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Ultrasonic rheological studies of a body lotion
Flow Measurement and Instrumentation 15 (2004) 139–144 www.elsevier.com/locate/flowmeasinst
Ultrasonic rheological studies of a body lotion Peter O. Brunn a, Thomas Wunderlich a,, Markus Mu¨ller b a
Universita¨t Erlangen-Nu¨rnberg, Lehrstuhl fu¨r Stro¨mungsmechanik, Cauerstraße 4, D-91058 Erlangen, Germany b PHYSICA Messtechnik GmbH, Vor dem Lauch 6, D-70567 Stuttgart, Germany
Abstract To demonstrate the advantage of the gradient ultrasound pulse Doppler method in laminar tube flow, a rather complex body lotion is tested. The results reveal that the fluid shows Newtonian behaviour up to some critical shear rate c_ c . For c_ > c_ c , a sudden phase separation can be detected: a Newtonian one close to the axis, followed by one in which the shear rate stays constant. Accompanying is this stress banding by slip. Customary means to account for this type of slip do not exist. # 2004 Published by Elsevier Ltd. Keywords: Gradient ultrasound pulse Doppler method (GUPD); Body lotion; Slip; Shear viscosity; Stress banding; Phase separation
1. Introduction Ultrasound is one possibility to determine rheologically relevant material properties by non-intrusive means [1–5]. It can be used in materials ranging from gases to solid bodies. Analysing the propagation of waves requires a distinction as to the nature of waves: are they Rayleigh waves, transversal (shear) waves or longitudinal waves? Rayleigh (or surface) waves require a free surface and they penetrate only a short distance into the material. They have thus far played no role in determining rheological material parameters. Shear waves allow one (in principle) to determine the complex modulus G while longitudinal waves would, in addition, provide information about the complex bulk modulus K, too, though K appears only in the combination K þ4=3G [6]. One can use continuous waves and measure the attenuation and the wave velocity [4] or use resonance methods [7], or one can use pulse methods, where usually the complex impedance is determined [8]. Ultrasound can also be used to measure the velocity of fluids. The article by Fiedler [9] is an excellent review of the state of the art till 1988. Replacing in tube flow of liquids the pointwise measurements Corresponding author. Tel.: +49-9131-8529474; fax: +49-91318529503. E-mail address: [email protected] (T. Wunderlich).
0955-5986/$ - see front matter # 2004 Published by Elsevier Ltd. doi:10.1016/j.flowmeasinst.2003.12.007
by (almost) continuous ones, thereby obtaining the complete velocity field, is rather new [10–22]. Here, the transmitter employs the pulsed mode and the Doppler effect (via back scattering) is utilized by the receiver (the ultrasound pulse Doppler (UPD) method). Although the flow can be turbulent, the rheologically relevant parameter, namely the shear viscosity, can be extracted only if laminar flow prevails. For this, flow measurements of the pressure drop Dp over a length L furnish the wall shear stress sw. The local shear stress at r (the distance from the axis) is then given by sðrÞ ¼
r r Dp sw ¼ R 2 L
ð1Þ
Approximating the measured data for the velocity by a continuous curve allows one to calculate the shear rate c_ , and subsequently the shear viscosity g ¼ s=_c
ð2Þ
Although the method has been employed successfully [23], there are two disturbing points. Firstly, to approximate experimental data by a continuous curve is, in general, rather time consuming and need not necessarily be unique [24]. Secondly, experimental data are always noisy to some extent. The UPD method uses pulse repetitions and calculates the average velocity hui which results from those repetitions at which a signal (echo scattered from a moving ‘‘particle’’) is received. This, then is used as input in an expression
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for the (average) shear rate, huðr þ DrÞiN2 huðrÞiN1 Dr
ð3Þ
If N is the total number of repetitions, then N1 ( N) is the number of repetitions at which a particle is found at r, while N2 ( N) is the number of repetitions at which a signal is received from r þ Dr. In general1, N1 6¼ N 2 , i.e. c_ is calculated from uncorrelated data. These deficiencies are overcome in the gradient ultrasound pulse Doppler (GUPD) method [22,25,26]. Here, one records signals only if they are (ideally simultaneously) received from r and from rþDr. If this happens N3 ( N) times, then huðr þ DrÞ uðrÞiN3 ð4Þ c_ ¼ Dr Thus, shear rates (and consequently by Eq. (2), shear viscosities) are measured directly. To demonstrate the advantage of this approach for a rather complex system is the subject of this study.
2. Experiments The liquid used is a body lotion, ‘‘TODAY Badedusch’’, of the company DALLI-Werke. Other body lotions given to us behaved similarly. No details about its composition were provided. What we do know is that such products are concentrated surfactant solutions containing, besides the surfactant, one or more cosurfactants, stabilizers, emulsifiers, conserving agents, perfumes and other ingredients (e.g. [27]). The viscometric measurements were performed in the HAAKE RV20-CV100 viscometer (CR-type) using the measuring device ME 15. This is a concentric cylinder device of the Mooney Ewart type with the dimensions: (stationary) bob diameter 13.91 mm, gap width 0.545 mm and bob length 12 mm. The UPD method utilizes the frequency shift a sound wave experiences if scattered from a moving particle. This frequency shift is directly proportional to the velocity of the particle. If, as in laminar tube flow, the flow direction is known, the sign of the frequency shift will always be the same. From the time span between emitting the pulse and receiving the echo, the position of the particle can be determined, provided the speed v of sound cs is known. In the body lotion at 22 C, we 1 found cs ¼ 1580 m s . Since the echoes are received continuously at definite time windows of about 0.5 ls, a complete flow field can be obtained. Further details 1
wall.
about the UPD method can be found in the relevant literature, e.g. [10–22]. The GUPD method is a variant of the UPD method. It employs an analogous trick by which laser Doppler velocimetry is changed to gradient laser Doppler velocimetry [18,28,29]. In the GUPD method, the shear rate c_ is determined by Eq. (4). To accomplish this via UPD, a second pulse is emitted and the echoes are recorded at time intervals shifted by dt (in our case, dt¼ 250 ns) relative to the first ones. This time shift is equivalent to a positional shift (Dr / dt). Thus, the shear rate distribution within the tube is known. This procedure is repeated 512 times to obtain the time average shear rate distribution (requiring about 2 min). Details of this method have been reported elsewhere [22,25,26]. The ultrasound device used is the 15 mW DOP 1000 (Co. SIGNAL Processing), operating at a pulse frequency of 4 MHz. The PVC tube (length 3 m) had an inner diameter (2R) of 16.6 mm (outer diameter 20 mm). Shear stresses s were determined by Eq. (1) by measuring the pressure drop Dp over a length L of 1 m, with the first pressure hole (diameter 0.2 mm) 1 m from the tube entrance. All ultrasonic measurements were v performed at T¼ 22 C. The closed flow loop is sketched in Fig. 1. It should be noted that the volume flow rate V_ can be obtained from the frequency f of the gear pump via calibration. This calibration curve is fluid specific, i.e. it has to be determined for the fluid used (although quite generally high frequencies imply high volume flow rates). Since in our case, no quantitative use of V_ is needed we refrained from determining the f V_ relation. Results will be reported for a given gear pump frequency rather than a given volume flow rate.
The ratio N 1 =N 2 will differ from 1 rather drastically close to the Fig. 1. The flow loop.
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3. Results The apparent shear viscosity ga (determined in the viscometer) as a function of apparent shear rate c_ a is shown in Fig. 2 with temperature T as parameter. Here, c_ a is the (ideal) shear rate a Newtonian fluid would have at the inner cylinder, 2j2 X ð5Þ j2 1 where X is the magnitude of the angular velocity of one cylinder relative to the other and j is the ratio of outer (R0) to inner (Ri) radius of the two cylinders (in our case, j¼ 1:078). c_ a ¼
R2 ð6Þ Ri Since the torque M is directly proportional to the shear stress, ga is given by j¼
M ð7Þ ga ¼ C X Here, C is a geometric constant, obtained by calibration with Newtonian fluids. Ideally, i.e. by neglecting any end effects, C¼
j2 1 4phR20
ð8Þ
corresponding to a shear stress si at the inner cylinder of si ¼
M 2phR2i
ð9Þ
where h is the length of the cylinder. Fig. 2 shows the apparent flow curve of our body lotion at various temperatures. It hints a pseudoplastic behaviour, especially at lower temperatures. The temperature range chosen is exactly that range which— according to the manufacturer—is encountered during production of the body lotion. While ga is essentially constant for low apparent shear rates (especially at high temperatures), pronounced shear thinning is detected at low temperatures.
Fig. 2. The apparent shear viscosity ga as a function of apparent shear rate c_ a , with temperature T as parameter.
Fig. 3. The apparent shear viscosity ga as a function of temperature T with c_ a as parameter.
Based on these results, the relation ga ¼ ga ðTÞ was obtained with c_ a as parameter (Fig. 3). Quite clearly, ga first increases with T before it starts to decrease. The temperature anomaly (ga increases with increasing T) occurs only when ga shows pseudoplastic behaviour. It is most pronounced when the power law index n (the slope of the logsi log_ca curve within the power law region) is very small (n 0). Below a certain shear rate, there is no anomaly and the same is true above a certain temperature. As long as ga stays essentially constant, the viscosity decreases with increasing temperature2. According to Fig. 2, the slope of log ga versus log c_ a approaches 1 if the rotation rate is high enough (implying constant shear stress). While shear banding could be the reason for this, another possibility could be slip. However, for constant s, customary methods to account for slip3 are not applicable. Application of such methods for another body lotion has indeed been totally unsuccessful [22]. To see what happens locally in tube flow, the GUPD method was employed. Fig. 4 shows a typical result4. It 2 For concentrated model surfactants of known composition with well defined phase behaviour, a maximum of the viscosity at a particular temperature has been reported. It is attributed to changes in the structural organisation of the surfactant molecules, e.g. a micelle-tolamella-to-reverse-micelle transition upon heating. 3 Here, one assumes that the slip velocity is a unique function of the wall shear stress. 4 The influence of the wall results from the finite size of the measuring volume. In the UPD method, the measured velocity value is always assigned to the centre of that volume. As long as this lies entirely within the fluid (in our case, for r < 6:3 mm), this seems legitimate. Yet, as soon as the measuring volume touches or intersects the tube wall, Doppler shifted echoes can be received only from that part of the volume which protrudes into the fluid (termed Vf). As far as velocity profiles are concerned (UPD method), it is straightforward to calculate the centre of Vf and to associate the data points to that centre. For Newtonian fluids and for polymer solutions, this furnishes rather good results [26,34]. Results for velocity profiles will be reported with this correction.
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becomes apparent from this observation that at low volume flow rates (small f), the shear rate c_ increases monotonically with increasing distance from the tube axis. Since this increase is essentially linear, the fluid shows Newtonian behaviour. This is to be contrasted with the behaviour at high volume flow rates (large f). Here, only shear rates close to the axis increase linearly with r. But from some critical distance rc—depending upon the volume flow rate V_ —on, a constant shear rate is encountered. Replotting the high f data (f ¼ 40 Hz) as s ¼ sð_cÞ (Fig. 5) shows this stress banding behaviour quite clearly. It can also be seen from that figure that the viscometric results si ¼ si ð_ca Þ can only be used as long as this relation is linear (Newtonian fluid). In this connection, it is interesting to note that dilute surfactant solutions (with wormlike micelles) also show stress banding. Here, one anticipates that the solution changes from a situation in which the micelles act individually into one in which they act collectively—the shear induced state—SIS [30–32]. The transition occurs
at a critical shear rate c_ c , which is rather specific for each surfactant solution, although it does depend upon the geometric details and dimensions of the measuring device as well as upon the time needed to obtain a flow curve [29,33]. For the concentrated surfactant solution studied here, this ‘‘explanation’’ cannot be used, since there will never be any action of individual ‘‘particles’’. All we can say is that some (unknown) micro-structural change has to occur. Another difference from the dilute case is the critical shear rate c_ c . This is not constant but rather depends upon the volume flow rate. For the V_ range studied, it increases from a value of about 10 s 1 (for f ¼ 20 Hz) to a value of nearly 20 s 1 (for f ¼ 45 Hz). The consequence of these results for the velocity profiles (which we have measured too) is as follows. For low volume flow rates (in our case, up to f ¼ 15 Hz), Newtonian behaviour is encountered, and the velocity profiles will be parabolic. At higher volume flow rates, this is still true close to the axis. This region is joined by a stress banding region, i.e. a region with constant shear rate, where the velocity varies linearly with r. This stress banding region increases with increasing f. Fig. 6 shows these findings. Also shown in this figure is some oddity which we observed as soon as stress banding occurs, namely extremely large velocities within a wall layer, whose thickness d seems to be independent of the volume flow rate. To account for this observation would require that some additional force (besides s) acts close to the wall. We know of no physical basis for such a force. Thus, we shall discard these wall layer data and regard them as artefacts of the UPD apparatus, the more so since d is of the order of one wave length of the ultrasound ( 0.4 mm). For completeness, we should point out that in the 8 years we have been working with ultrasound (two Ph.D. theses: [21,22]) we have never observed such an oddity, not with Newtonian fluids, not with polymer solutions
Fig. 5. The flow curve s ¼ sð_cÞ for f ¼ 40 Hz (A) in comparison to the apparent flow curve si ¼ si ð_ca Þ (o).
Fig. 6. The velocity profile as a function of distance from the axis for various flow rates.
Fig. 4. rates.
Shear rate distribution in tube flow for different volume flow
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and not with foodstuffs. Maybe stress banding causes reverberations of the ultrasonic pulse in a thin layer close to the wall that distorts the signal. Discarding these wall layer data and extrapolating the linear velocity profile up to the wall furnishes a slip velocity us. As Fig. 7 shows, us can rather well be approximated by the relation 0; for s < sc ð10Þ us ¼ aðs sc Þ; for s > sc with sc 25 Pa and a 1:54 10 3 m2 s=kg. In analogy to the dilute surfactant solution case, one expects quantitative differences of these results if different geometries are used. It is instructive, however, to speculate about apparent flow curves in circular Couette flow, if the capillary flow results carry over qualitatively. Anticipating this to be the case, let us assume that the fluid is entirely within the stress banding region, i.e. si =j2 > s1 , si < s2 (see Fig. 8). This being the case, the shear rate has to be constant across the gap, c_ ¼ c_ c
for Ri r jRi
ð11Þ
A logarithmically varying angular velocity x will be the consequence: x ¼ c_ c logr þ c
c_ a ¼ a þ bsi
ð13Þ
Eqs. (13) and (14) imply that the apparent shear rate as
The slip velocity as a function of the wall shear stress.
ð15Þ 5
where a and b are positive constants . This is in fundamental contrast to the true shear rate, which is constant (Eq. (11)). Solving Eq. (15) for the apparent viscosity ga (¼ si =_ca ) furnishes ga ¼
where xs is the angular slip velocity of the inner (xsi) and outer (xsa) cylinder, respectively. With xs ¼ us =R, we obtain, by recalling the form of Eq. (10), a 1 a 1 1 þ 3 si 1þ ð14Þ sc xsi þ xsa ¼ Ri j Ri j
Fig. 7.
given by Eq. (5) varies with si:
ð12Þ
Eq. (12) implies the identity X ðxsi þ xsa Þ ¼ c_ c logj
Fig. 8. A model of a s c_ relation which would lead to stress banding.
ð1 þ a=_ca Þ b
ð16Þ
This implies that within the range a=_ca 41, the concept of stress banding as directly observed in capillary flow would help to understand the apparent flow curves in circular Couette flow ðga / c_ 1 a Þ.
4. Summary and conclusion In this ,paper rheological results for a body lotion have been reported. Utilising the GUPD device in tube flow reveals that the fluid shows Newtonian behaviour when the flow rate is low enough. At higher volume flow rates, stress banding occurs as soon as some critical shear rate is exceeded, quite in accordance to what happens for dilute wormlike micellar solutions. In contrast to these latter solutions, c_ c is not constant but rather changes with the volume flow rate. Without knowing the details of the composition of our concentrated multicomponent fluid, it is not possible to list reasons for this micro-structural change which manifests itself in the phase separation observed. If the fluid’s rheology were to admit a non-uniqueness of the s c_ relation in terms of an S-shaped form (see Fig. 8), then the phase separation would correspond to a jump 5
The data for capillary flow hint at a > 0.
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at constant shear rate c_ c from the lower branch of the S to the upper branch (the range ds=d_c < 0 is unstable). Though totally speculative it would—anticipating that the range from c_ 1 to c_ 2 is unstable—allow a range of critical shear rates c_ c (within the limits c_ 1 c_ c c_ 2 ) at which stress banding can occur, and this is what we have observed. Finally, it should be noted that it is examples like the one studied that show the advantage of directly measuring velocity gradients. Quite clearly, had we tried to start from measured velocity profiles, and approximate these via a continuous curve (usually a polynomial fit), we would have failed to detect a linear region. Since the device is non-intrusive and easy to operate, it is hoped that it will gain acceptance in rheological studies. That it can be used online should prove useful for industrial practice.
References [1] H. Kolsky, Stress Waves in Solids, Dover Publication, 1963. [2] W.P. Mason, Physical Acoustics and the Properties of Solids, van Nostrand, 1958. [3] H.J. Mc Shiumin, in: W.P. Mason (Ed.), Physical Acoustics, vol. Ia Academic Press, 1964, pp. 271–334. [4] G. Harrison, The Dynamic Properties of Supercooled Liquid, Academic Press, 1976. [5] T.D. Ferry, Viscoelastic Properties of Polymers, second ed., Wiley, 1970. [6] R. Kono, The dynamic bulk viscosity of polystyrene and polymethyl methacrylate, J. Phys. Soc. Japan 15 (1960) 718. [7] J.L. Hunter, F.E. Fox, J. Acoust. Soc. Amer. 22 (1950) 238–243. [8] J. Laub, P. Linden, J. Acoust. Soc. Amer. 41 (1967) 1032. [9] O. Fiedler, in: W. Kraak, G. Schommartz (Eds.), Akustische Stro¨mungsmeßtechnik in angewandte Akustik 2, VEB Verlag Technik, Berlin, 1988. [10] P.A. Peronneau, F. Leger, Doppler ultrasonic pulse blood flowmeter, Proceedings of the Eighth International Conference on Medical and Biological Engineering, IEEE, Chicago, 1969. [11] D.W. Backer, Pulsed ultrasonic blood flow sensing, IEEE Trans. Sonics Ultrasound 17 (1970) 170. [12] J.J. Meister, These No. 84, Ecole Polytechnique Federale de Lausanne, Switzerland, 1983. [13] A. Hambach, Untersuchungen nicht-Newtonscher Fluide mit einem Ultraschall Geschwindigkeits Monitor, Diplomarbeit, Univ. Erlangen, 1993. [14] Y. Takeda, Velocity profile measurements by ultrasound Doppler shifted method, Int. J. Heat Fluid Flow 7 (1986) 313. [15] Y. Takeda, Measurement of velocity profile of mercury flow by ultrasound Doppler shift method, Nucl. Technol. 79 (1987) 120.
[16] Y. Takeda, Development of an ultrasound velocity profile monitor, Nucl. Eng. Design 126 (1991) 277. [17] Y. Takeda, E.W. Fischer, S. Sakakibara, Decomposition of the modulated waves in a rotating Couette System, Science 263 (1994) 502. [18] P.O. Brunn, J. Vorwerk, R. Steger, Optical and acoustic rheometers: three examples, Rheology 3 (1993) 20. [19] P.O. Brunn, J. Vorwerk, R. Steger, Ein akustisches Verfahren in der Rheometrie sowie zur online Qualita¨tskontrolle, Bericht u¨ber die 4. Tagung fu¨r LebensmittelrheologieGranum Verlag, Detmold, 1994, pp. 143. [20] R. Steger, J. Vorwerk, P.O. Brunn, Geschwindigkeitsmessungen mit einem online-Ultraschallsensor zur kontinuierlichen Prozesskontrolle komplexer Fluide, Chem. Ing. Tech. 65 (1993) 1087. [21] R. Steger, Optische und akustische Methoden in der Rheometrie, Diss., Univ. Erlangen, 1994. [22] M. Mu¨ller, Neue Methoden in der Prozeßviskosimetrie: Kugelschu¨ttungsstro¨mung und Gradienten Ultraschall Puls Doppler Verfahren, Diss., Univ. Erlangen, 1998. [23] T. Wunderlich, P.O. Brunn, Ultrasound pulse Doppler method as a viscometer for process monitoring, Flow Meas. Instrum. 10 (1999) 201–205. [24] J. Honerkamp, Ill posed problems in rheology, Rheol. Acta 28 (1989) 363. [25] M. Mu¨ller, P.O. Brunn, T. Wunderlich, New rheometric technique: the gradient-ultrasound pulse Doppler method, Appl. Rheol. 7 (1997) 204. [26] T. Wunderlich, Direkte Ermittlung des Geschwindigkeitsgradienten mit dem Ultraschall Puls Doppler Verfahren zur Ermittlung der Fließfunktion von Newtonschen und nichtNewtonschen Flu¨ssigkeiten, Diplomarbeit, Univ. Erlangen, 1996. [27] H. Stache, Die Tenside, Carl Hauser Verlag, Mu¨nchen, 1993. [28] P.O. Brunn, F. Durst, A. Wunderlich, Direct velocity gradient measurements using a dual point laser-Doppler anemometer, J. Non-Newtonian Fluid Mech. 22 (1987) 381. [29] A. Wunderlich, P.O. Brunn, The complex rheological behaviour of an aqueous surfactant solution investigated in a Couette type viscometer, Colloid Polym. Sci. 267 (1989) 627. [30] S. Hofmann, A. Rauscher, H. Hoffmann, Shear induced micellar structures, Ber. Bunsenges. Phys. Chem. 95 (1991) 153. [31] D. Ohlendorf, W. Interthal, H. Hoffmann, Surfactant systems for drag reduction: physico-chemical properties and rheological behaviour, Rheol. Acta 25 (1986) 468. [32] H. Rehage, I. Wunderlich, H. Hoffmann, Shear induced phase transition in dilute aqueous surfactant solutions, Progr. Colloid Polym. Sci. 72 (1986) 51. [33] H. Rehage, H. Hoffmann, Shear induced phase transition in highly dilute aqueous detergent solutions, Rheol. Acta 21 (1982) 561. [34] T. Wunderlich, P.O. Brunn, A wall layer correction for ultrasound measurements in tube flow: comparison between theory and experiments, Flow Meas. Instrum. 11 (2000) 63–69.
Ultrasonic rheological studies of a body lotion Peter O. Brunn a, Thomas Wunderlich a,, Markus Mu¨ller b a
Universita¨t Erlangen-Nu¨rnberg, Lehrstuhl fu¨r Stro¨mungsmechanik, Cauerstraße 4, D-91058 Erlangen, Germany b PHYSICA Messtechnik GmbH, Vor dem Lauch 6, D-70567 Stuttgart, Germany
Abstract To demonstrate the advantage of the gradient ultrasound pulse Doppler method in laminar tube flow, a rather complex body lotion is tested. The results reveal that the fluid shows Newtonian behaviour up to some critical shear rate c_ c . For c_ > c_ c , a sudden phase separation can be detected: a Newtonian one close to the axis, followed by one in which the shear rate stays constant. Accompanying is this stress banding by slip. Customary means to account for this type of slip do not exist. # 2004 Published by Elsevier Ltd. Keywords: Gradient ultrasound pulse Doppler method (GUPD); Body lotion; Slip; Shear viscosity; Stress banding; Phase separation
1. Introduction Ultrasound is one possibility to determine rheologically relevant material properties by non-intrusive means [1–5]. It can be used in materials ranging from gases to solid bodies. Analysing the propagation of waves requires a distinction as to the nature of waves: are they Rayleigh waves, transversal (shear) waves or longitudinal waves? Rayleigh (or surface) waves require a free surface and they penetrate only a short distance into the material. They have thus far played no role in determining rheological material parameters. Shear waves allow one (in principle) to determine the complex modulus G while longitudinal waves would, in addition, provide information about the complex bulk modulus K, too, though K appears only in the combination K þ4=3G [6]. One can use continuous waves and measure the attenuation and the wave velocity [4] or use resonance methods [7], or one can use pulse methods, where usually the complex impedance is determined [8]. Ultrasound can also be used to measure the velocity of fluids. The article by Fiedler [9] is an excellent review of the state of the art till 1988. Replacing in tube flow of liquids the pointwise measurements Corresponding author. Tel.: +49-9131-8529474; fax: +49-91318529503. E-mail address: [email protected] (T. Wunderlich).
0955-5986/$ - see front matter # 2004 Published by Elsevier Ltd. doi:10.1016/j.flowmeasinst.2003.12.007
by (almost) continuous ones, thereby obtaining the complete velocity field, is rather new [10–22]. Here, the transmitter employs the pulsed mode and the Doppler effect (via back scattering) is utilized by the receiver (the ultrasound pulse Doppler (UPD) method). Although the flow can be turbulent, the rheologically relevant parameter, namely the shear viscosity, can be extracted only if laminar flow prevails. For this, flow measurements of the pressure drop Dp over a length L furnish the wall shear stress sw. The local shear stress at r (the distance from the axis) is then given by sðrÞ ¼
r r Dp sw ¼ R 2 L
ð1Þ
Approximating the measured data for the velocity by a continuous curve allows one to calculate the shear rate c_ , and subsequently the shear viscosity g ¼ s=_c
ð2Þ
Although the method has been employed successfully [23], there are two disturbing points. Firstly, to approximate experimental data by a continuous curve is, in general, rather time consuming and need not necessarily be unique [24]. Secondly, experimental data are always noisy to some extent. The UPD method uses pulse repetitions and calculates the average velocity hui which results from those repetitions at which a signal (echo scattered from a moving ‘‘particle’’) is received. This, then is used as input in an expression
140
P.O. Brunn et al. / Flow Measurement and Instrumentation 15 (2004) 139–144
for the (average) shear rate, huðr þ DrÞiN2 huðrÞiN1 Dr
ð3Þ
If N is the total number of repetitions, then N1 ( N) is the number of repetitions at which a particle is found at r, while N2 ( N) is the number of repetitions at which a signal is received from r þ Dr. In general1, N1 6¼ N 2 , i.e. c_ is calculated from uncorrelated data. These deficiencies are overcome in the gradient ultrasound pulse Doppler (GUPD) method [22,25,26]. Here, one records signals only if they are (ideally simultaneously) received from r and from rþDr. If this happens N3 ( N) times, then huðr þ DrÞ uðrÞiN3 ð4Þ c_ ¼ Dr Thus, shear rates (and consequently by Eq. (2), shear viscosities) are measured directly. To demonstrate the advantage of this approach for a rather complex system is the subject of this study.
2. Experiments The liquid used is a body lotion, ‘‘TODAY Badedusch’’, of the company DALLI-Werke. Other body lotions given to us behaved similarly. No details about its composition were provided. What we do know is that such products are concentrated surfactant solutions containing, besides the surfactant, one or more cosurfactants, stabilizers, emulsifiers, conserving agents, perfumes and other ingredients (e.g. [27]). The viscometric measurements were performed in the HAAKE RV20-CV100 viscometer (CR-type) using the measuring device ME 15. This is a concentric cylinder device of the Mooney Ewart type with the dimensions: (stationary) bob diameter 13.91 mm, gap width 0.545 mm and bob length 12 mm. The UPD method utilizes the frequency shift a sound wave experiences if scattered from a moving particle. This frequency shift is directly proportional to the velocity of the particle. If, as in laminar tube flow, the flow direction is known, the sign of the frequency shift will always be the same. From the time span between emitting the pulse and receiving the echo, the position of the particle can be determined, provided the speed v of sound cs is known. In the body lotion at 22 C, we 1 found cs ¼ 1580 m s . Since the echoes are received continuously at definite time windows of about 0.5 ls, a complete flow field can be obtained. Further details 1
wall.
about the UPD method can be found in the relevant literature, e.g. [10–22]. The GUPD method is a variant of the UPD method. It employs an analogous trick by which laser Doppler velocimetry is changed to gradient laser Doppler velocimetry [18,28,29]. In the GUPD method, the shear rate c_ is determined by Eq. (4). To accomplish this via UPD, a second pulse is emitted and the echoes are recorded at time intervals shifted by dt (in our case, dt¼ 250 ns) relative to the first ones. This time shift is equivalent to a positional shift (Dr / dt). Thus, the shear rate distribution within the tube is known. This procedure is repeated 512 times to obtain the time average shear rate distribution (requiring about 2 min). Details of this method have been reported elsewhere [22,25,26]. The ultrasound device used is the 15 mW DOP 1000 (Co. SIGNAL Processing), operating at a pulse frequency of 4 MHz. The PVC tube (length 3 m) had an inner diameter (2R) of 16.6 mm (outer diameter 20 mm). Shear stresses s were determined by Eq. (1) by measuring the pressure drop Dp over a length L of 1 m, with the first pressure hole (diameter 0.2 mm) 1 m from the tube entrance. All ultrasonic measurements were v performed at T¼ 22 C. The closed flow loop is sketched in Fig. 1. It should be noted that the volume flow rate V_ can be obtained from the frequency f of the gear pump via calibration. This calibration curve is fluid specific, i.e. it has to be determined for the fluid used (although quite generally high frequencies imply high volume flow rates). Since in our case, no quantitative use of V_ is needed we refrained from determining the f V_ relation. Results will be reported for a given gear pump frequency rather than a given volume flow rate.
The ratio N 1 =N 2 will differ from 1 rather drastically close to the Fig. 1. The flow loop.
P.O. Brunn et al. / Flow Measurement and Instrumentation 15 (2004) 139–144
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3. Results The apparent shear viscosity ga (determined in the viscometer) as a function of apparent shear rate c_ a is shown in Fig. 2 with temperature T as parameter. Here, c_ a is the (ideal) shear rate a Newtonian fluid would have at the inner cylinder, 2j2 X ð5Þ j2 1 where X is the magnitude of the angular velocity of one cylinder relative to the other and j is the ratio of outer (R0) to inner (Ri) radius of the two cylinders (in our case, j¼ 1:078). c_ a ¼
R2 ð6Þ Ri Since the torque M is directly proportional to the shear stress, ga is given by j¼
M ð7Þ ga ¼ C X Here, C is a geometric constant, obtained by calibration with Newtonian fluids. Ideally, i.e. by neglecting any end effects, C¼
j2 1 4phR20
ð8Þ
corresponding to a shear stress si at the inner cylinder of si ¼
M 2phR2i
ð9Þ
where h is the length of the cylinder. Fig. 2 shows the apparent flow curve of our body lotion at various temperatures. It hints a pseudoplastic behaviour, especially at lower temperatures. The temperature range chosen is exactly that range which— according to the manufacturer—is encountered during production of the body lotion. While ga is essentially constant for low apparent shear rates (especially at high temperatures), pronounced shear thinning is detected at low temperatures.
Fig. 2. The apparent shear viscosity ga as a function of apparent shear rate c_ a , with temperature T as parameter.
Fig. 3. The apparent shear viscosity ga as a function of temperature T with c_ a as parameter.
Based on these results, the relation ga ¼ ga ðTÞ was obtained with c_ a as parameter (Fig. 3). Quite clearly, ga first increases with T before it starts to decrease. The temperature anomaly (ga increases with increasing T) occurs only when ga shows pseudoplastic behaviour. It is most pronounced when the power law index n (the slope of the logsi log_ca curve within the power law region) is very small (n 0). Below a certain shear rate, there is no anomaly and the same is true above a certain temperature. As long as ga stays essentially constant, the viscosity decreases with increasing temperature2. According to Fig. 2, the slope of log ga versus log c_ a approaches 1 if the rotation rate is high enough (implying constant shear stress). While shear banding could be the reason for this, another possibility could be slip. However, for constant s, customary methods to account for slip3 are not applicable. Application of such methods for another body lotion has indeed been totally unsuccessful [22]. To see what happens locally in tube flow, the GUPD method was employed. Fig. 4 shows a typical result4. It 2 For concentrated model surfactants of known composition with well defined phase behaviour, a maximum of the viscosity at a particular temperature has been reported. It is attributed to changes in the structural organisation of the surfactant molecules, e.g. a micelle-tolamella-to-reverse-micelle transition upon heating. 3 Here, one assumes that the slip velocity is a unique function of the wall shear stress. 4 The influence of the wall results from the finite size of the measuring volume. In the UPD method, the measured velocity value is always assigned to the centre of that volume. As long as this lies entirely within the fluid (in our case, for r < 6:3 mm), this seems legitimate. Yet, as soon as the measuring volume touches or intersects the tube wall, Doppler shifted echoes can be received only from that part of the volume which protrudes into the fluid (termed Vf). As far as velocity profiles are concerned (UPD method), it is straightforward to calculate the centre of Vf and to associate the data points to that centre. For Newtonian fluids and for polymer solutions, this furnishes rather good results [26,34]. Results for velocity profiles will be reported with this correction.
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becomes apparent from this observation that at low volume flow rates (small f), the shear rate c_ increases monotonically with increasing distance from the tube axis. Since this increase is essentially linear, the fluid shows Newtonian behaviour. This is to be contrasted with the behaviour at high volume flow rates (large f). Here, only shear rates close to the axis increase linearly with r. But from some critical distance rc—depending upon the volume flow rate V_ —on, a constant shear rate is encountered. Replotting the high f data (f ¼ 40 Hz) as s ¼ sð_cÞ (Fig. 5) shows this stress banding behaviour quite clearly. It can also be seen from that figure that the viscometric results si ¼ si ð_ca Þ can only be used as long as this relation is linear (Newtonian fluid). In this connection, it is interesting to note that dilute surfactant solutions (with wormlike micelles) also show stress banding. Here, one anticipates that the solution changes from a situation in which the micelles act individually into one in which they act collectively—the shear induced state—SIS [30–32]. The transition occurs
at a critical shear rate c_ c , which is rather specific for each surfactant solution, although it does depend upon the geometric details and dimensions of the measuring device as well as upon the time needed to obtain a flow curve [29,33]. For the concentrated surfactant solution studied here, this ‘‘explanation’’ cannot be used, since there will never be any action of individual ‘‘particles’’. All we can say is that some (unknown) micro-structural change has to occur. Another difference from the dilute case is the critical shear rate c_ c . This is not constant but rather depends upon the volume flow rate. For the V_ range studied, it increases from a value of about 10 s 1 (for f ¼ 20 Hz) to a value of nearly 20 s 1 (for f ¼ 45 Hz). The consequence of these results for the velocity profiles (which we have measured too) is as follows. For low volume flow rates (in our case, up to f ¼ 15 Hz), Newtonian behaviour is encountered, and the velocity profiles will be parabolic. At higher volume flow rates, this is still true close to the axis. This region is joined by a stress banding region, i.e. a region with constant shear rate, where the velocity varies linearly with r. This stress banding region increases with increasing f. Fig. 6 shows these findings. Also shown in this figure is some oddity which we observed as soon as stress banding occurs, namely extremely large velocities within a wall layer, whose thickness d seems to be independent of the volume flow rate. To account for this observation would require that some additional force (besides s) acts close to the wall. We know of no physical basis for such a force. Thus, we shall discard these wall layer data and regard them as artefacts of the UPD apparatus, the more so since d is of the order of one wave length of the ultrasound ( 0.4 mm). For completeness, we should point out that in the 8 years we have been working with ultrasound (two Ph.D. theses: [21,22]) we have never observed such an oddity, not with Newtonian fluids, not with polymer solutions
Fig. 5. The flow curve s ¼ sð_cÞ for f ¼ 40 Hz (A) in comparison to the apparent flow curve si ¼ si ð_ca Þ (o).
Fig. 6. The velocity profile as a function of distance from the axis for various flow rates.
Fig. 4. rates.
Shear rate distribution in tube flow for different volume flow
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and not with foodstuffs. Maybe stress banding causes reverberations of the ultrasonic pulse in a thin layer close to the wall that distorts the signal. Discarding these wall layer data and extrapolating the linear velocity profile up to the wall furnishes a slip velocity us. As Fig. 7 shows, us can rather well be approximated by the relation 0; for s < sc ð10Þ us ¼ aðs sc Þ; for s > sc with sc 25 Pa and a 1:54 10 3 m2 s=kg. In analogy to the dilute surfactant solution case, one expects quantitative differences of these results if different geometries are used. It is instructive, however, to speculate about apparent flow curves in circular Couette flow, if the capillary flow results carry over qualitatively. Anticipating this to be the case, let us assume that the fluid is entirely within the stress banding region, i.e. si =j2 > s1 , si < s2 (see Fig. 8). This being the case, the shear rate has to be constant across the gap, c_ ¼ c_ c
for Ri r jRi
ð11Þ
A logarithmically varying angular velocity x will be the consequence: x ¼ c_ c logr þ c
c_ a ¼ a þ bsi
ð13Þ
Eqs. (13) and (14) imply that the apparent shear rate as
The slip velocity as a function of the wall shear stress.
ð15Þ 5
where a and b are positive constants . This is in fundamental contrast to the true shear rate, which is constant (Eq. (11)). Solving Eq. (15) for the apparent viscosity ga (¼ si =_ca ) furnishes ga ¼
where xs is the angular slip velocity of the inner (xsi) and outer (xsa) cylinder, respectively. With xs ¼ us =R, we obtain, by recalling the form of Eq. (10), a 1 a 1 1 þ 3 si 1þ ð14Þ sc xsi þ xsa ¼ Ri j Ri j
Fig. 7.
given by Eq. (5) varies with si:
ð12Þ
Eq. (12) implies the identity X ðxsi þ xsa Þ ¼ c_ c logj
Fig. 8. A model of a s c_ relation which would lead to stress banding.
ð1 þ a=_ca Þ b
ð16Þ
This implies that within the range a=_ca 41, the concept of stress banding as directly observed in capillary flow would help to understand the apparent flow curves in circular Couette flow ðga / c_ 1 a Þ.
4. Summary and conclusion In this ,paper rheological results for a body lotion have been reported. Utilising the GUPD device in tube flow reveals that the fluid shows Newtonian behaviour when the flow rate is low enough. At higher volume flow rates, stress banding occurs as soon as some critical shear rate is exceeded, quite in accordance to what happens for dilute wormlike micellar solutions. In contrast to these latter solutions, c_ c is not constant but rather changes with the volume flow rate. Without knowing the details of the composition of our concentrated multicomponent fluid, it is not possible to list reasons for this micro-structural change which manifests itself in the phase separation observed. If the fluid’s rheology were to admit a non-uniqueness of the s c_ relation in terms of an S-shaped form (see Fig. 8), then the phase separation would correspond to a jump 5
The data for capillary flow hint at a > 0.
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at constant shear rate c_ c from the lower branch of the S to the upper branch (the range ds=d_c < 0 is unstable). Though totally speculative it would—anticipating that the range from c_ 1 to c_ 2 is unstable—allow a range of critical shear rates c_ c (within the limits c_ 1 c_ c c_ 2 ) at which stress banding can occur, and this is what we have observed. Finally, it should be noted that it is examples like the one studied that show the advantage of directly measuring velocity gradients. Quite clearly, had we tried to start from measured velocity profiles, and approximate these via a continuous curve (usually a polynomial fit), we would have failed to detect a linear region. Since the device is non-intrusive and easy to operate, it is hoped that it will gain acceptance in rheological studies. That it can be used online should prove useful for industrial practice.
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