- Email: [email protected]
Effective viscosity measurement of interfacial bubble and particle layers at high volume fraction
Flow Measurement and Instrumentation 41 (2015) 121–128
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Effective viscosity measurement of interfacial bubble and particle layers at high volume fraction Yuichi Murai a,n, Takahisa Shiratori a, Ichiro Kumagai b, Patrick A. Rühs c, Peter Fischer c a
Laboratory for Flow Control, Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan School of Science and Engineering, Meisei University, Hino 191-8506, Tokyo, Japan c Institute of Food, Nutrition and Health, ETH Zürich, 8092 Zürich, Switzerland b
art ic l e i nf o
a b s t r a c t
Available online 1 November 2014
An experimental method for measuring the effective viscosity of two dimensional dispersion systems is proposed. The method is based on interfacial rheology, which was originally developed to investigate surface active adsorption layers such as protein film formed at liquid–liquid interfaces. Bubbles or rigid particles at around 50% of volume fraction in liquid are positioned in the gap between a rotating disk and a stationary cylindrical container. With this configuration, shear-rate dependent effective viscosities of bubble and particle layers were investigated. Steep shear-thinning property is observed for spherical bubble systems in the shear rate regime from 10 1 to 102 s 1. This is explained by topological transition from regular to random arrangement of the bubbles at the interface. For rigid particle systems, the viscosity starts from high value due to solid contact friction, then decreases sharply due to fluidization process until inter-particle collision lead to an increase of the viscosity. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Bubbles Particles Multiphase flow Rheology Effective viscosity
1. Introduction The effective viscosity of liquids containing bubbles and particles is already an over one century old research topic since its first appearance in literature [1,2]. As the disperse phase is suspended in the continuous liquid phase of viscosity μ0, the effective viscosity of the dilute suspension μ is given by the following formula. ( f ¼ 5=2 ðsolidÞ μ ¼ 1 þ f α; ; ð1Þ f ¼1 ðgasÞ μ0 where α denotes volume fraction of the dispersion phase. This formula is valid for spherical dispersion at small volume fraction, α o0.05. Schowalter et al. [3] and Choi and Schowalter [4] obtained the following formula valid for deformable bubbles and for higher volume fraction 2 μ 1 þ ð1 þ 4αÞ 1 þ ð20=3Þα ð6=5ÞCa 5 ¼ 1 þ α þ α2 ; ð2Þ 2 2 μ0 2 1 þ 1 þð20=3Þα ð6=5ÞCa where Ca denoted the Capillary number defined by Ca ¼
μ0 γ_ r ; σ
n
Correspondence to: Laboratory for Flow Control, Division of Energy & Environmental System, Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan. Tel.: þ81 11 706 6372; fax: þ 81 11 706 6373. E-mail address: [email protected] (Y. Murai). http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.006 0955-5986/& 2014 Elsevier Ltd. All rights reserved.
ð3Þ
in which σ, γ_ , and r being surface tension coefficient, shear rate, and bubble radius. Temporal transition from spherical bubble regime to shear-yield bubble regime was theoretically modeled by Pal [5]. The validity of these theoretical equations for various bubbly liquids were confirmed with circular Couette flow by Rust and Manga [6], Müller-Fischer et al. [7], and Gutam and Mehandia [8]. The similar trend was reported for pipe flows by Llewellin and Manga [9]. Murai and Oiwa [10] studied the influence of non-equilibrium deformation of bubbles in unsteady shear flow and found significant increase in effective viscosity compared to steady shear rate values. With higher gas volume fractions, the effective viscosity is significantly controlled by the complex liquid film dominating the bulk properties [11,12]. Pronounced viscoelastic properties occur in foam flow, which is influenced by surfactants and electrochemical interfacial properties [13–19]. Back to the early research, Mooney [20] suggested the following formula to describe the increase of effective viscosity at high volume fraction in the case of spheres with non-slip surfaces, i.e. rigid spherical particles: μ Aα 1 ¼ exp ð4Þ ¼ 1 þ Aα þ AðA þ 2kÞα2 þ Oðα3 Þ; 2 1 kα μ0 where A and k denote the dimensionless values, which approximate the measured viscosity and the spatial arrangement. The value A is given by A¼1 for uncontaminated bubbles while A¼ 5/2 for rigid particles and well-contaminated bubbles. The value k is a factor that describes the influence of spatial arrangement pattern of the spheres.
122
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
Theoretically, the value k takes the range of 1.35oko1.91 in accordance with two limits between a face-centered cubic lattice and simple cubic arrangement. Comparison of the polynomial terms in Eq. (4) with Eq. (2) yields that k¼2 was employed in Eq. (2) at Ca¼0, which exceeds the highest limit of k for spherical dispersion. Batchelor and Green [21] theoretically obtained the proportional factor to α2 in case of rigid particle suspension to be 5.2 as the further higher order terms were neglected. This corresponds to k¼ 0.83, which is lower than the above-mentioned lowest limit. A common issue in these theoretical works is uncapability to address the spatial fluctuation of local dispersion arrangement that naturually happens to real systems. Numerical simulations by Kuwagi and Ozoe [22] adapted a representative value of the factor at k¼ 1.43 assuming a random arrangement. From the experiment of Darton and Harrison [23] and Tsuchiya et al. [24], average values of k for particle suspension at high volume fraction (0.45o α o0.55) were k¼ 1.2. This value is, on the other hand, being smaller than the theoretical smallest limit. Different from the theory, the spatial arrangement pattern of the spheres is hardly controlled in experiments. It varies with shear rate like a fluidization process of packed rigid spheres. Such transition of the arrangement produces dramatic shear rate dependence in viscosity whereas deformation of the dispersion is insignificant. Morris and Katyal [25] and Morris [26] found strong shear-thickening as a result of inhomogeneous particle distribution, which is triggered by direct contact among rigid spheres. Any case in the above previous works deals with 3D system of dispersion distribution subject to shear rate. Since 3D systems allow individual dispersions to move in three directions, shear-dependent viscosity correlates to transitions of three-dimensional arrangement pattern of dispersions. In contrast, rheological property of dispersion system confined in two dimensions has not been studied experimentally yet. In the present research, we investigate the effective viscosity of interfacial bubble and particle layers at high interfacial volume fraction, i.e. from 0.2 to the packing limit (inter-lock condition of spheres). We focus on spherical shapes of the bubbles where deformability is almost negligible, thus Cao0.05. These conditions are attractive to engineers who aim at flow control by means of small dispersions [27–29]. Especially, in turbulent boundary layers, the dispersions interact with coherent structures in turbulence and form clusters within the layers. The clusters can have local volume fraction much higher than 0.2 and often reach the closest packing limit. This happens because strong congregation force acts on the dispersion due to steep local pressure gradient within coherent structures [30]. Thus, effective viscosity of dispersion at high volume fraction takes a primary issue rather in turbulences than in laminar flows [31–33]. A theoretical and numerical approach to such densely arranged spheres in simplified flow geometry was reported by Kang et al. [34]. Direct numerical simulation by Yeo and Maxey [35] showed particles selfdiffusion which results in shear-thickening characteristics. In this paper, we desribe a new methodology to acquire the effective viscosity of interfacial bubble and particle layers at high interfacial volume fraction. The measurement principle is based on
interfacial rheometry which was established for shear viscosity assessment of interfacial adsorption layers. General configuration and application of interfacial rheometry is reported elsewhere [36– 40]. The original purpose of the method is to quantify the rheological properties of a very thin material layers formed at liquid interfaces such as a protein films and surfactant adsorption layers. The present study extends this technique to the measurement of the viscosity of bubble/particle multiphase layers just by changing the shape of a rotating cone. In this article, the working principle of the present rheometry, the advantages in experimental handling of bubble/particle suspension layers, and their applications to bubble/ particle multiphase layer are reported accompanied with a brief discussion on the measured results.
2. Interfacial rheometry Fig. 1 shows the schematic diagrams of the experimental setups. The measurement system is constructed by the combination of a rotating disk attached to a commercial rheometer (Physica MCR300, Anton Paar) and a cylindrical fluid container. The setup (a) is used to measure the interfacial rheology of adsorption layers. The working principle of interfacial rheology is reviewed in the literature [36]. The setup (b) used in the present study is applicable for bubble and particle layer with a finite thickness. In both types, shear flow is driven by a rotating disk that is separately installed from a stationary cylindrical container. Viscosity of the sub-phase liquid, μ0, filled in the container is accounted for in the full flow analysis performed by the rheometer software [36]. In the setup (b), the structure of the fluid interface is additionally observed and recorded by a video camera (Sony DFW V500) from the top and the side of the container. With this combination, spatial distribution and motion of floating dispersions are monitored to find their correlations to bulk viscosity to be measured. The dimensions, R1, R2, H, and δ in Fig. 1 can be modified in accordance with the target material of measurement as well as the range of shear rate to be studied. 2.1. Original interfacial rheometry Through a normal force transducer the biconical disk is positioned exactly at the interface. A ring-shaped gap between the rotating disk and the cylindrical container forms the free surface area to be measured. When the disk rotates, a circular Couette flow on the liquid surface within the gap is induced. In standard interfacial rheometry, the sharp edge of a rotating biconical disk drives the shear flow as shown in Fig. 1(a). The thickness of the material is ordinarily treated as unknown since this methodology is designed for adsorption layers with molecular dimensions. The interfacial viscosity has the dimension of [Pa s m]. Only in the case that the target layer is uniformly formed with a constant film thickness, the length scale multiplied in the dimension coincides with the film thickness. Dependent on the target material (e.g. surfactant, protein), the film
Fig. 1. Schematic diagram of the measurement system based on interfacial rheometry. Target fluid is set in a thin layer on the surface of liquid with known viscosity. (a) Rotating bi-cone with sharp edge for measurement of e.g. protein adsorption layers. (b) Rotating conical disk with a step for measurement of bubble/particle multiphase layer systems.
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
thickness shows spatio-temporal evolution, to which thermal, chemical, and electrical influences on the film property should be considered in a given environment. The definition of the interfacial viscosity allows one to include all these influences into a single value. 2.2. Interfacial rheometry for bubble/particle multiphase layer The setup (b) in Fig. 1 is the modified version adopted for the present study. The pointed edge of the biconial geometry is replaced by a vertical step which is higher than the thickness of the target layer δ. Dispersions (bubbles or particles) are adjusted to the height. By rotation, the dispersions move in the azimuthal direction so that a two-phase circular Couette flow is generated between the rotating cone and the cylindrical container. The target height δ is controlled to be within several diameters of the bubbles or particles. This allows the dispersion to behave nearly in two dimensions, i.e. dispersion motion is restricted in the horizontal plane. Additional advantage of this system is that the spatial arrangement of the dispersions can be directly observed from the top of the instrument. This is one of the most distinguishable points from the other types of rheometry for multiphase fluids purposed for high dispersion volume fraction. As the interfacial viscosity η is measured, the effective viscosity is estimated simply by division with the thickness of the dispersion layer, i.e.
μ¼
η δ
ð5Þ
This operation is valid only in the case that δ is controlled constant without change in both space and time. As δ is thin enough, η offers a new property of multiphase-layer of which internal motion of dispersion is restricted in two dimensions.
3. Bubble and sphere interfacial layer The present interfacial viscometry has been applied to two kinds of interfacial dispersions, i.e. bubbles and particulate layers floating at the air–water interface. There are two roles in buoyancy of dispersions, in which this setup is operated. First, the buoyancy keeps the dispersion layer stably at the initially set position. Second, the buoyancy accumulates dispersions densely inside the layer. For testing rigid particles, light rigid spheres of density smaller than water are used to maintain the interfacial particle layer.
123
For bubbles larger than 0.5 mm in diameter, we adopted hydrolysis through porous plates to generate bubbles into the cylindrical container. To change the mean bubble diameter from 0.5 mm to 3 mm, six different types of porous plates with different porosities were examined. A surfactant, SDS, was used to stabilize the generated bubbles and up to 2.0 g was added to the continuous aqueous phase. The surfactant helps us with generating mono-dispersed bubbles as well as to maintain their stability in shearing environment. In each type of porous plate, the amount of the surfactant sways the number of bubble layers, which influences the rheological properties as discussed in the next chapter. The diameter of bubbles was measured by video images of the top views. For microbubble experiments, we utilized water electrolysis which produces microbubbles less than 0.1 mm in mean diameter. The combined set-up with the electrolysis is shown in Fig. 2. In the target layer, hydrogen bubbles are accumulated due to their own buoyancy. The diameter of microbubbles was measured by photography enlarged around the cathode wire. Fig. 3 shows a typical arrangement pattern of spherical bubbles generated by a coarse porous plate with hydrolysis. The arrangement pattern is regulated in two dimensions on the water surface, and has honeycomb-like structures in the stationary state. Fig. 4 displays closer looks of the bubble arrangement as the mean bubble diameter changes. As shown in (a), bubbles with a mean diameter of 2.5 mm are regularly arranged with four to five rows in the gap between the rotating disk and the stationary outer cylinder. This regularized structure disappears for small bubbles (0.8 mm) as shown in (b). As microbubbles (d¼ 0.05 mm) are generated, milky clouds of microbubbles occupy the gap as shown in (c). On all the three conditions, we recorded the video images of the top view and confirmed no significant vanish of bubbles during the rotating operation which is less than one minute to perform the effective viscosity measurement. Also we confirmed that bubble motion inside the gap follows to a simple shear flow between the rotating disk and the stationary container. There was no shear localization observed such as in the vicinity of wall. Furthermore, we confirmed that the amount of SDS mainly affects the vertical thickness of bubble layer in steady rotating state. Since the surfactant itself does not significantly change the
3.1. Bubble interfacial layer The effective viscosity of spherical gas bubble arrangement is measured. By changing the method of bubble generation, three different bubble-layers are examined: large bubbles, small bubbles, and microbubbles. The details are summarized in Table 1. In all the conditions, capillary number defined by Eq. (3) is sufficiently smaller than unity, i.e. the maximum value is 0.05 at the highest shear rate. This means that bubbles are all spherical as they are dispersed freely.
Fig. 2. Schematic diagram of the measurement system coupled with water electrolysis for microbubble experiment. Platinum wire cathode set close to the bottom wall of the container produces hydrogen bubbles that naturally rise and reach the water surface to form the microbubble layer in the test section.
Table 1 Bubble-layer conditions. R1 R2
40 mm 50 mm
(See Fig. 1(a) and Fig. 2) (See Fig. 1(a) and Fig. 2)
Classification Large bubble Small bubble Microbubble
Gas Air Air Hydrogen
Bubble diameter d(σd)n 2.50 mm ( 0.07 mm) 0.80 mm ( 0.09 mm) 0.05 mm ( 0.02 mm)
n
d and σd denote arithmetic mean bubble diameter and standard deviation.
Method of bubble generation Hydrolysis with a coarse porous plate Hydrolysis with a fine porous plate Electrolysis (Cathode)
124
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
Fig. 3. Top view of spherical bubble arrangement formed on the water surface in the case of mean bubble diameter, d ¼2.5 mm. (a) Two-dimensional arrangement close to sphere’s packing limit observed in the container before installation of the rotating disk. (b) The same arrangement realized in the test section after installation of the rotating disk.
Fig. 4. Top view of local bubble arrangement patterns for three different conditions in bubble size. (a) Large bubble at d¼ 2.5 mm, (b) small bubbles at d ¼ 0.8 mm, and (c) microbubbles at d ¼ 0.05 mm.
viscosity of water, the interfacial viscosity of bubble layer can be attributed to the rheological property of bubble interfaces that exist densely within the bubble layer. Fig. 5 shows the side views of the bubble arrangement photographed during the rotation. In experiments for the bubbles larger than 1.0 mm, the number of bubble layers in the vertical direction is countable as shown in (a) to (d) since the arrangement is regularly stratified. For the bubbles smaller than 1.0 mm, the number of the rows increases more than four and their arrangement pattern starts to be irregular with three dimensional motions. We measure the total depth of the bubble layer, δ, from these video images, and use Eq. (5) to obtain the effective viscosity. 3.2. Rigid particle interfacial layer For rigid particle interfaces, light particles with density smaller than water were used. Therefore, the particles float on the water surface. The name of the particle is FLO-BEADS (CL-2507, Sumitomo Seika, Ltd), which is made of polyethylene formed in sphere. The particles are close to spherical, and have a high wettability. Three conditions in the particle distribution patterns are focused on in this demonstration as summarized in Table 2. The areal concentration called here is defined by the ratio of particle projection area to the
total area of the gap. It is measured from photography, of which samples are shown in Fig. 6. The areal concentration, C, is computed from the single particle area multiplied by the number of particles in a sampling area, using image processing software to detect individual particle centers. The particle areal concentration less than the packing limit always show inhomogeneous distribution in any geometry as long as the particles are neutrally distributed without mutual repulsion force. The bulk interfacial viscosity in such a situation is the present target of investigation. The condition (a) is low areal concentration at which the particles show stratified distribution in the radial direction. The radial stratification cannot be explained simply by the inertia effect (i.e. centrifugal acceleration), but should be coupled with particle-clustering rheology, which will be a point of discussion. The condition (b) has a slight difference in particle concentration but shows a wavy boundary of the particulate surface. On the condition (c), spatially continuous shearing motion of the particulate surface takes place entirely in the gap, allowing local smooth slip among particles. With further increase in the particle areal concentration, thin slits of particulate surface come out to serve large velocity differences between two neighbors as shown in (d). According to the video images, the slit moves unsteadily, and accompanies wavy fluctuation in the azimuthal direction. Under
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
125
Fig. 5. Side views of bubble arrangement patterns photographed from outside the cylindrical container. (a) One layer at d ¼ 2.5 mm, (b) two layers at d ¼ 1.8 mm, (c) three layers at d ¼ 1.6 mm, (d) four layers at d ¼ 1.3 mm, (e) five-to-six layers at d ¼0.7 mm, (f) more than six layers at d ¼ 0.7 mm.
Table 2 Particle-layering conditions. R1 R2 Density of rigid particle Diameter of rigid particle Condition (a) (b) (c) (d)
34 mm 44 mm 919 kg/m3 0.18 mm (with standard deviation of 0.01 mm) Areal concentration 0.20 0.22 0.43 0.76
Fig. 6. Top views of rigid particle distributions for four different concentrations.
(See Fig. 1(b)) (See Fig. 1(b))
Trend of particle distribution pattern Radial binary stratification Radial stratification with wavy boundary Global fluidized distribution Solidized surface separated by slit(s)
126
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
these curious conditions, we present in the next chapter how the bulk interfacial viscosity behaves with the increment in shear rate.
4. Results and discussions In this chapter, the measured relationship between the interfacial viscosity and the shear rate is presented. For each condition of dispersion layer, a brief discussion is given on the measured results. 4.1. Dense bubble layer Fig. 7 presents the interfacial viscosity of dense bubble layers in all the conditions. In most cases the interfacial viscosity declines monotonically with increase in shear rate. Thus, the dense bubble layer has a clear shear-thinning property. For the tested range in shear rate from 0.1 o 100 s 1, the magnitude of the interfacial viscosity changes over one digit. The highest interfacial viscosity is measured for small bubbles (d¼0.8 mm) while the layer of large bubbles (d¼2.5 mm) takes much lower values. This trend should be explained by multiple factors; one of them is the difference in bubble arrangement pattern as shown in Fig. 4. The correspondence to the photographs implies that a regular alignment of bubble rows produces shear stress smaller than the case of random arrangement. Another factor that we need to consider is interfacial area concentration, β, which is defined by A V
β¼ ¼
n πd 6α ¼ ; 2V d 2
α¼
n πd : 6V
Fig. 8 shows the same data evaluated by effective viscosity. The effective viscosity is calculated by Eq. (5), using the measured thickness of bubble layer. Since the thickness little changes with shear rate, the shear-thinning trend is obtained almost similarly to the previous graph on the interfacial viscosity. One of findings from this result is that small bubbles (d ¼ 0.8 mm) and microbubbles (d ¼0.05 mm) have large effective viscosity compared to large bubbles owing to the increase in the interfacial area concentration. Among them, the small bubbles of 1.0 g addition of SDS take the highest effective viscosity. On this condition, we observed from the side view camera that the number of bubble rows ranges from 2 to 3 in the vertical direction. This infers that the individual motion of bubbles within the bubble layer is regularly restricted in two dimensions, and such a state produces a shear stress higher than the case of 3D motion allowed in a thick bubble layer. For large bubble conditions (d ¼2.5 mm), single-layer arrangement (1.0 g SDS) creates higher effective viscosity at the low shear rate range, but turns into steep shear-thinning to have the viscosity
3
ð6Þ
where A and V are total area of bubble surface inside the bubble layer, and volume of the bubble layer. The interfacial area concentration describes the potential magnitude of surface tension of individual bubbles, which takes place as a strong shear resistance in the case of densely arranged bubbles. β is proportional to void fraction, α, but is inverse proportional to bubble diameter, d. This is why small bubbles have interfacial viscosity stronger than large bubbles. In the measured result, there can be seen a small influence of SDS percentage. This is not the direct effect of molecular or chemical property of the surfactant, but is a reflection to the depth of the bubble layer on the air–water surface when bubbles are formed before the rotation of the disk. For hydrogen microbubbles (d¼0.05 mm) produced by electrolysis, the shear-thinning stops at the shear rate exceeding 10 s 1.
Fig. 7. Interfacial viscosity of dense bubble layers.
Fig. 8. Effective viscosity of interfacial bubble layers.
Fig. 9. Relative viscosity to water plotted by Capillary number.
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
smaller than the multi-layer situations at high shear rate. This is considered as a feature of two-dimensional rheology of a singlediameter bubble layer. Fig. 9 summarizes all the data in dimensionless parameters using Capillary number (see Eq. (3)) on the abscissa, and relative viscosity to water in the ordinate. Since Capillary number is sufficiently smaller than 10 2 in any case, all the bubbles keep spherical shape. Therefore, the shear-thinning property in this flow configuration does not originate from bubble deformability. It is reasoned by bubble rheology in densely arranged states. The slope in the log–log representation is about 0.45. For small bubbles (2.0 g SDS), the relative viscosity starts at 2600 times great as water. Such a large value is explained only by a high void fraction close to contact limit. If the model of Eq. (4) is assumed to stand extendedly, the measured relative viscosity of 2600 corresponds to α ¼67% in void fraction for a face-centered cubic lattice arrangement that has a mechanical contact limit at α ¼74% (k¼1.35). However, Eq. (4) does not describe any influence of shear-thinning property directly. This yields that we need to attribute the present shear-thinning to the bubble arrangement pattern that transits from regular to random distribution as shear rate increases. In previous literatures on bubbly liquids, the steep shear-thinning trend with such a logarithmic impact has not been observed so far. Our work shows that as the volume fraction of bubbles approaches the packing limit (mechanical interlocking limit), shear thinning is observed. It is noted that the interpretation above is unaffected by the measurement uncertainties estimated for the effective viscosity and the relative viscosity. This is so because 20% of the measurement error for the bubble layer thickness δ, just slightly shifts the viscosity values 8% of a single digit along the ordinate in the log–log representation. Experimental repeatability fully relies on how the bubbles are formed on the water surface before rotation of the disk. Once the rotation starts for the measurement, the thickness and the internal motion of bubble layer reach at steady state with help of surfactant. Thereby our data were classified by the thickness of the bubble layer and the mean diameter of bubbles. On this context, we can conclude that the rheological characteristics of the bubble mixture are properly and validly measured by the present interfacial viscometry. 4.2. Rigid particle interface Following the measurements for bubble layers at the interface, we conducted the same approach for the interfaces with rigid particles. As shown in Fig. 6, the rigid particles tend to be distributed non-uniform. We intentionally chose such conditions, to see the relationship between the interfacial viscosity and the particle distribution pattern in the gap. Fig. 10 shows the measured interfacial viscosity of the particle interfacial layer. The data indicates that the interfacial viscosity starts from a very large value at the lowest shear rate, which is 20 [Pa s m] for the particle concentration, C¼ 0.76. This is apparently due to high contact friction among rigid particles. The increment in shear rate lowers the interfacial viscosity with a slope of about 1 in log–log representation. This remarkable shear-thinning effect originates from fluidization of densely arranged rigid particles from solid contact state [41]. We observed shear localization, i.e. fracture of the particle interfacial layer during the progress of fluidization (see Fig. 6(c)). Thus, the shear-thinning characteristics of the particulate surface are explained by particle clustering phenomenon in mezzo-scale, but not by local arrangement pattern of the spheres. The shear localization allows the fluid to have a large velocity difference where the local viscosity inside the shear localization is the viscosity of the aqueous sub-phase fluid. We can confirm that fluidization of rigid particles occurs non-uniformly as known in the field of powder science [26,42]. In the case of low particle concentrations (C o0.3), the interfacial viscosity takes low
127
Fig. 10. Interfacial viscosity of particulate surface.
values, which is consistently explained by the fluidization process. There observed a significant difference in the interfacial viscosity between C ¼0.20 and C ¼0.22. This is deduced from the interfacial particle behavior that takes different motions as shown in Fig. 6. A wavy boundary of the particulate surface promotes the momentum transfer while a completely stratified distribution produces little contribution to the stress. When the shear rate is increased over 5 s 1, the interfacial viscosity increases. This is mainly due to momentum transfer among the particles activated by two-dimensional particle motions, i.e. onset of the radial velocity component of the particles in the gap. In such a nature of particulate surface, there appear two shear rate values that match the single interfacial viscosity of 10 4 [Pa s m]. The multivalue characteristics support the status that high-shear regions and low-shear regions coexist in the same flow domain, being consistent to the observation of the shear localization. Also, it could be linked to shear banding phenomenon that is originated from multiphase fluid dynamics [43,44]. Concerning reproducibility of the data, we experienced certain initial-condition dependence in terms of the particle distribution in the gap. That is, the particles are randomly arranged as those are provided on the water surface at beginning, but they naturally structure a regular arrangement during the rotation especially in shear rate lower than 10 2 s 1. Therefore, the data in Fig. 10 excludes such initial condition dependence regimes.
5. Conclusions A method to measure the effective viscosity of bubble and particle layers at high interfacial volume fraction has been proposed. The method is based on interfacial rheometry, which was established originally for evaluating the rheological properties of surfactant, protein and particle adsorption layers. We have extended this principle to the measurement of bubbly liquid with a finite depth and particulate surface on liquid. The application of the present method to spherical bubble obtained a remarkable shear-thinning property within the investigated shear rate regime. This property is originated from transition of the bubble arrangement pattern from regularized structure at low shear rate to random arrangement at high shear rate. The effective viscosity decreases steeply whose slope is around 1/2 in the log–log diagram. This is a unique phenomenon when the volume fraction is close to the mechanical packing limit. Another
128
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
application to the interface covered with rigid particles also shows shear-thinning property with a slope of 1, but it also produces shearthickening effect at high shear rate. The shear-thickening behavior of particle layers were consistently explained with observed distributions of the particles. Through this study, we conclude that the present methodology of the effective viscosity measurement is suitably applicable to densely arranged interfacial dispersion and has a future potential to address unsolved rheological mechanism hidden in high-concentrated dispersive multiphase fluid. Acknowledgments The authors are grateful for financial support from the Japan Society for Promotion of Science (JSPS KAKENHI, No. 24246033). We also acknowledge E.J. Windhab, Y. Takeda, Y. Tasaka, Y. Oishi, Y. Aikawa, and T. Kimura for their technical supports and fruitful advises. References [1] Einstein A. Eine neue Bestimmung der Moleküldimensionen. Ann Phys 1906;19:289–306. [2] Batchelor GK. The viscosity of a dilute suspension of small particles: an introduction to fluid dynamics. Cambridge: Cambridge University Press; 1967; 246–255. [3] Schowalter WR, Chaffey CE, Brenner H. Rheological behavior of a dilute emulsion. J Colloid Interface Sci 1968;26:152–60. [4] Choi SJ, Schowalter WR. Rheological properties of nondilute suspensions of deformable particles. Phys Fluids 1975;18:420–7. [5] Pal R. Rheological behavior of bubble-bearing magmas. Earth Planet Sci Lett 2003;207:165–79. [6] Rust AC, Manga M. Effects of bubble deformation in the viscosity of dilute suspensions. J Non-Newtonian Fluid Mech 2002;104:53–63. [7] Müller-Fischer N, Tobler P, Dressler M, Fischer P, Windhab EJ. Single bubble deformation and breakup in simple shear flow. Exp Fluids 2008;45:917–26. [8] Gutam KJ, Mehandia V, Nott PR. Rheometry of granular materials in cylindrical Couette cells: anomalous stress caused by gravity and shear. Phys Fluids 2013;25:070602. [9] Llewellin EW, Manga M. Bubble suspension rheology and implications for conduit flow. J Volcanol Geotherm Res 2005;143:205–17. [10] Murai Y, Oiwa H. Increase of effective viscosity in bubbly liquids from transient bubble deformation. Fluid Dyn Res 2008;40:565–75. [11] Doi M, Ohta T. Dynamics and rheology of complex interfaces I. J Chem Phys 1991;95:1242–7. [12] Tisné P, Aloui F, Doubliez L. Analysis of wall shear stress in wet foam flows using the electrochemical method. Int J Multiphase Flow 2003;29:841–54. [13] Han CD, King RG. Measurement of the rheological properties of concentrated emulsions. J Rheol 1980;24(2):213–37. [14] Briceno MI, Joseph DD. Self-lubricated transport of aqueous foams in horizontal conduits. Int J Multiphase Flow 2003;29:1817–31. [15] Xu Q, Rossen WR. Effective viscosity of foam in periodically constricted tubes. Colloids Surf, A: Physicochem Eng Aspects 2003;216:175–94. [16] Hohler R, Cohen-Addad S. Rheology of liquid foam. J Phys: Condens Matter 2005;17:R1041–69. [17] Cervantes-Martinez A, Saint-Jalmes A, Maldonado A, Langevin D. Effect of cosurfactant on the free-drainage regime of aqueous foams. J Colloid Interface Sci 2005;292:544–7.
[18] Denkov ND, Tcholakova S, Golemanov K, Subramanian V, Lips A. Foam-wall friction: effect of air volume fraction for tangentially immobile bubble surface. Colloids Surf, A: Physicochem Eng Aspects 2006;282–283:329–47. [19] Lespiat R, Hoehler R, Biance A, Cohen-Addad S. Experimental study of foam jets. Phys Fluids 2010;22:033302. [20] Mooney M. The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 1951;6:162–70. [21] Batchelor GK, Green JT. The determination of the bulk stress in a suspension of spherical particles to order c2. J Fluid Mech 1972;56:401–27. [22] Kuwagi K, Ozoe H. Three-dimensional oscillation of bubbly flow in a vertical cylinder. Int J Multiphase Flow 1999;25:175–82. [23] Darton RC, Harrison D. The rise of single gas bubbles in liquid fluidized beds. Trans Inst Chem Eng 1974;52:301–6. [24] Tsuchiya K, Furumoto A, Fan L-S, Zhang J. Suspension viscosity and bubble rise velocity in liquid–solid fluidized beds. Chem Eng Sci 1997;52:3053–66. [25] Morris JF, Katyal B. Microstructure from simulated Brownian suspension flows at large shear rate. Phys Fluids 2002;14:19201937. [26] Morris JF. A review of microstructure in concentrated suspension and its implications for rheology and bulk flow. Rheol Acta 2009;48:909–23. [27] Legner HH. A simple model for gas bubble drag reduction. Phys Fluids 1984;27:2788–90. [28] Sato Y, Sekoguchi K. Liquid velocity distribution in two-phase bubble flow. Int J Multiphase Flow 1975;2:79–95. [29] Olmos E, Gentric C, Midoux N. Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model. Chem Eng Sci 2003;58:2113–21. [30] Murai Y. Frictional drag reduction by bubble injection. Exp Fluids 2014;55:1773 (p28). [31] Housiadas KD, Beris AN. Characteristic scales and drag reduction evaluation in turbulent channel flow of nonconstant viscosity viscoelastic fluids. Phys Fluids 2004;16:1581–6. [32] L’vov VS, Pomyalov A, Procaccia I, Tiberkevich V. Drag reduction by microbubbles in turbulent flows: the limit of minute bubbles. Phys Rev Lett 2005;94 (174502). [33] Fox RO. Large-eddy simulation tools for multiphase flows 2012;44:47–76Annu Rev Fluid Mech 2012;44:47–76. [34] Kang SY, Sangani AS, Tsao HK, Koch DL. Rheology of dense bubble suspensions. Phys Fluids 1997;9(6):1540–61. [35] Yeo K, Maxey MR. Dynamics and rheology of concentrated, finite-Reynoldsnumber suspensions in a homogeneous shear flow. Phys Fluids 2013;25 (053303). [36] Erni P, Fischer P, Windhab EJ, Kusnezov V, Stettin H, Läuger J. Stress- and strain-controlled measurements of interfacial shear viscosity and viscoelasticity at liquid/liquid and gas/liquid interfaces. Rev Sci Instrum 2003;74: 4916–4924. [37] Fischer P, Erni P. Emulsion drops in external flow fields—the role of liquid interfaces. Curr Opin Colloid Interface Sci 2007;12:196–205. [38] Sagis L. Dynamic properties of interfaces in soft matter: experiments and theory. Rev Mod Phys 2011;83:1367–403. [39] Erni P. Deformation modes of complex fluid interfaces. Soft Matter 2011;7:7586–600. [40] Krägel J, Derkatch SR. Interfacial shear rheology. Curr Opin Colloid Interface Sci 2010;15:246–55. [41] Stickel JJ, Powell RL. Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 2005;37:129–49. [42] Tsuji Y, Kawaguchi T, Tanaka T. Discrete particle simulation of twodimensional fluidized bed. Powder Technol 1993;77:79–87. [43] Cromer M, Villet MC, Fredrickson GH, Leal LG. Shear banding in polymer solutions. Phys Fluids 2013;25:051703. [44] Breugem WP. The effective viscosity of a channel-type porous medium. Phys Fluids 2007;19:103104.
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Effective viscosity measurement of interfacial bubble and particle layers at high volume fraction Yuichi Murai a,n, Takahisa Shiratori a, Ichiro Kumagai b, Patrick A. Rühs c, Peter Fischer c a
Laboratory for Flow Control, Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan School of Science and Engineering, Meisei University, Hino 191-8506, Tokyo, Japan c Institute of Food, Nutrition and Health, ETH Zürich, 8092 Zürich, Switzerland b
art ic l e i nf o
a b s t r a c t
Available online 1 November 2014
An experimental method for measuring the effective viscosity of two dimensional dispersion systems is proposed. The method is based on interfacial rheology, which was originally developed to investigate surface active adsorption layers such as protein film formed at liquid–liquid interfaces. Bubbles or rigid particles at around 50% of volume fraction in liquid are positioned in the gap between a rotating disk and a stationary cylindrical container. With this configuration, shear-rate dependent effective viscosities of bubble and particle layers were investigated. Steep shear-thinning property is observed for spherical bubble systems in the shear rate regime from 10 1 to 102 s 1. This is explained by topological transition from regular to random arrangement of the bubbles at the interface. For rigid particle systems, the viscosity starts from high value due to solid contact friction, then decreases sharply due to fluidization process until inter-particle collision lead to an increase of the viscosity. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Bubbles Particles Multiphase flow Rheology Effective viscosity
1. Introduction The effective viscosity of liquids containing bubbles and particles is already an over one century old research topic since its first appearance in literature [1,2]. As the disperse phase is suspended in the continuous liquid phase of viscosity μ0, the effective viscosity of the dilute suspension μ is given by the following formula. ( f ¼ 5=2 ðsolidÞ μ ¼ 1 þ f α; ; ð1Þ f ¼1 ðgasÞ μ0 where α denotes volume fraction of the dispersion phase. This formula is valid for spherical dispersion at small volume fraction, α o0.05. Schowalter et al. [3] and Choi and Schowalter [4] obtained the following formula valid for deformable bubbles and for higher volume fraction 2 μ 1 þ ð1 þ 4αÞ 1 þ ð20=3Þα ð6=5ÞCa 5 ¼ 1 þ α þ α2 ; ð2Þ 2 2 μ0 2 1 þ 1 þð20=3Þα ð6=5ÞCa where Ca denoted the Capillary number defined by Ca ¼
μ0 γ_ r ; σ
n
Correspondence to: Laboratory for Flow Control, Division of Energy & Environmental System, Faculty of Engineering, Hokkaido University, N13W8, Sapporo 060-8628, Japan. Tel.: þ81 11 706 6372; fax: þ 81 11 706 6373. E-mail address: [email protected] (Y. Murai). http://dx.doi.org/10.1016/j.flowmeasinst.2014.10.006 0955-5986/& 2014 Elsevier Ltd. All rights reserved.
ð3Þ
in which σ, γ_ , and r being surface tension coefficient, shear rate, and bubble radius. Temporal transition from spherical bubble regime to shear-yield bubble regime was theoretically modeled by Pal [5]. The validity of these theoretical equations for various bubbly liquids were confirmed with circular Couette flow by Rust and Manga [6], Müller-Fischer et al. [7], and Gutam and Mehandia [8]. The similar trend was reported for pipe flows by Llewellin and Manga [9]. Murai and Oiwa [10] studied the influence of non-equilibrium deformation of bubbles in unsteady shear flow and found significant increase in effective viscosity compared to steady shear rate values. With higher gas volume fractions, the effective viscosity is significantly controlled by the complex liquid film dominating the bulk properties [11,12]. Pronounced viscoelastic properties occur in foam flow, which is influenced by surfactants and electrochemical interfacial properties [13–19]. Back to the early research, Mooney [20] suggested the following formula to describe the increase of effective viscosity at high volume fraction in the case of spheres with non-slip surfaces, i.e. rigid spherical particles: μ Aα 1 ¼ exp ð4Þ ¼ 1 þ Aα þ AðA þ 2kÞα2 þ Oðα3 Þ; 2 1 kα μ0 where A and k denote the dimensionless values, which approximate the measured viscosity and the spatial arrangement. The value A is given by A¼1 for uncontaminated bubbles while A¼ 5/2 for rigid particles and well-contaminated bubbles. The value k is a factor that describes the influence of spatial arrangement pattern of the spheres.
122
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
Theoretically, the value k takes the range of 1.35oko1.91 in accordance with two limits between a face-centered cubic lattice and simple cubic arrangement. Comparison of the polynomial terms in Eq. (4) with Eq. (2) yields that k¼2 was employed in Eq. (2) at Ca¼0, which exceeds the highest limit of k for spherical dispersion. Batchelor and Green [21] theoretically obtained the proportional factor to α2 in case of rigid particle suspension to be 5.2 as the further higher order terms were neglected. This corresponds to k¼ 0.83, which is lower than the above-mentioned lowest limit. A common issue in these theoretical works is uncapability to address the spatial fluctuation of local dispersion arrangement that naturually happens to real systems. Numerical simulations by Kuwagi and Ozoe [22] adapted a representative value of the factor at k¼ 1.43 assuming a random arrangement. From the experiment of Darton and Harrison [23] and Tsuchiya et al. [24], average values of k for particle suspension at high volume fraction (0.45o α o0.55) were k¼ 1.2. This value is, on the other hand, being smaller than the theoretical smallest limit. Different from the theory, the spatial arrangement pattern of the spheres is hardly controlled in experiments. It varies with shear rate like a fluidization process of packed rigid spheres. Such transition of the arrangement produces dramatic shear rate dependence in viscosity whereas deformation of the dispersion is insignificant. Morris and Katyal [25] and Morris [26] found strong shear-thickening as a result of inhomogeneous particle distribution, which is triggered by direct contact among rigid spheres. Any case in the above previous works deals with 3D system of dispersion distribution subject to shear rate. Since 3D systems allow individual dispersions to move in three directions, shear-dependent viscosity correlates to transitions of three-dimensional arrangement pattern of dispersions. In contrast, rheological property of dispersion system confined in two dimensions has not been studied experimentally yet. In the present research, we investigate the effective viscosity of interfacial bubble and particle layers at high interfacial volume fraction, i.e. from 0.2 to the packing limit (inter-lock condition of spheres). We focus on spherical shapes of the bubbles where deformability is almost negligible, thus Cao0.05. These conditions are attractive to engineers who aim at flow control by means of small dispersions [27–29]. Especially, in turbulent boundary layers, the dispersions interact with coherent structures in turbulence and form clusters within the layers. The clusters can have local volume fraction much higher than 0.2 and often reach the closest packing limit. This happens because strong congregation force acts on the dispersion due to steep local pressure gradient within coherent structures [30]. Thus, effective viscosity of dispersion at high volume fraction takes a primary issue rather in turbulences than in laminar flows [31–33]. A theoretical and numerical approach to such densely arranged spheres in simplified flow geometry was reported by Kang et al. [34]. Direct numerical simulation by Yeo and Maxey [35] showed particles selfdiffusion which results in shear-thickening characteristics. In this paper, we desribe a new methodology to acquire the effective viscosity of interfacial bubble and particle layers at high interfacial volume fraction. The measurement principle is based on
interfacial rheometry which was established for shear viscosity assessment of interfacial adsorption layers. General configuration and application of interfacial rheometry is reported elsewhere [36– 40]. The original purpose of the method is to quantify the rheological properties of a very thin material layers formed at liquid interfaces such as a protein films and surfactant adsorption layers. The present study extends this technique to the measurement of the viscosity of bubble/particle multiphase layers just by changing the shape of a rotating cone. In this article, the working principle of the present rheometry, the advantages in experimental handling of bubble/particle suspension layers, and their applications to bubble/ particle multiphase layer are reported accompanied with a brief discussion on the measured results.
2. Interfacial rheometry Fig. 1 shows the schematic diagrams of the experimental setups. The measurement system is constructed by the combination of a rotating disk attached to a commercial rheometer (Physica MCR300, Anton Paar) and a cylindrical fluid container. The setup (a) is used to measure the interfacial rheology of adsorption layers. The working principle of interfacial rheology is reviewed in the literature [36]. The setup (b) used in the present study is applicable for bubble and particle layer with a finite thickness. In both types, shear flow is driven by a rotating disk that is separately installed from a stationary cylindrical container. Viscosity of the sub-phase liquid, μ0, filled in the container is accounted for in the full flow analysis performed by the rheometer software [36]. In the setup (b), the structure of the fluid interface is additionally observed and recorded by a video camera (Sony DFW V500) from the top and the side of the container. With this combination, spatial distribution and motion of floating dispersions are monitored to find their correlations to bulk viscosity to be measured. The dimensions, R1, R2, H, and δ in Fig. 1 can be modified in accordance with the target material of measurement as well as the range of shear rate to be studied. 2.1. Original interfacial rheometry Through a normal force transducer the biconical disk is positioned exactly at the interface. A ring-shaped gap between the rotating disk and the cylindrical container forms the free surface area to be measured. When the disk rotates, a circular Couette flow on the liquid surface within the gap is induced. In standard interfacial rheometry, the sharp edge of a rotating biconical disk drives the shear flow as shown in Fig. 1(a). The thickness of the material is ordinarily treated as unknown since this methodology is designed for adsorption layers with molecular dimensions. The interfacial viscosity has the dimension of [Pa s m]. Only in the case that the target layer is uniformly formed with a constant film thickness, the length scale multiplied in the dimension coincides with the film thickness. Dependent on the target material (e.g. surfactant, protein), the film
Fig. 1. Schematic diagram of the measurement system based on interfacial rheometry. Target fluid is set in a thin layer on the surface of liquid with known viscosity. (a) Rotating bi-cone with sharp edge for measurement of e.g. protein adsorption layers. (b) Rotating conical disk with a step for measurement of bubble/particle multiphase layer systems.
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
thickness shows spatio-temporal evolution, to which thermal, chemical, and electrical influences on the film property should be considered in a given environment. The definition of the interfacial viscosity allows one to include all these influences into a single value. 2.2. Interfacial rheometry for bubble/particle multiphase layer The setup (b) in Fig. 1 is the modified version adopted for the present study. The pointed edge of the biconial geometry is replaced by a vertical step which is higher than the thickness of the target layer δ. Dispersions (bubbles or particles) are adjusted to the height. By rotation, the dispersions move in the azimuthal direction so that a two-phase circular Couette flow is generated between the rotating cone and the cylindrical container. The target height δ is controlled to be within several diameters of the bubbles or particles. This allows the dispersion to behave nearly in two dimensions, i.e. dispersion motion is restricted in the horizontal plane. Additional advantage of this system is that the spatial arrangement of the dispersions can be directly observed from the top of the instrument. This is one of the most distinguishable points from the other types of rheometry for multiphase fluids purposed for high dispersion volume fraction. As the interfacial viscosity η is measured, the effective viscosity is estimated simply by division with the thickness of the dispersion layer, i.e.
μ¼
η δ
ð5Þ
This operation is valid only in the case that δ is controlled constant without change in both space and time. As δ is thin enough, η offers a new property of multiphase-layer of which internal motion of dispersion is restricted in two dimensions.
3. Bubble and sphere interfacial layer The present interfacial viscometry has been applied to two kinds of interfacial dispersions, i.e. bubbles and particulate layers floating at the air–water interface. There are two roles in buoyancy of dispersions, in which this setup is operated. First, the buoyancy keeps the dispersion layer stably at the initially set position. Second, the buoyancy accumulates dispersions densely inside the layer. For testing rigid particles, light rigid spheres of density smaller than water are used to maintain the interfacial particle layer.
123
For bubbles larger than 0.5 mm in diameter, we adopted hydrolysis through porous plates to generate bubbles into the cylindrical container. To change the mean bubble diameter from 0.5 mm to 3 mm, six different types of porous plates with different porosities were examined. A surfactant, SDS, was used to stabilize the generated bubbles and up to 2.0 g was added to the continuous aqueous phase. The surfactant helps us with generating mono-dispersed bubbles as well as to maintain their stability in shearing environment. In each type of porous plate, the amount of the surfactant sways the number of bubble layers, which influences the rheological properties as discussed in the next chapter. The diameter of bubbles was measured by video images of the top views. For microbubble experiments, we utilized water electrolysis which produces microbubbles less than 0.1 mm in mean diameter. The combined set-up with the electrolysis is shown in Fig. 2. In the target layer, hydrogen bubbles are accumulated due to their own buoyancy. The diameter of microbubbles was measured by photography enlarged around the cathode wire. Fig. 3 shows a typical arrangement pattern of spherical bubbles generated by a coarse porous plate with hydrolysis. The arrangement pattern is regulated in two dimensions on the water surface, and has honeycomb-like structures in the stationary state. Fig. 4 displays closer looks of the bubble arrangement as the mean bubble diameter changes. As shown in (a), bubbles with a mean diameter of 2.5 mm are regularly arranged with four to five rows in the gap between the rotating disk and the stationary outer cylinder. This regularized structure disappears for small bubbles (0.8 mm) as shown in (b). As microbubbles (d¼ 0.05 mm) are generated, milky clouds of microbubbles occupy the gap as shown in (c). On all the three conditions, we recorded the video images of the top view and confirmed no significant vanish of bubbles during the rotating operation which is less than one minute to perform the effective viscosity measurement. Also we confirmed that bubble motion inside the gap follows to a simple shear flow between the rotating disk and the stationary container. There was no shear localization observed such as in the vicinity of wall. Furthermore, we confirmed that the amount of SDS mainly affects the vertical thickness of bubble layer in steady rotating state. Since the surfactant itself does not significantly change the
3.1. Bubble interfacial layer The effective viscosity of spherical gas bubble arrangement is measured. By changing the method of bubble generation, three different bubble-layers are examined: large bubbles, small bubbles, and microbubbles. The details are summarized in Table 1. In all the conditions, capillary number defined by Eq. (3) is sufficiently smaller than unity, i.e. the maximum value is 0.05 at the highest shear rate. This means that bubbles are all spherical as they are dispersed freely.
Fig. 2. Schematic diagram of the measurement system coupled with water electrolysis for microbubble experiment. Platinum wire cathode set close to the bottom wall of the container produces hydrogen bubbles that naturally rise and reach the water surface to form the microbubble layer in the test section.
Table 1 Bubble-layer conditions. R1 R2
40 mm 50 mm
(See Fig. 1(a) and Fig. 2) (See Fig. 1(a) and Fig. 2)
Classification Large bubble Small bubble Microbubble
Gas Air Air Hydrogen
Bubble diameter d(σd)n 2.50 mm ( 0.07 mm) 0.80 mm ( 0.09 mm) 0.05 mm ( 0.02 mm)
n
d and σd denote arithmetic mean bubble diameter and standard deviation.
Method of bubble generation Hydrolysis with a coarse porous plate Hydrolysis with a fine porous plate Electrolysis (Cathode)
124
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
Fig. 3. Top view of spherical bubble arrangement formed on the water surface in the case of mean bubble diameter, d ¼2.5 mm. (a) Two-dimensional arrangement close to sphere’s packing limit observed in the container before installation of the rotating disk. (b) The same arrangement realized in the test section after installation of the rotating disk.
Fig. 4. Top view of local bubble arrangement patterns for three different conditions in bubble size. (a) Large bubble at d¼ 2.5 mm, (b) small bubbles at d ¼ 0.8 mm, and (c) microbubbles at d ¼ 0.05 mm.
viscosity of water, the interfacial viscosity of bubble layer can be attributed to the rheological property of bubble interfaces that exist densely within the bubble layer. Fig. 5 shows the side views of the bubble arrangement photographed during the rotation. In experiments for the bubbles larger than 1.0 mm, the number of bubble layers in the vertical direction is countable as shown in (a) to (d) since the arrangement is regularly stratified. For the bubbles smaller than 1.0 mm, the number of the rows increases more than four and their arrangement pattern starts to be irregular with three dimensional motions. We measure the total depth of the bubble layer, δ, from these video images, and use Eq. (5) to obtain the effective viscosity. 3.2. Rigid particle interfacial layer For rigid particle interfaces, light particles with density smaller than water were used. Therefore, the particles float on the water surface. The name of the particle is FLO-BEADS (CL-2507, Sumitomo Seika, Ltd), which is made of polyethylene formed in sphere. The particles are close to spherical, and have a high wettability. Three conditions in the particle distribution patterns are focused on in this demonstration as summarized in Table 2. The areal concentration called here is defined by the ratio of particle projection area to the
total area of the gap. It is measured from photography, of which samples are shown in Fig. 6. The areal concentration, C, is computed from the single particle area multiplied by the number of particles in a sampling area, using image processing software to detect individual particle centers. The particle areal concentration less than the packing limit always show inhomogeneous distribution in any geometry as long as the particles are neutrally distributed without mutual repulsion force. The bulk interfacial viscosity in such a situation is the present target of investigation. The condition (a) is low areal concentration at which the particles show stratified distribution in the radial direction. The radial stratification cannot be explained simply by the inertia effect (i.e. centrifugal acceleration), but should be coupled with particle-clustering rheology, which will be a point of discussion. The condition (b) has a slight difference in particle concentration but shows a wavy boundary of the particulate surface. On the condition (c), spatially continuous shearing motion of the particulate surface takes place entirely in the gap, allowing local smooth slip among particles. With further increase in the particle areal concentration, thin slits of particulate surface come out to serve large velocity differences between two neighbors as shown in (d). According to the video images, the slit moves unsteadily, and accompanies wavy fluctuation in the azimuthal direction. Under
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
125
Fig. 5. Side views of bubble arrangement patterns photographed from outside the cylindrical container. (a) One layer at d ¼ 2.5 mm, (b) two layers at d ¼ 1.8 mm, (c) three layers at d ¼ 1.6 mm, (d) four layers at d ¼ 1.3 mm, (e) five-to-six layers at d ¼0.7 mm, (f) more than six layers at d ¼ 0.7 mm.
Table 2 Particle-layering conditions. R1 R2 Density of rigid particle Diameter of rigid particle Condition (a) (b) (c) (d)
34 mm 44 mm 919 kg/m3 0.18 mm (with standard deviation of 0.01 mm) Areal concentration 0.20 0.22 0.43 0.76
Fig. 6. Top views of rigid particle distributions for four different concentrations.
(See Fig. 1(b)) (See Fig. 1(b))
Trend of particle distribution pattern Radial binary stratification Radial stratification with wavy boundary Global fluidized distribution Solidized surface separated by slit(s)
126
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
these curious conditions, we present in the next chapter how the bulk interfacial viscosity behaves with the increment in shear rate.
4. Results and discussions In this chapter, the measured relationship between the interfacial viscosity and the shear rate is presented. For each condition of dispersion layer, a brief discussion is given on the measured results. 4.1. Dense bubble layer Fig. 7 presents the interfacial viscosity of dense bubble layers in all the conditions. In most cases the interfacial viscosity declines monotonically with increase in shear rate. Thus, the dense bubble layer has a clear shear-thinning property. For the tested range in shear rate from 0.1 o 100 s 1, the magnitude of the interfacial viscosity changes over one digit. The highest interfacial viscosity is measured for small bubbles (d¼0.8 mm) while the layer of large bubbles (d¼2.5 mm) takes much lower values. This trend should be explained by multiple factors; one of them is the difference in bubble arrangement pattern as shown in Fig. 4. The correspondence to the photographs implies that a regular alignment of bubble rows produces shear stress smaller than the case of random arrangement. Another factor that we need to consider is interfacial area concentration, β, which is defined by A V
β¼ ¼
n πd 6α ¼ ; 2V d 2
α¼
n πd : 6V
Fig. 8 shows the same data evaluated by effective viscosity. The effective viscosity is calculated by Eq. (5), using the measured thickness of bubble layer. Since the thickness little changes with shear rate, the shear-thinning trend is obtained almost similarly to the previous graph on the interfacial viscosity. One of findings from this result is that small bubbles (d ¼ 0.8 mm) and microbubbles (d ¼0.05 mm) have large effective viscosity compared to large bubbles owing to the increase in the interfacial area concentration. Among them, the small bubbles of 1.0 g addition of SDS take the highest effective viscosity. On this condition, we observed from the side view camera that the number of bubble rows ranges from 2 to 3 in the vertical direction. This infers that the individual motion of bubbles within the bubble layer is regularly restricted in two dimensions, and such a state produces a shear stress higher than the case of 3D motion allowed in a thick bubble layer. For large bubble conditions (d ¼2.5 mm), single-layer arrangement (1.0 g SDS) creates higher effective viscosity at the low shear rate range, but turns into steep shear-thinning to have the viscosity
3
ð6Þ
where A and V are total area of bubble surface inside the bubble layer, and volume of the bubble layer. The interfacial area concentration describes the potential magnitude of surface tension of individual bubbles, which takes place as a strong shear resistance in the case of densely arranged bubbles. β is proportional to void fraction, α, but is inverse proportional to bubble diameter, d. This is why small bubbles have interfacial viscosity stronger than large bubbles. In the measured result, there can be seen a small influence of SDS percentage. This is not the direct effect of molecular or chemical property of the surfactant, but is a reflection to the depth of the bubble layer on the air–water surface when bubbles are formed before the rotation of the disk. For hydrogen microbubbles (d¼0.05 mm) produced by electrolysis, the shear-thinning stops at the shear rate exceeding 10 s 1.
Fig. 7. Interfacial viscosity of dense bubble layers.
Fig. 8. Effective viscosity of interfacial bubble layers.
Fig. 9. Relative viscosity to water plotted by Capillary number.
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
smaller than the multi-layer situations at high shear rate. This is considered as a feature of two-dimensional rheology of a singlediameter bubble layer. Fig. 9 summarizes all the data in dimensionless parameters using Capillary number (see Eq. (3)) on the abscissa, and relative viscosity to water in the ordinate. Since Capillary number is sufficiently smaller than 10 2 in any case, all the bubbles keep spherical shape. Therefore, the shear-thinning property in this flow configuration does not originate from bubble deformability. It is reasoned by bubble rheology in densely arranged states. The slope in the log–log representation is about 0.45. For small bubbles (2.0 g SDS), the relative viscosity starts at 2600 times great as water. Such a large value is explained only by a high void fraction close to contact limit. If the model of Eq. (4) is assumed to stand extendedly, the measured relative viscosity of 2600 corresponds to α ¼67% in void fraction for a face-centered cubic lattice arrangement that has a mechanical contact limit at α ¼74% (k¼1.35). However, Eq. (4) does not describe any influence of shear-thinning property directly. This yields that we need to attribute the present shear-thinning to the bubble arrangement pattern that transits from regular to random distribution as shear rate increases. In previous literatures on bubbly liquids, the steep shear-thinning trend with such a logarithmic impact has not been observed so far. Our work shows that as the volume fraction of bubbles approaches the packing limit (mechanical interlocking limit), shear thinning is observed. It is noted that the interpretation above is unaffected by the measurement uncertainties estimated for the effective viscosity and the relative viscosity. This is so because 20% of the measurement error for the bubble layer thickness δ, just slightly shifts the viscosity values 8% of a single digit along the ordinate in the log–log representation. Experimental repeatability fully relies on how the bubbles are formed on the water surface before rotation of the disk. Once the rotation starts for the measurement, the thickness and the internal motion of bubble layer reach at steady state with help of surfactant. Thereby our data were classified by the thickness of the bubble layer and the mean diameter of bubbles. On this context, we can conclude that the rheological characteristics of the bubble mixture are properly and validly measured by the present interfacial viscometry. 4.2. Rigid particle interface Following the measurements for bubble layers at the interface, we conducted the same approach for the interfaces with rigid particles. As shown in Fig. 6, the rigid particles tend to be distributed non-uniform. We intentionally chose such conditions, to see the relationship between the interfacial viscosity and the particle distribution pattern in the gap. Fig. 10 shows the measured interfacial viscosity of the particle interfacial layer. The data indicates that the interfacial viscosity starts from a very large value at the lowest shear rate, which is 20 [Pa s m] for the particle concentration, C¼ 0.76. This is apparently due to high contact friction among rigid particles. The increment in shear rate lowers the interfacial viscosity with a slope of about 1 in log–log representation. This remarkable shear-thinning effect originates from fluidization of densely arranged rigid particles from solid contact state [41]. We observed shear localization, i.e. fracture of the particle interfacial layer during the progress of fluidization (see Fig. 6(c)). Thus, the shear-thinning characteristics of the particulate surface are explained by particle clustering phenomenon in mezzo-scale, but not by local arrangement pattern of the spheres. The shear localization allows the fluid to have a large velocity difference where the local viscosity inside the shear localization is the viscosity of the aqueous sub-phase fluid. We can confirm that fluidization of rigid particles occurs non-uniformly as known in the field of powder science [26,42]. In the case of low particle concentrations (C o0.3), the interfacial viscosity takes low
127
Fig. 10. Interfacial viscosity of particulate surface.
values, which is consistently explained by the fluidization process. There observed a significant difference in the interfacial viscosity between C ¼0.20 and C ¼0.22. This is deduced from the interfacial particle behavior that takes different motions as shown in Fig. 6. A wavy boundary of the particulate surface promotes the momentum transfer while a completely stratified distribution produces little contribution to the stress. When the shear rate is increased over 5 s 1, the interfacial viscosity increases. This is mainly due to momentum transfer among the particles activated by two-dimensional particle motions, i.e. onset of the radial velocity component of the particles in the gap. In such a nature of particulate surface, there appear two shear rate values that match the single interfacial viscosity of 10 4 [Pa s m]. The multivalue characteristics support the status that high-shear regions and low-shear regions coexist in the same flow domain, being consistent to the observation of the shear localization. Also, it could be linked to shear banding phenomenon that is originated from multiphase fluid dynamics [43,44]. Concerning reproducibility of the data, we experienced certain initial-condition dependence in terms of the particle distribution in the gap. That is, the particles are randomly arranged as those are provided on the water surface at beginning, but they naturally structure a regular arrangement during the rotation especially in shear rate lower than 10 2 s 1. Therefore, the data in Fig. 10 excludes such initial condition dependence regimes.
5. Conclusions A method to measure the effective viscosity of bubble and particle layers at high interfacial volume fraction has been proposed. The method is based on interfacial rheometry, which was established originally for evaluating the rheological properties of surfactant, protein and particle adsorption layers. We have extended this principle to the measurement of bubbly liquid with a finite depth and particulate surface on liquid. The application of the present method to spherical bubble obtained a remarkable shear-thinning property within the investigated shear rate regime. This property is originated from transition of the bubble arrangement pattern from regularized structure at low shear rate to random arrangement at high shear rate. The effective viscosity decreases steeply whose slope is around 1/2 in the log–log diagram. This is a unique phenomenon when the volume fraction is close to the mechanical packing limit. Another
128
Y. Murai et al. / Flow Measurement and Instrumentation 41 (2015) 121–128
application to the interface covered with rigid particles also shows shear-thinning property with a slope of 1, but it also produces shearthickening effect at high shear rate. The shear-thickening behavior of particle layers were consistently explained with observed distributions of the particles. Through this study, we conclude that the present methodology of the effective viscosity measurement is suitably applicable to densely arranged interfacial dispersion and has a future potential to address unsolved rheological mechanism hidden in high-concentrated dispersive multiphase fluid. Acknowledgments The authors are grateful for financial support from the Japan Society for Promotion of Science (JSPS KAKENHI, No. 24246033). We also acknowledge E.J. Windhab, Y. Takeda, Y. Tasaka, Y. Oishi, Y. Aikawa, and T. Kimura for their technical supports and fruitful advises. References [1] Einstein A. Eine neue Bestimmung der Moleküldimensionen. Ann Phys 1906;19:289–306. [2] Batchelor GK. The viscosity of a dilute suspension of small particles: an introduction to fluid dynamics. Cambridge: Cambridge University Press; 1967; 246–255. [3] Schowalter WR, Chaffey CE, Brenner H. Rheological behavior of a dilute emulsion. J Colloid Interface Sci 1968;26:152–60. [4] Choi SJ, Schowalter WR. Rheological properties of nondilute suspensions of deformable particles. Phys Fluids 1975;18:420–7. [5] Pal R. Rheological behavior of bubble-bearing magmas. Earth Planet Sci Lett 2003;207:165–79. [6] Rust AC, Manga M. Effects of bubble deformation in the viscosity of dilute suspensions. J Non-Newtonian Fluid Mech 2002;104:53–63. [7] Müller-Fischer N, Tobler P, Dressler M, Fischer P, Windhab EJ. Single bubble deformation and breakup in simple shear flow. Exp Fluids 2008;45:917–26. [8] Gutam KJ, Mehandia V, Nott PR. Rheometry of granular materials in cylindrical Couette cells: anomalous stress caused by gravity and shear. Phys Fluids 2013;25:070602. [9] Llewellin EW, Manga M. Bubble suspension rheology and implications for conduit flow. J Volcanol Geotherm Res 2005;143:205–17. [10] Murai Y, Oiwa H. Increase of effective viscosity in bubbly liquids from transient bubble deformation. Fluid Dyn Res 2008;40:565–75. [11] Doi M, Ohta T. Dynamics and rheology of complex interfaces I. J Chem Phys 1991;95:1242–7. [12] Tisné P, Aloui F, Doubliez L. Analysis of wall shear stress in wet foam flows using the electrochemical method. Int J Multiphase Flow 2003;29:841–54. [13] Han CD, King RG. Measurement of the rheological properties of concentrated emulsions. J Rheol 1980;24(2):213–37. [14] Briceno MI, Joseph DD. Self-lubricated transport of aqueous foams in horizontal conduits. Int J Multiphase Flow 2003;29:1817–31. [15] Xu Q, Rossen WR. Effective viscosity of foam in periodically constricted tubes. Colloids Surf, A: Physicochem Eng Aspects 2003;216:175–94. [16] Hohler R, Cohen-Addad S. Rheology of liquid foam. J Phys: Condens Matter 2005;17:R1041–69. [17] Cervantes-Martinez A, Saint-Jalmes A, Maldonado A, Langevin D. Effect of cosurfactant on the free-drainage regime of aqueous foams. J Colloid Interface Sci 2005;292:544–7.
[18] Denkov ND, Tcholakova S, Golemanov K, Subramanian V, Lips A. Foam-wall friction: effect of air volume fraction for tangentially immobile bubble surface. Colloids Surf, A: Physicochem Eng Aspects 2006;282–283:329–47. [19] Lespiat R, Hoehler R, Biance A, Cohen-Addad S. Experimental study of foam jets. Phys Fluids 2010;22:033302. [20] Mooney M. The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 1951;6:162–70. [21] Batchelor GK, Green JT. The determination of the bulk stress in a suspension of spherical particles to order c2. J Fluid Mech 1972;56:401–27. [22] Kuwagi K, Ozoe H. Three-dimensional oscillation of bubbly flow in a vertical cylinder. Int J Multiphase Flow 1999;25:175–82. [23] Darton RC, Harrison D. The rise of single gas bubbles in liquid fluidized beds. Trans Inst Chem Eng 1974;52:301–6. [24] Tsuchiya K, Furumoto A, Fan L-S, Zhang J. Suspension viscosity and bubble rise velocity in liquid–solid fluidized beds. Chem Eng Sci 1997;52:3053–66. [25] Morris JF, Katyal B. Microstructure from simulated Brownian suspension flows at large shear rate. Phys Fluids 2002;14:19201937. [26] Morris JF. A review of microstructure in concentrated suspension and its implications for rheology and bulk flow. Rheol Acta 2009;48:909–23. [27] Legner HH. A simple model for gas bubble drag reduction. Phys Fluids 1984;27:2788–90. [28] Sato Y, Sekoguchi K. Liquid velocity distribution in two-phase bubble flow. Int J Multiphase Flow 1975;2:79–95. [29] Olmos E, Gentric C, Midoux N. Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model. Chem Eng Sci 2003;58:2113–21. [30] Murai Y. Frictional drag reduction by bubble injection. Exp Fluids 2014;55:1773 (p28). [31] Housiadas KD, Beris AN. Characteristic scales and drag reduction evaluation in turbulent channel flow of nonconstant viscosity viscoelastic fluids. Phys Fluids 2004;16:1581–6. [32] L’vov VS, Pomyalov A, Procaccia I, Tiberkevich V. Drag reduction by microbubbles in turbulent flows: the limit of minute bubbles. Phys Rev Lett 2005;94 (174502). [33] Fox RO. Large-eddy simulation tools for multiphase flows 2012;44:47–76Annu Rev Fluid Mech 2012;44:47–76. [34] Kang SY, Sangani AS, Tsao HK, Koch DL. Rheology of dense bubble suspensions. Phys Fluids 1997;9(6):1540–61. [35] Yeo K, Maxey MR. Dynamics and rheology of concentrated, finite-Reynoldsnumber suspensions in a homogeneous shear flow. Phys Fluids 2013;25 (053303). [36] Erni P, Fischer P, Windhab EJ, Kusnezov V, Stettin H, Läuger J. Stress- and strain-controlled measurements of interfacial shear viscosity and viscoelasticity at liquid/liquid and gas/liquid interfaces. Rev Sci Instrum 2003;74: 4916–4924. [37] Fischer P, Erni P. Emulsion drops in external flow fields—the role of liquid interfaces. Curr Opin Colloid Interface Sci 2007;12:196–205. [38] Sagis L. Dynamic properties of interfaces in soft matter: experiments and theory. Rev Mod Phys 2011;83:1367–403. [39] Erni P. Deformation modes of complex fluid interfaces. Soft Matter 2011;7:7586–600. [40] Krägel J, Derkatch SR. Interfacial shear rheology. Curr Opin Colloid Interface Sci 2010;15:246–55. [41] Stickel JJ, Powell RL. Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 2005;37:129–49. [42] Tsuji Y, Kawaguchi T, Tanaka T. Discrete particle simulation of twodimensional fluidized bed. Powder Technol 1993;77:79–87. [43] Cromer M, Villet MC, Fredrickson GH, Leal LG. Shear banding in polymer solutions. Phys Fluids 2013;25:051703. [44] Breugem WP. The effective viscosity of a channel-type porous medium. Phys Fluids 2007;19:103104.