Unsteady Carbopol Flows VPF 2013 -...

www.complexfluids.euLTN VPF 2013, Paris

Teo Burghelea, CR1

Unsteady laminar flows of a Carbopol gel

1Laboratoire de Thermocinétique, Nantes, CNRS1Laboratoire de Thermocinétique, Nantes, CNRS

Antoine Poumaere1, Miguel Moyers-Gonzalez2, Cathy Castelain1, !

Teo Burghelea1

2Department of Mathematics and Statistics, University of Canterbury, New Zealand

http://www.complex

http://www.complex


Teo Burghelea, CR1

Why unsteady flows of Carbopol gels?

http://www.complex

http://www.complex


Teo Burghelea, CR1

Mars 3 (ThermoFischer Scientific) +

Nano - Torque module

The common knowledge: !

Carbopol gels are “model” yield stress fluid properly described by the Herschel Bulkley model

The steady state dynamics - well documented (for Carbopol gels as model yield stress fluids)

http://www.complex

http://www.complex


Teo Burghelea, CR1

Does the steady state yielding picture suffice? (can it accurately describe any type of realistic flows?)

http://www.complex

http://www.complex


Teo Burghelea, CR1

Nope. They don’t. Carbopol gels can still surprise us!

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p

Measured Flow Pattern around a sphere

freely falling in Carbopol

Numerical simulation by A. Beris within

the “model” framework

Settling of an isolated spherical particle in a yield stress shearthinning fluid

A. M. V. Putz,1,a! T. I. Burghelea,1,b! I. A. Frigaard,1,2,c! and D. M. Martinez3,d!

1Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver,British Columbia V6T 1Z2, Canada2Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane,Vancouver, British Columbia V6T 1Z4, Canada3Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall,Vancouver, British Columbia V6T 1Z3, Canada

!Received 16 May 2007; accepted 7 January 2008; published online 12 March 2008"

We visualize the flow induced by an isolated non-Brownian spherical particle settling in a shearthinning yield stress fluid using particle image velocimetry. With Re!1, we show a breaking of thefore-aft symmetry and relate this to the rheological properties of the fluid. We find that the shape ofthe yield surface approximates that of an ovoid spheroid with its major axis approximately five timesgreater than the radius of the particle. The disagreement of our experimental findings with previousnumerical simulations is discussed. © 2008 American Institute of Physics.#DOI: 10.1063/1.2883937$

I. INTRODUCTION

The focus of the present work is an experimental studyof the flow induced by the motion of an isolated non-Brownian glass sphere settling in a yield stress fluid at lowReynolds number; i.e., Re!1. Although the settling processis found in a number of industrial and natural settings, thereare still many fundamental, unanswered questions regardingthe critical force required to initiate motion of particles inthis complex medium. This is mainly due to changes in themolecular organization of the fluid when under stress. Themacroscopic effect of this is the coexistence of yielded !flu-idized" and unyielded !solidlike" zones in the fluid domain.The motivation of this work stems from an interest in twoindustrial applications; namely, the separation of particles ofdifferent densities and the transport of small particles in oiland gas well construction operations.

Understanding the motion of a particle in a complex orstructured fluid is difficult. Insight into this phenomenon canbe gained by first examining the simpler case of the motionof particles settling slowly in a Newtonian fluid. For a singledense particle falling at low Reynolds number !Re"1" in aless dense solution, particles settle at the Stokes velocity1,2

Us =2#$R2

9%g , !1"

where R is the sphere radius, #$=$p−$ f is the differencebetween the density of the particles and that of the fluid, % isthe fluid viscosity, and g is the acceleration due to gravity. Itis clear from the abundant literature in this area that the flowaround the particle is symmetrical and that the sphere expe-riences a drag force that is proportional to the terminal

velocity.3,4 With yield stress fluids, this problem is morecomplex: the presence of a yield stress implies that settlingcan only occur if the net gravitational force is greater thanthe resistive force due to the yield stress of the material.Despite the simplicity of the problem, the mechanism bywhich the particle settles is poorly understood. Clearly, acritical force is required to initiate the motion of the particle.This force is proportional to the magnitude of the yield stressand related to the shape of the yielded envelope, but its exactvalue remains an open question for general particle shapes.

Before summarizing the existing literature, let us definethe problem under consideration mathematically. We willconsider the motion of a spherical particle of radius R anddensity $p settling in an quiescent fluid with a characteristicvelocity U. The fluid behaves !somewhat" like a Herschel–Bulkley fluid, which is traditionally characterized by threeparameters: the consistency k, a power law index n, and theyield stress &y. The density of the fluid $ f is constrained suchthat $p'$ f. When scaled using these parameters, the equa-tions of motion for both the fluid and the particle become!see Refs. 5 and 6 and references contained therein"

Re% "ui

"t+ uj

"ui

"xj& =

"&ij

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"p

"xi,

"ui

"xi= 0, !2a"

Red!Up"i

dt= Re Ri!1 − $q"qi + $q'

"P(ijnjdS , !2b"

Re( ")!

"t− !J=)! " * )! ) = $q'

"Pr! * !(= n! "dS , !2c"

where u! denotes the fluid velocity; p the pressure and &= theextra stress tensor; U! p denotes the velocity of the center ofmass of the particle, )! the angular rotation around the centerof mass, J= the inertia tensor, and r! denotes a material point ofthe particle. The usual Cartesian summation convention is

a"Electronic mail: [email protected]"Electronic mail: [email protected]"Electronic mail: [email protected]"Electronic mail: [email protected].

PHYSICS OF FLUIDS 20, 033102 !2008"

1070-6631/2008/20#3!/033102/11/$23.00 © 2008 American Institute of Physics20, 033102-1

Downloaded 12 Mar 2008 to 142.103.197.122. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

Example 1(Phys. Fluids 2008)

Similar issues for the motion of bubbles in Carbopol, see keynote presentation by Prof. Tsamopoulos on Monday

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Teo Burghelea, CR1

Example 2:

the talk by Zinedinne Kebiche on startup convective flows in Carbopol gels (last talk on TUE)

The thermal convection is initiated around the yield point so elasticity might also play a significant role. Bingham? Herschel-Bulkley?

(The frame rate does not reflect the real flow speed!)

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http://www.complex


Teo Burghelea, CR1

Example 2:

the talk by Zinedinne Kebiche on startup convective flows in Carbopol gels (last talk on TUE)

The thermal convection is initiated around the yield point so elasticity might also play a significant role. Bingham? Herschel-Bulkley?

(The frame rate does not reflect the real flow speed!)

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http://www.complex


Teo Burghelea, CR1

Example 3: withdrawal of a solid plate from a Carbopol 980 bath(Landau - Levich flow)

The negative wake effect can not be predicted by the classical and elasticity free steady state pictures

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Teo Burghelea, CR1

Example 3: withdrawal of a solid plate from a Carbopol 980 bath(Landau - Levich flow)

The negative wake effect can not be predicted by the classical and elasticity free steady state pictures

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Teo Burghelea, CR1

All these three flows bear two common features !

THEY OCCUR AROUND THE YIELD POINT, THEY ARE UNSTEADY

!While moving along a Lagrangian trajectory, the material

elements DO NOT wait for a steady state of stressing to be reached

Many real flows involving yield stress fluids are unsteady: (coating flows, startup flows of waxy crude oil, magma volcanic

flows)

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Teo Burghelea, CR1

Unsteady rheometric flow of a Carbopol gel

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Teo Burghelea, CR1

Summary of the rheological tests (2008 - 2013)

DATE CARBOPOL GRADE RHEOMETER T(C)

APROXIMATE!NUMBER OF

FLOW CURVESCOUNTRY

2006!-!

2008940 Malvern

(CVOR, CS) 23 120 CA

2008 -

2010980

TA Instruments, (Malvern

Gemini, AR-G2)

11-55 50 DE

2010!-!

2013980 Thermo-Haake,

(MARS III) 23 60 (and raising!) FR

⇡ 230

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Teo Burghelea, CR1

(a) (b)

Figure 3: (a) Flow curves measured with a polished geometry (!, ") and with a rough geometry (⋄, #). The full/empty symbols refer to theincreasing/decreasing branch of the flow curves. The full line is a Herschel-Bulkley fit that gives τy = 0.64± 0.003(Pa), K = 0.4± 0.002(Pasn),n = 0.54±0.001. The controlled stress unsteady stress ramp is schematically illustrated in the insert. The symbols marking the highlighted regionsdenote the deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) - fluid. (b) Dependence of thehysteresis area on the characteristic forcing time t0 measured with a smooth geometry (!) and a rough one ("). The full line (−) is a guide for theeye, P ∝ t−0.63

0 and the dash dotted line (−.−) is a guide for the eye, P ∝ t−0.030 .

The irreversibility of the deformation states observed within the solid and the intermediate deformation regimesand initially reported by Putz et al in [27] has been recently confirmed by others, [11, 9, 10]. A similar irreversiblerheological behaviour has been found in rheological studies performed on various grades of Carbopol R⃝ gels usingvarious rheometers (and various geometries) at various temperatures, [17, 33].

We note that, during each of the rheological measurements reported in this manuscript and in our previous studies[27, 17, 33], the Reynolds number was smaller than unity, therefore the experimentally observed irreversibility of thedeformation states illustrated in Fig. 3(a) should not be associated with inertial effects.

To gain further insights into the role of wall slip in the unsteady yielding of the Carbopol R⃝ gel, we have performedthe same type of rheological measurements with smooth parallel plates. The main effect of the wall slip is to shiftthe solid-fluid coexistence regime to lower values of the applied stresses (see the squares (!, ") in Fig. 3(a)). Theapparent yield stress measured in the presence of wall slip τa

y is nearly an order of magnitude smaller than the yieldstress measured in the absence of slip, τy. The data acquired with and without wall slip overlap only within the fluidregime, corresponding to shear rates γ > γc (with γc ≈ 0.8s−1 see Fig. 3(a)). This critical value of the shear rate abovewhich the slip and no slip data overlap is comparable to the critical shear rate found by Bertola and his coworkersbelow which no steady state flow could be observed by MRI, [3]. The existence of the critical shear rate γc is alsoconsistent with the measurements of the velocity profiles performed by Salmon and his coworkers with a concentratedemulsion sheared in a small gap Couette geometry, [31].

Additionally, we note that it is solely within the viscous deformation regime that all the data sets can be reli-ably fitted by the Herschel-Bulkley model (the full line in Fig. 3(a)) which reinforces the idea that only within thisdeformation range the Carbopol R⃝ gel behaves like a simple or ”model” yield stress fluid.

A next fundamental question that deserves being addressed is how the irreversible character of the rheologicalflow curves observed within the solid-fluid coexistence regime in the form of a hysteresis loop is related to the rateat which the deformation energy is transferred to the material or, in other words, to the characteristic forcing time t0.It is equally important to understand if (and how) the presence of wall slip influences this dependence. To addressthis points we compare measurements of the area of the hysteresis observed in the flow curves presented in Fig. 3(a)P =

∫

γud∆τu −∫

|γd |d∆τd performed for various values of t0 for both slip and no-slip cases. Here the indices ”u,d”refer to the increasing/decreasing stress branches of the flow ramp. As already discussed in Ref. [27] the area P hasthe dimensions of a deformation power deficit per unit volume of sheared material. The results of these comparativemeasurements are presented in Fig. 3(b). In the absence of wall slip, the deformation power deficit scales with thecharacteristic forcing time as P ∝ t−0.63

0 (the full squares (") in Fig. 3(b)). A similar scaling has been found in the

6

No slip, increasing stressesNo slip, decreasing stresses

Slip, increasing stressesSlip, decreasing stresses

absence of wall slip for several Carbopol R⃝ solutions with various concentrations and at various temperatures in Ref.[27], P ∝ t−1

0 . The difference in the scaling exponent found in the present study and the one initially reported Ref.[27] may be related to differences in the grade of the Carbopol R⃝.

The existence of a rheological hysteresis for Carbopol R⃝ gels characterised by a decrease of the power deficit withthe characteristic forcing time consistent with both our initial finding reported in Ref. [27] and the data presentedin Fig. 3(b) was afterwards confirmed by others in independent experiments performed with different Carbopol R⃝

solutions and using different experimental protocols, [10].The next question we address is what is the influence of the wall slip phenomenon on the rheological hysteresis

observed during stepped stress ramps and on its scaling with the characteristic forcing time t0. To address this pointssimilar measurements of the hysteresis area P for various values of t0 have been performed in the presence of wallslip (the empty squares (!) in Fig. 3(b)). It is found that the wall slip affects both the magnitude of the deformationpower deficit and its scaling with the characteristic forcing time by nearly suppressing it: P ∝ t−0.03

0 (the dash-dottedline (−.−) in Fig. 3(b)).

This new scaling of the rheological hysteresis loses with the degree of flow steadiness provides the first quantitativeevidence that a true steady state of deformation is practically impossible to achieve in the presence of wall slip. Indeed,the low values of the scaling exponent indicates that reaching a steady state of the flow in the presence of wall slip(the hysteresis vanishes) practically requires huge waiting times t0 which are significantly larger than any time scaleassociated to a rheological test.

3.2. Unsteady yielding in a laminar unsteady pipe flow in the presence of wall

Following the comparative investigation of the yielding of a Carbopol R⃝ gel in a rheometric flow in the presenceand in the absence of wall slip presented in Sec. 3.1, the main question that arises is to what extent the findings onthe solid-fluid transition investigated in a classical rheometric flow could be transferred to flows that are more relevantfrom a practical perspective, such as a laminar unsteady pipe flow in the presence of wall slip.

To characterise the solid-fluid transition in the unsteady channel flow in the presence of the wall slip, a timeseries of flow fields has been acquired during a controlled increasing/decreasing pressure ramp (see Fig. 2(b)). Thiscontrolled pressure ramp closely mimics the controlled stress ramps used to characterise the solid-fluid transition in arheometric flow.

Choosing a large value of the characteristic forcing time t0 allows one to study the limiting steady state case. Fort0 = 300 s a typical laminar viscoplastic plug-like flow is observed, Fig. 4(a).

The velocity fluctuations observed in the steady case are solely of an instrumental nature and do not exceed severalpercents of the time averaged velocity in the bulk but approach 10% in the boundary, Fig. 4(b). The transverse profileof the time averaged velocity displayed in Fig. 4(c) also reproduces well the laminar and steady flow behaviour ofa viscoplastic fluid in a tube. Following [8] and in the framework of the Herschel-Bulkley model the profile can beformally 2 fitted by:

U(r) =Us +

(

n

n+ 1

)(

1

2K·

∆p

L

)1n

(R−Rp)1n+1

[

1−

(

r−Rp

R−Rp

)1n+1

]

(1)

Here Us is the slip velocity and Rp is the radius of the un-yielded plug. The slip velocity was measured by eitherextrapolating the fit given by Eq. 1 at the wall when a reliable fit could be obtained (i.e. at larger driving pressureswhen the viscoplastic profile is developed) or by extrapolation of a spline interpolation function. The transverseprofiles of the time averaged velocity are reproducible upon increasing/decreasing the pressure drop consistently witha reversible flow regime.

To characterise the flow response to a unsteady forcing (finite values of t0) we first monitor the time series of theabsolute value of the plug velocity

∣

∣Up

∣

∣ , Fig. 5 measured during a controlled pressure ramp for a finite value of thecharacteristic forcing time, t0 = 7.5 s.

The spikes visible in the velocity time series presented in Fig. 5 are related to the switch of the driving pressurerealised by the displacement of the vertical translational stage (TS in Fig. 2(a)) which carries the inlet fluid container

2Note that the slip term Us is not accounted for in [8] which discusses the slip-free case and has been formally added here to describe themeasured transverse velocity profiles.

7













U(r) =Us +

(

n

n+ 1

)(

1

2K·

∆p

L

)1n

(R−Rp)1n+1

[

1−

(

r−Rp

R−Rp

)1n+1

]

(1)



∣

∣Up

∣




7

NO SLIP

SLIP

Scaling of the hysteresis losses with the degree of flow steadiness

- Rheological hysteresis !

- Gradual Solid-Fluid transition coupled to the wall slip !

- NO STEADY STATE REACHABLE IN THE PRESENCE OF WALL SLIP!

(a) (b)









∫

γud∆τu −∫



6(a) (b)

Figure 3: (a) Flow curves measured with a polished geometry (!, ") and with a rough geometry (⋄, #). The full/empty symbols refer to theincreasing/decreasing branch of the flow curves. The full line is a Herschel-Bulkley fit that gives τy = 0.64± 0.003(Pa), K = 0.4± 0.002(Pasn),n = 0.54± 0.001. A magnified view of the solid-fluid transition is presented in the upper insert. The controlled stress unsteady stress ramp isschematically illustrated in the lower insert. The symbols marking the highlighted regions denote the deformation regimes and are explained in thetext: (S) - solid, (S+F) - solid- fluid coexistence, (F) - fluid. (b) Dependence of the hysteresis area on the characteristic forcing time t0 measuredwith a smooth geometry (!) and a rough one ("). The full line (−) is a guide for the eye, P ∝ t−0.63

0 and the dash dotted line (−.−) is a guide for

the eye, P ∝ t−0.030 .

Within this regime the deformation states are not reversible upon increasing/decreasing applied stresses whichtranslates into different elastic moduli: Gu > Gd . Here the indices denote the increasing/decreasing stress ramps,respectively.

For large values of the applied stresses ( τ ≥ 0.64Pa) a reversible fluid regime is observed. One has to emphasisethat the transition from a irreversible solid like deformation regime to a reversible fluid one is not direct, but mediatedby an intermediate deformation regime which can not be associated with neither a solid like behaviour nor a fluid onebut with a coexistence of the two phases. A similar smooth solid-fluid transition has been recently observed in a lowReynolds number steady pipe flow, [29].

An interesting flow feature can be observed on the decreasing branch of the pressure ramp in the form of a cuspof the dependence of |γ| on the applied stress (the empty rhombs in Fig. 3(a)). Corresponding to this point, the shearrate changes its sign. To better see this effect, we re-plot a magnified view of the cusp region in a linear scale in theupper insert. This effect may be understood in terms of an elastic recoil manifested through a change in the directionof rotation of the top disk.



To gain further insights into the role of wall slip in the unsteady yielding of the Carbopol R⃝ gel, we have performedthe same type of rheological measurements with smooth parallel plates (the squares in Fig. 3(a)). The main effect ofthe wall slip is to shift the solid-fluid coexistence regime to lower values of the applied stresses in Fig. 3(a)). Theapparent yield stress measured in the presence of wall slip τa

y is nearly an order of magnitude smaller than the yieldstress measured in the absence of slip, τy. The data acquired with and without wall slip overlap only within the fluidregime, corresponding to shear rates γ > γc (with γc ≈ 0.8s−1 see Fig. 3(a)). This critical value of the shear rate abovewhich the slip and no slip data overlap is comparable to the critical shear rate found by Bertola and his coworkers

6

http://www.complex

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Teo Burghelea, CR1

The rheological hysteresis for a Carbopol gel is a new (and still subject of some controversy) observation:

Rheol ActaDOI 10.1007/s00397-009-0365-9

ORIGINAL CONTRIBUTION

The solid–fluid transition in a yield stressshear thinning physical gel

Andreas M. V. Putz · Teodor I. Burghelea

Received: 28 May 2008 / Accepted: 23 April 2009© Springer-Verlag 2009

Abstract We present an experimental investigation ofthe solid–fluid transition in a yield stress shear thin-ning physical gel (Carbopol® 940) under shear. Upona gradual increase of the external forcing, we observethree distinct deformation regimes: an elastic solid-likeregime (characterized by a linear stress–strain depen-dence), a solid–fluid phase coexistence regime (char-acterized by a competition between destruction andreformation of the gel), and a purely viscous regime(characterized by a power law stress-rate of strain de-pendence). The competition between destruction andreformation of the gel is investigated via both system-atic measurements of the dynamic elastic moduli (asa function of stress, the amplitude, and temperature)and unsteady flow ramps. The transition from solid be-havior to fluid behavior displays a clear hysteresis uponincreasing and decreasing values of the external forcing.We find that the deformation power corresponding tothe hysteresis region scales linearly with the rate atwhich the material is being forced (the degree of flow

A. M. V. PutzDepartment of Mathematics,University of British Columbia,1984 Mathematics Road, Vancouver,British Columbia, Canada, V6T 1Z2e-mail: [email protected]

T. I. Burghelea (B)Institute of Polymer Materials,University of Erlangen-Nürnberg,Martensstrasse 7, 91058 Erlangen, Germanye-mail: [email protected]

unsteadiness). In the asymptotic limit of small forcingrates, our results agree well with previous steady stateinvestigations of the yielding transition. Based on theseexperimental findings, we suggest an analogy betweenthe solid–fluid transition and a first-order phase tran-sition, e.g., the magnetization of a ferro-magnet whereirreversibility and hysteresis emerge as a consequenceof a phase coexistence regime. In order to get furtherinsight into the solid–fluid transition, our experimentalfindings are complemented by a simple kinetic modelthat qualitatively describes the structural hysteresisobserved in our rheological experiments. The modelis fairly well validated against oscillatory flow databy a partial reconstruction of the Pipkin space of thematerial’s response and its nonlinear spectral behavior.

Keywords Yield stress fluids · Solid–fluid transition ·Hysteresis

Introduction

During the past few decades, physical gels have foundan increasing number of applications in both industry(cosmetics, food processing, pharmaceutics, etc.) andfundamental research (targeted drug delivery, biotech-nology, etc.). More recently, injectable physical gelsare used for medical implants, tissue regeneration, andnoninvasive intervertebral disc repair (Hou et al. 2004).From the rheological point of view, such gels are usuallyreferred to as yield stress materials, that is they areable to sustain finite deformations prior to flowing. Atthe microscopic level, such materials are made of high-molecular-weight constituents (typically in the range

http://www.complex

http://www.complex


Teo Burghelea, CR1

The rheological hysteresis for a Carbopol gel is a new (and still subject of some controversy) observation:

Rheol ActaDOI 10.1007/s00397-009-0365-9

ORIGINAL CONTRIBUTION

The solid–fluid transition in a yield stressshear thinning physical gel

Andreas M. V. Putz · Teodor I. Burghelea

Received: 28 May 2008 / Accepted: 23 April 2009© Springer-Verlag 2009

Abstract We present an experimental investigation ofthe solid–fluid transition in a yield stress shear thin-ning physical gel (Carbopol® 940) under shear. Upona gradual increase of the external forcing, we observethree distinct deformation regimes: an elastic solid-likeregime (characterized by a linear stress–strain depen-dence), a solid–fluid phase coexistence regime (char-acterized by a competition between destruction andreformation of the gel), and a purely viscous regime(characterized by a power law stress-rate of strain de-pendence). The competition between destruction andreformation of the gel is investigated via both system-atic measurements of the dynamic elastic moduli (asa function of stress, the amplitude, and temperature)and unsteady flow ramps. The transition from solid be-havior to fluid behavior displays a clear hysteresis uponincreasing and decreasing values of the external forcing.We find that the deformation power corresponding tothe hysteresis region scales linearly with the rate atwhich the material is being forced (the degree of flow

A. M. V. PutzDepartment of Mathematics,University of British Columbia,1984 Mathematics Road, Vancouver,British Columbia, Canada, V6T 1Z2e-mail: [email protected]

T. I. Burghelea (B)Institute of Polymer Materials,University of Erlangen-Nürnberg,Martensstrasse 7, 91058 Erlangen, Germanye-mail: [email protected]

unsteadiness). In the asymptotic limit of small forcingrates, our results agree well with previous steady stateinvestigations of the yielding transition. Based on theseexperimental findings, we suggest an analogy betweenthe solid–fluid transition and a first-order phase tran-sition, e.g., the magnetization of a ferro-magnet whereirreversibility and hysteresis emerge as a consequenceof a phase coexistence regime. In order to get furtherinsight into the solid–fluid transition, our experimentalfindings are complemented by a simple kinetic modelthat qualitatively describes the structural hysteresisobserved in our rheological experiments. The modelis fairly well validated against oscillatory flow databy a partial reconstruction of the Pipkin space of thematerial’s response and its nonlinear spectral behavior.

Keywords Yield stress fluids · Solid–fluid transition ·Hysteresis

Introduction

During the past few decades, physical gels have foundan increasing number of applications in both industry(cosmetics, food processing, pharmaceutics, etc.) andfundamental research (targeted drug delivery, biotech-nology, etc.). More recently, injectable physical gelsare used for medical implants, tissue regeneration, andnoninvasive intervertebral disc repair (Hou et al. 2004).From the rheological point of view, such gels are usuallyreferred to as yield stress materials, that is they areable to sustain finite deformations prior to flowing. Atthe microscopic level, such materials are made of high-molecular-weight constituents (typically in the range

Other groups observed it (independently) as well!

From stress-induced fluidization processes to Herschel-Bulkley behaviour insimple yield stress fluids

Thibaut Divoux,*a Catherine Barentinb and S!ebastien Mannevillea

Received 6th April 2011, Accepted 25th May 2011

DOI: 10.1039/c1sm05607g

Stress-induced fluidization of a simple yield stress fluid, namely a carbopol microgel, is addressed

through extensive rheological measurements coupled to simultaneous temporally and spatially resolved

velocimetry. These combined measurements allow us to rule out any bulk fracture-like scenario during

the fluidization process such as that suggested in [Caton et al., Rheol Acta, 2008, 47, 601–607]. On the

contrary, we observe that the transient regime from solid-like to liquid-like behaviour under a constant

shear stress s successively involves creep deformation, total wall slip, and shear banding before

a homogeneous steady state is reached. Interestingly, the total duration sf of this fluidization process

scales as sff 1/(s! sc)b, where sc stands for the yield stress of the microgel, and b is an exponent which

only depends on the microgel properties and not on the gap width or on the boundary conditions.

Together with recent experiments under imposed shear rate [Divoux et al., Phys. Rev. Lett., 2010, 104,

208301], this scaling law suggests a route to rationalize the phenomenological Herschel-Bulkley (HB)

power-law classically used to describe the steady-state rheology of simple yield stress fluids. In

particular, we show that the steady-state HB exponent appears as the ratio of the two fluidization

exponents extracted separately from the transient fluidization processes respectively under controlled

shear rate and under controlled shear stress.

1 Introduction

Yield stress fluids (YSF) are widely involved in manufactured

products such as creams, gels, or shampoos. These materials are

characterized by a transition from solid-like to liquid-like above

the yield stress sc, which is of primary importance at both the

manufacturing stage and the end-user level.1 Recently it was

recognized that simple YSF, which mainly consist in emulsions,

foams, and carbopol microgels, can be clearly distinguished from

thixotropic YSF:2 in steady state the former ones can flow

homogeneously at vanishingly small shear rates under controlled

stress3,4 while the latter exhibit a finite critical shear rate.5,6 Still,

in spite of its importance for applications, the transient fluid-

ization process of simple YSF has remained largely unexplored

and previous works have focused either on global rheometry

under an applied stress1,7,8 or on time-resolved local velocimetry

under controlled shear rate.9–11 Thus, detailed local information

concerning the fluidization of a simple YSF under applied shear

stress are still lacking, which prevents to make clear connections

with observations under imposed shear rate and with steady-

state rheology.

In this article we report a temporally and spatially resolved

study of the stress-induced fluidization of carbopol microgels

through ultrasonic echography. Our aim is to address the

following basic questions: (i) What is the fluidization scenario

of such a simple YSF under imposed shear stress? (ii) How

does it compare to imposed shear rate experiments? (iii) Can

one make a connection between these transient fluidization

processes and the steady-state rheology, which is well described

by the Herschel-Bulkley (HB) law? 9,11–14 Here, we show using

an ultrasonic velocimetry technique that carbopol microgels

submitted to a constant shear stress s under rough boundary

conditions successively exhibit creep deformation, total wall

slip, and shear banding before reaching a homogeneous steady

state. A close inspection of the backscattered ultrasonic signals

allows us to rule out a scenario involving bulk fracture of the

material. The duration of the fluidization process decreases as

a power-law with the reduced shear stress s ! sc. This power

law only depends on the sample preparation protocol and not

on boundary conditions or on the cell gap. Together with

recent experiments under imposed shear rate,11 this provides for

the first time a direct link between the yielding dynamics of

a simple YSF and the HB law which accounts for its steady-

state rheology.

aUniversit!e de Lyon, Laboratoire de Physique, !Ecole Normale Sup!erieurede Lyon, CNRS UMR 5672 - 46 All!ee d’Italie, 69364 Lyon cedex 07,FrancebLaboratoire de Physique de la Mati!ere Condens!ee et Nanostructures,Universit!e de Lyon, Universit!e Claude Bernard Lyon I, CNRS UMR5586 - 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex,France

This journal is ª The Royal Society of Chemistry 2011 Soft Matter

Dynamic Article LinksC<Soft Matter

Cite this: DOI: 10.1039/c1sm05607g

www.rsc.org/softmatter PAPER

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Rheological hysteresis in soft glassy materials

Thibaut Divoux,1 Vincent Grenard,1 and Sebastien Manneville1, 2

1Universite de Lyon, Laboratoire de Physique, Ecole Normale Superieure de Lyon,CNRS UMR 5672, 46 Allee d’Italie, 69364 Lyon cedex 07, France.

2Institut Universitaire de France(Dated: July 18, 2012)

The nonlinear rheology of a soft glassy material is captured by its constitutive relation, shear stressvs shear rate, which is most generally obtained by sweeping up or down the shear rate over a finitetemporal window. For a huge amount of complex fluids, the up and down sweeps do not superimposeand define a rheological hysteresis loop. By means of extensive rheometry coupled to time-resolvedvelocimetry, we unravel the local scenario involved in rheological hysteresis for various types ofwell-studied soft materials. Building upon a systematic experimental protocol, we introduce twoobservables that quantify the hysteresis in macroscopic rheology and local velocimetry respectively,as a function of the sweep rate δt−1. Strikingly, both observables present a robust maximum withδt, which defines a single material-dependent timescale that grows continuously from vanishinglysmall values in simple yield stress fluids to large values for strongly time-dependent materials. Inline with recent theoretical arguments, these experimental results hint at a universal timescale-basedframework for soft glassy materials, where inhomogeneous flows characterized by shear bands and/orwall slip play a central role.

PACS numbers: 83.60.La, 83.50.Ax, 83.50.Rp

When submitted to an external stress, soft glassy ma-terials such as colloidal gels, clay suspensions, concen-trated emulsions, and foams, display a fascinating varietyof behaviors because the applied strain may disrupt andrearrange the microstructure over a wide range of spa-tial and temporal scales leading to heterogeneous flowproperties [1, 2]. For more than a decade, flow dynamicshave been probed by combining standard rheology, e.g.through the determination of the “constitutive relation”between the shear stress σ and the shear rate γ, and localstructural or velocity measurements [3, 4]. While muchprogress has been made on steady-state flow properties,the relevance of transient phenomena has been recog-nized only recently [5–7]. Still, in practice, it can beargued that any experimental determination of the flowcurve σ(γ) is effectively transient since it is obtained bysweeping up or down γ over a finite temporal window.In other words the measured flow curve coincides withthe steady-state relation σ(γ) only if the sweep rate isslow enough compared to any intrinsic timescale of thefluid. On the other hand, when the microstructure dy-namics are governed by long timescales, one expects hys-teresis loops in σ(γ) measurements performed by sweep-ing up then down the shear rate (or vice versa). Thisphenomenon, referred to as “rheological hysteresis,” hasindeed been commonly observed in a host of complexfluids for about 70 years [8, 9]. However, to date, thisubiquitous signature of the interplay between timescalesin complex fluids has not been quantitatively studied bymeans of local measurements.

In this Letter, we use time-resolved velocimetry to un-veil the local scenario involved in rheological hysteresisin various types of well-studied soft materials. Build-ing upon a systematic experimental protocol, we intro-

duce two observables, Aσ and Av, that quantify the am-plitude of the hysteresis phenomenon as a function ofthe sweep rate δt−1 in macroscopic rheology and localvelocity respectively. Both Aσ and Av go through amaximum with δt, pointing to the existence of a char-acteristic timescale θ for the microstructure dynamics.In thixotropic (laponite) suspensions and (carbon black)gels, θ is of the order of several hundreds of seconds,while it becomes hardly measurable for simple yieldstress fluids such as carbopol and concentrated emulsions.Velocity profiles allow us to understand this evolutionby clearly differentiating a succession of homogeneous,shear-banded, and arrested flows depending on the fluidand on the sweep rate, thus providing a local interpreta-tion of rheological hysteresis.

Experimental set-up and protocol.- Experiments areperformed in a polished Plexiglas Couette geometry (typ-ical roughness 15 nm, height 28 mm, rotating innercylinder of radius 24 mm, fixed outer cylinder of ra-dius 25 mm, gap e = 1 mm) equipped with a home-made lid to minimize evaporation. Rheological data arerecorded with a stress-controlled rheometer (MCR 301,Anton Paar). Two flow curves are successively recorded,first by decreasing the shear rate γ from high shear(γmax = 103 s−1) to low shear (γmin = 10−3 s−1) throughN = 90 successive logarithmically-spaced steps of dura-tion δt each, and then by immediately increasing γ backfrom γmin up to the initial value γmax following the sameN steps in reverse order. In general the downward andupward flow curves, σdown(γ) and σup(γ) do not coincideand define a hysteresis loop [see Fig. 1(a) as an example].Simultaneously to the flow curves, the azimuthal velocityv is measured as a function of the radial distance r to therotor, at about 15 mm from the cell bottom, and with

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2 Experimental

2.1 Sample preparation

2.1.1 Working system: carbopol microgels. Our working

system is a microgel made of carbopol ETD 2050 which

comprises homo- and co-polymers of acrylic acid highly cross-

linked with a polyalkenyl polyether.13,15 The microgel is tradi-

tionally prepared in two steps: (i) the polymer is dispersed in

water leading to a suspension of carbopol aggregates and (ii)

a neutralizing agent is added (in our case sodium hydroxyde)

leading to polymer swelling and to microgel formation. The

microstructure of such microgels consists in an assembly of soft

jammed swollen polymer particles, with typical size ranging

roughly from a few microns to roughly 20 microns16–18 depending

on the type of carbopol,15 its concentration,13 the final value of

the pH,19 the type of neutralizing agent17 and, last but not least,

the stirring speed during neutralization.15 Carbopol microgels

exhibit good temperature stability.20 They are also known in the

literature to be non-aging, non-thixotropic simple YSF3,14,21,22

and their steady-state flow curve nicely follows the HB law:

s ¼ sc + ~h _gn, (1)

where _g is the shear rate and n¼ 0.3–0.6 depending on the type of

carbopol and its concentration.3,11,13,18

2.1.2 Sample preparation protocol. For our study we prepare

two kinds of samples: traditional samples on the one hand, and

samples that are seeded with micronsized glass spheres on the

other hand, in order to use ultrasonic speckle velocimetry

(USV)23 simultaneously to standard rheological measurements.

The detailed protocol for seeded samples is as follows: we first

add 0.5% wt. of hollow glass spheres (Potters, Sphericel, mean

diameter 6 mm, density 1.1) in ultrapure water; pH increases

roughly from 7 to 8. As carbopol is hydrosoluble only for pH <7,

we add one or two drops of concentrated sulfuric acid (H2SO4,

96%) to make the pH drop to roughly 5. The glass sphere

suspension is heated at 50 "C and the carbopol powder is care-

fully dispersed under magnetic stirring at 300 rpm for 40 min, at

a weight fraction C, with C ranging from 0.5 to 3% wt. The

mixture is then left to rest at room temperature for another 30

min, after which pH x 3. Finally we neutralize the solution with

sodium hydroxide (NaOH, concentration 10 mol L#1) until pH¼7.0 $ 0.5 while stirring manually. This leads to a carbopol

microgel which is finally centrifuged for 10 min at 2500 rpm to

get rid of trapped bubbles. As for traditional samples without

glass beads, the protocol starts directly by adding the carbopol

powder to a heated volume of ultrapure water and continues as

explained above.

2.1.3 Influence of the preparation protocol on the batch

properties. We emphasize that the final microgel macroscopic

properties are quite sensitive to the preparation protocol. In

particular, it is well known in the literature that during the

neutralization step, (i) the way the base is added (drop by drop or

all at once) as well as (ii) the exact final value of the pH and (iii)

the stirring speed during the neutralization process influence the

values of the parameters of the HB model.15,24 This derives from

the fact that these three parameters control the particle size

distribution of the microgel.15,19 In this paper, we take good care

to neutralize our samples in a reproducible fashion. Nonetheless,

from batch to batch, the final pH value of the microgel varies in

the range 6.5 < pH < 7.5. Therefore, when comparing different

geometries, gaps, or boundary conditions, we pay special atten-

tion to use results obtained on a single batch, so that the prep-

aration protocol does not introduce any bias. We will mainly

report data obtained on two different batches of carbopol weight

fraction C ¼ 1% wt. and seeded with glass spheres, noted batch 1

and batch 2, prepared separately but following the same

protocol. We will also discuss the influence of the carbopol

concentration C on four different traditional unseeded batches

prepared separately: C ¼ 0.5, 0.7, 1, and 3% wt.

2.1.4 Influence of the seeding glass spheres. Linear viscoelas-

ticity measurements show that the addition of hollow glass

spheres generally slightly stiffens the system (by at most 10%).25

However, we shall check throughout the whole manuscript that

traditional samples and seeded samples exhibit the exact same

rheological trends, which demonstrates that the acoustic contrast

agents play no significant role in the fluidization scenario under

imposed shear stress.

2.2 Experimental setup and protocol

2.2.1 Rheological setup. Rheological measurements are per-

formed with a stress-controlled rheometer (Anton Paar

MCR301). Two different small-gap Couette cells were used to

test the influence of the boundary conditions (BC) on the

fluidization process: a rough cell (surface roughness d x 60 mm

obtained by gluing sandpaper to both walls, rotating inner

cylinder radius Rint ¼ 23.9 mm, gap width e ¼ 1.1 mm, and

height h ¼ 28 mm) and a smooth Plexiglas cell [d x 15 nm

(AFM measurements), Rint ¼ 24 mm, e ¼ 1 mm, and h ¼ 28

mm]. Also, to test the influence of both the geometry and the

gap, experiments were performed with a plate-plate geometry

(radius 21 mm, gap width e ¼ 1 and e ¼ 3 mm) with two

different boundary conditions: rough (glued sandpaper, d x 46

mm) and smooth [glass, d x 6 nm (AFM measurements)].

Finally, note that for both geometries, we use a solvent trap

including a cover and a small water tank so as to efficiently

prevent evaporation.

2.2.2 Local velocity measurements. Velocity profiles are

measured at about 15 mm from the cell bottom through ultra-

sonic speckle velocimetry (USV) as described in details by

Manneville et al.23 In brief, USV relies on the analysis of ultra-

sonic speckle signals that result from the interferences of the

backscattered echoes of successive incident pulses of central

frequency 36 MHz generated by a high-frequency piezo-polymer

transducer (Panametrics PI50-2) connected to a broadband

pulser-receiver (Panametrics 5900PR with 200 MHz bandwidth).

The speckle signals are sent to a high-speed digitizer (Acqiris

DP235 with 500 MHz sampling frequency) and stored on a PC

for postprocessing. A cross-correlation algorithm yields the local

displacement from one pulse to another as a function of the

radial position across the gap with a spatial resolution of 40 mm.

After a calibration step using a Newtonian fluid, tangential

velocity profiles are then obtained by averaging over 10 to 1000

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as tc f 1/s1/a. This can also be read as tc f 1/(s ! sc)1/a with sc ¼

0. Using an original time stress superposition principle, the

authors have shown that a and n are equal within error bars.

Here, our approach allows us to predict that acidic solution of

the same type I collagen should also fluidize under an applied

shear rate _g after a lag time tcwhich should scale as tcf 1/ _gbwith

b ¼ 1, so that if one imposes the lag time for both fluidization

processes to be propotional, one would recover the steady-state

rheology with n ¼ a. Such a prediction remains to be experi-

mentally verified and could reinforce our claim that the two

fluidization timescales are proportional.

Last but not least, in a recent work on bidimensional wet

foams, Katgert et al.40 also proposed an interpretation of the HB

exponent n. The authors made a connection between the way the

average drag force on a bubble scales with the velocity and the

power-law behaviour of the viscous stress s ! sc in the HB

model. In this framework, n appears as a direct measure of the

average forces at the bulk level taking into account both the local

interbubble drag and the disorder induced by the flow. This is

also compatible with our results, as we have observed that the

fluidization exponents a and b are sensitive to the size distribu-

tion and/or the spatial organization of the soft particles that

constitute the microgel, through the preparation protocol and

the carbopol concentration (Fig. 4).

4.5 Time-dependent effects and hysteresis cycles

In this subsection, we provide the reader with further evidence

showing that it would be artificial to invoke any other charac-

teristic time or any hidden dynamical variable besides sf to

describe the fluidization process of carbopol microgels. This

point is strongly related to the fact that these microgels are simple

YSF.

Thixotropic and simple YSF are usually placed in two cate-

gories which preclude one another.2 If satisfying at first sight,

such a rough description remains qualitative and has resulted in

defining useless sub-categories such as ‘‘unusual yield stress

fluids’’.41 In order to overcome those difficulties, Coussot and

Ovarlez42 recently proposed an interesting reunification of these

two categories within a single theoretical framework by intro-

ducing the ratio D of two timescales: a characteristic relaxation

time h0/G0 (built on the viscosity h0 and the elastic modulus G0 of

the fluid) and the restructuring time q of the system. A fluid with

D ¼ h0/(G0q) close to 1 would be a simple YSF for which the

restructuring time is indeed roughly equal to the relaxation time,

whereas a fluid with D # 1 would correspond to a thixotropic

material, concomitantly presenting aging effects and restructure

over long durations. The key point of such a description is that

one goes continuously from one type of fluid to the other simply

by tuning the timescale ratio D.

A simple way to probe the relevant timescales consists in

performing successive decreasing and increasing ramps of

controlled shear rates. Fixing the shear rate range (here, _gmin ¼10!3 < _g < _gmax ¼ 102 s!1) and the number of experimental data

points (here, 15 points per decade), the only control parameter is

the waiting time per point tw spent at each imposed value of _g. Inother words, tw

!1 is the rate at which we scan the flow curve s( _g).In Fig. 8, we report flow curves obtained with four different

values of tw on a 1% wt. carbopol microgel. As already briefly

mentioned in,11 for a given value of tw, we observe a slight

hysteresis between decreasing and increasing shear-rate sweeps.

Let us emphasize here that such an effect is also noticeable but

not discussed in previous literature3,21 and that it is thus not

particular to the type of carbopol that we are using.

Repeating the shear rate sweep for different waiting times, we

observe that the area A of the hysteresis loop, defined as

Ahðlog _gmax

log _gmin

s½logð _g0Þ'd½logð _g0Þ'; (5)

Fig. 8 Flow curves, shear stress s vs shear rate _g, obtained by decreasing _g from 100 to 10!3 s!1 (B) and then increasing _g from 10!3 to 100 s!1 (red line)

for various waiting times per point: (a) tw ¼ 2 s, (b) tw ¼ 10 s, (c) tw ¼ 30 s, and (d) tw ¼ 70 s. Note that the hysteresis loop for _g ( 1 s!1 decreases for

increasing waiting times. The total duration of the longest measurements shown in (d) is 2.1$104 s. Experiments performed in a plate-plate geometry (e¼1 mm) with rough BC (d ¼ 46 mm) on a 1% wt. carbopol microgel without any seeding glass spheres.

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decreases as power law of tw with an exponent 0.36 [Fig. 9]. First

note that this result is robust: the same trend is observed for

carbopol microgels of three different mass concentrations (C¼ 1,

2, and 3%) and, when rescaled by the elastic modulus G0 of the

microgel, A is roughly independent of C. Second, in light of the

work by Coussot and Ovarlez42 detailed above, this result is (i)

compatible with a simple YSF, ruling out any thixotropic

behaviour, and (ii) suggests that no other timescale than the

fluidization time sf is necessary to describe the flow behaviour

of carbopol microgels. Indeed, in the case of a simple YSF, for

tw T 1 s, tw is generally large enough compared to the charac-

teristic relaxation time of the fluid so that tw [ h0/G0 " q. Thus,

the larger the imposed value of tw, the less the effects of the fluid

relaxation and restructuration will be probed and so the smaller

the area of the hysteresis loop. In other words, the larger tw, the

more the carbopol microgel ‘‘forgets’’ about its shear history.

In the case of a thixotropic YSF, the restructuring timescale

would be of hundreds of seconds or more so that h0/G0 # q and

tw would be of the same order of magnitude as q. Thus, in the

thixotropic case, at least two timescales, q and tw, would be

involved in the material dynamics during shear rate sweeps and

one would expect a more complex behaviour of A vs tw than

a simple decreasing function. In particular, it is anticipated that

the hysteresis loop grows larger as long as tw < q and that it

decreases (or at least saturates) for tw [ q when the restructu-

ration timescale is no longer relevant. An extensive study of the

behaviour of hysteresis cycles for both simple and thixotropic

YSF is currently underway to fully test these ideas. In any case,

to us, the results shown in Fig. 9 provide a strong confirmation

that (i) carbopol microgels are simple YSF which exhibit negli-

gible hysteresis when tw is large enough and (ii) no other time-

scale or hidden parameter is needed to describe the microgel

rheology as the hysteresis can be accounted for only by a tran-

sient shear banding phenomenon which is fully described by the

timescale sf.

5 Summary, open questions, and outlook

5.1 Summary

We have performed a temporally and spatially resolved study of

the stress-induced fluidization of a simple yield stress fluid. The

fluidization is a four step process that successively involves

Andrade-like creep deformation, a total wall slip regime, and

a transient shear banding phenomenon that leads to a homoge-

neous flow in steady state. The time to reach a linear velocity

profile is a robust decreasing power law of the applied shear

stress which neither depends on the boundary conditions nor on

the gap width, while the exponent is a function of the microgel

microstructure. One of the key results of this article is that the

exponent n in the HB model which describes the steady-state

rheology naturally appears as the ratio a/b of two fluidization

exponents derived from independent experiments under

controlled stress and under controlled shear rate. To our

knowledge, this provides for the first time a clear link between the

transient regime of the fluidization process and the steady-state

rheology.

5.2 Open questions and outlook

We would like to speculate that this last result is general for

simple YSF and future experiments will focus on measuring

s(s)f and s( _g)f in emulsions and wet foams so as to extract the value

of the exponent a and b and test their link with the steady-state

rheology. Concerning Carbopol microgels, it would also be of

valuable interest to unambiguously link the microscopic prop-

erties of the microgel, in particular the size of the microstructure,

to the value of the fluidization exponents.

The transient shear banding scenario, common to both applied

shear stress and shear rate experiments, also remains to be

characterized at a microscopic scale. In the case of traditional

steady-state shear banding, the two flowing bands present two

different microstructures.43 In the case of wormlike micelle

solutions for instance, the highly sheared band presents

a nematic-like order whereas the micelles are more entangled in

the weakly sheared band. Here, for the transient shear banding

observed during the fluidization of carbopol microgels, one may

wonder if there is any structural difference between the flowing

band and the arrested region.

Another puzzling issue comes up when one compares the

fluidization laws of two different soft systems: carbopol micro-

gels whose fluidization time decreases as a power law of the

viscous stress, and weakly attractive carbon black gels whose

fluidization time decreases exponentially with the applied

stress.30While the latter system is a fractal colloidal gel with a low

volume fraction, carbopol microgels are constituted of jammed

swollen particles. How and why does such a structural difference

lead to different stress-induced fluidization law? Could one tune

continuously the system properties to switch from one fluidiza-

tion behaviour to the other?

Finally, we wish to emphasize that it would be very interesting

to compare the present experimental data on stress-induced

fluidization to theoretical predictions. Unfortunately, to the best

of our knowledge, most recent theoretical works on shear

banding in yield stress materials have focused on stationary

states only.42,44One may think of using standard models for time-

Fig. 9 Area A of the hysteresis between the decreasing and the

increasing flow curve vs the waiting time per point tw for various carbopol

weight fractions (symbol, % wt. carbopol): (C, 1%); (#, 2%); (,, 3%).A

decreases as a power law of the waiting time: A=G0 ¼ 0:41=t0:36w , where

G0 is the elastic modulus of the microgel. Data obtained in a plate-plate

geometry (e ¼ 1 mm) with rough BC (d ¼ 46 mm) on carbopol microgels

without any seeding glass spheres.


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From stress-induced fluidization processes to Herschel-Bulkley behaviour insimple yield stress fluids

Thibaut Divoux,*a Catherine Barentinb and S!ebastien Mannevillea

Received 6th April 2011, Accepted 25th May 2011

DOI: 10.1039/c1sm05607g

Stress-induced fluidization of a simple yield stress fluid, namely a carbopol microgel, is addressed

through extensive rheological measurements coupled to simultaneous temporally and spatially resolved

velocimetry. These combined measurements allow us to rule out any bulk fracture-like scenario during

the fluidization process such as that suggested in [Caton et al., Rheol Acta, 2008, 47, 601–607]. On the

contrary, we observe that the transient regime from solid-like to liquid-like behaviour under a constant

shear stress s successively involves creep deformation, total wall slip, and shear banding before

a homogeneous steady state is reached. Interestingly, the total duration sf of this fluidization process

scales as sff 1/(s! sc)b, where sc stands for the yield stress of the microgel, and b is an exponent which

only depends on the microgel properties and not on the gap width or on the boundary conditions.

Together with recent experiments under imposed shear rate [Divoux et al., Phys. Rev. Lett., 2010, 104,

208301], this scaling law suggests a route to rationalize the phenomenological Herschel-Bulkley (HB)

power-law classically used to describe the steady-state rheology of simple yield stress fluids. In

particular, we show that the steady-state HB exponent appears as the ratio of the two fluidization

exponents extracted separately from the transient fluidization processes respectively under controlled

shear rate and under controlled shear stress.

1 Introduction

Yield stress fluids (YSF) are widely involved in manufactured

products such as creams, gels, or shampoos. These materials are

characterized by a transition from solid-like to liquid-like above

the yield stress sc, which is of primary importance at both the

manufacturing stage and the end-user level.1 Recently it was

recognized that simple YSF, which mainly consist in emulsions,

foams, and carbopol microgels, can be clearly distinguished from

thixotropic YSF:2 in steady state the former ones can flow

homogeneously at vanishingly small shear rates under controlled

stress3,4 while the latter exhibit a finite critical shear rate.5,6 Still,

in spite of its importance for applications, the transient fluid-

ization process of simple YSF has remained largely unexplored

and previous works have focused either on global rheometry

under an applied stress1,7,8 or on time-resolved local velocimetry

under controlled shear rate.9–11 Thus, detailed local information

concerning the fluidization of a simple YSF under applied shear

stress are still lacking, which prevents to make clear connections

with observations under imposed shear rate and with steady-

state rheology.

In this article we report a temporally and spatially resolved

study of the stress-induced fluidization of carbopol microgels

through ultrasonic echography. Our aim is to address the

following basic questions: (i) What is the fluidization scenario

of such a simple YSF under imposed shear stress? (ii) How

does it compare to imposed shear rate experiments? (iii) Can

one make a connection between these transient fluidization

processes and the steady-state rheology, which is well described

by the Herschel-Bulkley (HB) law? 9,11–14 Here, we show using

an ultrasonic velocimetry technique that carbopol microgels

submitted to a constant shear stress s under rough boundary

conditions successively exhibit creep deformation, total wall

slip, and shear banding before reaching a homogeneous steady

state. A close inspection of the backscattered ultrasonic signals

allows us to rule out a scenario involving bulk fracture of the

material. The duration of the fluidization process decreases as

a power-law with the reduced shear stress s ! sc. This power

law only depends on the sample preparation protocol and not

on boundary conditions or on the cell gap. Together with

recent experiments under imposed shear rate,11 this provides for

the first time a direct link between the yielding dynamics of

a simple YSF and the HB law which accounts for its steady-

state rheology.

aUniversit!e de Lyon, Laboratoire de Physique, !Ecole Normale Sup!erieurede Lyon, CNRS UMR 5672 - 46 All!ee d’Italie, 69364 Lyon cedex 07,FrancebLaboratoire de Physique de la Mati!ere Condens!ee et Nanostructures,Universit!e de Lyon, Universit!e Claude Bernard Lyon I, CNRS UMR5586 - 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex,France


Dynamic Article LinksC<Soft Matter

Cite this: DOI: 10.1039/c1sm05607g

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Teo Burghelea, CR1

Unsteady laminar pipe flow of a Carbopol !

(no more rheometry for today!)

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Experimental Setup. Modus operandi.

Figure 1: Schematic view of the experimental apparatus: FT - fish tank, I2 - water inlet, O2 - water outlet, FC - flow channel, I1 - working fluidinlet, O1 - working fluid outlet, L - solid state laser, CO - cylindrical optics block, LS - laser sheet, CCD - digital camera, PC - personal computer.

Figure 2: (a) Schematic view of the flow control system: TS - vertical translational stage, M - stepping motor, GB - gear box, GB - micro stepcontroller, PC - personal computer, IFC - inlet fluid container, OFC - outlet fluid container, FC - flow channel, FC - flow channel, I1 - workingfluid inlet, O1 - working fluid outlet, B - digital balance. (b) Schematical representation of the hydrostatic pressure ramp. t0 is the characteristicforcing time (the averaging time per stress point) and N is the total number of steps. The symbols marking the highlighted regions denote thedeformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) - fluid.

channel also provides a good matching of the refractive index which minimises the optical distortions induced by thecurved surface of the flow channel.

A green laser beam (λ = 514 nm) with a power of 500 mW emitted by the solid state laser L (from ChangchunIndustries, Model LD-WL206) is deflected by a mirror through a cylindrical optics block CO which reshapes it into ahorizontal laser sheet, Fig. 1.

The thickness of the generated laser sheet is roughly 80µm in the beam waist region which was carefully positionedat the centre line of the flow channel. The working fluid was seeded with an amount of 200 parts per million (ppm) ofpolyamide particles with a diameter of 60 µm (from Dantec Dynamics). A time series of flow images is acquired witha 12 bit quantisation digital camera CCD (Thorlabs, model DCU224C) through a 12X zoom lens (from La Vision).The resolution of the acquired images is 1280X1024 pixels which insures a spatial resolution of 2.96µm per pixel.

A slow controlled pressure flow has been generated by controlling the hydrostatic pressure difference between theinlet and the outlet of the flow channel as schematically illustrated in Fig. 2. The working fluid is held in a containerIFC with a large free surface rigidly mounted on a vertical translational stage TS. The translational stage is verticallydriven by a step motor M via a gear box GB.

The position and the speed of the stepping motor are controlled by a micro-step controller (model IT 116 Flashfrom Isel Gmbh) connected to a personal computer PC via a RS 232 serial interface. To control the motion of thestepping motor, a graphical user interface (GUI) has been developed under Matlab R⃝. The vertical position of the inlet

4

Rpipe = 2 mm

(Note that we are not “micro”, we are … “milli”, that is far beyond the scale of the gel)

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Teo Burghelea, CR1








4

Unsteady pressure driven slow (inertia free) flow ramps

As the flow pipe is glass made, the flows are SLIPPERY

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Figure 4: (a) Time averaged (over 100 instantaneous fields) velocity field. The false colour map refers to the modulus of the velocity. (b) Reducedvelocity fluctuations, Urms/U . (c) Time averaged velocity profiles measured for ∆p = 1300 Pa on the increasing (full symbols) and decreasing(empty symbols) branch of the pressure ramp. The error bars are defined by the root mean square deviation (rms) of the velocity. The full line is afit by the analytical solution defined by Eq. 1. The velocity data were acquired after a steady state was achieved, t0 = 300 s.

8













U(r) =Us +

(

n

n+ 1

)(

1

2K·

∆p

L

)1n

(R−Rp)1n+1

[

1−

(

r−Rp

R−Rp

)1n+1

]

(1)



∣

∣Up

∣




7

The steady state flow case (the patience is “the key”!)

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Teo Burghelea, CR1

The unsteady state flow case

The elastic recoil effects (similar somehow to the negative wake) are present in the unsteady pipe flow as well!Figure 5: Time series of the absolute value of the plug velocity Up = U

∣

∣

∣

∣

r=0

. The insert presents a magnified view of the time series plotted in a

linear scale after the elastic recoil effect is observed (see the text for the explanation). The characteristic forcing time was t0 = 7.5s.

We now turn our attention to the case when the forcing is unsteady. To characterise the flow response to a unsteadyforcing (finite values of t0) we first monitor the time series of the absolute value of the plug velocity

∣

∣Up

∣

∣ , Fig. 5measured during a controlled pressure ramp for a finite value of the characteristic forcing time, t0 = 7.5 s.

The spikes visible in the velocity time series presented in Fig. 5 are related to the switch of the driving pressurerealised by the displacement of the vertical translational stage (TS in Fig. 2(a)) which carries the inlet fluid containerIFC. Their magnitude depends on the inertia of the mechanical system and the acceleration of the stepping motorM. To avoid accounting for this response of the mechanical system which controls the pressure drop along the flowchannel, the flow data acquired during these transients (extended over roughly 0.5s in each of the experiments reportedhere) has been always discarded and no conclusions were drawn based on it.

For low values of the driving pressure on the increasing branch of the ramp the time (in the range t/t0 < 3)dependence of the centreline velocity does not reach a steady state during the characteristic forcing time t0 = 7.5s.

A further increase of the driving pressure (corresponding to t/t0 ∈ [3,6] in Fig. 5) reveals an interesting featureof the velocity time series in the form of a non-monotonic time dependence: upon the change of the pressure drop,the centreline velocity first decreases and then increases. A phenomenological explanation of this observation may begiven in terms of a strong heterogeneity of the flow, i.e. a dynamic coexistence and competition between yielding ofsolid material elements (or bands) and their re-combination which, as illustrated by the rheometric data presented inFig. 3(a), occurs within an intermediate range of the applied stresses (pressure drops). Indeed, as the driving pressuredrop ∆p is further increased, this non-monotonic behaviour is no longer observed and the plug velocity reaches asteady state during a characteristic time smaller than t0. This indicates that, within this range of driving pressure, asteady yielded state is achieved, Fig. 5.

The time series presented in Fig. 5 may also provide a first indication on the reversibility of the flow states uponincreasing/decreasing the driving pressure drops. Thus, one can note that the data is not symmetric with respect to thevertical line t/t0 = 10. This observation is consistent with the irreversibility of the deformation states observed for therheological data acquired at low and intermediate values of the applied stresses, Fig.3(a). An even clearer signatureof the irreversibility of the deformation states can be noted if one monitors the time series data acquired during thelast four steps of the decreasing branch of the pressure ramp and highlighted in the insert of Fig. 5. During thesesteps of the pressure ramp the velocity fluctuates strongly and, corresponding to the the last steps, a flow reversal isapparent: the plug velocity velocity changes sign as the pressure drop is decreased to its smallest value. This ratherunexpected effect can be associated to a elastic recoil effect similar to the elastic recoil observed during the rheologicalmeasurements on the decreasing stress branch and manifested by the cusp visible in Fig. 3(a).

Profiles of the absolute value of the time averaged flow velocity U and the reduced velocity fluctuation Urms/U

measured for several values of the driving pressure ∆p on both the increasing and the decreasing branch of the flow

9








4

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The unsteady state flow case

The elastic recoil effects (similar somehow to the negative wake) are present in the unsteady pipe flow as well!Figure 5: Time series of the absolute value of the plug velocity Up = U

∣

∣

∣

∣

r=0

. The insert presents a magnified view of the time series plotted in a

linear scale after the elastic recoil effect is observed (see the text for the explanation). The characteristic forcing time was t0 = 7.5s.

We now turn our attention to the case when the forcing is unsteady. To characterise the flow response to a unsteadyforcing (finite values of t0) we first monitor the time series of the absolute value of the plug velocity

∣

∣Up

∣

∣ , Fig. 5measured during a controlled pressure ramp for a finite value of the characteristic forcing time, t0 = 7.5 s.

The spikes visible in the velocity time series presented in Fig. 5 are related to the switch of the driving pressurerealised by the displacement of the vertical translational stage (TS in Fig. 2(a)) which carries the inlet fluid containerIFC. Their magnitude depends on the inertia of the mechanical system and the acceleration of the stepping motorM. To avoid accounting for this response of the mechanical system which controls the pressure drop along the flowchannel, the flow data acquired during these transients (extended over roughly 0.5s in each of the experiments reportedhere) has been always discarded and no conclusions were drawn based on it.

For low values of the driving pressure on the increasing branch of the ramp the time (in the range t/t0 < 3)dependence of the centreline velocity does not reach a steady state during the characteristic forcing time t0 = 7.5s.

A further increase of the driving pressure (corresponding to t/t0 ∈ [3,6] in Fig. 5) reveals an interesting featureof the velocity time series in the form of a non-monotonic time dependence: upon the change of the pressure drop,the centreline velocity first decreases and then increases. A phenomenological explanation of this observation may begiven in terms of a strong heterogeneity of the flow, i.e. a dynamic coexistence and competition between yielding ofsolid material elements (or bands) and their re-combination which, as illustrated by the rheometric data presented inFig. 3(a), occurs within an intermediate range of the applied stresses (pressure drops). Indeed, as the driving pressuredrop ∆p is further increased, this non-monotonic behaviour is no longer observed and the plug velocity reaches asteady state during a characteristic time smaller than t0. This indicates that, within this range of driving pressure, asteady yielded state is achieved, Fig. 5.

The time series presented in Fig. 5 may also provide a first indication on the reversibility of the flow states uponincreasing/decreasing the driving pressure drops. Thus, one can note that the data is not symmetric with respect to thevertical line t/t0 = 10. This observation is consistent with the irreversibility of the deformation states observed for therheological data acquired at low and intermediate values of the applied stresses, Fig.3(a). An even clearer signatureof the irreversibility of the deformation states can be noted if one monitors the time series data acquired during thelast four steps of the decreasing branch of the pressure ramp and highlighted in the insert of Fig. 5. During thesesteps of the pressure ramp the velocity fluctuates strongly and, corresponding to the the last steps, a flow reversal isapparent: the plug velocity velocity changes sign as the pressure drop is decreased to its smallest value. This ratherunexpected effect can be associated to a elastic recoil effect similar to the elastic recoil observed during the rheologicalmeasurements on the decreasing stress branch and manifested by the cusp visible in Fig. 3(a).

Profiles of the absolute value of the time averaged flow velocity U and the reduced velocity fluctuation Urms/U

measured for several values of the driving pressure ∆p on both the increasing and the decreasing branch of the flow

9








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Figure 6: (a) Profiles of the absolute value of the mean flow velocity Uav. The full lines are the fit functions by the analytical solution defined by Eq.1. (b) Profiles of the reduced root mean square variation of the velocity, Urms/Uav. The symbols refer to different values of the pressure drop: (!,")- ∆p = 310.8Pa, (◦,•) - ∆p = 621.6Pa, (△,#) - ∆p = 932.3Pa . The full/empty symbols refer to the data acquired on the increasing/decreasingbranch of the pressure ramp, respectively. The characteristic forcing time is t0 = 10s.

the driving pressure (∆p = 932.3Pa and ∆p = 621.6Pa, (the triangles (△,#) and the circles (◦,•) in Fig. 6 (a)))the velocity profiles measured on the increasing/decreasing branches of the ramp are consistent with a reversible andflow behaviour. Within this flow regime the velocity fluctuations quantified by Urms/Uav are the smallest and they aremainly localised near the channel wall which is another indicator for the steadiness of the flow (see Fig. 6 (b)).

At low driving pressures the flow reversibility is broken: the profile measured on the decreasing branch (the emptysquares, !, in Fig. 6 (a)) falls clearly below the profile measured on the increasing pressure branch.

It is equally worth noting that, corresponding to this driving pressure, the mean flow profile is less smooth aroundthe centreline of the channel which indicates a rather large level of velocity fluctuations. It also indicates that smallerthe driving pressure is further the flow is from a steady state.

To probe this hypothesis we turn our attention to the transverse profiles of the reduced velocity fluctuationsUrms/Uav, Fig. 6 (b)). For large driving pressure (the triangles in Fig. 6 (b)) the reduced velocity variations ac-count for roughly 6% of the mean flow velocity in the bulk region of the flow (around the unyielded plug) and theymay reach up to 20% of the mean flow velocity in the wall region (due to the smallness of the measured velocity).The rather large values of the fluctuations observed near the channel walls are instrumental and they are due to thesmallness of the measured velocity within this regions (here, the DPIV heavily relies on the sub-pixel interpolationbecause of the very small displacements of the flow tracers).

On the decreasing branch of the controlled pressure ramp and for low driving pressures a localisation of thevelocity fluctuations can be observed in the form of a local maximum centred around the middle of the channel(r/R = 0) can be observed (the empty circles (◦), the empty triangles (△) and the empty squares (!) in Fig. 6(b)).The localisation of the velocity fluctuations near the centre line of the flow channel may be interpreted in terms oflarge fluctuations of an elastic (unyielded) plug which is consistent with the experimental findings for the case of aoscillatory pipe flow, [23].

The measurements of the time averaged transverse profiles of the axial velocity allow one to quantify the extend ofthe elastic solid plug. To do this, the transverse velocity profiles have been formally interpolated by spline functionswhich allowed an accurate calculation of their derivatives. The plug regions have been defined in relation to theinstrumental error of the DPIV measurements by the loci of the points for which the absolute value of the numericalderivative does not exceed 5% of its maximal value (typically measured near the solid wall of the flow channel). Thedependence of the plug radius Rp on the driving pressure drop ∆p measured for various values of the characteristicforcing time t0 on both the increasing and the decreasing branch of the controlled pressure ramp is presented in Fig. 7.At low values of the pressure drops on the increasing branch or the flow ramp the plug radius is practically independent

10

Unsteady flow profiles

(1) The flow is reversible only in the fluid regime where the H-B picture holds valid. (2) Significant wall slip is observed. (2) Strong velocity fluctuations of the rigid plug are observed in the solid and solid-fluid regimes (and this is not the instrumental noise of our measurements!).

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Unsteady measurements of the plug radius for various degrees of flow steadiness (t0)

Figure 7: Dependence of the plug radius Rp on the driving pressure drop ∆p for various values of the characteristic forcing time t0 : (△,!)- t0 = 4s, (▽,") - t0 = 7.5s, (!,#) - t0 = 15s. The full/empty symbols refer to the data acquired on the increasing/decreasing branch of thepressure ramp, respectively. The full line is the Herschel - Bulkley prediction, Rp =

0.64∆p . The symbols marking the highlighted regions denote the

deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) - fluid.

F), the flow profiles can be fitted by Eq. 1 and the same procedure of determining the slip velocity and plug size usedfor the steady state case can be applied. The extrapolation of the spline functions at the walls of the channel allowsone to estimate the magnitude of the slip velocity. The plug regions have been defined in relation to the instrumentalerror of the DPIV measurements by the loci of the points for which the absolute value of the numerical derivative doesnot exceed 5% of its maximal value (typically measured near the solid wall of the flow channel). The dependenceof the plug radius Rp on the driving pressure drop ∆p measured for various values of the characteristic forcing timet0 on both the increasing and the decreasing branch of the controlled pressure ramp is presented in Fig. 7. At lowvalues of the pressure drops on the increasing branch or the flow ramp the plug radius is practically independent onthe driving pressure. This is consistent with a rigid body downstream motion of the core un-yielded fluid: the shearstresses associated with the yielded wall layers can not overcome the bulk yield stress of the fluid and the size of theplug is insensitive to the changes in the driving pressure. Beyond a critical value of the pressure drop ∆py a monotonedecrease of the plug radius is observed and, according to [9], this dependence can be fitted by Rp = 2Lτa

y /∆p whichleads to τa

y ≈ 0.27Pa. This value of the apparent yield stress is close to the apparent yield stress measured in arheometric flow in the presence of wall slip, Fig. 3. The plug radius data acquired on the decreasing branch of thepressure ramp reproduces the data acquired on the increasing branch only within the yielded regime, ∆p > ∆py. Thisonce again indicates that the flow states are irreversible around the yield point, in agreement with the rheologicalmeasurements presented in Fig. 3. On the decreasing branch of the flow ramp (the empty symbols in Fig. 7), thevalues of the plug radius pass through a local maximum before reaching an elastic solid plateau. This effect may berelated to the elastic recoil effect primarily observed on the decreasing stress branch of the rheological flow ramp (thecusp in Fig. 3) and thereon highlighted in the insert of Fig. 5.

Within the framework of the Herschel-Bulkley model and in the absence of the wall slip, the mean flow velocityUav is related to the driving pressure drop via:

Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)

see e.g. [5]. The dimensionless parameter ξ is defined:

ξ =R∆p

2Lτy(3)

11

Full Symbols - UP !Empty Symbols - DOWN








Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)


ξ =R∆p

2Lτy(3)

11

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Teo Burghelea, CR1

Unsteady measurements of the plug radius for various degrees of flow steadiness (t0)








Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)


ξ =R∆p

2Lτy(3)

11

Full Symbols - UP !Empty Symbols - DOWN








Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)


ξ =R∆p

2Lτy(3)

11

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Teo Burghelea, CR1

Figure 7: Dependence of the plug radius Rp on the driving pressure drop ∆p for various values of the characteristic forcing time t0 : (△,!) -t0 = 4s, (◦,•) - t0 = 5s (▽,") - t0 = 7.5s, (∗,⋆) - t0 = 10s, (!,$) - t0 = 15s, (%,&) - t0 = 20s . The full/empty symbols refer to the data acquiredon the increasing/decreasing branch of the pressure ramp, respectively. The full line is the Herschel - Bulkley prediction, Rp =

0.64∆p . The symbols

marking the highlighted regions denote the deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) -fluid.

on the driving pressure (the full symbols in Fig. 7). This is consistent with a rigid body downstream motion of thecore un-yielded fluid: the shear stresses associated with the yielded wall layers can not overcome the bulk yield stressof the fluid and the size of the plug is insensitive to the changes in the driving pressure. Beyond a critical value of thepressure drop ∆py a monotone decrease of the plug radius is observed and, according to [8], this dependence can befitted by Rp = 2Lτa

y /∆p which leads to τay ≈ 0.27Pa. This value of the apparent yield stress is close to the apparent

yield stress measured in a rheometric flow in the presence of wall slip, Fig. 3. The plug radius data acquired on thedecreasing branch of the pressure ramp reproduces the data acquired on the increasing branch only within the yieldedregime, ∆p > ∆py. This once again indicates that the flow states are irreversible around the yield point, in agreementwith the rheological measurements presented in Fig. 3. On the decreasing branch of the flow ramp (the empty symbolsin Fig. 7), the values of the plug radius pass through a local maximum before reaching an elastic solid plateau. Thiseffect may be related to the elastic recoil effect primarily observed on the decreasing stress branch of the rheologicalflow ramp (the cusp in Fig. 3) and thereon highlighted in the insert of Fig. 5.


Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)


ξ =R∆p

2Lτy(3)

The Eq. 2 relies heavily on the Herschel-Bulkley yielding picture, the flow steadiness and the absence of velocity slipat the channel’s wall. Although we expect that none of these assumptions is fully fulfilled by the flow we investigate,we still believe it is reasonable to use it in order to get just an indicator for order of magnitude for the appliedpressure drop that corresponds to yielded states, ∆py, which would ultimately indicate us a suitable range of thedriving pressure drops ∆p that needs to be explored during our experiments in order to systematically characterise thesolid-fluid transition for this type of unsteady pipe flow.

11

Figure 7: Dependence of the plug radius Rp on the driving pressure drop ∆p for various values of the characteristic forcing time t0 : (△,!) -t0 = 4s, (◦,•) - t0 = 5s (▽,") - t0 = 7.5s, (∗,⋆) - t0 = 10s, (!,$) - t0 = 15s, (%,&) - t0 = 20s . The full/empty symbols refer to the data acquiredon the increasing/decreasing branch of the pressure ramp, respectively. The full line is the Herschel - Bulkley prediction, Rp =

0.64∆p . The symbols

marking the highlighted regions denote the deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) -fluid.

on the driving pressure (the full symbols in Fig. 7). This is consistent with a rigid body downstream motion of thecore un-yielded fluid: the shear stresses associated with the yielded wall layers can not overcome the bulk yield stressof the fluid and the size of the plug is insensitive to the changes in the driving pressure. Beyond a critical value of thepressure drop ∆py a monotone decrease of the plug radius is observed and, according to [8], this dependence can befitted by Rp = 2Lτa

y /∆p which leads to τay ≈ 0.27Pa. This value of the apparent yield stress is close to the apparent

yield stress measured in a rheometric flow in the presence of wall slip, Fig. 3. The plug radius data acquired on thedecreasing branch of the pressure ramp reproduces the data acquired on the increasing branch only within the yieldedregime, ∆p > ∆py. This once again indicates that the flow states are irreversible around the yield point, in agreementwith the rheological measurements presented in Fig. 3. On the decreasing branch of the flow ramp (the empty symbolsin Fig. 7), the values of the plug radius pass through a local maximum before reaching an elastic solid plateau. Thiseffect may be related to the elastic recoil effect primarily observed on the decreasing stress branch of the rheologicalflow ramp (the cusp in Fig. 3) and thereon highlighted in the insert of Fig. 5.


Uav =

⎧

⎨

⎩

0, ξ ≤ 1.[

R∆pLK

]1/nR

2+1/n

[

1− 1ξ

]1+1/n [

1+ 1ξ (1+1/n)

]

, ξ > 1.(2)


ξ =R∆p

2Lτy(3)


11

(1) Just like the rheometric flow, the

unsteady pipe flow is irreversible upon

increasing/decreasing flow forcing.

!(2) A cusp related to an elastic recoil is equally

observed.

Figure 8: Dependence of the absolute value of the mean flow velocity Uav on the applied pressure drop ∆p. The characteristic forcing time wast0 = 4s. The full/empty symbols refer to the increasing/decreasing branch of the linear controlled stress ramp. The symbols marking the highlightedregions denote the deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) - fluid.


Using the yielding criterion ξ ≈ 1 the pressure drop corresponding to the yielded (flowing) states can be estimatedas ∆py ≈ 774Pa.

Measurements of the absolute value of the mean flow velocity Uav performed for increasing/decreasing values ofthe driving pressure ∆p and several values of the characteristic forcing time are presented in Fig. 8. The mean flow ve-locity Uav has been calculated by numerical integration of the DPIV measured velocity profiles, Uav = 2π

∫ R0 rU(r)dr

1.The mean velocity data acquired for increasing/decreasing driving pressures overlap only within the fluid regime

∆p > ∆py. This indicates that only corresponding to this range the Carbopol R⃝ gel is fully yielded. Below thiscritical value of the driving pressure a hysteresis of the flow states is observed. These observations are fully consistentwith the rheological hysteresis observed in the presence of wall slip (the squares (!) in Fig. 3(a)). As previouslymentioned through our paper, the cusp visible on the decreasing pressure branch corresponds to a elastic recoil effectmanifested by a reversal of the flow direction. This indicates that, as in the case of the solid-fluid transition observedin a rheometric flow [32], the elasticity cannot be ignored while studying the unsteady yielding in a slow pipe flow ofa Carbopol R⃝ gel.

The dependence of the of the hysteresis observed in the dependence of the mean flow velocity Uav on the drivingpressure ∆p defined by A =

∫

Uuavd∆pu −

∫

|Uav|dd∆pd , illustrated in Fig. 8, is presented in Fig. 9. The absolute valueappearing in the second term of the expression above is explained by the fact that, on the decreasing branch of theramp, the velocity may become negative due to an elastic recoil effect manifested within the solid-fluid coexistenceregime.

As in the case of the rheological hysteresis in the presence of slip illustrated in Fig. 3(b) a weak dependence ofthe hysteresis area on the characteristic forcing time is observed, A ∝ t−0.1

0 .This fact indicates that the presence of wall slip modifies the irreversibility of the flow curves within the solid-fluid

1The axial symmetry of the flow has been assumed.

12

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Teo Burghelea, CR1

Figure 9: Dependence of the area of the hysteresis of the dependencies of the mean flow velocity on the driving pressure drop presented in Fig. 8on the characteristic forcing time t0. The full line is a guide for the eye, A ∝ t−0.1

0 .

Figure 10: Dependence of the slip velocity Us on the applied pressure drop ∆p . The characteristic forcing time was t0 = 20s. The full/emptysymbols refer to the increasing/decreasing branches of the stress ramp, respectively. The full line is guides for the eye, Us ∝ ∆p2. The symbolsmarking the highlighted regions denote the deformation regimes and are explained in the text: (S) - solid, (S+F) - solid- fluid coexistence, (F) -fluid.

13

A true steady state can not be achieved during a finite time in a pipe flow in the presence of wall slip regardless how slowly we force the flow.


0 .


13

Scaling of the hysteresis loses with the degree of flow steadiness (t0)

Carbopol gels slip “very well” on glass surfaces!





∫ R0 rU(r)dr




∫

Uuavd∆pu −

∫





12

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Teo Burghelea, CR1


0 .


13

A true steady state can not be achieved during a finite time in a pipe flow in the presence of wall slip regardless how slowly we force the flow.


0 .


13

Scaling of the hysteresis loses with the degree of flow steadiness (t0)

(a) (b)









∫

γud∆τu −∫



6

Carbopol gels slip “very well” on glass surfaces!





∫ R0 rU(r)dr




∫

Uuavd∆pu −

∫





12

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Teo Burghelea, CR1


0 .


13

A hysteresis is equally observed in the slip velocity

Figure 11: Dependence of the slip velocity Us on the wall velocity gradients. The full line is a guide for the eye, Us ≈ 3 · 10−4 ∂U∂ r

∣

∣

∣

∣

r=±R

. The

symbols are: (△, !) - t0 = 4 (s), (◦, •) - t0 = 7.5 (s), (", #) - t0 = 10 (s), (▽, !) - t0 = 15 (s), (▹, %) - t0 = 20 (s).

As in the case of the pressure dependence of the mean flow velocity Uav illustrated in Fig. 8 and discussed above,one can clearly see in Fig. 10 that the dependence of the slip velocity on the applied pressure drop ∆p is not reversibleupon increasing/decreasing pressure over the entire range of pressures. The results holds for all the values of thecharacteristic forcing time t0 (data not show here) and represents another indicator that the slip behaviour can not beentirely decoupled from the irreversible solid-fluid transition observed during the rheological measurements presentedin Fig. 3(a).

Regardless the value of the characteristic forcing time t0, within the fluid flow regime (∆p > 500Pa) the slipvelocity scales with the applied pressure drop as Us ∝ ∆p2 (see the full lines in Fig. 10). This scaling law differsfrom the one found by Gonzalez and his coworkers in [24], Us ∝ ∆p0.876 (see Fig. 6 in Ref. [24] in connection totheir Eq. (3)). As the scaling law we found is practically insensitive to the characteristic forcing time, we believethat this discrepancy is not related to the degree of steadiness of the pipe flow investigated in [24] but to the differentrheological properties of the Carbopol R⃝ solutions they have used (nearly an order of magnitude difference in both theyield stress and the consistency). Based on this comparison, it is apparent that even within the reversible fluid regime(τ > τy) the wall slip behaviour remains correlated to the rheological properties of the solution.

To get a deeper insight into this correlation, we focus on the dependence of the slip velocity Us on the wall

velocity gradients ∂U∂ r

∣

∣

∣

∣

r=±R

, Fig. 11. The measurements presented in Fig. 11 are performed for various values of the

characteristic forcing time t0 on both the increasing and the decreasing branch of the pressure ramp (see Fig. 2 (b)).For each of the data sets presented, the wall velocity gradients have been obtained by numerical differentiation

of either the fit defined by Eq. 1 or the interpolation of the time averaged velocity profiles near the wall and the slipvelocity has been obtained by extrapolating the same fitted profile or its interpolation (see Fig. 6 (a)) at the wall,r/R = ±1. To increase the accuracy of the numerical differentiation, a central difference differentiation scheme hasbeen used.

Previous studies have reported a decrease of the slip velocity with the applied stresses (velocity gradients) beyondthe yield-point, [19]. Contrarily to this, our experimental findings indicate a monotone increase of the slip velocitywith the wall velocity gradients, regardless the deformation regime (solid, fluid or solid-fluid) and the degree of flowsteadiness (the value of t0). This conclusion is consistent, however, with the experimental findings for the case of asteady pipe flow, [24].

Regardless the deformation regime, the value of the characteristic forcing time t0 and the type of the pressureramp (increasing or decreasing pressures) a universal dependence of the slip velocity Us on the wall velocity gradients

14

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Teo Burghelea, CR1

Figure 11: Dependence of the slip velocity Us on the wall velocity gradients. The full line is a guide for the eye, Us ≈ 3 · 10−4 ∂U∂ r

∣

∣

∣

∣

r=±R

. The

symbols are: (△, !) - t0 = 4 (s), (◦, •) - t0 = 7.5 (s), (", #) - t0 = 10 (s), (▽, !) - t0 = 15 (s), (▹, %) - t0 = 20 (s).

As in the case of the pressure dependence of the mean flow velocity Uav illustrated in Fig. 8 and discussed above,one can clearly see in Fig. 10 that the dependence of the slip velocity on the applied pressure drop ∆p is not reversibleupon increasing/decreasing pressure over the entire range of pressures. The results holds for all the values of thecharacteristic forcing time t0 (data not show here) and represents another indicator that the slip behaviour can not beentirely decoupled from the irreversible solid-fluid transition observed during the rheological measurements presentedin Fig. 3(a).

Regardless the value of the characteristic forcing time t0, within the fluid flow regime (∆p > 500Pa) the slipvelocity scales with the applied pressure drop as Us ∝ ∆p2 (see the full lines in Fig. 10). This scaling law differsfrom the one found by Gonzalez and his coworkers in [24], Us ∝ ∆p0.876 (see Fig. 6 in Ref. [24] in connection totheir Eq. (3)). As the scaling law we found is practically insensitive to the characteristic forcing time, we believethat this discrepancy is not related to the degree of steadiness of the pipe flow investigated in [24] but to the differentrheological properties of the Carbopol R⃝ solutions they have used (nearly an order of magnitude difference in both theyield stress and the consistency). Based on this comparison, it is apparent that even within the reversible fluid regime(τ > τy) the wall slip behaviour remains correlated to the rheological properties of the solution.

To get a deeper insight into this correlation, we focus on the dependence of the slip velocity Us on the wall

velocity gradients ∂U∂ r

∣

∣

∣

∣

r=±R

, Fig. 11. The measurements presented in Fig. 11 are performed for various values of the

characteristic forcing time t0 on both the increasing and the decreasing branch of the pressure ramp (see Fig. 2 (b)).For each of the data sets presented, the wall velocity gradients have been obtained by numerical differentiation

of either the fit defined by Eq. 1 or the interpolation of the time averaged velocity profiles near the wall and the slipvelocity has been obtained by extrapolating the same fitted profile or its interpolation (see Fig. 6 (a)) at the wall,r/R = ±1. To increase the accuracy of the numerical differentiation, a central difference differentiation scheme hasbeen used.

Previous studies have reported a decrease of the slip velocity with the applied stresses (velocity gradients) beyondthe yield-point, [19]. Contrarily to this, our experimental findings indicate a monotone increase of the slip velocitywith the wall velocity gradients, regardless the deformation regime (solid, fluid or solid-fluid) and the degree of flowsteadiness (the value of t0). This conclusion is consistent, however, with the experimental findings for the case of asteady pipe flow, [24].

Regardless the deformation regime, the value of the characteristic forcing time t0 and the type of the pressureramp (increasing or decreasing pressures) a universal dependence of the slip velocity Us on the wall velocity gradients

14

A linear and universal scaling of the slip velocity with the wall velocity gradients is found regardless the flow steadiness and regardless the

flow regime.

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Teo Burghelea, CR1

Some flows of Carbopol gels DO depart from the ideal yield stress “model” picture by:

!- elastic effects !- gradual yielding (coexistence of solid and fluids bands) !- irreversibility of deformation states around yielding !- highly non trivial slip dynamics coupled to the yielding

IN THE PRESENCE OF WALL SLIP, THERE IS NO CURE FOR THE FLOW UNSTEADINESS AND IRREVERSIBILITY!

To sum up the story

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Teo Burghelea, CR1

Thanks

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http://www.complex

http://www.complex

http://www.complex

Date post:	07-Jun-2021
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Unsteady Carbopol Flows VPF 2013 -...

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