Sedimentation of Cylindrical Particles in a Viscoelastic Liquid: Shape-Tilting

Transcription

1 Sedimentation of Cylindrical Particles in a Viscoelastic Liquid: Shape-Tilting J. Wang R. Bai C. Lewandowski G. P. Galdi D. D. Joseph Abstract Aluminum and Teflon cylindrical particles with flat ends are dropped in aqueous Polyox solutions. The terminal equilibrium orientation of the particles is characterized by the tilt angle, α, formed by the major axis of the cylinder with the horizontal. It is observed thatαis a function of the aspect ratiol=l/d, wherelis the length anddis the diameter of the cylinder and that it varies continuously from a certain angle, α 0, to 90 0, as L increases toward a valuel 0. For a given shape, both α 0 and L 0 depend on the density of the cylinder and the properties of the liquid used. For the particles we have considered the value ofl 0 is of the order of 2. This tilt-angle phenomenon disappears as soon as the ends of the cylinder are round. Specifically, cylinders of the same density and with the same aspect ratio but with round ends, when dropped in the same polymeric solution will reach a final orientation withα=90 0. Therefore, this tilt-angle phenomenon seems to be tightly related to the shape of the particle. Keywords. Sedimentation; Tilt angle; Orientation; Viscoelastic liquid. Introduction As is well-known, the orientation of long bodies 1 in liquids of different nature is a fundamental issue in many problems of practical interest. These problems cover a wide range of applications, including manufacturing of short-fiber composites [1], [12], separation of macromolecules by electrophoresis, [7, 8, 18, 17], flow-induced microstructures [9], models of blood flow [16], and particle-laden materials [3]. A first, fundamental step in understanding the orientation of long bodies in liquids is to investigate experimentally their free fall behavior, both in Newtonian and viscoelastic liquids [11], [4], [5], [13]. It is a well-established experimental fact that homogeneous bodies of revolution around an axis k (say) with fore-and-aft symmetry (like cylinders, round ellipsoids, etc., of constant density), when dropped in a quiescent viscous liquid will eventually orient themselves (with respect to the horizontal,h, say) in a way that depends on the weight of the body, on its geometric properties (like being prolate or oblate in shape), and on the physical properties of the liquid (viscosity, inertia, non-newtonian characteristics, etc.). In particular, if the liquid is viscous and Newtonian, due to the inertia of the liquid, (homogeneous) cylinders or prolate spheroids will always reach an equilibrium orientation withkparallel toh, no matter what their initial orientation; see [15], [2] and Fig. 1(A). In contrast, if the fluid is viscoelastic, the situation changes dramatically and the final orientation may be completely different than that observed for a Newtonian liquid at nonzero Reynolds number. Detailed experimental studies were performed by Liu and Joseph [13] and Chiba et al. [4]. In particular, in [13], cylinders with round ends of different materials were dropped in a 2% solution of Polyox in water. It was then observed that when viscoelastic stresses are Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis MN, U.S.A. Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh PA, U.S.A. Corresponding author 1 Loosely, a long body is a body where one dimension is larger than the other two. 1

2 Figure 1: Snapshots of aluminum cylinders sedimenting in (A) Newtonian liquid (60% glycerine in water); (B) Viscoelastic liquid, negligible inertia (2% Polyox in water) [9]. g denotes the direction of gravity. predominant over the inertia of the liquid the final orientation of the particles was always with their broadside perpendicular to h; see Fig. 1(B). In [4] similar experiments where performed with cylinders with flat ends of length ranging from 30mm to 1.7cm and with aspect ratios ranging from 30 to 100, sedimenting in aqueous solution of polyacrylamide of different concentrations. Viscoelastic stresses are again predominant on the inertia of the liquid. The final orientation of all particles turns out to be with their broadside parallel to gravity. Another series of remarkable experiments is reported in the work of Cho et al. [5], on the orientation of small cylinders with flat ends in aqueous solutions of polyacrylamide. The cylinders are of length of a few mm, with aspect ratios ranging from 10 to 100. One of the objectives of the paper [5] is to investigate the dependence of the orientation of cylinders on the shear-thinning property of the polymer. As in the work of Chiba et al., also in this one the effect of inertia of the liquid can be neglected compared to the effect of viscoelastic stresses. It is observed that the orientation of the particles changes continuously from broadsideperpendicular-to-gravity to broadside-parallel-to-gravity by increasing the concentration of polymer [5], p This phenomenon, usually referred to as tilt-angle phenomenon, is attributed by the authors, to the significant shear-thinning properties of the liquid (polyacrylamide). The objective of the current paper is to present a number of experiments that show another interesting and puzzling property of sedimentation of cylinders in a viscoelastic liquid. Specifically, we drop squared-off cylinders with flat ends of diameter d and length L in an aqueous polymeric solution of fixed concentration. The cylinders are aluminum and Teflon, while the liquid is aqueous Polyox solution of different concentrations. Denote by α Figure 2: Definition of the tilt angle. g denotes the direction of gravity. the tilt angle that the major axis of the cylinder forms with the horizontal in its equilibrium terminal orientation; see Fig. 2. It is then observed that the tilt angle α varies continuously from a certain angle,α 0, to 90 0, as the aspect ratiol=l/d increases from 1 to a valuel 0. Bothα 0 andl 0 depend on the density of the cylinder and the properties of the liquid. In particular, the value ofl 0 is of the order of 2, for the particles we have used. This explains why such a phenomenon is not observed in the cited works of Chiba et al. and Cho et al., where, as noticed,lis at least 10. Another interesting and maybe unexpected characteristic of our experiments is that this tilt-angle phenomenon disappears if the ends of the cylinder are round. Expressly, cylinders of the same material and same aspect ratio but with round ends, when dropped in the same polymeric solution will always reach a final orientation withα=90 0. Therefore, this tilt-angle phenomenon seems to be tightly related to the shape of the particle and, therefore, we call it shape-tilting phenomenon. Before ending this introductory section, we would like to spend a few words about possible theoretical explanation of the shape-tilting phenomenon. Galdi et al. [6] have performed a mathematical analysis of particle sedimentation in a second-order viscoelastic fluid. This analysis is in a very good agreement with the experimental results of Liu and Joseph [13] and, in particular, it explains the observed orientation of cylinders with round ends sedimenting in Newtonian and viscoelastic liquids. However, the analysis requires the particle to be smooth and, as a consequence, it is unable to interpret the shape-tilting phenomenon that involves non-smooth cylinders with flat ends. Therefore, the shape-tilting phenomenon does not have, to date, any theoretical explanation. 1. Methods and Results Methods. Particles are dropped in a channel filled with an aqueous polymeric solution and snapshots of the settling particles are taken when they reach a steady state. Details are given below. Liquid. Aqueous Polyox solution of different concentrations (1.5%, 1%, 0.75%). P articles. Four sets of cylindrical particles with flat ends are used in the experiments. The material used is 2

3 aluminum (density=2.7 g/cm 3 ) and Teflon (density=2.1 g/cm 3 ). The diameter and length of a particle aredand L, respectively. The particles in each set have the same diameter but different aspect ratio L/d. Set 1: nine aluminum particles withd=0.635cm, and1.0 L/d 2.4; Set 2: ten aluminum particles withd=0.9525cm and0.8 L/d 2.4; Set 3: seven Teflon particles withd=0.635cm and1.0 L/d 2.0; Set 4: two Teflon particles withd=0.9525cm andl/d=1.0 and2.0. Channel. The dimension of the channel is 1.27cm 20.32cm 101.6cm. The gravity is directed along the 101.6cm side. Experimental P rocedure. Cylindrical particles are dropped in the channel and several pictures of the particle are taken when it reaches the steady state. The tilt angleα(see Fig. 2) is then measured from the photos using a protractor. We also measured the time for the steadily falling particles to pass a 12.7cm span, and computed the sedimentation speeds. For each particle, both tilt angle and sedimentation speed are measured for several times (typically, from 3 to 7 times) in order to ensure a small error margin. The results presented here are average values. Results. Sedimentationina1.5%Solution. In Table 1 are reported the values of the tilt angleαfor particles of different materials and different aspect ratios L/d sedimenting in a 1.5% aqueous Polyox solution. In Fig. 3 a plot ofαversusl/d is given for different materials. From these results we see that the particles tilt whenl L/d is small and fall with their broadside parallel to gravity whenlexceeds a critical value,l 0. We find thatl 0 is about 1.6 for Teflon particles, while it is around 2.4 for aluminum particles. Aluminum d = 0.25in Aluminum d = 0.375in Teflon d = 0.25in Teflon d = 0.375in L/d α L/d α L/d α L/d α Table 1. Tilt angleα(in degrees) for the four sets of particles in a 1.5% solution. Figure 3: Tilt angle α versus the aspect ratio L/d for three sets of particles in a 1.5% solution. The α versus L/d curves for the two sets of aluminum particles almost overlap, as can be seen in Fig. 3. Moreover, in Table 1, we can also see that two Teflon particles with the same aspect ratio L/d but different values of d have similar tilt angles. These results indicate that the tilt angle depends on the aspect ratiol/d and the density of the particle, but not on the weight or the size. Another important parameter in our sedimentation experiments is the terminal speed,u, of the settling particles. In Table 2 we report the measured values of u for the four sets of particles. Moreover, in Fig. 4, we provide curves of u versus the aspect ratio L/d, for different particles. 3

4 Aluminum d = 0.25in Aluminum d = 0.375in Teflon d = 0.25in Teflon d = 0.375in L/d u L/d u L/d u L/d u Table 2. Sedimentation speedu(in cm/s) for the four sets of particles in a 1.5% solution. Figure 4: Sedimentation speed u versus the aspect ratio L/d for three sets of particles in a 1.5% solution. formula α= The u versus L/d curves show that u is a strictly increasing function of the aspect ratio. In fact, we found that a power-law correlation formula of the type: u=k 1 (L/d) k2, (1.1) gives best curve-fitting. In (1.1)k 1 andk 2 are fitting parameters depending on the particle. Values of k 1 and k 2 for our particles are reported in Table 3. A correlation law for the tilt angleαversusl/d seems more complicated than power-law. Actually, following the logistic dose-response curve fitting described in the Appendix of the paper by Patankar et. al. [14], we found that the data are well fit by the following a [ ( ) c ] b (1.2) L/d 1+ t wherea,b,candtare fitting parameters. Values of these parameters for four sets of particles are given in Table 4. Since the data for the two sets of aluminum particles (d=0.635cm and d=0.9525cm) are very close (see Fig. 3 and Table 1), a single logistic curve is used to fit the two sets of data. k 1 k 2 Aluminum (d = 0.25in) Aluminum (d = 0.375in) Teflon (d = 0.25in) Table 3. Fitting parametersk 1 andk 2 in the power-law formula (1.1) for experiments using a 1.5% solution. a b c t Aluminum (d = 0.25in and d = 0.375in) Teflon (d = 0.25in) Table 4. Fitting parametersa,b,candtin the correlation formula (1.2) for experiments using a 1.5% solution. 4

5 Sedimentation in a 1% Solution. The results are qualitatively very similar to those found for the 1.5% solution. Variation of the tilt angle and the sedimentation speed with the aspect ratio L/d are reported in Fig. 5 and Fig.6. Data-fitting curves of the type (1.1) and (1.2) are also obtained and the corresponding values of the parameters are given in Table 5 and Table 6. Figure 5: Tilt angleαversus the aspect ratiol/d for three sets of particles in a 1% solution. Figure 6: Sedimentation speed u versus the aspect ratio L/d for three sets of particles in a 1% solution. k 1 k 2 Aluminum (d = 0.635cm) Aluminum (d = cm) Teflon (d = 0.635cm) Table 5. Fitting parametersk 1 andk 2 in the power-law formula (1.1) for experiments using a 1% solution. a b c t aluminum (d = 0.635cm) aluminum (d = cm) Teflon (d = 0.635cm) Table 6. Fitting parametersa,b,candtin the correlation formula (1.2) for experiments using a 1% solution. In Fig. 5, there is an obvious difference between the two tilt angle vs. L/d curves for the two sets of aluminum particles with different diameters, which is unlike the case of a 1.5% solution, wherein such two curves almost overlap (Fig. 3). This could be attributed to the fact that the viscoelastic effects are not as predominant in a 1% solution as they are in a 1.5% solution. The inertia in a 1% solution becomes important; the size and weight of the particles are relevant. Sedimentation in a 0.75% Solution. In comparison with the trials conducted in the 1.5% solution, the data obtained here proved to be more sporadic. The particles, especially those with large L/d ratios and diameters, were often attracted to the wall even if they were released at the centerline of the channel (see [10] for discussions of particle-wall interactions in viscoelastic fluids). The increased particle-wall interactions distorted the tilt angle and the outcome is that the trends are much less apparent. The cm diameter aluminum cylinders proved to be most inconsistent. On the other hand, the other particles appear to be leveling off and possibly approaching the asymptotic limit,90 0 ; see Fig. 7. 5

6 Figure 7: Tilt angleαversus the aspect ratiol/d for three sets of particles in a 0.75% solution. Figure 8: Tilt angleα versus the aspect ratiol/d for the cm aluminum cylinders in a 0.75% solution. In Fig. 8 is the tilt angle vs. L/d graph of the cm aluminum cylinders. The data acquired is highly suspect due to the strong affinity between the particles and walls. Cylinders with relatively small L/d ratios demonstrate the previous leveling pattern observed in 1.5% and 1% solutions. Thereafter, the angles become increasingly questionable as no coherent relationship is observed. For the four cylinders with the largest L/d ratios, no angles could be measured due to a combination of rapid speeds and the aforementioned wall interactions. Figure 9: Sedimentation speed u versus the aspect ratio L/d for three sets of particles in a 0.75% solution. Figure 10: Sedimentation speed u versus the aspect ratio L/d for the cm aluminum cylinders in a 0.75% solution. Illustrated in Fig. 9 is the sedimentation speed plotted against the aspect ratio for three sets of particles. The results appear relatively linear, and it is clear that an escalation of L/d ratio and weight generates a simultaneous increase in speed. It should be emphasized that it is a combination of the ratio and weight that produces such an effect. Note that the Teflon particle (d = cm) with an aspect ratio of has a weight of 1.46g. This is larger than the weight of any of the 0.635cm diameter aluminum particles; however, six of the aluminum cylinders are moving at faster rates than the Teflon particle. In Fig. 10 it is shown the speed vs. L/d curve of the cm diameter aluminum cylinders. Throughout the trials, these particles demonstrated considerable affinity for the channel walls. Moreover, it was observed that larger aspect ratios resulted in greater interactions between the particles and channel walls. Consequently, for larger ratios, the graph appears to level off due to the fact that the cylinders repeatedly collide with the walls, therefore augmenting the fall times. The upward twist at the end of the curve is due to minimized/negligible collisions during the trial of three of the cylinders with aspect ratio 2.4. This lessened the fall time and produced a greater speed. Were a wider channel be available, it is probable that the speeds of the larger particles would have been larger, and the resulting plot would have likely resembled the curves shown for the 1.5% and 1% solutions. 6

7 Figure 11: Sedimentation of an aluminum cylinder with round ends in a 1% aqueous Polyox solution. The particle has d = 0.635cm, L/d=2.2 and a weight of 1.00g. It is dropped horizontally and then it turns as it falls down. At the steady state, it lines up with the longest line in the body parallel to the gravity. The timetis in sec. Figure 12: Sedimentation of the same cylinder of Figure 11 in a 0.75% aqueous Polyox solution. It is dropped horizontally and, eventually, at the steady state, it lines up with the longest line in the body parallel to the gravity. The timetis in sec. Sedimentation of Cylindrical P articles with Round Ends. As we mentioned in the introduction, the shape-tilting phenomenon disappears if the cylinders have round ends and they fall with their broadside parallel to gravity. Snapshots of these experiments are shown in Fig. 11 and Fig. 12 for the two different 1% and 0.75% Polyox solutions. These experiments are in perfect agreement with the theory developed in [6]. A Heuristic Interpretation of Shape Tilting. In the experiments performed by Liu and Joseph [13] it was shown that flat objects, like plates and disks, when dropped in a viscoelastic liquid tend to orient themselves with their face parallel to gravity (the object cuts the liquid), whenever viscoelastic stresses are predominant on the inertia of the liquid. This is just the opposite of what happens in a Newtonian liquid where the final orientation of flat objects is with their face perpendicular to gravity (maximum drag). A disk can be viewed as a cylindrical particle with an aspect ratiol/d 1. On the other extreme, we know that a slender cylinder havingl/d 1 falling in a viscoelastic liquid where, again, viscoelastic stresses are predominant on inertia, will eventually orient itself with its broadside parallel to gravity [11], [4]. If we keepdfixed and increaselcontinuously, it might be then reasonable to expect that the orientation Figure 13: Orientation of cylinders in a viscoelastic liquid as a function of the cylinder will likewise change in such a way that of the aspect ratiol/d. The diameterdis fixed and the lengthlis the tilt angle increases from0 0 to 90 0 ; see Fig. 13. varied. The common feature of all these equilibrium orientations is that the particles have the tendency to orient themselves with the longest line in the body parallel to gravity. 2. Conclusions Cylindrical particles with flat ends of aluminum and Teflon of different aspect ratio L have been dropped in Polyox solutions of various concentrations. It is observed that forlless than a critical valuel 0 (depending on particle and liquid properties) the particles eventually orient themselves by reaching a configuration characterized by the tilt angle, α, that their major axis forms with the horizontal. α may range between0 0 and90 0. It is found that α < 90 0 if L < L 0, whereas α = 90 0 if L > L 0. In a 1.5% Polyox solution, it is measured L for aluminum particles, whilel for Teflon particles. Approximately, the cylinder always tends to line up with the longest line in the body parallel to the gravity. It is also observed that α and the sedimentation (terminal) 7

8 speed u are increasing functions of L. This tilt-angle phenomenon disappears if the cylinders have round ends, and, in this case, the particle will always reach a final orientation withα=90 0. Therefore, it is inferred that the phenomenon is tightly related to the shape of the particle. References [1] Advani, A.S., 1994, Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam [2] Becker, H.A., 1959, The Effects of Shape and Reynolds Number on Drag in the Motion of a Freely Oriented Body in an Infinite Fluid, Can. J. Chem. Eng., 37, [3] Chhabra R.P., Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press 1993 [4] Chiba, K., Song, K., and Horikawa, A., 1986, Motion of a Slender body in a Quiescent Polymer Solution, Rheol. Acta, 25, [5] Cho, K., Cho, Y.I., and Park, N.A., 1992, Hydrodynamics of a Vertically Falling Thin Cylinder in non- Newtonian Fluids, J. Non-Newtonian Fluid Mech., 45, [6] Galdi, G.P., Pokorny, M., Vaidya, A., Joseph, D.D. and Feng, J., Orientation of symmetric bodies falling in a second-order liquid at non-zero Reynolds number, Math. Models Methods Appl. Sci. 12 (2002), no. 11, [7] Grossman, P.D., and Soane, D.S., 1990, Orientation Effects on the Electrophoretic Mobility of Rod-Shaped Molecules in Free Solution, Anal. Chem., 62, [8] Hames, B.D., and Rickwood, D., Eds., 1984, Gel Electrophoresis of Proteins, IRL Press, Washington, D.C. [9] Joseph, D.D., 2000, Interrogations of Direct Numerical Simulation of Solid-Liquid Flow, [10] Joseph, D.D., Flow induced microstructure in Newtonian and viscoelastic fluids, in Proceedings of the 5th World Congress of Chemical Engineering, Particle Technology Track, 6, American Institute of Chemical Engineers, San Diego Keynote presentation (Paper no. 95a, Second Particle Technology Forum). San Diego, California (1996). [11] Leal, L.G., 1975, The Slow Motion of Slender Rod-Like Particles in a Second-Order Fluid, J. Fluid Mech., 69, [12] Lee, S.C., Yang, D.Y., Ko, J., and You, J.R.,1997, Effect of Compressibility on Flow Field and Fiber Orientation During the Filling Stage of Injection Molding, J Mater. Process. Tech., 70, [13] Liu, Y.J., and Joseph, D.D., 1993, Sedimentation of Particles in Polymer Solutions, J. Fluid Mech., [14] Patankar, N.A., Joseph D.D., Wang J., Barree R.D., Conway M. and Asadi M., 2002, Power law correlations for sediment transport in pressure driven channel flows Int. J. Mult. Flow [15] Pettyjohn, E.S., and Christiansen, E.B., 1948, Effect of Particle Shape on Free-Settling Rates of Isometric Particles, Chem. Eng. Prog., 44, [16] Schmid-Schonbein, H., and Wells,R., 1969, Fluid Drop-Like Transition of Erythrocytes under shear, Science, 165, (3890), [17] Tinland, B., Meistermann, L., Weill, G., 2000, Simultaneous Measurements of Mobility, Dispersion, and Orientation of DNA During Steady-Field Gel Electrophoresis Coupling a Fluorescence Recovery after Photobleaching Apparatus with a Fluorescence Detected Linear Dichroism Setup, Phys. Rev. E, 61 (6)

9 [18] Trainor, G.L., 1990, DNA Sequencing, Automation and Human Genome, Anal. Chem., 62, Acknowledgments. The work of G. P. Galdi was partially supported by the NSF grant DMS ; the work of J. Wang, R. Bai, C. Lewandowski and D. D. Joseph was partially supported by the grant of NSF/CTS and the Department of Basis Energy Science at DOE. 9