1-s2.0-S0093641303000673-main

MECHANICS

Mechanics Research Communications 30 (2003) 651–662

www.elsevier.com/locate/mechrescom

RESEARCH COMMUNICATIONS

Drag forces of interacting spheres in power-law fluids

Chao Zhu a,*, Kit Lam b, Hoi-Hung Chu b, Xu-Dong Tang c, Guangliang Liu a

a Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USAb Department of Mechanical Engineering, Hong Kong Polytechnic University, Hong Kongc Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China

Received 18 December 2001

Abstract

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag

forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles

include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the

particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were

used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6

to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a

pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the

power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly

depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a

particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.

� 2003 Published by Elsevier Ltd.

1. Introduction

An extensive and rapidly increased number of industrial applications of multiphase flows involve particle

motions in non-Newtonian flows. Typical cases are exemplified by crude oil flows with rocks, sands or

natural gas; bubble entrainment and migration in plastic casting processes; polymeric flows with catalytic

particles; biofluid flows in three-phase fluidized beds; and aseptic processing of particulate food in liquid.All of the above applications call for an in-depth understanding of particle dynamics in non-Newtonian

flows, especially the information of drag forces of interacting particles.

The earliest theoretical work on the force experienced by a sphere suspended in a Newtonian fluid was

dated back in 1901 when Stokes (1901) rigorously derived the famous Stokes� law to calculate the drag force

* Corresponding author. Tel.: +1-973-642-7624; fax: +1-973-642-4282.

E-mail address: [email protected] (C. Zhu).

0093-6413/$ - see front matter � 2003 Published by Elsevier Ltd.doi:10.1016/S0093-6413(03)00067-3

mail to: [email protected]

652 C. Zhu et al. / Mechanics Research Communications 30 (2003) 651–662

on a single rigid sphere in an unbounded creeping Newtonian flow. Drag forces of multi-spheres in a

viscous fluid flow have been extensively investigated. Stimson and Jeffery (1926) presented a complete

solution for the slow motion of two spheres parallel to their line of centers in an unbounded viscous fluid

while Kynch (1958) delivered a theoretical treatment of Stokes motions of three or more spheres. Payne andPell (1960) solved the Stokes equations for the slow motions of several axially symmetric bodies in an

unbounded viscous fluid. Zick and Homsy (1982) numerically investigated the drag forces on the simple

and periodic arrays of spheres in a Stokes flow by formulating the problem as a set of two-dimensional

integral equations. In order to account for the wall effect, Greenstein (1980) numerically computed the drag

forces on two spherical particles translating in a cylindrical tube filled with an incompressible viscous fluid.

From his calculation, interaction and wall correction factors based on the distance between particles and

the distance between the particles and the tube wall were proposed.

Kaneda and Ishii (1982) pointed that, if particles are randomly located in a particulate fluid system, themost important hydrodynamic interactions are those between a pair of particles. However, in the case of a

pair of particles sedimenting vertically one above the other in an unbounded fluid, the difference between

the forces on the leading and trailing spheres cannot be explained by an analysis based on the Stokes

equation. Hence, Kaneda and Ishii further proposed an asymptotic analysis to treat the hydrodynamic

interaction of two spheres moving in an unbounded fluid at small but finite Reynolds number, in which the

inertia effect was taken into account. However, these treatments can only be applied to cases with large the

separation distance between particles in small Reynolds numbers. It is realized that the typical distance of

strong interaction of a pair of particles is less than twice particle diameter (Zhu et al., 1994). In addition, formost applications in particulate multiphase flows, the particle Reynolds number based on the isolated-

particle terminal velocity and particle diameter is typically in a range from tens to several hundreds. For

example, at a particle Reynolds number from 10 to 200, the corresponding sizes of glass beads vary from

230 lm to 1.1 mm in water and from 130 to 650 lm in air. Hence, the Reynolds number range of practicalsignificance to a multiphase flow system may be far beyond the Stokes regime. For such a case, inertia effect

and wake effect must be taken into account.

Happel and Pfeffer (1960) experimentally investigated on the two spheres falling in viscous liquids. They

observed that the terminal velocity of two spheres was greater than that of an isolated sphere, whichsuggests that the drag force of any one of the two spheres is reduced by the sphere interaction. Realizing the

importance of the direct measurements of drag force of interacting particles, Rowe and Henwood (1961),

Lee (1979) and Tsuji et al. (1982) used the pendulum method and water channel flow to measure the drag

force at a particle Reynolds number ranged from 500 to 10,000. However, the data were so scattered that

only a general trend of the interactions could be reflected. Zhu et al. (1994) developed a micro-force

measuring system to directly measure the drag forces on two interacting particles at a particle Reynolds

number varying from 20 to 130. It was found that the particle Reynolds number affects not only the

magnitude of the drag forces of an interacting particle but also its variation with the separation distance. Byusing Zhu�s experimental technique, Liang et al. (1996) and Chen and Wu (2000) have also measured thedrag forces of interacting spheres in Newtonian fluids.

The early study of particle dynamics in non-Newtonian flows can be dated back in early 1960s. However,

the study of particulate non-Newtonian multiphase flows was not very active until 1990s. Slattery and Bird

(1961) performed terminal velocity experiments of single spheres in various sized tubes filled with carbo-

xymethycellulose (CMC) solutions to determine the drag coefficients of a single sphere in such kinds of pipe

flows. Acharya et al. (1976) presented an approximated momentum integral boundary layer analysis to

determine the drag coefficient of a slow moving sphere in the creeping flow regime through a power-lawnon-Newtonian fluid in the presence of a flat wall. An active yet extensive study of a single sphere in various

non-Newtonian fluids can be found in several recent reports (Briscoe et al., 1993; Graham and Jones, 1994;

Ribeiro et al., 1994; Machac and Lecjaks, 1995). Settling of a single but non-spherical particle in a non-

Newtonian fluid was also reported (Madhav and Chhabra, 1994; Feng et al., 1995).

C. Zhu et al. / Mechanics Research Communications 30 (2003) 651–662 653

Similar to Newtonian multiphase flows, groups of multi-particles do not behave like those of isolated

particles unless the flow is extremely dilute. Hence the study of the multi-particle group behavior and

hydrodynamic interactions among particles is of great importance. Kawase and Ulbrecht (1981) used a free

surface model and the boundary layer theory to estimate the motion of an assemblage of spheres moving ina power-law non-Newtonian fluid at high Reynolds numbers. Staish and Zhu (1992) and Jaiswal et al.

(1993) numerically investigated unbounded slow flows of a power-law non-Newtonian fluid through an

assemblage of spheres. Considering the effect of wall and the flow disturbance induced by upstream and

downstream particle groups, Subramaniam and Zuritz (1994) determined the drag force on multiple as-

semblies of spheres suspended in CMC solutions, from which an averaged drag force on each particle was

deduced. However, this averaged drag force should not be used in the calculation of dynamic motions of

particles with strong wake interactions. As measured by Zhu et al. (1994), for a pair of particles, the drag

force of the trailing particle can be less than 20% of that of the leading particle. An averaged drag force ofthe two would lead to a biased account of the dynamic motion of the pair, especially of the trailing particle.

So far, there appears no report on direct drag force measurements of a single particle inside an assemblage

in Newtonian and/or power-law flows. In addition, due to the strong particle–particle interaction, it is

unlikely to have a stable structure of particle assemblage in actual flows. Hence, a direct drag force

measurement of a single particle in various positions inside an assemblage would be of interest.

The objectives in this study were to investigate the drag forces of interacting particles suspended in

power-law flows and to understand the effect of power-law index on the drag forces. These objectives were

approached by both direct measurements and numerical computations on drag forces of interacting par-ticles suspended in power-law pipe flows. In conjunction to the drag force measurements, flow patterns and

velocity fields of power-law flows over a pair of interacting spheres were also obtained from laser assisted

flow visualization and numerical simulations.

2. Experimental methodology

In order to perform the direct measurements of drag forces and flow velocity fields, an experimental

setup was built up, which consisted of a flow-circulating loop; a micro-force measuring system; and a PIV

flow visualization and velocity measurement system, as shown in Fig. 1. In addition, selection and cali-

bration of easy-to-make, not highly viscous, transparent, simple (in viscosity formulation), and time-in-dependent non-Newtonian fluids were also crucial to this experiment.

The flow-circulating loop aims to provide a steady and isothermal flow for designated measurements

without time restraint. The loop includes a gear pump (Oriental, Model GB-200); two controlling valves; a

temperature control system; a flow reduction section atop of which the test section (a vertical rectangular

column) is located; a fluid reservoir; and an overflow end section with a flow return pipe for reducing the

bubble generation. Hence, a steady upward flow is yielded in the vertical test section where the interacting

spheres are suspended. Various flow velocities in the test section can be obtained by adjusting the con-

trolling valves.The drag force of an interacting particle can be directly measured using the micro-force measuring

system, as illustrated in Fig. 2. The test particle (whose drag force is to be measured) is connected to a frame

with a thin rod and the frame is directly put on top of a precision electronic balance. Under the flow

condition, the drag force on the test particle and the connecting rod leads to a reduction in the loading

weight of the frame on the balance, as shown in Fig. 2(a). Hence, by deducting the drag force of the rod, the

drag force of the test particle can be obtained. In this experiment, a precision top-loading electronic balance

(Sartorius, Model LC211S) with the accuracy of 1 mg and response frequency of 2 Hz is used. The balance

is further connected to an IBM PC586 for data acquisition and analysis. It must be noted that the dragforce on the thin rod which is partially submerged into the fluid is not negligibly small (about 25–40% of the

Fig. 1. Schematic diagram of experimental apparatus.

Stainless Steel frame

(a) (b)

Balance

Fig. 2. Micro-force measuring system.


total drag in our experiments). Therefore, in order to yield the drag force of the particle, drag forces of therod without particle connected must be independently measured, as shown in Fig. 2(b). Schematic de-

scription of drag force measurements for cases where the test particle is located inside or in the downstream

region of a cubic matrix of multi-particle assemblage is provided in Fig. 3.

Flow visualizations and velocity field measurements of flows over a pair of interacting particles are

carried out by using an optical measuring system, as depicted in Fig. 1. The laser sheet, for the purpose of

illuminating the measurement area of interest, is generated by a laser beam sweeping unit whose sweeping

range and frequency are governed by a scanning beam box and controller. In this study, the sweeping range

is 150 mm and frequency is up to 2000 Hz. The laser sheet thickness, adjustable by the collimating optics,

Fig. 3. Cubically arranged multi-particle assemblage.


can be as thin as 1 mm. The image capturing and analysis system consists of a digital CCD camera with a

resolution of 1k by 1k, an image processing system (Matrox), an IBM PC586, and the PIV software

(Optical Flow Systems, VidPIV) for velocity field analysis.

In this experiment, the non-Newtonian fluids are desired to be easy-to-make, not highly viscous,

transparent, simple (in viscosity formulation), and time-independent. A reasonably low viscosity of the fluid

is preferred because of the limitations in the available power of the gear pump for flow circulation.Transparency of fluid is required for the sake of flow visualization and optical measurement techniques

involved. Simplicity in fluid viscosity formulation is desired for the benefit of simple mathematical for-

mulations in numerical computations. The time-independence is sought due to the requirement in re-

peatability of experimental measurements and time duration needed for taking the measurements. The

simplest non-Newtonian fluids are power-law fluids. For convenience, CMC fluids that are transparent,

power-law, shear thinning and time-independent are selected. The CMC solutions of various concentra-

tions are prepared by mixing and dissolving different amounts of dry CMC powders into water. A powerful

blender (Toshiba, Model BMP-150B) is used for the mixing and a precision viscometer (Brookfield, ModelLDV II+) is used for the determination of power-law index and consistency coefficient of a CMC fluid.

3. Fundamentals of numerical modeling

Numerical simulations of power-law flows over a pair of interacting spheres suspended on the axis in a

circular duct are performed to provide theoretical explanation and extend understanding of the experi-mental results. These simulations are aimed at the calculations of drag forces of the spheres and flow field

under the particle interactions. The calculated drag forces and flow fields are then compared with the

experimental measurements and visualizations.

In the numerical modeling approach, it is assumed that the flows are laminar, steady, incompressible,

and axisymmetric. The body force, i.e., gravity, is omitted because its effect has already been deducted from

the experimental measurements. The constitutive relationship between the shear stress and the rate of shear

deformation for one-dimensional shear flow of power-law liquids is determined from the viscometer

measurements as follows:

s ¼ kð _ccÞn ð1Þ

where s and c are shear stress and the shear strain respectively, while k and n are consistency coefficient andpower-law index respectively. The above scalar expression can be expressed by (William, 1978)

Fig. 4. Grid distributions.


s$ ¼ lA ¼ k

1

2trA2

� �ðn�1Þ=n

ð2Þ

where s$now represents the shear stress tensor and A is the shear strain tensor, and trA2 denotes for the

trace of the dot product of A and A.For steady, incompressible, axisymmetric and laminar flows, the continuity and momentum equations

are expressed as

r � ðqV*

Þ ¼ 0 ð3Þ

r � ðqV*

V*

Þ ¼ �rp þr � s$ þ f

*

ð4Þ

where q presents the density of fluid, V*

is the flow velocity, p is the pressure, and f*

implies the body force.

Computational domain and grid distributions for numerical simulations are illustrated in Fig. 4. For the

convenience of computation, the entire domain consists of four sub-domains, respectively for entrance, exit,

near field, and far field (wall side). In the near field sub-domain, the boundary fitting grids are used to

match perfectly the contour of the interacting spheres. The inlet velocity distributions are set either ac-cording to experimental measurements (if available) or assuming fully developed flow conditions. At the

exit, it is always assumed that the flow reaches fully developed. On the central line, the axisymmetric

condition is used while non-slip condition is applied to the surfaces of spheres. On the pipe wall, the non-

slip condition is applied for the simulation of pipe flows while slip condition is adopted when simulating

unbounded flows (with a large ratio of pipe diameter to sphere diameter). It is noted that, if only the ratio of

drag coefficient concerned, the computational results are weakly dependent on the size of computational

domain and inlet velocity distributions. Hence, the results obtained could be extended to more general

applications. From the non-dimensional governing equations and their boundary conditions, it is shownthat there are only three major parameters affecting on the ratio of drag forces, namely, the particle

Reynolds number, the power-law index and the relative separation distance between the two interacting

particles. The effects of these parameters on the ratio of drag forces will be discussed later.

4. Results and discussions

In this study, investigations of drag forces of interacting spheres in power-law flows were conducted via

both direct measurements and numerical simulations. The power-law flow range of the experimental in-

vestigation is considerably limited due to two primary restraints, which are the limited pump power and the

problems of bubble entrainment at high velocity. The bubble entrainment from various free surfaces in ahighly viscous fluid flow would create a serious problem to all experimental measurements concerned in this

study. For example, the bubble attachment to the test spheres yields various errors in drag force mea-


surements. The bubble-contained fluid would cause a reduced and uncontrollable apparent fluid viscosity,

which may significantly change the flow conditions of the experiment. In addition, bubbles would strongly

affect the PIV measurements, even make the flow visualization impossible. To make things worse, in the

flow circulation process, the pump would further break up the large bubble into many tiny bubbles that arehard to get rid of. As a result, the experiments were performed using CMC solutions with the range of

power-law index from 0.9 to 1.0 and consistency coefficient from 0.089 to 0.375 Pa sn. The flow velocity

varies from 0.009 to 0.07 m/s in the test section, which results in a particle Reynolds number range from 0.7

to 23. In order to expand the investigation range and to provide a theoretical comparison, a numerical

study (which can overcome such technical difficulties in experiments) of power-law flows over a pair of

interacting spheres in tandem was also performed, from which drag forces of interacting spheres were

calculated. In the numerical computation, the power-law index varies from 0.6 to 1.0.

The comparison of drag coefficient between our experimental results and reported results (Chhabra,1990; Anubav et al., 1994) for an isolated sphere in power-law flows is presented in Fig. 5. It is shown that

the experimental results are bounded within the standard curve of n ¼ 1 (Newtonian flows) and the nu-merical computational curve of n ¼ 0:6. As consistent with the literature results, the effect of power-lawindex appears to be strong and is particle Reynolds number dependent. The drag coefficients of power-law

fluids are higher than that of Newtonian flows when Re is less than a critical Reynolds number (say,Re ¼ 2–3) while the drag coefficient are less than that of Newtonian flows when Re is beyond the criticalvalue. The higher the power-index deviation, the larger the difference.

Drag force measurements of a pair of interacting spheres in tandem were carried to illustrate the effect ofpower-law index on the particle–particle interaction. As shown in Fig. 6, the drag reduction of the trailing

sphere due to the particle wake interaction is exponentially correlated to the separation distance and the

particle Reynolds number. It is interesting to further examine the effect of power-law index on this drag

reduction. At the similar Re conditions, little effect of power-law index on the curve(s) of drag coefficientratio of the trailing sphere is observed, as illustrated in Fig. 7. However, it should be realized that, due to

Fig. 5. Single sphere drag coefficient.

Fig. 7. Effect of power-law index on drag coefficient ratio.

Fig. 6. Drag coefficient ratio versus separation distance.


the experimental limitations, the variation range of power-law index of the experiments might be toonarrow to yield conclusive results. In order to offset this experimental limitation, a wider range of inves-

tigation on power-law index is completed through the numerical simulation. As shown in Fig. 7, the nu-

merical results with the power-law index from 0.6 to 1.0 also suggest that the drag coefficient ratio is very

weakly dependent on the power-law index. Hence, for engineering application purposes, it may be deduced

that the ratio of drag coefficient in power-law flows is exponentially dependent on the separation distance

and is strongly related to the particle Reynolds number but nearly independent of the power-law index

within the range of our investigation.

As mentioned before, there appears no report on direct drag force measurements of a single particleinside an assemblage in Newtonian and/or power-law flows. In addition, to reveal the unstable nature of the

assemblage structure in actual flows, a direct drag force measurement of a single particle in various po-

sitions inside an assemblage is of interest. Drag forces of a single particle at various positions inside or in

the downstream of a cubic assemblage (refer to Fig. 3) are presented in Fig. 8. It is shown that the drag

force of a particle in an assemblage is position dependent. Similar to the case of two-particle interaction, the

Fig. 8. Drag coefficient of a sphere in an assemblage: (a) inside an assemblage and (b) downstream of an assemblage.

Fig. 9. Velocity field from PIV measurement.


Fig. 10. Comparison of velocity field between PIV and simulation.


drag force reaches the minimum at contact with the leading particle. There exists a position where the drag

force of the particle inside the assemblage reaches the maximum due to the combined influence of both

leading particle and downstream particle. Within the range of the experiment, the maximum difference

among drag forces at various positions is up to 38% in Fig. 8(a) and up to 30% in Fig. 8(b). This significant

difference in drag forces of a particle at various positions inside an assemblage clearly illustrates that the

drag forces of particles in a transient particle group in an actual flow could be quite different from theaveraged drag force at a node of an assemblage. Hence, an accurate account of transient dynamic

movements of particles in multiphase flows should be based on the transient drag force information rather

than based on the averaged drag force.

In this study, flow visualization, PIV measurements and numerical computation are attempted all to-

gether to investigate the flow filed of a pair of interacting spheres in tandem. Fig. 9 illustrates a set of typical

flow patterns in the wake region between two interacting particles with various interacting distances. At low

particle Reynolds numbers, no significant flow separation or vortex formation is observed in the wake

region. However, velocities in the wake region are greatly reduced. It is noted that when the trailing particleis away from the leading particle, the velocity between the two interacting particles increases. This partially

explains why the drag force of trailing particle increases with the increase of separation distance. The

comparison between the PIV measurements and numerical simulations in power-law flows, as given in Fig.

10, shows the same velocity tendency in the flow field.

5. Conclusions

In the present study, drag forces of interacting particles at the particle Reynolds number from 0.7 to 23

were directly measured in flows with power-law index from 0.9 to 1.0. In order to extend the investigation

range of power-law index, numerical simulation method is also performed for flows with power-law index

from 0.6 to 1.0. Both experimental and numerical studies suggest that the ratio of drag coefficients in

power-law flows is exponentially dependent on the separation distance and is strongly related to the particleReynolds number but nearly independent of the power-law index within the range of our investigation.


Drag forces of a single particle at node location inside a simple cubic assemblage are directly measured.

This drag force can be regarded as averaged drag force of particles in the assemblage. However, the drag

force measurements of a single particle at various locations inside the assemblage illustrate that the drag

force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.This significant position dependency suggests that the drag forces of particles in actual transient structure of

particle assemblage may be quite different from the averaged drag force at the node. Consequently, ad-

vanced modeling of transient dynamic movements of particles in multiphase flows should be based on the

transient drag information rather than based on the averaged drag force of an assemblage.

Study on flow field of interacting spheres by flow visualization and numerical computation indicates that

the effect of power-law index on flow pattern and hence drag force may be weak, which is in agreement with

our drag force measurements.

Acknowledgements

The authors wish to thank the Research Committee and Department of Mechanical Engineering at theHong Kong Polytechnic University for their support. This project was substantially supported by a grant

from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. 350/692).

Advisory comments from Professor Z.C. Zhu at Tsinghua University are also deeply appreciated.

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