Radial segregation patterns in rotating granular mixtures: Waviness selection

Physical Review Letters

Volumn 93, Issue 22, 2004, Pages

  Hill, K M ^a   Gioia, G ^a   Amaravadi, D ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

AMPLIFICATION; FOURIER TRANSFORMS; FUNCTIONS; GRANULAR MATERIALS; MATHEMATICAL MODELS; OSCILLATIONS;

ANGULAR VELOCITY; CUTOFF WAVINESS; RADIAL SEGREGATION PATTERNS; THAWING FLUX;

MIXTURES;

EID: 19944428740     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevLett.93.224301     Document Type: Article

References (23)

1. A. Rosato, K.J. Strandburg, F. Prinz, and R.H. Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J.B. Knight, H. M. Jaeger, and S.R. Nagel, ibid 70, 3728 (1993); O. Zik, D. Levine, S. G. Shtrikman, and J. Stavans, ibid 73, 644 (1994); S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988); H.A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Nature (London) 386, 379 (1997).
2. A. Rosato, K.J. Strandburg, F. Prinz, and R.H. Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J.B. Knight, H. M. Jaeger, and S.R. Nagel, ibid 70, 3728 (1993); O. Zik, D. Levine, S. G. Shtrikman, and J. Stavans, ibid 73, 644 (1994); S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988); H.A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Nature (London) 386, 379 (1997).
3. A. Rosato, K.J. Strandburg, F. Prinz, and R.H. Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J.B. Knight, H. M. Jaeger, and S.R. Nagel, ibid 70, 3728 (1993); O. Zik, D. Levine, S. G. Shtrikman, and J. Stavans, ibid 73, 644 (1994); S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988); H.A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Nature (London) 386, 379 (1997).
4. A. Rosato, K.J. Strandburg, F. Prinz, and R.H. Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J.B. Knight, H. M. Jaeger, and S.R. Nagel, ibid 70, 3728 (1993); O. Zik, D. Levine, S. G. Shtrikman, and J. Stavans, ibid 73, 644 (1994); S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988); H.A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Nature (London) 386, 379 (1997).
5. A. Rosato, K.J. Strandburg, F. Prinz, and R.H. Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J.B. Knight, H. M. Jaeger, and S.R. Nagel, ibid 70, 3728 (1993); O. Zik, D. Levine, S. G. Shtrikman, and J. Stavans, ibid 73, 644 (1994); S. B. Savage and C. K. K. Lun, J. Fluid Mech. 189, 311 (1988); H.A. Makse, S. Havlin, P. R. King, and H. E. Stanley, Nature (London) 386, 379 (1997).
6. This might be the first form of granular segregation ever noted in a scientific paper; A. Schoklitsch, Akad. Wiss. 142, 343 (1933).
7. The armour layer has often been ascribed to selective erosion [e.g., R. Bettess and A. Frangipane, J. Hydraul. Res. 41, 179 (2003)], but field data pointing to the importance of granular segregation has been known for a long time [e.g., L. B. Leopold, M.G. Wolman, and J. P. Miller, Fluvial Processes in Ceomorphology (W. H. Freeman and Co., San Francisco, 1964), Chap. 7].
8. The armour layer has often been ascribed to selective erosion [e.g., R. Bettess and A. Frangipane, J. Hydraul. Res. 41, 179 (2003)], but field data pointing to the importance of granular segregation has been known for a long time [e.g., L. B. Leopold, M.G. Wolman, and J. P. Miller, Fluvial Processes in Ceomorphology (W. H. Freeman and Co., San Francisco, 1964), Chap. 7].
9. T. Shinbrot and F. J. Muzzio, Phys. Today 53, No. 3, 25 (2000).
10. E. Clement, J. Rajchenbach, and J. Duran, Europhys. Lett. 30, 7 (1995); F. Cantelaube and D. Bideau, ibid 30, 133 (1995); D.V. Khakhar, J. J. McCarthy, and J.M. Ottino, Phys. Fluids 9, 3600 (1997).
11. E. Clement, J. Rajchenbach, and J. Duran, Europhys. Lett. 30, 7 (1995); F. Cantelaube and D. Bideau, ibid 30, 133 (1995); D.V. Khakhar, J. J. McCarthy, and J.M. Ottino, Phys. Fluids 9, 3600 (1997).
12. E. Clement, J. Rajchenbach, and J. Duran, Europhys. Lett. 30, 7 (1995); F. Cantelaube and D. Bideau, ibid 30, 133 (1995); D.V. Khakhar, J. J. McCarthy, and J.M. Ottino, Phys. Fluids 9, 3600 (1997).
13. K.M. Hill et al., Proc. Natl. Acad. Sci. U.S.A. 96, 11701 (1999).
14. D.V. Khakhar, A.V. Orpe, and J.M. Ottino, Powder Technol. 116, 232 (2001).
15. If air is the interstitial fluid instead of water, similar patterns form, and the sun patterns occur only in a narrow range of fill levels [6,7], just as in Fig. 2(c). This fact suggests that our analysis could be broadly applicable, in particular, to granular flows in air. [9] We use NIH Image, which computes the length of the boundary between dark and light areas in a given picture. To calibrate this computation, we use the picture of a moon pattern, for which p can be calculated analytically.
16. N. Jain, J. M. Ottino, and R. M. Lueptow, Phys. Fluids 14, 572 (2002); D. Bonamy, F. Daviaud, and L. Laurent, ibid 14, 1666 (2002); K. M. Hill, G. Gioia, and V. V. Tola, Phys. Rev. Lett. 91, 064302 (2003).
17. N. Jain, J. M. Ottino, and R. M. Lueptow, Phys. Fluids 14, 572 (2002); D. Bonamy, F. Daviaud, and L. Laurent, ibid 14, 1666 (2002); K. M. Hill, G. Gioia, and V. V. Tola, Phys. Rev. Lett. 91, 064302 (2003).
18. N. Jain, J. M. Ottino, and R. M. Lueptow, Phys. Fluids 14, 572 (2002); D. Bonamy, F. Daviaud, and L. Laurent, ibid 14, 1666 (2002); K. M. Hill, G. Gioia, and V. V. Tola, Phys. Rev. Lett. 91, 064302 (2003).
19. n to be constant on the segment Δx of the freezing (or thawing) line.
20. In our model, a moon pattern becomes wavy when the interface between the black and the white beads, initially a circular arc, becomes undulated with a wavelength λ. Thus the volume of the white stripes is the same as the volume of the black stripes, regardless of the average composition of the mixture.
21. To estimate w we write p = 2x + L + 2nΔx, where n is the number of stripes, and L = (π + 2θ)x secθ [Fig. 4(d)]. By substituting n = 2L/λ we arrive to the formula w = (p - 2x - L)/2L. From the picture of the pattern we obtain p, x, and L, and then estimate w using this formula. For a moon pattern p = 2x + L, and w = 0.
22. u ∼ 5, which is in reasonable accord with the peak value of w measured in those experiments (Fig. 5).
23. a > 1.