Chaotic granular mixing

Chaos

Volumn 9, Issue 3, 1999, Pages 611-620

  Shinbrot, Troy ^a   Alexander, Albert ^a   Moakher, Maher ^a   Muzzio, Fernando J ^a  

^a The State University of New Jersey   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033196387     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166434     Document Type: Article

References (58)

1. R. P. Killmeyer, US Dept. of Energy report, "National Coal Preparation Process Facility (1994)."
2. J. Van Cleef, "Powder Technology," Am. Sci. 79, 304-315 (1991).
3. C. E. Harwood, "Powder segregation due to vibration," Powder Technol. 16, 51-57 (1977); J. C. Williams, "The segregation of particulate materials. A review," ibid. 15, 245-251 (1976) J. Knight, H. Jaeger, and S. Nagel, "Vibration-induced size separation ingranular media: The convention connection," Phys. Rev. Lett. 70, 3728 (1993) J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and S. Sokolowski, "Molecular dynamics simulation of size segregation in three dimensions," J. Stat. Phys. 82, 443-450 (1996); T. Shinbrot and F. J. Muzzio, "Reverse buoyancy in shaken granular beds," Phys. Rev. Lett. 81, 4365-4368 (1998).
4. C. E. Harwood, "Powder segregation due to vibration," Powder Technol. 16, 51-57 (1977); J. C. Williams, "The segregation of particulate materials. A review," ibid. 15, 245-251 (1976) J. Knight, H. Jaeger, and S. Nagel, "Vibration-induced size separation ingranular media: The convention connection," Phys. Rev. Lett. 70, 3728 (1993) J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and S. Sokolowski, "Molecular dynamics simulation of size segregation in three dimensions," J. Stat. Phys. 82, 443-450 (1996); T. Shinbrot and F. J. Muzzio, "Reverse buoyancy in shaken granular beds," Phys. Rev. Lett. 81, 4365-4368 (1998).
5. C. E. Harwood, "Powder segregation due to vibration," Powder Technol. 16, 51-57 (1977); J. C. Williams, "The segregation of particulate materials. A review," ibid. 15, 245-251 (1976) J. Knight, H. Jaeger, and S. Nagel, "Vibration-induced size separation ingranular media: The convention connection," Phys. Rev. Lett. 70, 3728 (1993) J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and S. Sokolowski, "Molecular dynamics simulation of size segregation in three dimensions," J. Stat. Phys. 82, 443-450 (1996); T. Shinbrot and F. J. Muzzio, "Reverse buoyancy in shaken granular beds," Phys. Rev. Lett. 81, 4365-4368 (1998).
6. C. E. Harwood, "Powder segregation due to vibration," Powder Technol. 16, 51-57 (1977); J. C. Williams, "The segregation of particulate materials. A review," ibid. 15, 245-251 (1976) J. Knight, H. Jaeger, and S. Nagel, "Vibration-induced size separation ingranular media: The convention connection," Phys. Rev. Lett. 70, 3728 (1993) J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and S. Sokolowski, "Molecular dynamics simulation of size segregation in three dimensions," J. Stat. Phys. 82, 443-450 (1996); T. Shinbrot and F. J. Muzzio, "Reverse buoyancy in shaken granular beds," Phys. Rev. Lett. 81, 4365-4368 (1998).
7. C. E. Harwood, "Powder segregation due to vibration," Powder Technol. 16, 51-57 (1977); J. C. Williams, "The segregation of particulate materials. A review," ibid. 15, 245-251 (1976) J. Knight, H. Jaeger, and S. Nagel, "Vibration-induced size separation ingranular media: The convention connection," Phys. Rev. Lett. 70, 3728 (1993) J. A. C. Gallas, H. J. Herrmann, T. Pöschel, and S. Sokolowski, "Molecular dynamics simulation of size segregation in three dimensions," J. Stat. Phys. 82, 443-450 (1996); T. Shinbrot and F. J. Muzzio, "Reverse buoyancy in shaken granular beds," Phys. Rev. Lett. 81, 4365-4368 (1998).
8. C. Bizon, M. D. Shattuck, J. R. deBruyn, J. B. Swift, W. D. McCormick, and H. L. Swinney, "Convection and diffusion in patterns in oscillated granular media," J. Stat. Phys. 93, 449-454 (1998).
9. G. Metcalfe, T. Shinbrot, J. McCarthy, and J. M. Ottino, "Avalanche mixing of granular solids," Nature (London) 374, 39-41 (1995); J. Rajchenbach, "Flow in powders: From discrete avalanches to continuous regime," Phys. Rev. Lett. 65, 2221-2224 (1990); F. Cantelaube, Y. I. Duparcmeur, D. Bideau, and G. H. Ristow, "Geometrical analysis of avalanches in a 2d drum," J. Phys. I 5, 581-587 (1995); F. Cantelaube and D. Bideau, "Radial segregation in a 2d drum: experimental analysis," Europhys. Lett. 30, 133-136 (1995); and O. Zik, D. Levine, S. G. Lipson, S. Shrikman, and J. Stavans, "Rotationally induced segregation of granular materials," Phys. Rev. Lett. 73, 644-647 (1994).
10. G. Metcalfe, T. Shinbrot, J. McCarthy, and J. M. Ottino, "Avalanche mixing of granular solids," Nature (London) 374, 39-41 (1995); J. Rajchenbach, "Flow in powders: From discrete avalanches to continuous regime," Phys. Rev. Lett. 65, 2221-2224 (1990); F. Cantelaube, Y. I. Duparcmeur, D. Bideau, and G. H. Ristow, "Geometrical analysis of avalanches in a 2d drum," J. Phys. I 5, 581-587 (1995); F. Cantelaube and D. Bideau, "Radial segregation in a 2d drum: experimental analysis," Europhys. Lett. 30, 133-136 (1995); and O. Zik, D. Levine, S. G. Lipson, S. Shrikman, and J. Stavans, "Rotationally induced segregation of granular materials," Phys. Rev. Lett. 73, 644-647 (1994).
11. G. Metcalfe, T. Shinbrot, J. McCarthy, and J. M. Ottino, "Avalanche mixing of granular solids," Nature (London) 374, 39-41 (1995); J. Rajchenbach, "Flow in powders: From discrete avalanches to continuous regime," Phys. Rev. Lett. 65, 2221-2224 (1990); F. Cantelaube, Y. I. Duparcmeur, D. Bideau, and G. H. Ristow, "Geometrical analysis of avalanches in a 2d drum," J. Phys. I 5, 581-587 (1995); F. Cantelaube and D. Bideau, "Radial segregation in a 2d drum: experimental analysis," Europhys. Lett. 30, 133-136 (1995); and O. Zik, D. Levine, S. G. Lipson, S. Shrikman, and J. Stavans, "Rotationally induced segregation of granular materials," Phys. Rev. Lett. 73, 644-647 (1994).
12. G. Metcalfe, T. Shinbrot, J. McCarthy, and J. M. Ottino, "Avalanche mixing of granular solids," Nature (London) 374, 39-41 (1995); J. Rajchenbach, "Flow in powders: From discrete avalanches to continuous regime," Phys. Rev. Lett. 65, 2221-2224 (1990); F. Cantelaube, Y. I. Duparcmeur, D. Bideau, and G. H. Ristow, "Geometrical analysis of avalanches in a 2d drum," J. Phys. I 5, 581-587 (1995); F. Cantelaube and D. Bideau, "Radial segregation in a 2d drum: experimental analysis," Europhys. Lett. 30, 133-136 (1995); and O. Zik, D. Levine, S. G. Lipson, S. Shrikman, and J. Stavans, "Rotationally induced segregation of granular materials," Phys. Rev. Lett. 73, 644-647 (1994).
13. G. Metcalfe, T. Shinbrot, J. McCarthy, and J. M. Ottino, "Avalanche mixing of granular solids," Nature (London) 374, 39-41 (1995); J. Rajchenbach, "Flow in powders: From discrete avalanches to continuous regime," Phys. Rev. Lett. 65, 2221-2224 (1990); F. Cantelaube, Y. I. Duparcmeur, D. Bideau, and G. H. Ristow, "Geometrical analysis of avalanches in a 2d drum," J. Phys. I 5, 581-587 (1995); F. Cantelaube and D. Bideau, "Radial segregation in a 2d drum: experimental analysis," Europhys. Lett. 30, 133-136 (1995); and O. Zik, D. Levine, S. G. Lipson, S. Shrikman, and J. Stavans, "Rotationally induced segregation of granular materials," Phys. Rev. Lett. 73, 644-647 (1994).
14. D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M. Ottino, "Transverse flow and mixing of granular materials in a rotating cylinder," Phys. Fluids 9, 31-43 (1997).
15. H. Campbell and W. C. Bauer, "Cause and cure of demixing in solid-solid mixers," Chem. Eng. September, 179-185 (1966).
16. K. M. Hill, A. Caprihan, and J. Kakalios, "Bulk segregation in rotated granular material measured by magnetic resonance imaging," Phys. Rev. Lett. 78, 50-53 (1997).
17. K. Choo, T. C. A. Molteno, and S. W. Morris, "Traveling granular segregation patterns in a long drum mixer," Phys. Rev. Lett. 79, 2975-2978 (1997); I. Aranson and L. Tsimring, "Dynamics of axial separation in long rotating drums," xyz.lanl.gov preprint patt-sol/9810004 (1998).
18. K. Choo, T. C. A. Molteno, and S. W. Morris, "Traveling granular segregation patterns in a long drum mixer," Phys. Rev. Lett. 79, 2975-2978 (1997); I. Aranson and L. Tsimring, "Dynamics of axial separation in long rotating drums," xyz.lanl.gov preprint patt-sol/9810004 (1998).
19. C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," Powder Technol. 84, 231-240 (1995); T. Shinbrot, A. Alexander, and F. J. Muzzio, "Spontaneous Chaotic Granular Mixing," Nature (London) 397, 675-678 (1999).
20. C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," Powder Technol. 84, 231-240 (1995); T. Shinbrot, A. Alexander, and F. J. Muzzio, "Spontaneous Chaotic Granular Mixing," Nature (London) 397, 675-678 (1999).
21. S. Nasuno, A. Kudrolli, and J. P. Gollub, "Friction in granular layers: Hysteresis and precursors," Phys. Rev. Lett. 79, 949-952 (1997); B. Miller, C. O'Hern, and R. P. Behringer, "Stress fluctuations for continuously sheared granular materials," ibid. 77, 3110-3113 (1996); and S. Dippel and D. E. Wolf, "Rapid granular flow on a rough inclined plane," to appear in Traffic and Granular Flow 1997, edited by M. Schreckenberg and D. E. Wolf (Springer, Singapore, 1998).
22. S. Nasuno, A. Kudrolli, and J. P. Gollub, "Friction in granular layers: Hysteresis and precursors," Phys. Rev. Lett. 79, 949-952 (1997); B. Miller, C. O'Hern, and R. P. Behringer, "Stress fluctuations for continuously sheared granular materials," ibid. 77, 3110-3113 (1996); and S. Dippel and D. E. Wolf, "Rapid granular flow on a rough inclined plane," to appear in Traffic and Granular Flow 1997, edited by M. Schreckenberg and D. E. Wolf (Springer, Singapore, 1998).
23. S. Nasuno, A. Kudrolli, and J. P. Gollub, "Friction in granular layers: Hysteresis and precursors," Phys. Rev. Lett. 79, 949-952 (1997); B. Miller, C. O'Hern, and R. P. Behringer, "Stress fluctuations for continuously sheared granular materials," ibid. 77, 3110-3113 (1996); and S. Dippel and D. E. Wolf, "Rapid granular flow on a rough inclined plane," to appear in Traffic and Granular Flow 1997, edited by M. Schreckenberg and D. E. Wolf (Springer, Singapore, 1998).
24. G. Baumann, I. Janosi, and D. E. Wolf, "Surface properties and flow of granular material in a 2D rotating drum model," Phys. Rev. E 51, 1879-1887 (1995); D. V. Ktitarev and D. E. Wolf, "Stratification of granular matter in a rotating drum: Cellular automaton modeling," Gran. Matt. 1, 141-145 (1998); E. Clement, J. Rajchenbach, and J. Duran, "Mixing of a granular material in a bidimensional rotating drum," Europhys. Lett. 30, 7-11 (1995).
25. G. Baumann, I. Janosi, and D. E. Wolf, "Surface properties and flow of granular material in a 2D rotating drum model," Phys. Rev. E 51, 1879-1887 (1995); D. V. Ktitarev and D. E. Wolf, "Stratification of granular matter in a rotating drum: Cellular automaton modeling," Gran. Matt. 1, 141-145 (1998); E. Clement, J. Rajchenbach, and J. Duran, "Mixing of a granular material in a bidimensional rotating drum," Europhys. Lett. 30, 7-11 (1995).
26. G. Baumann, I. Janosi, and D. E. Wolf, "Surface properties and flow of granular material in a 2D rotating drum model," Phys. Rev. E 51, 1879-1887 (1995); D. V. Ktitarev and D. E. Wolf, "Stratification of granular matter in a rotating drum: Cellular automaton modeling," Gran. Matt. 1, 141-145 (1998); E. Clement, J. Rajchenbach, and J. Duran, "Mixing of a granular material in a bidimensional rotating drum," Europhys. Lett. 30, 7-11 (1995).
27. M. Nakagawa, S. A. Altobelli, A. Caprihan, E. Fukushima, and E.-K. Jeong, "Non-invasive measurements of granular flows by magnetic resonance imaging," Exp. Fluids 16, 54-60 (1993).
28. J. Bridgwater, "Fundamental powder mixing mechanisms," Powder Technol. 15, 215-236 (1976).
29. R. Hogg and D. W. Fuerstenau, "Transverse mixing in rotating cylinders," Powder Technol. 6, 139-148 (1972).
30. A. W. Chester, J. A. Kowalski, M. E. Coles, E. L. Muegge, F. J. Muzzio, and D. Brone, "Mixing dynamics in catalyst impregnation in twin-cone blenders," Powder Technol. 102, 85-94 (1999).
31. M. Moakher, T. Shinbrot, and F. J. Muzzio, "Experimentally validated computations of flow mixing and segregation of non-cohesive grains in 3D tumbling blenders," Powder Technol. (in press).
32. Indications are that mixing dynamics in the V and tote blenders are substantially different from those in the drum or double-cone blenders (see Ref. 17).
33. D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M. Ottino, "Transverse flow and mixing of granular materials in a rotating cylinder," Phys. Fluids 9, 31-43 (1997); R. Hogg and D. W. Fuerstenau, "Transverse mixing in rotating cylinders," Powder Technol. 6, 139-148 (1972).
34. D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M. Ottino, "Transverse flow and mixing of granular materials in a rotating cylinder," Phys. Fluids 9, 31-43 (1997); R. Hogg and D. W. Fuerstenau, "Transverse mixing in rotating cylinders," Powder Technol. 6, 139-148 (1972).
35. H. Aref, "Stirring by chaotic advection," J. Fluid Mech. 143, 1-21 (1984); S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994), p. 210.
36. H. Aref, "Stirring by chaotic advection," J. Fluid Mech. 143, 1-21 (1984); S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994), p. 210.
37. S. Wiggins, Chaotic Transport in Dynamical Systems (Springer, New York, 1992).
38. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
39. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
40. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
41. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
42. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
43. M. Alonso, M. Satoh, and K. Miyanami, "Influence of rocking motion on the mixing of powders," Powder Technol. 59, 65-67 (1989); C. Wightman, P. R. Mort, F. J. Muzzio, R. E. Riman, and E. K. Gleason, "The structure of mixtures of particles generated by time-dependent flows," ibid. 84, 231-240 (1995); D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," ibid. 91, 165-172 (1997); C. Wightman, F. J. Muzzio, and J. Wilder, "A quantitative image analysis method for characterizing mixtures of granular materials," ibid. 89, 165-176 (1996); C. Wightman and F. J. Muzzio, "Mixing of granular material in a drum mixer undergoing rotational and rocking motions. I. Uniform particles." ibid. 98, 113-124 (1998); "Mixing of granular material in a drum mixer undergoing rotational and rocking motions, II. Segregating particles," 98, 125-135 (1998).
44. C. M. Dury and G. H. Ristow, "Axial particle diffusion in rotating cylinders," Gran. Matt. 1, 151-160 (1999).
45. Details appear in Ref. 10.
46. R. S. MacKay, J. D. Meiss, and I. C. Percival, "Transport in Hamiltonian systems," Physica D 13, 55-81 (1984); J. M. Greene, R. S. McKay, and J. Stark, "Boundary circles for area-preserving maps," ibid. 21, 267-295 (1986).
47. R. S. MacKay, J. D. Meiss, and I. C. Percival, "Transport in Hamiltonian systems," Physica D 13, 55-81 (1984); J. M. Greene, R. S. McKay, and J. Stark, "Boundary circles for area-preserving maps," ibid. 21, 267-295 (1986).
48. M. Rudolph, T. Shinbrot, and R. M. Lueptow, "A model of mixing and transport in wavy Taylor-Couette Flow," Physica D 121, 163-174 (1998).
49. M. Shinbrot, "Fixed-point theorems," Sci. Am. 214, 105-110 (1966).
50. cf. D. R. Scott, "Seismicity and stress rotation in a granular model of the brittle crust," Nature (London) 381, 592-595 (1996).
51. T. Pöschel and V. Buchholtz, "Complex flow of granular material in a rotating cylinder," Chaos, Solitons and Fractals 5, 1901-1912 (1995).
52. F. J. Muzzio, M. M. Alvarez, S. Cerbelli, M. Giona, and A. Adrover, "The intermaterial area density and striation thickness distribution generated by time- and spatially-periodic 2D chaotic flows" (in press); D. M. Hobbs, M. M. Alvarez, and F. J. Muzzio, "Mixing in globally chaotic flows: A self-similar process," Fractals 5, 395-428 (1997).
53. F. J. Muzzio, M. M. Alvarez, S. Cerbelli, M. Giona, and A. Adrover, "The intermaterial area density and striation thickness distribution generated by time- and spatially-periodic 2D chaotic flows" (in press); D. M. Hobbs, M. M. Alvarez, and F. J. Muzzio, "Mixing in globally chaotic flows: A self-similar process," Fractals 5, 395-428 (1997).
54. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov exponents from a time series," Physica D 16, 285-317 (1985).
55. For example, J. M. N. T. Gray, "Granular flow in partially filled slowly rotating drums" (unpublished); D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M. Ottino, "Transverse flow and mixing of granular materials in a rotating cylinder," Phys. Fluids 9, 31-43 (1997).
56. For example, J. M. N. T. Gray, "Granular flow in partially filled slowly rotating drums" (unpublished); D. V. Khakhar, J. J. McCarthy, T. Shinbrot, and J. M. Ottino, "Transverse flow and mixing of granular materials in a rotating cylinder," Phys. Fluids 9, 31-43 (1997).
57. D. Brone, C. Wightman, K. Connor, A. Alexander, F. J. Muzzio, and P. Robinson, "Using flow perturbations to enhance mixing of dry powders in V-blenders," Powder Technol. 91, 165-172 (1997).
58. E. Spartalis, A. Alexander, T. Shinbrot, and F. J. Muzzio, "Multiple segregation patterns in double-cone blender" (unpublished).