Effective viscosity measurement of interfacial bubble and particle layers at high volume fraction

https://doi.org/10.1016/j.flowmeasinst.2014.10.006Get rights and content

Highlights

  • A new concept for measuring the viscosity of bubbly liquid that has a high void fraction, close to the packing limit has been suggested and applied.

  • Effective viscosity of bubbly liquid in three different classes of bubble size from, microbubble to large bubbles has been obtained, which are much larger than those, measured in dilute situations.

  • The viscosity of a thin layer consisted of solid particles has been also measured, which, has shown intrinsic flow curve that is provided by shear localization events.

Abstract

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.

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.μμ0=1+fα,{f=5/2(solid)f=1(gas),where α denotes volume fraction of the dispersion phase. This formula is valid for spherical dispersion at small volume fraction, α<0.05. Schowalter et al. [3] and Choi and Schowalter [4] obtained the following formula valid for deformable bubbles and for higher volume fractionμμ0=1+(1+4α)(1+(20/3)α)((6/5)Ca)21+(1+(20/3)α)2((6/5)Ca)2(1+α+52α2),where Ca denoted the Capillary number defined byCa=μ0γ̇rσ,in which σ, γ̇, 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], [14], [15], [16], [17], [18], [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:μμ0=exp(Aα1kα)=1+Aα+12A(A+2k)α2+O(α3),where A and k denote the dimensionless values, which approximate the measured viscosity and the spatial arrangement. The value A is given by A=1 for uncontaminated bubbles while A=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. Theoretically, the value k takes the range of 1.35<k<1.91 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 k=2 was employed in Eq. (2) at Ca=0, 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 k=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 k=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 (0.45<α<0.55) were k=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 Ca<0.05. These conditions are attractive to engineers who aim at flow control by means of small dispersions [27], [28], [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], [32], [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 self-diffusion 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], [37], [38], [39], [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.

Section snippets

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

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

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.

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

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.

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