Rheol Acta (2012) 51:937–946 DOI 10.1007/s00397-012-0651-9

ORIGINAL CONTRIBUTION

Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall José Pérez-González · Juan Javier López-Durán · Benjamín M. Marín-Santibáñez · Francisco Rodríguez-González

Received: 26 April 2012 / Revised: 10 August 2012 / Accepted: 4 September 2012 / Published online: 27 September 2012 © Springer-Verlag 2012

Abstract An analysis of the yielding and flow behavior of a model yield-stress fluid, 0.2 wt% Carbopol gel, in a capillary with slip at the wall has been carried out in the present work. For this, a study of the flow kinematics in a capillary rheometer was performed with a twodimensional particle image velocimetry (PIV) system. Besides, a stress-controlled rotational rheometer with a vane rotor was used as an independent way to measure the yield stress. The results in this work show that in the limit of resolution of the PIV technique, the flow behavior agrees with the existence of a yield stress, but there is a smooth solid–liquid transition in the capillary flow curve, which complicates the determination of the yield stress from rheometrical data. This complication, however, is overcome by using the solely velocity profiles and the measured wall shear stresses,

J. Pérez-González (B) · J. J. López-Durán Laboratorio de Reología, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, U. P. Adolfo López Mateos Edif. 9, Col. San Pedro Zacatenco, 07738, México City, México e-mail: [email protected] B. M. Marín-Santibáñez Sección de Estudios de Posgrado e Investigación, Escuela Superior de Ingeniería Química e Industrias Extractivas, Instituto Politécnico Nacional, U. P. Adolfo López Mateos Edif. 8, Col. San Pedro Zacatenco, 07738, México City, México F. Rodríguez-González Departamento de Biotecnología, Centro de Desarrollo de Productos Bióticos, Instituto Politécnico Nacional, Col. San Isidro, 62731 Yautepec, Morelos, México

from which the yield-stress value is reliably determined. The main details of the kinematics in the presence of slip were all captured during the experiments, namely, a purely plug flow before yielding, the solid–liquid transition, as well as the behavior under flow, respectively. Finally, it was found that the slip velocity increases in a power-law way with the shear stress. Keywords Capillary rheometer · PIV · Yield stress · Wall slip · Herschel–Bulkley

Introduction Yield-stress fluids are defined as materials that require the application of a critical shear stress (τy ) to initiate the flow. In this view, such materials exhibit a solid– like behavior for shear stresses below τy and then flow for shear stresses above τy . The existence of a yield stress has been a matter of debate for a long time, since its measured value depends on the experimental conditions, as sample preparation, as well as on the sensitivity of the rheometer utilized (Barnes and Walters 1985; Nguyen and Boger 1992; Barnes 1999; Watson 2004). From the practical point of view, however, the concept of yield stress has been widespread and has become very useful in industry. For example, the yield stress is a useful parameter for the assessment of shelf-life of paints and other consumer products. Yield-stress fluids have been long studied by theoretical and experimental methods, and the main results are summarized in a series of reviews by different authors (see for example Cheng 1986; Yoshimura et al. 1987; Nguyen and Boger 1992; Denn and Bonn 2011). The

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simplest yield-stress fluid, also known as the Bingham fluid, is described by the constitutive equation: τ = τy + ηp γ˙ , τ > τy and γ˙ = 0, τ ≤ τy

(1)

where τ is the shear stress, τy is the yield stress, ηp is known as the plastic viscosity, and γ˙ is the shear rate. Equation 1 predicts a Newtonian behavior once the fluid starts flowing. In practice, however, most fluids with a yield stress are shear thinning. Thus, generalizations to account for the effect of shear thinning have been introduced. A widespread model is the Hershel– Bulkley’s one, given by τ = τy + Kγ˙ n , τ > τy

(2)

where K and n have the typical meaning of consistency and shear-thinning index, respectively. Torsional rheometers have been widely used for measuring the steady shear properties of yield-stress fluids. Provided that slip is restricted, a stresscontrolled rotational rheometer can give a relatively fast and meaningful value of the yield stress (Keentok 1982). On the contrary, pressure-driven rheometers, as the capillary one, do not have the acceptation of their torsional counterparts for yield-stress fluids characterization. The main reason for this is that experiments with capillaries are very time consuming and the results may be affected by slip at the capillary wall. The slip at solid boundaries of rheometers has been one of the main problems in the characterization of yield-stress fluids, since it complicates the determination of a reliable yield-stress value (Kalyon 2005). Thus, proper evaluation of the slip velocity becomes relevant for a reliable characterization of a yield-stress fluid. The calculation of the slip velocity in drag- and pressure-induced flows has been typically performed by using the Mooney method (1931). Yilmazer and Kalyon (1988) have suggested that the mechanisms for wall slip are the same for both types of flow. In the case of polymer solutions, polymer melts with external additives, suspensions, and gels, apparent slip may take place, which has been attributed to the development of a lowviscosity slip layer at the rheometer wall. This layer is typically micrometric in size and acts as a lubricant for the fluid in the bulk. The characteristic flow curve of a yield-stress fluid contains a region of true flow preceded by another region without flow, but in which slip may be present. The transition between such regions, i.e., the yielding behavior, depends on whether slip is present or not (see for example Fig. 4 in Nguyen and Boger 1992). In the presence of slip in a capillary rheometer of given length (L) to diameter (D) ratio (L/D), the flow curve

depends on the capillary diameter and the transition between both regions is expected to be sharper for bigger diameters in shear-thinning fluids. This is due to the predominance of shear over slip flow for large diameters. Thus, the yield stress is determined by finding the critical shear stress at the transition between the two regions. The accuracy on the determination of such a value will depend on the sharpness of the transition as well as on the density of experimental points. Nevertheless, the yield stress, being a property of the material, should be independent of the capillary diameter. This fact has been corroborated by Kalyon (2005) for concentrated suspensions. In addition, Kalyon has suggested that the transition from plug to flow, namely, the yield stress, for concentrated suspensions in capillaries coincides with the shear stress at which the ratio between the flow rate due to slip (Qs ) and the total flow rate (Q), becomes less than one, i.e., Qs /Q < 1. This way to evaluate the yield stress assumes pure pluglike flow in the first region, which in most studies is not corroborated. On the other hand, as it happens with other rheological systems, most of the analysis of the flow of yield-stress fluids has been mainly done by rheometrical (mechanical) measurements, but the study of their kinematics in different geometries has received limited attention. Early reports by Sisko (1958) and Keentok (1982) refer to a photographic study by Mahncke and Tabor to obtain velocity profiles of grease in a tube (D = 0.0254 m). Such velocity profiles contained an apparent pluglike region and were roughly fitted by a Bingham equation. However, Sisko (1958) included non-Newtonian shear-thinning characteristics of the fluid and provided a better fit to the data. Recently, the use of velocimetry techniques has brought new light on the behavior of yields-stress fluids under flow. Velocimetry techniques may be useful to locate the solid–liquid transition region in a given fluid. In particular, magnetic resonance imaging (MRI) techniques and ultrasound Doppler velocimetry (UDV) have been used to study the flow of nontransparent pasty materials (Fukushima 1999; Götz et al. 2002; Raynaud et al. 2002; Ouriev and Windhab 2002; Bonn et al. 2008; Coussot et al. 2009; Rabideau et al. 2010; Derakhshandeh et al. 2010a, b, 2012). Ouriev and Windhab (2002) carried out a study of the flow of concentrated suspensions in a tube by using UDV. The authors analyzed cornstarch shear-thinning and shear-thickening suspensions and obtained velocity profiles in part of the cross section of the tube. For a shear-thickening suspension (ϕ = 0.4 in glucose syrup), these authors calculated slip velocities from extrapolation of the partial velocity profiles at the die

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wall. The slip velocity increased meanwhile the flow behavior changed from shear thickening to Newtonian with increasing flow rate. Also, for another suspension (ϕ = 0.4 in silicon oil), Ouriev and Windhab fitted the velocity profiles to shear-thinning and Herschel– Bulkley functions without slip by using rheometrical data and reported a decrease of the plug region with increasing flow rate. Coussot et al. (2009) performed MRI measurements with a Carbopol gel in a Couette rheometer and showed that local velocimetry measurements were in good agreement with rheometrical ones. Also, these authors showed that the Carbopol gel behaves as a simple yieldstress fluid characterized by a Herschel–Bulkley constitutive equation. Later, Rabideau et al. (2010) used MRI along with finite element simulation to analyze the extrusion of a model yield-stress fluid (Carbopol at 0.4 wt%). The authors obtained the velocity profiles in the contraction region as well as in the tube, which were well fitted by a Herschel–Bulkley model without slip. Derakhshandeh et al. (2010a) calculated the yield stress of pulp fiber suspensions from UDV in a vane rheometer. These authors used the velocity profiles to determine the radius of the shearing zone, which, along with the torque reading, allowed for the calculation of the yield stress. In a subsequent work with similar suspensions, these authors (Derakhshandeh et al. 2010b) reported the existence of significant wall slip at the vane surface and used the Herschel–Bulkley model to fit the velocity profiles. In a more recent work, Derakhshandeh et al. (2012) studied the thixotropy and yielding of pulp fiber suspensions by means of UDV coupled with a vane rheometer. These authors reported thixotropy of their suspensions as well as a discontinuity of the velocity profiles in the vicinity of the yielding position, which was attributed either to shear banding or slip among fiber flocks. Finally, as stated by Magnin and Piau (1990), it is “unthinkable to carry out rheometrical tests on yieldstress fluids without monitoring the real kinematic field, which can differ considerably from the theoretical kinematic field deduced from the movement of the mechanical parts.” Therefore, in the present work, a detailed analysis of the flow of a model yield-stress fluid, 0.2 wt% Carbopol gel, in a capillary has been carried out by using particle image velocimetry along with rheometrical measurements (Rheo-PIV), which, to our knowledge, has not been made. PIV is a powerful noninvasive technique to describe the flow kinematics in transparent fluids. Also, PIV is a whole-field method that allows for the determination of instantaneous velocity maps in a flow region. This last approach, of

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common use in fluid mechanics, has been gradually implemented for the analysis of the flow behavior of complex fluids (Perez-Gonzalez et al. 2012). Thus, by using PIV, we have been able to capture the main details of the flow development of a yield-stress fluid in the presence of slip at the wall, namely, the purely pluglike flow before yielding, the solid–liquid transition, as well as the behavior under flow. The results in this work show that in the limit of resolution of the PIV technique, the behavior of the fluid agrees well with the existence of a yield stress, which can be reliably determined from the solely velocity profiles and the measured wall shear stress.

Experimental procedure The system studied was a Carbopol gel at a temperature of 25 ◦ C. A solution was first prepared by dissolving 0.2 wt% of Carbopol 940 (BF Goodrich) in tri-distilled water. Then, hollow glass spheres (as tracers for PIV) of 12 μm of average diameter (Sphericel 110 P8, Potters Industries), in a concentration of 0.2 wt%, were dispersed in the solution, which was further neutralized, according to the Carbopol content, with sodium hydroxide to obtain the gel. It was found that the addition of the spheres modified the final pH and provided an alkaline character to the gel (pH = 8.26). This pH value resulted in decreased viscosity and yield stress as compared to a well-neutralized gel without particles. Rheological measurements were performed in a pressure-controlled capillary rheometer with a borosilicate glass capillary of L = 0.298 m and D = 0.00294 m (L/D = 101.4). This large L/D capillary makes unnecessary pressure corrections for end effects. The reservoir from which the fluid was fed as well as the capillary were kept in a controlled temperature water bath at 25 ± 0.2 ◦ C, but one part of the capillary remained out of the bath for PIV measurements. The pressure drop between capillary ends (p) was measured with a Validyne® differential pressure transducer and the flow rate (Q) was determined by collecting and measuring the ejected mass as a function of time. From these data, the wall shear stress (τw ) and the apparent shear rate γ˙app were calculated as p  L 4 D

τw = 

γ˙app =

32Q . π D3

(3)

(4)

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Flow steadiness was assessed for each point in the flow curve before PIV measurements. First, there was a waiting time for the electronics of the transducer to read a constant pressure value for each flow condition; then, the corresponding flow rate was obtained. Only variations of around 1 % in the flow rate were allowed for a given flow condition to consider the flow as steady. Once the steady state was reached, the velocity maps were acquired for each flow condition. On the other hand, a stress-controlled Paar Physica UDS 200 rotational rheometer with the FL 100 vane geometry was also utilized as an independent way to measure the yield stress. The study of the flow kinematics in the capillary was performed with a two-dimensional PIV Dantec Dynamics system as sketched in Fig. 1. This system has been previously used for the analysis of the flow kinematics of micellar solutions (Méndez-Sánchez et al. 2003; Marín-Santibañez et al. 2006) and polymer melts (Rodríguez-González et al. 2009, 2010, 2011) in capillaries. All the details of the PIV set up to analyze capillary flow may be found elsewhere (Perez-Gonzalez et al. 2012) and only a brief description is presented here. The PIV system consists of a high-speed and high-sensitivity HiSense MKII CCD camera of 1.35 megapixels, two coupled Nd:YAG lasers of 50 mJ with λ = 532 nm, and the Dantec Dynamic Studio 2.1 software. A light sheet was reduced in thickness up to less than 200 μm by using a biconvex lens with 0.05 m of focal distance and sent through the center plane of the capillary. An InfiniVar® continuously focusable video microscope CFM-2/S was attached to the CCD camera in order to increase the spatial resolution. The images taken by the PIV system covered an area of 0.002942 × 0.0023 m which was located around an axial position of z = 75D downstream from the contraction. Series of 50 image pairs were obtained for each flow condition, and all the image pairs were correlated

Results and discussion Determination of the yield stress from rheometrical data Figure 2 shows the capillary flow curve for the gel (hollow symbols), along with the one obtained from the integration of the velocity profiles (filled symbols) according to 2π R Q=

rv (r) drdθ. 0

(5)

0

Validation of the PIV measurements is performed by their direct comparison with rheometrical data in the flow curve. In this case, the data obtained from the velocity profiles agree well with the rheometrical ones; the maximum difference in the volumetric flow rates obtained by using the two methods was 6.5 %, which shows the reliability of the PIV technique to describe the behavior of the gel in capillary flow.

1

4

3

to obtain the corresponding velocity map. Then, the 50 velocity maps were averaged in time to obtain a single one, from which the velocity as a function of the radial position (velocity profile) was obtained for the desired axial position.

2

To PC

9 8

5

6

7

Fig. 1 Experimental setup. 1 Water bath, 2 glass capillary, 3 CCD camera, 4 microscope, 5 Nd:YAG Laser, 6 Cylindrical lens, 7 prism, 8 biconvex lens, 9 aberration corrector

Fig. 2 Wall shear stress vs. volumetric flow rate for the Carbopol gel. Open and f illed symbols correspond to rheometrical and PIV measurements, respectively. The flow curve was divided into three regions: (I) solid-like behavior (pure slip), (II) transition, and (III) flow with slip. The continuous and dashed lines represent the fitting to models

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The flow curve covers a volumetric flow rate range from 10−8 to 10−6 m3 /s (corresponding to an apparent shear rate range of about 300 s−1 ) and has been divided into three regions (I, II, and III); two of them (I and III) appear linear in the log–log plot and have, therefore, been fitted by power-law equations (see the equations inserted in Fig. 2). The other region (II) is a fairly smooth transition connecting I and III. The volumetric flow rate-limiting regions I and II were established as those at the intersections of the straight lines in regions I and III with the curve in region II. It is noteworthy at this point that the only flow curve does not provide indication for the existence of a yield stress nor for the presence of slip. Moreover, the flow curve does not give information on the characteristics of the different flow regimes. Then, additional information is required for a full description of the flow behavior of the gel. In order to initiate the analysis of the flow curve, and in accordance with the existence of a yield stress for the gel, let us assume that region I arises from a purely pluglike flow (it will be shown in the next section that it is indeed the case). Then, the critical shear stress at the intersection of regions I and II marks the transition from plug to flow and may be considered as the yield stress. This was calculated from the crossing point of the linear function describing region I and a polynomial fitting for regions II and III together, which gives a value of 21.62 ± 5.96 Pa. Note that the bottom scale in Fig. 2 has been chosen as the volumetric flow rate (raw data) since it includes data that do not strictly correspond to shear flow (those in region I). In spite of this, the apparent shear rate has been included in the top scale for comparison. A different approach to the determination of the yield stress (which is commonly used in practical calculations) would be to find the stress at the intersection of the two extrapolated lines in Fig. 2, which corresponds to 31.87 ± 0.98 Pa. Nevertheless, from the data in Fig. 2, it should be clear that the yield stress calculated via this approach is overestimated (in about 47.4 % in this case), since the true yield value must lie well at the beginning of region II. The validity of this last statement must be supported by experimental evidence of purely pluglike flow in region I and shear flow in region II, and/or by an alternative measurement of the yield stress Figure 3 shows the results of an analysis of the gel by a slow shear stress ramp in the vane rheometer. In this case, we have made use of the fact that for a solidlike behavior, the relationship between the shear stress and shear deformation (γ) should be linear (Hookean behavior). Hence, the torque vs. strain data were recur-

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Fig. 3 Torque vs. strain for the Carbopol gel. The continuous line represents the fitting to the Hooke model

rently fitted by considering different amounts of points, until a slope with a value of 1 was obtained (R2 = 0.995). It may be observed that the torque changes linearly with γ up to a value of 0.07 and then deviates from the linearity. The first experimental point considered as lying out of the straight line had a deviation of 6.3 % in the torque value. The previous experimental point, which had a deviation of only 4.75 % with respect to the solid-like behavior, was used to calculate the yield stress for the gel as 20.63 ± 0.98 Pa, which is more accurate and lies inside the range 21.62 ± 5.96 Pa value calculated above. Description of the flow kinematics Figure 4 shows the velocity maps for some of the flow conditions investigated in this work. The velocity maps clearly exhibit the changes in the velocity field with increasing the wall shear stress, namely, the profiles change from completely pluglike to shear-thinning-like with a non-yielded region and slip at the wall. These characteristics are better appreciated in Fig. 5a–c, which exhibit the velocity profiles for the different flow conditions investigated in this work. In addition, it may be seen from Fig. 4 that there is no significant change of the velocity profiles inside the observation region, while the radial component of the velocity was at least 3 orders of magnitude smaller than the axial one in each case. Thus, the flow was unidirectional and the velocity field is simply given by vz = vz (r) for the different shear stresses, as it is expected for a fully developed shear flow. The velocity profiles in Fig. 5a–c represent the average of 50 consecutive profiles for a given position.

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Fig. 4 Velocity maps for different wall shear stresses for the Carbopol gel

FLOW DIRECTION l = 0.0023 m

l = 0.0023 m

a) 18.1 Pa

b) 27.2 Pa

c) 32.2 Pa

d) 42.2 Pa

e) 54.2 Pa

f) 64.2 Pa

g) 71.6 Pa

h) 80.2 Pa

-0.001471 m r z

0

0.001471 m -0.001471 m r z

0

0.001471 m -0.001471 m r z

0

0.001471 m -0.001471 m r z

0

0.001471 m

The average of the standard deviations for each profile was around 1 %, which is of the size of the symbols used to represent the data and shows that the flow was also steady. For the sake of clarity, error bars have been included only in the profile corresponding to 80.2 Pa in Fig. 5c to illustrate this fact. Figure 5a shows that for shear stresses in region I (below the yield value), the velocity profiles are characteristic of a solid-like behavior, i.e., they appear completely flat or pluglike, in agreement with the assumption made in the division of the flow curve in Fig. 2. Under these conditions, the flow rate is only due to slip of the gel at the capillary wall. At the shear stress of 26.8 Pa, it is possible to appreciate a slight development of flow near the capillary wall (see the highest profile in Fig. 5a), even though the velocity profile appears almost flat. Thus, this shear stress value, which is the first experimental point taken in region II, cannot be considered as corresponding to the yield stress, since it already includes a flow development. Consequently,

an accurate determination of the yield stress by using the pure rheometrical data would require of a large number of experimental points, which would be very impractical. One way to face this problem is discussed in the next section. Beyond the yield stress, the velocity profiles in Fig. 5b, c show the pluglike region around the capillary axis and shear thinning, both are characteristics of a yield-stress fluid. In addition, the velocity profiles clearly exhibit a nonzero velocity at the wall, i.e., the gel slips at the capillary wall with a velocity (vs ) that increases along with the wall shear stress (Fig. 6). Many works on the study of yield-stress fluids report the reduction and almost disappearance of the slip velocity once the yield value is surpassed (Nguyen and Boger 1992). In contrast with those reports, the slip velocity for this gel in contact with the glass capillary does not decrease once the yield stress has been surpassed. On the contrary, vs increases with increasing shear stress. Assuming the origin of slip as the development

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Fig. 6 Slip velocity vs. wall shear stress for the Carbopol gel. The continuous line represents the fitting to a power-law model

shear stress. Unfortunately, the characteristic changes in the rheological properties of the layer cannot be detected by the present experimental protocol, since as it has been previously mentioned, the layer is typically micrometric in thickness, or even thinner, which lies below the resolution range of these experiments. More work is necessary to clarify this point. Determination of the yield stress from velocity profiles Figure 7 shows a plot of Qs /Q as a function of the wall shear stress, i.e., the contribution of slip to the total

Fig. 5 Velocity profiles for different wall shear stresses for the Carbopol gel: a 11.8, 18.1, and 26.8 Pa; b 27.2, 29.7, 32.2, and 33.2 Pa; c 42.2, 54.2, 64.2, 71.6, and 80.2 Pa

of a depleted layer (Kalyon 2005), the increase in vs under flow conditions may be related to changes in the rheological properties of the layer with increasing the

Fig. 7 Ratio of volumetric flow rate due to slip (Qs ) and total flow rate (Q) as a function of the wall shear stress

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flow rate. Qs has been calculated as Qs = π vs D2 /4, with vs obtained directly from the velocity profiles. Kalyon (2005) has suggested that Qs /Q < 1 indicates the yielding of the fluid. However, this sort of analysis carries the same type of uncertainty in calculating the yield stress as the one that uses the rheometrical data, i.e., it requires a large number of experimental points to obtain an accurate evaluation. This problem can be overcome by calculating the yield stress directly from the velocity profiles via their first derivative. According to the shape of a velocity profile, its first derivative, which also represents the true shear rate for a unidirectional flow, must become zero at the position where the shear stress reaches the yield value. Figure 8 shows the first derivative of the velocity as a function of the radial position (dv/dr) for the profile obtained at τw = 71.6 Pa. The position at which dv/dr = 0 in Fig. 8 is found as r0 = 0.000424 ± 0.000021 m, which corresponds to a shear stress value of 20.62 ± 1 Pa, in agreement with the τy calculated from rheometrical data. If the same analysis is performed with all the velocity profiles in Fig. 5b, c, considering the symmetry of the profiles (i.e., the values obtained at both sides of the profiles), a value of 20.45 ± 0.13 Pa is obtained. Note that the yielding position r0 is a function of the wall shear stress (r0 = τy R/τw ) and decreases with increasing τw . This means that more fluid in the capillary yields as τw is increased and the yielding region approaches asymptotically to the capillary center; such a trend may be observed for the data in Fig. 9.

Fig. 8 First derivative of the velocity as a function of the radial position for the profile obtained at τw = 71.6 Pa. Vertical lines indicate the position at which dv/dr = 0 (r0 = 0.000424 ± 0.000021 m), which corresponds to τy = 20.62 ± 1 Pa

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Reconstruction of the true flow curve from velocity profiles The true flow curve (free of slip) for the gel may be obtained from the velocity profiles and the measured wall shear stresses. This is done by considering the radial distribution of the shear stress in capillaries, i.e., τ = τ (r), as well as the shear rate as a function of the radial position. The local shear rate may be calculated from the numerical derivative of the velocity profiles with respect to the radial position. Such a calculation was performed in this work by a central difference approximation as shown below:     1 vi+1 − vi vi − vi−1 ∂vz = + (6) ∂r i 2 ri+i − ri ri − ri−1 where v and r represent the local velocity and radial position, respectively. Meanwhile, the corresponding shear stress was calculated from the measured τ w by τ (r) = τw

r . R

(7)

The flow curve calculated by using the velocity profiles corresponding to 26.8, 27.2, 29.7, 32.2, 33.2, 42.2, 54.2, 64.2, 71.6, and 80.2 Pa, respectively, is displayed in Fig. 10a along with that obtained by applying the Rabinowitsch and slip correction to the rheometrical data. The segments of the flow curve reconstructed from the velocity profiles are represented with different symbols for the sake of clarity. Note that the data obtained from the velocity profiles follow the trend of the rheometrical ones very well. Moreover, the PIV data extend to lower shear rate values than the rheometrical ones. This is a valuable fact, since it allows one to

Fig. 9 Plot of r0 as a function of the wall shear stress

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a

b

Fig. 10 a Free of slip flow curve for the Carbopol gel obtained from velocity profiles, the continuous line represents the fitting to the Herschel–Bulkley model. b Master curve of normalized

velocity profiles as a function of dimensionless radial position; the continuous line represents the fitting to the normalized Herschel– Bulkley model

reach low-shear rate values that are not accessible by using the macroscopic physical quantities provided by capillary measurements, namely, τw and γ˙w . In addition, the wider shear rate range covered by the data in Fig. 10a permits a good fitting of the flow curve by the Hershel–Bulkley model (see the inserted equation and the continuous line in Fig. 10a). The fitting of the Herschel–Bulkley model to these data represent another independent method to determine the yield stress. The value of 19.82 ± 0.24 Pa calculated from this method has been also included in Table 1. The difference between this value and that found directly from the velocity profiles is 7.6 %. The agreement of the flow behavior of the gel with the Herschel–Bulkley model is also observed in a plot of the normalized velocity profiles (Fig. 10b) as a function of the dimensionless radial position (r = r/r0 ) as suggested by Coussot (2005):

where Vmax is the velocity of the plug flow section of the vz (r) profile and r0 is the radial position associated with the yield stress τy . The indices k and n are the parameters of the Herschel–Bulkley model. It is clear from this figure that the normalized velocity profiles may be reduced to a single master curve described by the Herschel–Bulkley equation. Summarizing, the yield-stress values obtained from the different analysis are presented in Table 1. From this, it is clear that the more reliable values are obtained from the velocity profiles and vane measurements, respectively. The yield values obtained from the rheometrical capillary data, although in agreement with the former two, carry higher uncertainty since they are very sensitive to the number of experimental points. Finally, it is worth to highlight the flat shape of the velocity profiles in region I. To our knowledge, it is not common to observe this sort of profiles in rheological fluids, even in the presence of slip. Although many authors have suggested that the velocity profiles may become pluglike in the presence of slip at high shear stresses, there is no reason for this to be expected. Recently, Rodríguez-González et al. (2010) have shown, that even under strong slip flow of linear polyethylenes (Qs /Q > 0.7), the velocity profiles do not become pluglike. In other words, there must always be a shear contribution in the bulk that leads to the development of the velocity profile (unless the fluid shows a yield stress, which is not the case for unfilled polymer melts). Thus, the appearance of completely pluglike velocity profiles must be ubiquitous of yield-stress fluids with slip at the wall before yielding.

1 1+ n  τy  n1   Vmax − vz (r) = r −1 n r0 n+1 k

(8)

Table 1 Yield-stress values obtained from different methods Technique

Yield-stress value (Pa)

Two lines method Intersection of functions describing regions I and II Vane rheometry PIV profiles (first derivative) Fitting to Herschel–Bulkley model

31.87 ± 0.98 21.62 ± 5.96 20.63 ± 0.98 20.45 ± 0.13 19.82 ± 0.24

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Conclusions The yielding and flow behavior of a model yield-stress fluid, 0.2 wt% Carbopol gel, in a capillary with slip at the wall was investigated in this work by Rheo-PIV. The flow behavior consists of a purely pluglike flow before yielding, followed by the solid–liquid transition and shear thinning at relatively high shear rates, respectively. There is a smooth solid–liquid transition in the capillary flow curve, which makes the determination of the yield stress from rheometrical data difficult. However, the velocity profiles allow for the reliable determination of the yield stress. Finally, the slip velocity was found to increase in a power-law way with the shear stress; meanwhile, the flow curve and the velocity profiles free of slip were well fitted by the Herschel– Bulkley model. Acknowledgements This research was supported by SIP-IPN (No. Reg. 20121261). J. J. L-D had a PIFI-IPN scholarship to perform this work, and J. P. -G, B. M. M. -S and F. R-G are COFAA-EDI fellows.

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