Mechanics of Time-Dependent Materials 2: 171–193, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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Time-Dependent and Flow Properties of Foams K. BEKKOUR and O. SCRIVENER Institut de Mécanique des Fluides, UMR 7507 ULP-CNRS, Université Louis Pasteur, 2 Rue Boussingault, 67000 Strasbourg, France (Received 27 January 1997; accepted in revised form 18 November 1998) Abstract. Foams have been prepared from water added with a surfactant and a polymer. A controlled stress rheometer was used to study the changes of their rheological properties during ageing by the mean of different rheological tests: shear viscosity measurements, creep compliance tests at a constant low shear stress and dynamic experiments have been performed. It has been observed that apparent viscosity decreases with ageing. A thixotropic behaviour was also found, loading and unloading curves present an hysteresis. Then, the choice of the stress ascent time is of primary importance to study the time-dependent properties of the foam. The viscosity was found to decrease with the stress ascent time while the thixotropic area decreases. Creep flow and harmonic tests have shown important viscoelastic properties of the foams. The second part of this work is concerned with the study of the flow properties of the foams. Pressure drops were measured during flow in capillary tubes in laminar regime. Pronounced wall slip effects were found. While the Mooney method for slip correction is not applicable, the Oldroyd– Jastrzebski method leads to satisfactory results. The Metzner and Reed correlation method was found to be applicable in the case of the corrected data. Key words: capillary flow, foams, friction factor, Reynolds number, rheology, slip correction methods, slip velocity, thixotropy, viscoelasticity

1. Introduction Foams are in two-phase gas-liquid dispersions with the liquid as the continuous phase and the gas as the dispersed phase. Foam behaviour is of great interest due to its numerous applications in many industrial sectors (chemical, cosmetic, food, fire-fighting technology, etc.). Foams are also widely used in petroleum industry as drilling fluids and in enhanced oil recovery (EOR) processes (Burly and Shakarin, 1992; Heller and Kuntamukkula, 1987). The knowledge of the rheological behaviour of foamed fluids is therefore essential in engineering applications, product formulation, development of measurements techniques or quality control. A literature survey shows that most of the published papers are dealing with the structure of foams, topological models and surface chemistry of the interfacial films. Less papers are concerned with experimental research on rheological properties and flow of this two-phase material (Khan et al., 1988; Princen, 1985; Princen and Kiss, 1986, 1989).

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Figure 1. Foam generator.

The first part of this work is devoted to an experimental study of steady and unsteady shear flow properties and to time-dependent behaviour of foams. In the second part, the flow development of foamed solutions in capillary rheometry is presented. It is shown that a diameter dependence of the flow curves of the studied foams is obtained. Methods developed by Mooney (1931) and Jastrzebski (1967) are applied to correct the data for slip effects. 2. Experiment 2.1. F OAM P REPARATION Foams were prepared with distilled water added with 0.02 g/g of Sodium-DodecylSulfate (SDS) as surfactant and 0, 0.05, 0.1 or 0.3% (wt/wt) Poly-Ethylene-Oxide (PEO) as stabiliser in order to reduce gravitational drainage. PEO was chosen with a low molecular weight so as to prevent non-Newtonian behaviour of the liquid. Thus, any observed non-Newtonian behaviour could be directly attributed to the structure of the two phase fluid. Several polymer concentrations were used and allowed to study the stabilising effect of the polymer. Indeed, the viscosity of the interfacial liquid films increases with concentration while the superficial tension acting on these films remains unchanged. The solution was agitated during 24 hours and then heated at 50◦ C during one hour in order to avoid formation of aggregates. A generator described in Figure 1 was designed to produce the foam. It consists essentially of a pack of porous medium 25 mm in length and 3 cm in diameter into which both air and surfactant solution could be simultaneously injected. The porous medium was made with glass beads of different diameters ranging between 0.8 and 1.4 mm so as to produce foams of different textures. The foaming solution is injected in the generator at a pressure slightly higher than that of air. The foam is generated when the mixture passes through the glass beads pack. The foam produced is very uniform both in flow rate and in bubble size distribution. Different

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Figure 2. Drainage curves of foams at different PEO concentrations.

qualities and different textures can be produced by varying the air and liquid injection pressures and the porous medium granulometry. The produced foams have a very stable structure and are reproducible. Different qualities of the foam (defined as the ratio of the gas volume to the total foam volume) were produced by changing the relative air and foaming solution flow rates. Several sets of experiments were carried out with foam qualities ranging between 0.9 and 0.98. In this paper, only the results concerned with a foam quality of 0.97 are presented.

2.2. F OAM S TABILITY It is of primary importance to analyse the stability of the foam and the factors which could increase it. The foam stability is estimated by measuring the volume of the liquid drained during the foam destabilisation. We used a transparent glass column sealed at its bottom by a porous disk through which the drained liquid can pass. The solution is collected in a graduated test tube allowing to measure the volume of drained liquid and the corresponding flow times. The drainage curves, for several PEO concentrations, are plotted in Figure 2 and the corresponding halflife times (time necessary for drainage of half of the volume of the liquid dispersed in the foam) are listed in Table I. Differences in foam stability are clearly observed. For all PEO concentrations drainage curves present the same trend. They exhibit an ascent linear part followed by an asymptotic value corresponding to the total volume of the liquid present in the foam. It can be observed that the characteristic time t1/2 and, thus, the stability of the foam increases slightly with the concentration

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Table I. Foam characteristics. PEO Con. (%)

0

0.05

0.1

0.3

Viscosity (mPa.s) t1/2 (min.)

1.39 12’30

1.59 12’30

1.86 12’43

2.94 13’58

of PEO. Also, one may observe that the total drained volume is sensitive to the PEO concentration. 2.3. R HEOLOGICAL M EASUREMENTS The rheological measurements have been carried out by means of a controlled stress rheometer (Carri-Med CSL 100) equipped with a plate and cone measuring device (6 cm, 4 degrees). The cone is made from acrylic material and allows us to visualise the evolution of the foam structure during the shearing experiments. Preliminary tests have confirmed the reproducibility of the measurements for shear rates ranging between 0 and 350 s−1 . At higher shear rates the results are less reproducible due to the breakdown of the foam structure. The analysis of our experimental data will, therefore, be restricted to this range of shear rates. All measurements were performed at room temperature (20◦ C). To prevent changes in composition during measurements due to water evaporation a humidification chamber was placed around the cone and plate device. All capillary experiments were carried out at room temperature, using a constant-speed piston driven capillary rheometer which will be described in Section 4. 3. Viscosity Measurement Results 3.1. T IME -D EPENDENCE

OF THE

F OAM R HEOLOGICAL B EHAVIOUR

Loading and unloading experiments have shown that the behaviour of the foam is time-dependent which means that it exhibits a viscosity-time relationship. Among these time-dependent fluids, thixotropic fluids show a viscosity which decreases with the time of shearing, but finding again the initial value after a time of rest (Cheng, 1980). On the opposite, anti-thixotropy shows an increase of viscosity with time of application of shear and is also reversible. These time-dependent behaviours are resulting from changes occurring in the inner structure of the fluid: changes in the sizes of the dispersion, bubble coalescence or breaking, flocculation, etc. The time-dependent behaviour of foams is of a non-reversible kind because of the drainage process and the irreversible breakdown of the inner structure under shearing. A practical consequence for industrial processes is that it is important to

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Figure 3. Flow curves of 0.05 and 0.3% PEO based foam.

known these non-reversible time-effects during the manufacturing and at the end of the process. Several flow tests were performed, in changing the fluid parameters and the experimental procedures. Loading and unloading experiments at constant stress rate, by increasing the applied stress from zero to a prefixed maximum value and thereafter decreasing it from this value to zero, were carried out systematically. In order to take into account the time-dependent rheological properties of the foams, equilibrium flow measurements were carried out by dividing the shear stress ramp into a series of finite steps. For each shear stress applied, the software waits until a corresponding constant stationary state of the shear rate is attained. Figure 3 shows typical flow curves for two PEO concentrations (0.05 and 0.3%). It is noticeable that the viscosity of the foam is much higher, about five times, than the Newtonian viscosity of the PEO-based solution used to prepare foam (respectively 1.59 and 2.94 mPA.s for 0.05 and 0.3% PEO concentrations). It was found that foams behave like a non-Newtonian fluid with a small yield stress (nearly 2 Pa). Princen and Kiss (1989) measured yield-stresses comprised between 5 and 45 Pa with paraffin oil in water emulsions and Khan et al. (1988) obtained yield stress value of 13.5 Pa with polymer-surfactant-based foams having a quality of 0.98. In a review of experimental work on emulsions and foams, Princen (1983) specified that yield stresses of typical emulsions are of the order of a few ten to a few hundred Pa and much smaller for foams and that the yield stress increases with decreasing bubble size. The smaller value of the yield stress obtained in our experiments could be explained by the larger cell size (of order of 200 µm in diameter) compared with

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the cell size of the foams used by Khan et al. (100 µm). The viscosity is shear rate dependent and increases with PEO concentration. A composite behaviour is observed: thixotropic (or more precisely, rheomalaxis) at low shear rates and antithixotropic at high shear rates indicating a degradation of the inner structure below a certain shear rate and its building-up for the higher shear rates. At shear rates lower than 250 and 350 s−1 for respectively 0.3 and 0.05% PEO concentrations, the rheological behaviour is shear thinning for both up and down curves. At higher shear rates a shear thickening behaviour is observed for loading curves while it remains shear thinning for the unloading curves. Visualisation of the samples during the measurements allows us to detect structural changes related to the shear conditions. The characterisation of the microstructure was carried out by use of video image analysis (Bekkour et al., 1994). It was observed that the rheological modifications described above are related to structural changes of foam. The different regions observed in the loading curves suggest the following sequence of events: − a yield stress region where bubbles are deformed under the influence of the shear stress before the appearance of the movement (Figure 4a); − an unstable region where the foam displays a shear thinning behaviour (corresponding to the setting in motion of the bubbles and the arrangement of the structure; see Figure 4b); − a shear thickening behaviour region where the increase of the shear strain causes a bubble break-up and thereby a shift of the bubble size distribution to smaller diameters. The resulting increased bubble number leads to an increase of the interacting forces (Figure 4c). This scheme is corroborated by the unloading curve: at the end of the rising curve an equilibrium structure is attained and the fluid exhibits then a viscoplastic shear thinning behaviour. The effect of shear stress increasing/decreasing time and of ageing on timedependent behaviour of the foam was studied. Five sets of experiments have been carried out, at a constant stress rate, with increasing stress times varying from 2 to 10 minutes and a maximal stress of 50 Pa (Figure 5). The flow curves obtained for different duration times of the experiments resemble that of Figure 3: a composite curve is observed; that is, the rheological behaviour changes from thixotropic to rheopectic (anti-thixotropic). Note that this inversed behaviour is not observed for short duration times (2 and 4 minutes), probably due to the fact that the maximum value of the applied shear stress (50 Pa) is the same for all experiments. Consequently, for short ascent times, the range of shear rates attained is not large enough to allow the observation of the so-called inverse behaviour. Also, it can be noted that the viscosity decreases with increased ascent times. An alternative way to characterise a time-dependent fluid is to represent the thixotropic surface (the surface between the ascent and the down curves) as a

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(a)

(b)

(c) Figure 4. Evolution of the foam structure under shearing for a 0.3% PEO foam. (a) Beginning of the recording, (b) shear rate of 100 s−1 , (c) end of the upwards curve at 350 s−1 .

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Figure 5. Flow curves for different stress ascent times (0.3% PEO based foam).

Figure 6. Thixotropic surface as a function of the stress ascent time (0.3% PEO based foam).

function of the stress ascent time. Figure 6 shows that the thixotropic area depends on the duration of the experiment. The negative values mean that the area under the down curve is more important than that under the up curve. This shows that the anti-thixotropic behaviour of foam is as pronounced as the ascent time decreases.

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Figure 7. Comparison of flow curves for different ageing time (0.3% PEO based foam).

The evolution of the rheological properties of foam with ageing time was investigated in order to get information on its lifetime and its dynamic structure at rest and in flow conditions. To perform these tests, the foam was kept at rest during 0 to 10 minutes before starting the next experiment. Drastic changes in bubble size and shape were observed. A qualitative analysis of the structural changes brought us to an estimation of the relative effect of bubble sizes on the behaviour of foam during viscometric flow tests. It can be observed in Figure 7 that the flow curves shift towards the shear rate axis when the storage time increases, showing a decrease of the foam viscosity due to an increase of the size of the bubbles. The results have shown that the hysteresis loops are reduced in size when the storage time increases. This causes a decrease of the thixotropic surface (Figure 8). For practical applications in some industrial processes, foam ageing appears as an excellent way to reduce thixotropic effects. Such thixotropic behaviour is caused by a degradation of the inner structure and in the case of the studied foams by flocculation and coalescence of bubbles.

3.2. L INEAR V ISCOELASTIC M EASUREMENTS Dynamic and creep flow tests are used to detect eventual elastic properties of the material. The linear viscoelastic properties of foam were first studied in terms of complex dynamic properties using small amplitude oscillatory shear experiments. A sinusoidal stress τ = τ0 sin ωt is applied to the sample whereas its corresponding

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Figure 8. Thixotropic surface as a function of ageing time (0.3% PEO based foam).

sinusoidal strain γ = γ0 sin(ωt + δ) is measured, ω is the frequency of oscillation in rad/s, γ0 is the strain amplitude, τ0 is the stress amplitude, δ is the phase angle between the stress and the strain (i.e., the loss angle) and t is the time. The shear storage and shear loss moduli, respectively G0 = (τ0 /γ0 ) cos δ and G00 = (τ0 /γ0 ) sin δ, were calculated from the measured outputs, the strain amplitude γ0 and the loss angle δ. The effect of frequency was investigated with a constant deformation amplitude of 10%. Fresh foams were used for each data point so as to prevent any effect of drainage of the liquid from one test to the next. The modules G0 and G00 are plotted in Figure 9 as functions of frequency for a 0.3% PEO based foam. It can be observed that both G0 and G00 are frequency independent. This shows a pronounced elastic behaviour of the foam. Furthermore, the storage modules G0 is larger than the loss modules G00 , showing that the foam elastic behaviour is dominant compared to the viscous behaviour. This result is in good agreement with those obtained by other authors (Khan et al., 1988). In addition to the transient properties discussed previously, creep flow tests were carried out. A creep curve is obtained by applying at time t0 a stress held constant and measuring the time evolution of the resultant deformation followed by a strain recovery curve obtained after removal of the constant stress. The elastic properties are defined by correlating the results with the classical viscoelastic models of Maxwell or Kelvin–Voigt which can be physically described by assemblies of dashpots and springs. From our experiments it was found that the viscoelastic behaviour of the foams could be represented by the generalised Kelvin–Voigt model

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Figure 9. Variation of G0 and G00 with frequency for a 0.3% PEO based foam.

which combines a Maxwell unit (Newtonian dashpot in series with purely elastic spring) in serial with one or more Kelvin–Voigt units (dashpots in parallel with springs). The creep compliance for the foams is then defined as: f (t) = J0 +

n X i=1

Ji (1 − e−t /θi ) +

t , ηN

(1)

where f (t) is the overall creep compliance at the time t during the test, J0 the instantaneous elastic compliance; Ji the contribution to retarded elastic compliance of the ith component with a retardation time θi = (Ji ηi ) and a viscosity ηi and ηN is the Newtonian viscosity. A typical creep and recovery curve, of the type obtained for many viscoelastic foams, is illustrated in Figure 10. The mechanical behaviour is represented by the inset mechanical model in Figure 10 which consists of a Maxwell unit in series with a Kelvin–Voigt unit. The viscoelastic parameters were derived from the creep compliance-time curves by use of the well-known method initially developed by Inokuchi (1955; see also Warburton and Barry, 1968). In the analysis procedure, the instantaneous compliance J0 is set from the curve intercept on the compliance axis of the graph, and the Newtonian viscosity ηN is derived from the slope of the linear portion of the curve which represents the Newtonian shear of the sample. When these terms are removed from Equation (1), only the exponential terms remain. Thereafter, a stepwise analysis of the retarded elasticity region of each curve yields discrete pairs of values of Ji , θi for n viscoelastic Kelvin–Voigt units. An exhaustive study of a foam may require the determination of its viscoelastic properties over a wide frequency scale. Unfortunately, due to drainage process, the

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Figure 10. Typical creep compliance and recovery curve for a viscoelastic foam (0.3%PEO). A mechanical model which may be used to describe the creep results is also shown (a Maxwell unit in series with a Kelvin–Voigt unit).

experiments are limited in time. However, the high-frequency (short-time) range may be investigated by a dynamic method such as oscillatory while the long-time range (low-frequency) is accessible by a transient method such as creep because a periodic experiment at frequency ω is qualitatively equivalent to a creep test at time 1/ω. Mathematical methods (Barry, 1975) may then be used to unify data so as to describe the viscoelastic behaviour over the frequency range using only one type of response function. That is, it was possible to complete the dynamic response in the low-frequency range (long-time) using data from the transient response in creep. Creep compliance curves, analysed by the discrete spectral method, may be employed to calculate storage and loss compliances (J 0 and J 00 ) using the Fourier transformation. The exact interrelationships obtained are given by (Ferry, 1970; Tschoegl, 1997) J 0 (ω) = J0 +

n X

Ji /(1 + ω2 τi2 ),

(2)

i=1

00

J (ω) =

n X i=1

Ji ωτi /(1 + ω2 τi2 ) +

t . ηN

(3)

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Figure 11. Unified plots of the variation of storage modulus G0 and dynamic viscosity η0 [η0 = (τ0 /ωγ0 ) cos δ] with frequency for a 0.3% PEO foam. Low frequency values are derived by transforming creep data, and high frequency values are measured by oscillatory measurements with a Carri-Med rheometer.

Values for G0 and G00 may be deduced using Equations (4) and (5): G0 =

J0 , J 02 + J 002

(4)

G00 =

J 00 . J 02 + J 002

(5)

The results in terms of variation of the storage modulus G0 with frequency are illustrated in Figure 11. The high-frequency values are derived from oscillatory measurements with the Carri-Med rheometer and the low-frequency values are obtained by transforming creep data. One may observe that the transformed dynamic functions compare well with the functions obtained directly from oscillatory measurements. 4. Capillary Experiments 4.1. E XPERIMENTAL S ET-U P A capillary-tube rheometer designed in our laboratory and described elsewhere (Scrivener et al., 1979) was used to investigate the flow properties of the foams.

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Figure 12. Capillary tube rheometer.

Originally, this kind of instrumentation was developed by Hoyt (1971) and later by Sellin and Loeffler (1977) to investigate the drag reducing properties of polymer solutions. In our system (see Figure 12), a 50 ml foam sample is discharged by a brass piston activated by gas pressure through a vertical steel capillary. The piston used to charge the cylinder follows the liquid level closely when the fluid is discharged under gas pressure. The pipe flow velocity, and hence the Reynolds number values, are based on a timing device operated by two blades moving with the cylinder in the slit of a light barrier. The position of the two blades (monitoring respectively the beginning and the end of the time measure) are chosen so that only the middle part of the fluid discharge is timed. This avoids problems with driving air out of the pipe and uniform flow establishment at the start of the stroke. The pressure drop is measured over the whole tube length, including entrance effects. A pressure gauge allows the measure of the pressure at the entrance of the capillary. Tests with a very short tube of the same diameter showed that errors in the average friction coefficient values (λ) introduced by the entrance effects are negligible. Changes in kinetic energy and losses at a sharp-edged pipe entrance are also included in the computer analysis of the data. The length to diameter ratios of the capillaries are also chosen large enough so as to minimise the corrections

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for end effects. In fact, these corrections are difficult to determine owing to the non-Newtonian behaviour of the fluid. Four tubes of different diameter ranging from 2.5 to 4 mm and length 1 m were used. The effective gas pressure range, 0.2–1.8 bar, gives Reynolds number values for flow of foam from 0.001 to 100. In order to ensure reproducibility, each data point presented in the following figures was obtained by averaging repeated runs with freshly prepared foam for each run. 4.2. P RESSURE D ROP C OEFFICIENT The scaling of experimental pipe flow data obtained using Newtonian fluids is well understood and a common way of showing such data is a λ–Re plot. The Reynolds number of the flow is defined by the well-known relationship Re = ρuD/µ, in which u is the mean velocity of flow in the capillary, D is the capillary diameter, ρ is the density and µ is the viscosity of the fluid. The pressure drop coefficient λ is expressed by: λ=

1P 1 D 2 1/2ρu L

(6)

in which L denotes the pipe length and 1P is the total pressure drop along the capillary. Note that λ as defined here is equal to 4f , the Fanning friction factor. For a Newtonian fluid flowing through a smooth pipe, the pressure drop coefficient can be expressed as a single and universal function of the Reynolds number. The corresponding curve allows the calculation of pressure drop in pipes of any diameter. For non-Newtonian fluids the problem is much more complicated, the viscosity, and thus the Reynolds number, being shear rate dependent. Among different methods proposed to solve this problem, one of the most convenient is that of Metzner and Reed (1955). This correlation is based on the definition of a generalised Reynolds number for power law fluids, applying the concept of apparent viscosity measured in laminar flow: Reg =

ρuD , µ0c

where µ0c denotes the apparent viscosity in the pipe given by: τw µ0c = γ˙a

(7)

(8)

in which τw = R1P /2L is the wall shear stress and γ˙a = 8u/D is the apparent shear rate. The application of the above-mentioned method leads to the following expression for µ0c for a power-law fluid (Fam et al., 1987):     3n + 1 n 8u n−1 (9) µ0c = K 4n D

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Figure 13. Pressure drop coefficient λ (Fanning friction factor) for laminar flow as a function of generalised Reynolds number for different pipe diameters (0.3% PEO foam).

Figure 14. Pressure drop coefficient λ (Fanning friction factor) for laminar flow as a function of generalised Reynolds number for different PEO concentrations (pipe diameter of 3.5 mm).

in which K and n are the consistency and the flow structure index of the fluid, respectively. The pressure drop coefficient is shown as a function of the generalised Reynolds number for a 0.3% PEO foam for different diameters of the capillary tube (Figure 13) and different PEO concentrations foams in a 3.5 mm pipe diameter

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Figure 15. Flow curves of a 0.3% PEO foam for different capillary diameters.

(Figure 14). These results confirm the validity of the use of the rheological model and that the representation of the pressure drop coefficient as a function of the generalised Reynolds number is appropriate for these foams. Nevertheless, in both cases, for generalised Reynolds numbers lower than 1, the experimental points line up with the Poiseuille law λ = 64/Reg . At higher Reynolds numbers a deviation of the experimental points from the Poiseuille straight line is observed. This deviation could be due to wall slip effects. This assumption is confirmed in Figure 15 which depicts typical flow curves for a 0.3% PEO foam for three capillary tubes having different diameters (2.5, 3.5 and 4 mm). A strong dependence on the diameter of the tube can be seen. This puts in evidence the existence of a slip velocity Vs at the wall. Similar phenomena have been observed by several authors (Enzendorfer et al., 1995; de Vargas et al., 1993; Heller and Kuntamukkula, 1987). This effect was also reported for flows of solid suspension and other dispersed systems in capillary viscometers (Cohen and Metzner, 1986). The shift of the curves toward lower stresses when the diameter of the tube decreases could be well explained by a discontinuity in the velocity profile near the wall of the tube. It is generally admitted that the existence of a wall slip velocity is related to the presence of a thin layer of fluid near the wall, the viscosity of which is much lower than the viscosity of the dispersion. It is important to take into account this ‘effective slip velocity’ if one needs to predict the flow rates of dispersed media in tubes of different diameters.

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4.3. A PPLICATION

OF

WALL S LIP C ORRECTION M ETHODS

If we assume that the velocity at the wall of the pipe is different from zero as assumed for Newtonian fluids, the contribution to the total flow rate can be divided in two parts: the ‘true’ flow rate and a flow rate corresponding to the slip velocity: Q = Qtrue + Qslip .

(10)

Mooney (1931) obtained the following relationship for capillary flow: 8Vs (τw ) 4 8u = + 3 D D τw

Zτw τ 2 f (τ ) dτ.

(11)

0

It is assumed that the slip velocity depends only on the wall stress Vs = f (τw ). For constant values of τw if 8u/D is plotted as a function of the reciprocal diameter 1/D, the wall slip velocity Vs is deduced from the slopes of the straight lines obtained. The best fit curve of Vs versus τw can then be used to correct the wall slip effects. Hence, Equation (11) can be expressed as:   8(u − VS ) 8u . (12) = D c D This procedure was applied to our data (see Figure 16). Unfortunately, it was found that extrapolation to infinite diameter (free of slip velocity) of many of the curves leads to negative values of the wall shear rate. Thus, the basic assumptions and the Mooney correction method does not apply in the case of our experiments with foams. This is also in accordance with results established by other authors (Burley and Shakarin, 1992; Enzendorfer et al., 1995; Mourniac et al., 1992). A second attempt to calculate the slip velocity was made using the method proposed by Jastrzebski (1967). This work predicts that the slip velocity depends not only on the wall stress but also on the pipe diameter according to Vs = S(τw )/D. With this assumption, Equation (11) can be rewritten as: 

8u D



8S(τw ) 4 = + 3 2 D τw

Zτw τ 2 f (τ ) dτ.

(13)

0

A plot of (8u/D) versus 1/D 2 at different fixed wall shear stresses gives straight lines from which S(τw ) is determined by dividing the slopes by 8. The corrected value of (8u/D) becomes then:     S(τw ) 8 8u u− (14) = D c D D which is used for true shear rate calculation. This method was applied to the same data as previously and the results are presented in Figure 17. It can be observed that,

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Figure 16. Application of the Mooney method for a 0.3% PEO foam (shear rate as a function of 1/D at given shear stress). A straight line was fit to the experimental data points for each wall stress using the least squares method.

Figure 17. Oldroyd–Jastrzebski plots for 0.3% PEO foam (shear rate as a function of 1/D 2 at given shear stress). A straight line was fit to the experimental data points for each wall stress using the least squares method.

with this method, the intercepts of the straight lines with the apparent shear rate axis are all positive and vary consistently with the wall stress. Clearly, this shows that the correcting method of Oldroyd–Jastrzebski can be applied successfully to our foamed polymer solutions. This validation of the correcting method brought us to determine S(τw ) and to plot the slip velocity Vs as a function of the wall shear stress τw for each capillary diameter (Figure 18). It may be observed that the slip velocity decreases as the capillary diameter increases. Also, the slip velocity increases linearly as τw increases for any diameter in the range of shear stresses studied. Therefore, it may be concluded that the slip velocity determined by the

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Figure 18. Slip velocities as a function of wall shear stress for a 0.3% PEO foam.

Figure 19. Corrected flow curves for a 0.3% PEO foam.

Oldroyd–Jastrzebski method is a function of both the wall shear stress and the capillary diameter, as expected. Figure 19 shows a plot of the wall shear stress as a function of the shear rate corrected for slip. It can be seen that the experimental results obtained with different capillaries are fitted by a unique master flow curve. A best fit analysis of the corrected data brought us to the conclusion that the rheological behaviour of our foams can be modelled by a power law constitution model. The values of the flow index n and the consistency k derived from this curve were used to

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Figure 20. Pressure drop coefficient λ (friction factor) as a function of corrected generalised Reynolds number for a 0.3% PEO foam and different pipes ( , : 4 mm; , : 3.5 mm; , : 2.5 mm) and comparison with non-corrected data.





#



calculate the corrected generalised Reynolds number. Figure 20 shows the results obtained. It appears that, compared to non-corrected data, the corrected data are closer to the Poiseuille curve for Reynolds numbers ranging between 0.1 and 100. On the opposite, for Reynolds numbers lower than 0.1 the correlation seems to be better when non-corrected data are used. These results indicate that within the low velocities range, the slip effects are negligible. 5. Conclusion The rheological experiments executed on foams revealed a shear thinning behaviour at low shear rates and a shear thickening behaviour at high shear rates. The viscosity of the foam is over five times higher than the Newtonian viscosity of the basic foaming solution. As observed by visualisation, the shear stress produces changes in the inner structure of the foam. Related to this, loading and unloading tests show differences in the rheological behaviour of the dispersion: shear thinning followed by shear thickening behaviour for the upwards curve, shear thinning for the downwards curve. Also, a composite behaviour is observed: thixotropic at low-shear rates and anti-thixotropic at high-shear rates. The size of the corresponding hysteresis loops are reduced in size with ageing. This shows that the time-dependent properties of the foams are strongly influenced by storage time. Transient and harmonic shear flow experiments have put in evidence a strong viscoelastic behaviour of the fluid which can be represented by the mechanical Kelvin–Voigt model. It was also observed that the flow curves measured using a

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