direction *z*. Error bars show the standard deviation over *z*. Grey lines are the theoretical predictions for a Newtonian fluid [Eq. (10) ]. Inset: relative deviation δ*v*/*v* computed as the standard deviation of *v*(*r*, *z*) over *z* relative to its mean value *v*(*r*). Experiments performed in a Newtonian suspension of hollow glass spheres at 1 wt.% in water.

Tangential velocity profiles *v*(*r*) deduced from the calibration procedure [Eqs. (5) and (6) ] with *y* _{0} = 28.66 mm and ϕ = 5.2° for different applied shear rates: (•), 10 (□), 15 (•), and 20 s^{−1} (▼). Data are averaged over 199 successive correlations and over the vertical direction *z*. Error bars show the standard deviation over *z*. Grey lines are the theoretical predictions for a Newtonian fluid [Eq. (10) ]. Inset: relative deviation δ*v*/*v* computed as the standard deviation of *v*(*r*, *z*) over *z* relative to its mean value *v*(*r*). Experiments performed in a Newtonian suspension of hollow glass spheres at 1 wt.% in water.

Start-up of shear in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} in the laminar regime. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 50 pulses sent every millisecond (see also the supplementary material for movie 3). ^{ 32 } (b) Stress response σ(*t*) recorded simultaneously to the velocity maps. The symbols indicate the times corresponding to the images shown in (a). Inset: magnification of the stress response σ(*t*) (in black) together with the instantaneous shear rate (in red) imposed by the rheometer. (c) Velocity profiles ⟨*v*(*r*, *z*, *t*)⟩_{ z } averaged over the whole height of the transducer array and shown at *t* = 0 (□), 0.075 (⋄), 0.15 (△), 0.225 (▽), 0.375 (○), 0.7 (◁), and 1.275 s (▷) consistently with (a) and (b). The grey line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ]. Inset: time evolution of ⟨*v*(*r*, *z*, *t*)⟩_{ z } at *r* = 0.01, 0.46, 0.98, 1.43, and 1.95 mm from top to bottom.

Start-up of shear in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} in the laminar regime. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 50 pulses sent every millisecond (see also the supplementary material for movie 3). ^{ 32 } (b) Stress response σ(*t*) recorded simultaneously to the velocity maps. The symbols indicate the times corresponding to the images shown in (a). Inset: magnification of the stress response σ(*t*) (in black) together with the instantaneous shear rate (in red) imposed by the rheometer. (c) Velocity profiles ⟨*v*(*r*, *z*, *t*)⟩_{ z } averaged over the whole height of the transducer array and shown at *t* = 0 (□), 0.075 (⋄), 0.15 (△), 0.225 (▽), 0.375 (○), 0.7 (◁), and 1.275 s (▷) consistently with (a) and (b). The grey line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ]. Inset: time evolution of ⟨*v*(*r*, *z*, *t*)⟩_{ z } at *r* = 0.01, 0.46, 0.98, 1.43, and 1.95 mm from top to bottom.

Start-up of shear in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} in a vortex flow regime. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 50 pulses sent every 0.5 ms (see also the supplementary material for movie 4). ^{ 32 } (b) Stress response σ(*t*) (in black) recorded simultaneously to the velocity maps together with the instantaneous shear rate (in red) imposed by the rheometer. The symbols indicate the times corresponding to the images shown in (a). Inset: apparent viscosity . (c) Velocity profiles *v*(*r*, *z*, *t*) at *t* = 2.375 s and for *z* = 19.25 (white symbols, outflow boundary), 20.5 (gray symbols, in between outflow and inflow), and 21.5 mm (black symbols, inflow boundary). The gray line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ].

Start-up of shear in a Newtonian suspension of hollow glass spheres at 1 wt.% in water and sheared at s^{−1} in a vortex flow regime. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 50 pulses sent every 0.5 ms (see also the supplementary material for movie 4). ^{ 32 } (b) Stress response σ(*t*) (in black) recorded simultaneously to the velocity maps together with the instantaneous shear rate (in red) imposed by the rheometer. The symbols indicate the times corresponding to the images shown in (a). Inset: apparent viscosity . (c) Velocity profiles *v*(*r*, *z*, *t*) at *t* = 2.375 s and for *z* = 19.25 (white symbols, outflow boundary), 20.5 (gray symbols, in between outflow and inflow), and 21.5 mm (black symbols, inflow boundary). The gray line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ].

Start-up of shear in a solution of wormlike micelles ([CTAB] = 0.3 M, [NaNO_{3}] = 0.34 M) seeded with hollow glass spheres at 1 wt.% and sheared at s^{−1} in the shear-banding regime with Taylor-like vortices. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 100 pulses sent every 1 ms (see also the supplementary material for movie 6). ^{ 32 } Each of the *N* _{seq} = 200 sequences of *N* = 100 pulses is separated from the next one by 0.5 s. (b) Stress response σ(*t*) (in black) imposed by the rheometer recorded simultaneously to the velocity maps together with the instantaneous shear rate (in red). The symbols indicate the times corresponding to the images shown in (a). Inset: apparent viscosity . (c) Velocity profiles ⟨*v*(*r*, *z*, *t*)⟩_{ z } averaged over the whole height of the transducer array and shown at *t* = 1.8 (□), 2.3 (∗), 2.8 (+), and 9.8 s (△). The gray line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ]. Inset: velocity profiles *v*(*r*, *z*, *t*) at *t* = 39.8 s and for *z* = 10.5 (white symbols, outflow boundary), 9.5 (gray symbols, in between outflow and inflow), and 8.5 mm (black symbols, inflow boundary).

Start-up of shear in a solution of wormlike micelles ([CTAB] = 0.3 M, [NaNO_{3}] = 0.34 M) seeded with hollow glass spheres at 1 wt.% and sheared at s^{−1} in the shear-banding regime with Taylor-like vortices. (a) Velocity maps *v*(*r*, *z*, *t*) at different times *t* indicated on the top row. Each map corresponds to an average over 100 pulses sent every 1 ms (see also the supplementary material for movie 6). ^{ 32 } Each of the *N* _{seq} = 200 sequences of *N* = 100 pulses is separated from the next one by 0.5 s. (b) Stress response σ(*t*) (in black) imposed by the rheometer recorded simultaneously to the velocity maps together with the instantaneous shear rate (in red). The symbols indicate the times corresponding to the images shown in (a). Inset: apparent viscosity . (c) Velocity profiles ⟨*v*(*r*, *z*, *t*)⟩_{ z } averaged over the whole height of the transducer array and shown at *t* = 1.8 (□), 2.3 (∗), 2.8 (+), and 9.8 s (△). The gray line shows the velocity profile expected for a Newtonian fluid in the laminar regime [Eq. (10) ]. Inset: velocity profiles *v*(*r*, *z*, *t*) at *t* = 39.8 s and for *z* = 10.5 (white symbols, outflow boundary), 9.5 (gray symbols, in between outflow and inflow), and 8.5 mm (black symbols, inflow boundary).

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