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An experimental investigation of the settling behavior of two spheres in a power law fluid
Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
Contents lists available at SciVerse ScienceDirect
Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm
An experimental investigation of the settling behavior of two spheres in a power law fluid Abbas H. Sulaymon a, Catherine A.M.E. Wilson b, Abeer I. Alwared a,⇑ a b
Environmental Engineering Department, College of Engineering, Baghdad University, Baghdad, Iraq Hydro-environmental Research Centre, Environmental Hydraulics, Cardiff School of Engineering, Cardiff, United Kingdom
a r t i c l e
i n f o
Article history: Received 25 March 2012 Received in revised form 23 June 2012 Accepted 17 September 2012 Available online 20 November 2012 Keywords: Two spheres Settling behavior Sphere interaction Power law fluid Flow visualization
a b s t r a c t Experiments were conducted to examine the settling motion of two stainless steel spheres at two sphere arrangements in a power law fluid (carboxymethylcellulose solution) with different concentrations (0.7%, 0.9% and 1.1% weight to volume). Both ‘in line’ and ‘side by side’ sphere arrangements were investigated. The impact of the initial separation distance between spheres; defined as a ratio of the distance between the spheres centers to their diameter, on their behavior was examined. The velocity ratio, defined by the velocity of the two interacting spheres compared to the velocity of a single isolated sphere, was determined. When the spheres are touching in their initial position, the velocity ratio was equal to 2.14, however as the separation distance increased, the velocity ratio decreased until a value of 1.14 at a separation distance of 5.41 indicating that the spheres behaved as isolated spheres for this condition. The behavior for both the in line and side by side sphere arrangements in the power law fluid was found to be very similar to that observed in a Newtonian fluid, with the exception of the case of the two spheres side by side. The behavior of the side by side spheres diverged from that observed for a Newtonian fluid at a separation distance of between two and five, implying that at some stage within this range a stable equilibrium distance may be reached. While when the initial separation distance was less than two, the two spheres repelled each other that are similar to the behavior in a Newtonian fluid, when the separation distance increased to five, the two spheres kept the same distance between them for a while before coming together. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the sedimentation of particles in polymeric fluids has been a subject of growing interest. In such media, the settling behavior is strongly modified since the flow properties are non-linear even at very low Reynolds number (Re < 0.5). When two identical spheres fall along their line of centers, their behavior is affected strongly by the rheology of the suspending fluid. If the fluid is Newtonian, the trailing sphere is drafted into the wake of the leading sphere, and the sphere touch and tumble due to dominance of the inertial effects associated with wake [1]. This can occur only if the two spheres are interacting and the centers of spheres are up to five or six diameters apart. The behavior will be dependent on the Reynolds number, at low Reynolds numbers (Re < 0.5) it is observed that the sphere to sphere separation remains constant irrespective of the initial separation distance [1].
⇑ Corresponding author. E-mail addresses: [email protected] (C.A.M.E. Wilson), abeerwared@ yahoo.com (A.I. Alwared). 0377-0257/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2012.09.011
In contrast, different behavior has been observed in polymer solutions and this depends on the fluid rheological properties as well as the initial separation distance of the spheres. Bot et al. [2] investigated the motion of two identical spheres, both accelerating from rest at the same time along the center line of a cylindrical tube filled with a Boger fluid. They concluded that the two spheres attract for large initial distances but separate for small initial distances and that this behavior eventually resulted in a finite stable distance between the spheres which was found to be independent of the initial distance. Riddle et al. [3] investigated the interaction between two identical spheres settling in line through a shear-thinning viscoelastic fluid. They found that there was a critical value of initial separation distance of sphere, that determined whether the spheres came closer together or moved further apart. Complex effects in terms of an effective ‘attraction’ at short distances and ‘repulsion’ between particles at larger initial distances have been observed by Gheissary and Brule [4] who performed similar experiments. However they did not observe a critical separation distance at which the behavior of the settling spheres changed. They found that in shear-thinning fluids the two spheres always came together
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A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
and in viscoelastic fluids with a constant shear viscosity, the spheres always separated when they were initially close. These results seem to suggest that fluid elasticity causes the spheres to separate while aggregation of the spheres must be associated with shear-thinning effects. An explanation for this behavior is the formation of a corridor of reduced viscosity in the wake of the leading particle, where the trailing sphere settles faster than the leading sphere when the initial separation distance is small and eventually the particles chain. Likewise for large initial separation distances the particles behave as in a Newtonian fluid and they both settle at the velocity of a single isolated particle [5,6]. Daugan et al. [6] investigated the settling velocity of two particles in line in three types of shear thinning fluid of varying power law index (0.42, 0.49 and 0.35). They concluded that this problem is made more complex by the large variety of rheological behaviors of fluids that they investigated and that generalizations cannot be made from their results. They reported that the critical separation distance is always large compared to the length of the influence of the hydrodynamic interactions in a Newtonian fluid and could be up to 35 times the particle diameter. Gheissary and Brule [4] investigated the behavior of two spheres settling in line in an inelastic shear thinning fluid. They did not observe a critical separation distance as observed by others (e.g. [6,3,5]) and reported that the two spheres always came together, even for a very large initial separation distance (i.e. >50d). For spheres in a side by side arrangement, in a Newtonian fluid, the two spheres do not touch, but can be spaced to produce stability; Fortes et al. [1] determined the spacing to be of the order of one sphere. While in a polymer viscoelastic solution, Joseph and Lui [5] defined two distinct distances denoted by the ‘critical’ and the ‘touching’ distance for the settling of two spheres side by side. They observed that if the initial distance between the centers of the two spheres was such that they were touching, the spheres attracted, touched, turned and fell in a vertical tumble. If the spheres were between a ‘touching’ and a ‘critical’ distance, the spheres attracted initially and the axis of their line of centers changed from being horizontal to vertical, the spheres then separated as they settled. They observed that for values larger than the critical distance, the spheres did not interact. In their tests conducted by using a highly shear thinning fluid (aqueous Carbopol solution) Gheissary and Brule [4] observed that for side by side arrangements and for initial separation distances from 1d (touching) to more than 50d, the spheres are attracted towards each other until they touch, at this point the line of centers becomes vertical and the spheres separate until they reach a
non-interacting distance. This sequence of events is the contrary of the drafting, touching and tumbling motions observed in Newtonian fluids. This paper will investigate in more detail the settling motion of two spheres in two arrangements, falling in line (along their line of centers) and side by side, in three different concentrations of a power law fluid (carboxymethylcellulose, CMC). A range of initial separation distances will be considered (1–8.35 for spheres in line; 1–5 for spheres side by side) and we will define a range of critical separation distances at which the behavior of the spheres change from having influence upon each other and act as independent isolated spheres.
2. Experimental apparatus and measurement systems 2.1. Experimental apparatus The experimental apparatus, which is shown schematically in Fig. 1a consisted of the following: A plastic cylinder of 100 cm height and 16 cm diameter made of perplex plastic which was filled with the test fluid (carboxymethylcellulose), which was placed inside a square PVC box (29.5 29.5 94.5) cm which was filled with water in order to reduce optical distortion of the images. Release mechanism (see Fig. 1b), which consists of an adjusted beam, stainless rods and aluminum beam. A high speed camera with 500–25000 frame per second type (Photron, FASTCAM – APX RS 250K) was used, for the experiments reported herein 1000 frames per second was adopted. A Computer linked to the high speed camera. Stainless spheres of density 7762.05 kg/m3 and of different diameters (9.52, 12.78, 15.87) mm. Carboxymethylcellulose (CMC) solution was supplied by Aldrich Chemicals was used as a test fluid, at three different concentrations (0.7%, 0.9%, and 1.1% weight to volume). Tests were also conducted with water. The average molecular mass of the CMS solution is approximately 250,000 Da. Kelessidis and Mpandelis [7] used the CMC solution supplied by the same company as used in our experiments herein. The rheological properties for the CMC concentrations used in this paper were measured by Kelessidis and Mpandelis [7] and physical property data from their study is presented in Table 1. A magnetic pickup tool for returning the spheres to their starting position after a test is completed.
Computer Release mechanism Adjusted beam
Adjusted beam
Cylinder
Aluminum beam
Aluminum beam
High speed camera
Ruler Rectangular box Spheres
Stainless Spheres rods
Side by side
(a) Fig. 1. Schematic diagram of flow visualization experimental apparatus.
Stainless rods
In line
(b)
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A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36 Table 1 Physical properties of the CMC solution, data taken from Kelessidis and Mpandelis [7]. CMC concentration (%)
Power law index, n
Consistency index, k (dynes s2 n/cm2)
Density (q) (g/cm3)
0.7 0.9 1.1
0.7449 0.8610 0.9099
1.1521 0.8648 0.8492
0.9996 1.0008 0.9999
Table 2 Test spheres and initial conditions. Spheres diameter (mm)
CMC concentration (%)
In line arrangement Reynolds no. range
Side by side arrangement Reynolds no. range
0.7 0.9 1.1
11.86–185.44 11.22–189.84 34.29–200.74
13.11–358.08 4.35–282.55 2.31–138.47
3. Results and discussion
Two spheres along their line of Two spheres side by side centers Initial distance between spheres (center to center), l (mm)
l/d
Initial distance between spheres (center to center), l (mm)
l/d
9.52 35.52 43.52 79.52
1 3.73 4.57 8.353
9.52 19.04 47.6
1 2 5
12.78
12.78 39.56 46.78 82.78
1 12.78 3.0344 25.56 3.66 63.9 6.477
1 2 5
15.87
15.87 41.87 49.87 85.87
1 2.638 3.14 5.41
1 2 5
9.52
Table 3 Range of Reynolds number values for each sphere arrangement and CMC concentration.
15.87 31.74 79.35
2.2. Experimental procedure For the two different sphere arrangements (side by side and in line of centers) the settling motion through the cylindrical tube contained solution of CMC, was recorded by using the high speed camera at a rate of thousand frames per second. Regardless of the arrangement, the spheres were simultaneously released and accelerated from rest through the CMC solution. The distance between the spheres in both arrangements was varied, for the two spheres in line of centers arrangement the separation ratio l/d (distance between the sphere’s centers divided by the sphere’s diameter) varied between 1 and 8.4, while for the spheres side by side arrangement the separation ratio l/d varied between 1 and 5. Three different concentrations of the CMC solution in the cylindrical tube were examined (0.7%, 0.9%, and 1.1% weight to volume). The distance between the two spheres and the time were determined using the digitalizing software (Phortrn FASTCAM Viewer), and then the velocity was determined. The two spheres were placed on the two stainless rods (side by side) of the release mechanism (see Fig. 1b) and the initial distance between them was varied (see Table 2). The release mechanism was lowered into the cylinder so that the spheres were completely covered with liquid. This prevented any splashing and precluded the spheres from falling through the air before they entered the CMC fluid. In order to release the spheres along their line of centers, another two stainless rods were added to the release mechanism at different distances from the first two rods (see Fig. 1b), and then the two spheres were placed at the centers of these rods at the center of the cylinder and the above procedure was repeated. Each run was repeated more than ten times to ensure experimental repeatability. The interval period between any two runs was approximately 10 min; this was the time necessary to collect the spheres and for the fluid to calm and release any bubbles that were created when the spheres were falling.
The results will be presented in the following order: the position of the spheres as a function of time, the velocity of the spheres, and the interaction of the spheres in terms of the spheres’ velocity compared to that of a single isolated sphere and several video stills that show the behavior of the spheres. The range of Reynolds at which the experiments were conducted is presented in Table 3. 3.1. Position of sphere For the two spheres along their line of centers, an example where the initial separation is relatively small (l/d = 1) and relatively large (l/d = 5.41) is presented in Fig. 5. When the initial separation distance is small (Fig. 2a), the trailing sphere tumbles with the downstream sphere, then the two spheres settle by side by side (see Fig. 2a) and keep the same position along the cylinder. For a relatively large separation distance (l/d = 5.41) there is no interaction between the spheres and the spheres maintain their separation throughout the distance traveled (see Fig. 2b). For the side by side sphere arrangement, for differing initial separation distances (Fig. 3) the spheres with a small separation distance (l/d = 1), the spheres separate as they settle through the fluid cylinder, while the two spheres with a relatively large separation distance (l/d = 5) move towards each other. 3.2. Velocity of spheres A typical variation of the velocities as function of time for two spheres along their line of centers is shown in Fig. 4, which shows that for a relatively small initial separation distance, the two spheres keep the same velocity for a while, then the velocity of the upper sphere increases until it tumbles with the lower sphere until the spheres settle side by side at the same velocity (see Fig. 4a). As would be expected from the observations shown in Fig. 4b, for a relatively large initial separation distance the two spheres maintain the same velocity (see Fig. 4b). For the side by side arrangement, for both separation distances (one and 5.41), the two spheres maintain the same velocity implying that the two spheres are repelling each other (see Fig. 5). 3.3. Sphere interaction In order to show whether a sphere influences another sphere, the velocity of a sphere in a two spheres arrangement is compared to that of a single isolated sphere and is defined as the ‘velocity ratio’. When the velocity ratio is equal to one, the two spheres are not interacting while when the velocity ratio is less or greater than one the spheres are under the influence of each other and hence interacting. The velocity ratio as a function of time is presented in Fig. 6, for the in line spheres where v1 refers to the velocity of the upper sphere and v2 refers to the velocity of the lower sphere and in Fig. 7 for the side by side spheres (v1 for first sphere and v2 for second sphere). It can be seen that the high velocity ratio is for spheres chaining or tumbling (in contact), the chained spheres move at a velocity twice of that experienced by a single sphere but when
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Fig. 2. Position as a function of time for two spheres in line for the condition (a) 0.9% weight to volume CMC solution; diameter = 9.52 mm; and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm. The instantaneous positions are shown in the top left hand-side of each plot.
Fig. 3. Position as a function of time for two spheres side by side for the condition (a) 0.7% weight to volume CMC solution; diameter = 12.78 mm and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm. The instantaneous positions are shown in the top left hand-side of each plot.
Fig. 4. Velocity as a function of time for two spheres in line for the condition (a)1.1% w/v CMC solution; diameter 15.87 mm and (b) 0.7% w/v CMC solution; diameter 15.87 mm.
Fig. 5. Velocity as a function of time for two spheres side by side for the conditions (a) 0.7% w/v in CMC solution; diameter = 15.87 mm; l/d = 1 and (b) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 5.
A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
33
Fig. 6. Variation in velocity ratio as a function of time for two spheres in line for conditions (a) 1.1% w/v CMC solution; diameter = 15.87; l/d = 1 and (b) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 5.41. In the legend v1 is the velocity of the upper sphere and v2 is the velocity of for lower sphere.
Fig. 7. Variation in velocity ratio as a function of time for two spheres side by side for the condition (a) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 1; and (b) 0.7% w/v CMC solution; diameter = 9.52 mm; l/d = 5.
Fig. 8. Velocity ratio of sphere in interaction to single for two spheres in line for the condition (a) 1.1% weight to volume CMC solution; diameter = 15.87 mm, l/d = 1; and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm, l/d = 5.41.
Fig. 9. Instantaneous velocity ratio of sphere in interaction to single for two spheres side by side in 0.7% w/v CMC solution (diameter = 15.87 mm) (a) l/d = 1; and (b) l/d = 5.
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(1)
(2)
(3)
(a)
(1)
(2)
(3)
(b) Fig. 10. Flow visualization for (a) two steel spheres in line d = 15.87 mm, l/d = 3.14) in 0.7% w/v CMC solution and (b) two spheres in line (d = 15.87 mm, l/d = 5.41) in 0.7% w/v CMC solution.
the initial separation distance between them increases, the velocity of each sphere becomes similar to that of a single isolated sphere (see Figs. 6 and 7). Similar magnitudes of velocity ratio can be observed for relatively small initial separation distances (l/d = 1) whether the arrangement is in line or side by side (Figs. 6a and 7a). The velocity ratio as a function of the instantaneous distance between the spheres relative to the sphere diameter is presented in Figs. 8 and 9, for the in line spheres (v1 for upper sphere and v2 for lower sphere) and the side by side spheres (v1 for first sphere and v2 for second sphere) respectively. For spheres in line and for initial separation distance of between unity and less than five the velocity ratio is complicated by the tumbling nature of the spheres. The velocity ratio is around unity at the start of their motion, increases due to the tumbling motion and then returns to a value of around unity as the spheres settle side by side (see Fig. 8). This is the case for initial separation distances of up to five. At larger separation distances (separation distances equal and greater than 5) the velocity ratio is maintained around one and the spheres behave as single isolated spheres (see Fig. 8b). For spheres side by side when the initial separation distance is equal to unity the velocity ratio is around two at the commence-
ment of the fall and then decreases until a value of unity is reached indicating that the spheres are non-interacting (see Fig. 9a). However for initial separation distances of the order of five, the velocity ratio of sphere is maintained around unity throughout the fall (see Fig. 9b). The critical value of separation distance at which there is a change in settling behavior was found to be between 2 and 5. 3.4. Flow visualization The behavior of the spheres was found to be typical to that observed in a Newtonian fluid except for the side by side arrangement for initial separation distances equal to and greater than five. For an in line arrangement, the velocity of the upper sphere was faster than the lower one and was drafted into the wake of the downstream sphere until they touched each other then they tumbled and continued settling side by side at the same velocity (see Fig. 10a). For initial separation distances greater than five, the two spheres settled at the same velocity and maintained the same separation distance until they reached the base of the cylinder (see Fig. 10b). These results are similar to the experimental results obtained by Riddle et al. [3], Joseph and Lui [5] and Daugan et al. [6] for the initial separation distance but differ in the limits of the distance and this is probably due to the different polymer fluids used.
A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
(1)
(2)
35
(3)
(a)
(1)
(2)
(3)
(b) Fig. 11. Flow visualization for (a) Two steel spheres side by side (d = 15.87 mm, l/d = 1) in 0.7% w/v CMC solution (b) Two spheres side by side (d = 15.87 mm, l/d = 5.41) in 0.7% w/v CMC solution.
For two spheres settling side by side in CMC solution at different separation distances, it can be seen that they fall with the same velocity and repel each other when the separation distance (l/d) is between one and two (see Fig. 11a). The same results were obtained by Kim et al. [8] and Nguyen and Ladd [9] for the settling behavior in a Newtonian fluid. For spheres side by side for initial separation distances of five and greater, the two spheres keep the same distance between them for a while and then the distance between them decreases but they do not touch and continue falling until they reach the base of the cylinder (see Fig. 11b). In contrast to the behavior in a Newtonian fluid, the spheres fall at the same velocity and keep the same distance between them. Finally the same settling behavior was observed for the three different fluid concentrations that were examined and as would be expected, the motion of the spheres became slower with increasing fluid density. 4. Conclusions 1. The settling behavior of two spheres in line in a carboxymethylcellulose solution at different initial separation distances can be summarized as follows:
For separation distances from one to five, their behavior was typical to that in Newtonian fluid; the velocity of the upper sphere was faster than the lower one until they touched, tumbled and continued dropping side by side at the same velocity. For separation distances of more than five the two spheres settled at the same velocity and maintained the same distance between them until they reached the base of the cylinder. 2. The settling behavior of the two spheres side by side in a CMC solution at different initial separation distances can be summarized as: For separation distances from one to two, the spheres settle at the same velocity and repel each other, which is similar behavior to that observed in a Newtonian fluid. For separation distances equal to or greater than five, the two spheres keep the same distance between them for a while and then the distance decreases but they do not touch and continue falling until they reach the base of the cylinder. This is in contrast to what has been observed in a Newtonian fluid. 3. The critical separation distance at which the settling behavior changes is between two and five. This implies that at some stage within this range a stable equilibrium distance may have been reached.
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A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
4. The velocity ratio of the spheres decreases with increasing initial separation distance; a value of 2.14 was found for spheres that were initially touching compared to a value of 1.14 for a initial separation distance of 5.41, implying that the spheres were not interacting at larger separation distances. Acknowledgments We are indebted to Dr. Peter Kay and Professor Phil Bowen of Cardiff University for the use of the high speed camera and for their assistance in using the camera. References [1] A.F. Fortes, D.D. Joseph, T.S. Lundgren, Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech. 177 (1987) 467–483.
[2] E.T.J. Bot, M.A. Hulsen, B.H. Van den Brule, The motion of two spheres falling along their line of centre in a Boger fluid, J. Non-Newton. Fluid Mech. 79 (1998) 191–212. [3] M.J. Riddle, C. Narvaez, R.B. Bird, Interactions between two spheres falling along their line of centers in a viscoelastic fluid, J. Non-Newton. Fluid Mech. 2 (1977) 23–35. [4] G. Gheissary, V.B.H. Brule, Unexpected phenomena observed in particle settling in non-Newtonian media, J. Non-Newton. Fluid Mech. 67 (1996) 1–18. [5] D.D. Joseph, Y.J. Liu, Orientation of long bodies falling in a viscoelastic liquid, J. Rheol. 37 (1993) 1–22. [6] S. Daugan, L. Talini, B. Herzhaft, C. Allain, Aggregation of particles settling in shear thinning fluids, Part 1, two particle aggregation, Eur. Phys. J. 7 (2002) 73– 81. [7] V.C. Kelessidis, G. Mpandelis, Measurements and prediction of terminal velocity of solid spheres falling through stagnant pseudo plastic liquids, Powder Technol. 147 (2004) 117–125. [8] I. Kim, S. Elghobashi, W.A. Sirignano, Three-dimensional flow over two spheres placed side by side, J. Fluid Mech. 246 (1993) 465–488. [9] N.-Q. Nguyen, A.J.C. Ladd, Sedimentation of hard-sphere suspensions at low Reynolds number, J. Fluid Mech. 525 (2005) 73–104.
Contents lists available at SciVerse ScienceDirect
Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm
An experimental investigation of the settling behavior of two spheres in a power law fluid Abbas H. Sulaymon a, Catherine A.M.E. Wilson b, Abeer I. Alwared a,⇑ a b
Environmental Engineering Department, College of Engineering, Baghdad University, Baghdad, Iraq Hydro-environmental Research Centre, Environmental Hydraulics, Cardiff School of Engineering, Cardiff, United Kingdom
a r t i c l e
i n f o
Article history: Received 25 March 2012 Received in revised form 23 June 2012 Accepted 17 September 2012 Available online 20 November 2012 Keywords: Two spheres Settling behavior Sphere interaction Power law fluid Flow visualization
a b s t r a c t Experiments were conducted to examine the settling motion of two stainless steel spheres at two sphere arrangements in a power law fluid (carboxymethylcellulose solution) with different concentrations (0.7%, 0.9% and 1.1% weight to volume). Both ‘in line’ and ‘side by side’ sphere arrangements were investigated. The impact of the initial separation distance between spheres; defined as a ratio of the distance between the spheres centers to their diameter, on their behavior was examined. The velocity ratio, defined by the velocity of the two interacting spheres compared to the velocity of a single isolated sphere, was determined. When the spheres are touching in their initial position, the velocity ratio was equal to 2.14, however as the separation distance increased, the velocity ratio decreased until a value of 1.14 at a separation distance of 5.41 indicating that the spheres behaved as isolated spheres for this condition. The behavior for both the in line and side by side sphere arrangements in the power law fluid was found to be very similar to that observed in a Newtonian fluid, with the exception of the case of the two spheres side by side. The behavior of the side by side spheres diverged from that observed for a Newtonian fluid at a separation distance of between two and five, implying that at some stage within this range a stable equilibrium distance may be reached. While when the initial separation distance was less than two, the two spheres repelled each other that are similar to the behavior in a Newtonian fluid, when the separation distance increased to five, the two spheres kept the same distance between them for a while before coming together. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the sedimentation of particles in polymeric fluids has been a subject of growing interest. In such media, the settling behavior is strongly modified since the flow properties are non-linear even at very low Reynolds number (Re < 0.5). When two identical spheres fall along their line of centers, their behavior is affected strongly by the rheology of the suspending fluid. If the fluid is Newtonian, the trailing sphere is drafted into the wake of the leading sphere, and the sphere touch and tumble due to dominance of the inertial effects associated with wake [1]. This can occur only if the two spheres are interacting and the centers of spheres are up to five or six diameters apart. The behavior will be dependent on the Reynolds number, at low Reynolds numbers (Re < 0.5) it is observed that the sphere to sphere separation remains constant irrespective of the initial separation distance [1].
⇑ Corresponding author. E-mail addresses: [email protected] (C.A.M.E. Wilson), abeerwared@ yahoo.com (A.I. Alwared). 0377-0257/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2012.09.011
In contrast, different behavior has been observed in polymer solutions and this depends on the fluid rheological properties as well as the initial separation distance of the spheres. Bot et al. [2] investigated the motion of two identical spheres, both accelerating from rest at the same time along the center line of a cylindrical tube filled with a Boger fluid. They concluded that the two spheres attract for large initial distances but separate for small initial distances and that this behavior eventually resulted in a finite stable distance between the spheres which was found to be independent of the initial distance. Riddle et al. [3] investigated the interaction between two identical spheres settling in line through a shear-thinning viscoelastic fluid. They found that there was a critical value of initial separation distance of sphere, that determined whether the spheres came closer together or moved further apart. Complex effects in terms of an effective ‘attraction’ at short distances and ‘repulsion’ between particles at larger initial distances have been observed by Gheissary and Brule [4] who performed similar experiments. However they did not observe a critical separation distance at which the behavior of the settling spheres changed. They found that in shear-thinning fluids the two spheres always came together
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A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
and in viscoelastic fluids with a constant shear viscosity, the spheres always separated when they were initially close. These results seem to suggest that fluid elasticity causes the spheres to separate while aggregation of the spheres must be associated with shear-thinning effects. An explanation for this behavior is the formation of a corridor of reduced viscosity in the wake of the leading particle, where the trailing sphere settles faster than the leading sphere when the initial separation distance is small and eventually the particles chain. Likewise for large initial separation distances the particles behave as in a Newtonian fluid and they both settle at the velocity of a single isolated particle [5,6]. Daugan et al. [6] investigated the settling velocity of two particles in line in three types of shear thinning fluid of varying power law index (0.42, 0.49 and 0.35). They concluded that this problem is made more complex by the large variety of rheological behaviors of fluids that they investigated and that generalizations cannot be made from their results. They reported that the critical separation distance is always large compared to the length of the influence of the hydrodynamic interactions in a Newtonian fluid and could be up to 35 times the particle diameter. Gheissary and Brule [4] investigated the behavior of two spheres settling in line in an inelastic shear thinning fluid. They did not observe a critical separation distance as observed by others (e.g. [6,3,5]) and reported that the two spheres always came together, even for a very large initial separation distance (i.e. >50d). For spheres in a side by side arrangement, in a Newtonian fluid, the two spheres do not touch, but can be spaced to produce stability; Fortes et al. [1] determined the spacing to be of the order of one sphere. While in a polymer viscoelastic solution, Joseph and Lui [5] defined two distinct distances denoted by the ‘critical’ and the ‘touching’ distance for the settling of two spheres side by side. They observed that if the initial distance between the centers of the two spheres was such that they were touching, the spheres attracted, touched, turned and fell in a vertical tumble. If the spheres were between a ‘touching’ and a ‘critical’ distance, the spheres attracted initially and the axis of their line of centers changed from being horizontal to vertical, the spheres then separated as they settled. They observed that for values larger than the critical distance, the spheres did not interact. In their tests conducted by using a highly shear thinning fluid (aqueous Carbopol solution) Gheissary and Brule [4] observed that for side by side arrangements and for initial separation distances from 1d (touching) to more than 50d, the spheres are attracted towards each other until they touch, at this point the line of centers becomes vertical and the spheres separate until they reach a
non-interacting distance. This sequence of events is the contrary of the drafting, touching and tumbling motions observed in Newtonian fluids. This paper will investigate in more detail the settling motion of two spheres in two arrangements, falling in line (along their line of centers) and side by side, in three different concentrations of a power law fluid (carboxymethylcellulose, CMC). A range of initial separation distances will be considered (1–8.35 for spheres in line; 1–5 for spheres side by side) and we will define a range of critical separation distances at which the behavior of the spheres change from having influence upon each other and act as independent isolated spheres.
2. Experimental apparatus and measurement systems 2.1. Experimental apparatus The experimental apparatus, which is shown schematically in Fig. 1a consisted of the following: A plastic cylinder of 100 cm height and 16 cm diameter made of perplex plastic which was filled with the test fluid (carboxymethylcellulose), which was placed inside a square PVC box (29.5 29.5 94.5) cm which was filled with water in order to reduce optical distortion of the images. Release mechanism (see Fig. 1b), which consists of an adjusted beam, stainless rods and aluminum beam. A high speed camera with 500–25000 frame per second type (Photron, FASTCAM – APX RS 250K) was used, for the experiments reported herein 1000 frames per second was adopted. A Computer linked to the high speed camera. Stainless spheres of density 7762.05 kg/m3 and of different diameters (9.52, 12.78, 15.87) mm. Carboxymethylcellulose (CMC) solution was supplied by Aldrich Chemicals was used as a test fluid, at three different concentrations (0.7%, 0.9%, and 1.1% weight to volume). Tests were also conducted with water. The average molecular mass of the CMS solution is approximately 250,000 Da. Kelessidis and Mpandelis [7] used the CMC solution supplied by the same company as used in our experiments herein. The rheological properties for the CMC concentrations used in this paper were measured by Kelessidis and Mpandelis [7] and physical property data from their study is presented in Table 1. A magnetic pickup tool for returning the spheres to their starting position after a test is completed.
Computer Release mechanism Adjusted beam
Adjusted beam
Cylinder
Aluminum beam
Aluminum beam
High speed camera
Ruler Rectangular box Spheres
Stainless Spheres rods
Side by side
(a) Fig. 1. Schematic diagram of flow visualization experimental apparatus.
Stainless rods
In line
(b)
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A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36 Table 1 Physical properties of the CMC solution, data taken from Kelessidis and Mpandelis [7]. CMC concentration (%)
Power law index, n
Consistency index, k (dynes s2 n/cm2)
Density (q) (g/cm3)
0.7 0.9 1.1
0.7449 0.8610 0.9099
1.1521 0.8648 0.8492
0.9996 1.0008 0.9999
Table 2 Test spheres and initial conditions. Spheres diameter (mm)
CMC concentration (%)
In line arrangement Reynolds no. range
Side by side arrangement Reynolds no. range
0.7 0.9 1.1
11.86–185.44 11.22–189.84 34.29–200.74
13.11–358.08 4.35–282.55 2.31–138.47
3. Results and discussion
Two spheres along their line of Two spheres side by side centers Initial distance between spheres (center to center), l (mm)
l/d
Initial distance between spheres (center to center), l (mm)
l/d
9.52 35.52 43.52 79.52
1 3.73 4.57 8.353
9.52 19.04 47.6
1 2 5
12.78
12.78 39.56 46.78 82.78
1 12.78 3.0344 25.56 3.66 63.9 6.477
1 2 5
15.87
15.87 41.87 49.87 85.87
1 2.638 3.14 5.41
1 2 5
9.52
Table 3 Range of Reynolds number values for each sphere arrangement and CMC concentration.
15.87 31.74 79.35
2.2. Experimental procedure For the two different sphere arrangements (side by side and in line of centers) the settling motion through the cylindrical tube contained solution of CMC, was recorded by using the high speed camera at a rate of thousand frames per second. Regardless of the arrangement, the spheres were simultaneously released and accelerated from rest through the CMC solution. The distance between the spheres in both arrangements was varied, for the two spheres in line of centers arrangement the separation ratio l/d (distance between the sphere’s centers divided by the sphere’s diameter) varied between 1 and 8.4, while for the spheres side by side arrangement the separation ratio l/d varied between 1 and 5. Three different concentrations of the CMC solution in the cylindrical tube were examined (0.7%, 0.9%, and 1.1% weight to volume). The distance between the two spheres and the time were determined using the digitalizing software (Phortrn FASTCAM Viewer), and then the velocity was determined. The two spheres were placed on the two stainless rods (side by side) of the release mechanism (see Fig. 1b) and the initial distance between them was varied (see Table 2). The release mechanism was lowered into the cylinder so that the spheres were completely covered with liquid. This prevented any splashing and precluded the spheres from falling through the air before they entered the CMC fluid. In order to release the spheres along their line of centers, another two stainless rods were added to the release mechanism at different distances from the first two rods (see Fig. 1b), and then the two spheres were placed at the centers of these rods at the center of the cylinder and the above procedure was repeated. Each run was repeated more than ten times to ensure experimental repeatability. The interval period between any two runs was approximately 10 min; this was the time necessary to collect the spheres and for the fluid to calm and release any bubbles that were created when the spheres were falling.
The results will be presented in the following order: the position of the spheres as a function of time, the velocity of the spheres, and the interaction of the spheres in terms of the spheres’ velocity compared to that of a single isolated sphere and several video stills that show the behavior of the spheres. The range of Reynolds at which the experiments were conducted is presented in Table 3. 3.1. Position of sphere For the two spheres along their line of centers, an example where the initial separation is relatively small (l/d = 1) and relatively large (l/d = 5.41) is presented in Fig. 5. When the initial separation distance is small (Fig. 2a), the trailing sphere tumbles with the downstream sphere, then the two spheres settle by side by side (see Fig. 2a) and keep the same position along the cylinder. For a relatively large separation distance (l/d = 5.41) there is no interaction between the spheres and the spheres maintain their separation throughout the distance traveled (see Fig. 2b). For the side by side sphere arrangement, for differing initial separation distances (Fig. 3) the spheres with a small separation distance (l/d = 1), the spheres separate as they settle through the fluid cylinder, while the two spheres with a relatively large separation distance (l/d = 5) move towards each other. 3.2. Velocity of spheres A typical variation of the velocities as function of time for two spheres along their line of centers is shown in Fig. 4, which shows that for a relatively small initial separation distance, the two spheres keep the same velocity for a while, then the velocity of the upper sphere increases until it tumbles with the lower sphere until the spheres settle side by side at the same velocity (see Fig. 4a). As would be expected from the observations shown in Fig. 4b, for a relatively large initial separation distance the two spheres maintain the same velocity (see Fig. 4b). For the side by side arrangement, for both separation distances (one and 5.41), the two spheres maintain the same velocity implying that the two spheres are repelling each other (see Fig. 5). 3.3. Sphere interaction In order to show whether a sphere influences another sphere, the velocity of a sphere in a two spheres arrangement is compared to that of a single isolated sphere and is defined as the ‘velocity ratio’. When the velocity ratio is equal to one, the two spheres are not interacting while when the velocity ratio is less or greater than one the spheres are under the influence of each other and hence interacting. The velocity ratio as a function of time is presented in Fig. 6, for the in line spheres where v1 refers to the velocity of the upper sphere and v2 refers to the velocity of the lower sphere and in Fig. 7 for the side by side spheres (v1 for first sphere and v2 for second sphere). It can be seen that the high velocity ratio is for spheres chaining or tumbling (in contact), the chained spheres move at a velocity twice of that experienced by a single sphere but when
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Fig. 2. Position as a function of time for two spheres in line for the condition (a) 0.9% weight to volume CMC solution; diameter = 9.52 mm; and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm. The instantaneous positions are shown in the top left hand-side of each plot.
Fig. 3. Position as a function of time for two spheres side by side for the condition (a) 0.7% weight to volume CMC solution; diameter = 12.78 mm and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm. The instantaneous positions are shown in the top left hand-side of each plot.
Fig. 4. Velocity as a function of time for two spheres in line for the condition (a)1.1% w/v CMC solution; diameter 15.87 mm and (b) 0.7% w/v CMC solution; diameter 15.87 mm.
Fig. 5. Velocity as a function of time for two spheres side by side for the conditions (a) 0.7% w/v in CMC solution; diameter = 15.87 mm; l/d = 1 and (b) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 5.
A.H. Sulaymon et al. / Journal of Non-Newtonian Fluid Mechanics 192 (2013) 29–36
33
Fig. 6. Variation in velocity ratio as a function of time for two spheres in line for conditions (a) 1.1% w/v CMC solution; diameter = 15.87; l/d = 1 and (b) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 5.41. In the legend v1 is the velocity of the upper sphere and v2 is the velocity of for lower sphere.
Fig. 7. Variation in velocity ratio as a function of time for two spheres side by side for the condition (a) 0.7% w/v CMC solution; diameter = 15.87 mm; l/d = 1; and (b) 0.7% w/v CMC solution; diameter = 9.52 mm; l/d = 5.
Fig. 8. Velocity ratio of sphere in interaction to single for two spheres in line for the condition (a) 1.1% weight to volume CMC solution; diameter = 15.87 mm, l/d = 1; and (b) 0.7% weight to volume CMC solution; diameter = 15.87 mm, l/d = 5.41.
Fig. 9. Instantaneous velocity ratio of sphere in interaction to single for two spheres side by side in 0.7% w/v CMC solution (diameter = 15.87 mm) (a) l/d = 1; and (b) l/d = 5.
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(1)
(2)
(3)
(a)
(1)
(2)
(3)
(b) Fig. 10. Flow visualization for (a) two steel spheres in line d = 15.87 mm, l/d = 3.14) in 0.7% w/v CMC solution and (b) two spheres in line (d = 15.87 mm, l/d = 5.41) in 0.7% w/v CMC solution.
the initial separation distance between them increases, the velocity of each sphere becomes similar to that of a single isolated sphere (see Figs. 6 and 7). Similar magnitudes of velocity ratio can be observed for relatively small initial separation distances (l/d = 1) whether the arrangement is in line or side by side (Figs. 6a and 7a). The velocity ratio as a function of the instantaneous distance between the spheres relative to the sphere diameter is presented in Figs. 8 and 9, for the in line spheres (v1 for upper sphere and v2 for lower sphere) and the side by side spheres (v1 for first sphere and v2 for second sphere) respectively. For spheres in line and for initial separation distance of between unity and less than five the velocity ratio is complicated by the tumbling nature of the spheres. The velocity ratio is around unity at the start of their motion, increases due to the tumbling motion and then returns to a value of around unity as the spheres settle side by side (see Fig. 8). This is the case for initial separation distances of up to five. At larger separation distances (separation distances equal and greater than 5) the velocity ratio is maintained around one and the spheres behave as single isolated spheres (see Fig. 8b). For spheres side by side when the initial separation distance is equal to unity the velocity ratio is around two at the commence-
ment of the fall and then decreases until a value of unity is reached indicating that the spheres are non-interacting (see Fig. 9a). However for initial separation distances of the order of five, the velocity ratio of sphere is maintained around unity throughout the fall (see Fig. 9b). The critical value of separation distance at which there is a change in settling behavior was found to be between 2 and 5. 3.4. Flow visualization The behavior of the spheres was found to be typical to that observed in a Newtonian fluid except for the side by side arrangement for initial separation distances equal to and greater than five. For an in line arrangement, the velocity of the upper sphere was faster than the lower one and was drafted into the wake of the downstream sphere until they touched each other then they tumbled and continued settling side by side at the same velocity (see Fig. 10a). For initial separation distances greater than five, the two spheres settled at the same velocity and maintained the same separation distance until they reached the base of the cylinder (see Fig. 10b). These results are similar to the experimental results obtained by Riddle et al. [3], Joseph and Lui [5] and Daugan et al. [6] for the initial separation distance but differ in the limits of the distance and this is probably due to the different polymer fluids used.
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(1)
(2)
35
(3)
(a)
(1)
(2)
(3)
(b) Fig. 11. Flow visualization for (a) Two steel spheres side by side (d = 15.87 mm, l/d = 1) in 0.7% w/v CMC solution (b) Two spheres side by side (d = 15.87 mm, l/d = 5.41) in 0.7% w/v CMC solution.
For two spheres settling side by side in CMC solution at different separation distances, it can be seen that they fall with the same velocity and repel each other when the separation distance (l/d) is between one and two (see Fig. 11a). The same results were obtained by Kim et al. [8] and Nguyen and Ladd [9] for the settling behavior in a Newtonian fluid. For spheres side by side for initial separation distances of five and greater, the two spheres keep the same distance between them for a while and then the distance between them decreases but they do not touch and continue falling until they reach the base of the cylinder (see Fig. 11b). In contrast to the behavior in a Newtonian fluid, the spheres fall at the same velocity and keep the same distance between them. Finally the same settling behavior was observed for the three different fluid concentrations that were examined and as would be expected, the motion of the spheres became slower with increasing fluid density. 4. Conclusions 1. The settling behavior of two spheres in line in a carboxymethylcellulose solution at different initial separation distances can be summarized as follows:
For separation distances from one to five, their behavior was typical to that in Newtonian fluid; the velocity of the upper sphere was faster than the lower one until they touched, tumbled and continued dropping side by side at the same velocity. For separation distances of more than five the two spheres settled at the same velocity and maintained the same distance between them until they reached the base of the cylinder. 2. The settling behavior of the two spheres side by side in a CMC solution at different initial separation distances can be summarized as: For separation distances from one to two, the spheres settle at the same velocity and repel each other, which is similar behavior to that observed in a Newtonian fluid. For separation distances equal to or greater than five, the two spheres keep the same distance between them for a while and then the distance decreases but they do not touch and continue falling until they reach the base of the cylinder. This is in contrast to what has been observed in a Newtonian fluid. 3. The critical separation distance at which the settling behavior changes is between two and five. This implies that at some stage within this range a stable equilibrium distance may have been reached.
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4. The velocity ratio of the spheres decreases with increasing initial separation distance; a value of 2.14 was found for spheres that were initially touching compared to a value of 1.14 for a initial separation distance of 5.41, implying that the spheres were not interacting at larger separation distances. Acknowledgments We are indebted to Dr. Peter Kay and Professor Phil Bowen of Cardiff University for the use of the high speed camera and for their assistance in using the camera. References [1] A.F. Fortes, D.D. Joseph, T.S. Lundgren, Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech. 177 (1987) 467–483.
[2] E.T.J. Bot, M.A. Hulsen, B.H. Van den Brule, The motion of two spheres falling along their line of centre in a Boger fluid, J. Non-Newton. Fluid Mech. 79 (1998) 191–212. [3] M.J. Riddle, C. Narvaez, R.B. Bird, Interactions between two spheres falling along their line of centers in a viscoelastic fluid, J. Non-Newton. Fluid Mech. 2 (1977) 23–35. [4] G. Gheissary, V.B.H. Brule, Unexpected phenomena observed in particle settling in non-Newtonian media, J. Non-Newton. Fluid Mech. 67 (1996) 1–18. [5] D.D. Joseph, Y.J. Liu, Orientation of long bodies falling in a viscoelastic liquid, J. Rheol. 37 (1993) 1–22. [6] S. Daugan, L. Talini, B. Herzhaft, C. Allain, Aggregation of particles settling in shear thinning fluids, Part 1, two particle aggregation, Eur. Phys. J. 7 (2002) 73– 81. [7] V.C. Kelessidis, G. Mpandelis, Measurements and prediction of terminal velocity of solid spheres falling through stagnant pseudo plastic liquids, Powder Technol. 147 (2004) 117–125. [8] I. Kim, S. Elghobashi, W.A. Sirignano, Three-dimensional flow over two spheres placed side by side, J. Fluid Mech. 246 (1993) 465–488. [9] N.-Q. Nguyen, A.J.C. Ladd, Sedimentation of hard-sphere suspensions at low Reynolds number, J. Fluid Mech. 525 (2005) 73–104.