On Riemann problems and front tracking for a model of sedimentation of poly disperse suspensions

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 87, Issue 10, 2007, Pages 665-691

  Berres, Stefan ^a   Bürger, Raimund ^a  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

Conservation law; Front tracking; Liu entropy condition; Polydisperse suspensions; Riemann problem

Indexed keywords

EID: 36048986358     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200710343     Document Type: Article

References (53)

1. N. Andrianov and G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys. 195, 434-464 (2004).
2. F. Ancona and A. Marson, A note on the Riemann problem for general n × n conservation laws, J. Math. Anal. Appl. 260, 279-293 (2001).
3. S. Benzoni-Gavage and R. Colombo, An n-populations model for traffic flow, Eur. J. Appl. Math. 14, 587-612 (2003).
4. S. Berres, R. Bürger, and H. Frid, Neumann problems for quasi-linear parabolic systems modelling polydisperse suspensions, SIAM J. Math. Anal. 38, 557-573 (2006).
5. S. Berres, R. Bürger, K. H. Karlsen, and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math. 64, 41-80 (2003).
6. S. Bianchini, On the Riemann problem for non-conservative hyperbolic systems, Arch. Rational Mech. Anal. 166, 1-26 (2003).
7. S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161, 223-342 (2005).
8. A. Bressan, Hyperbolic Systems of Conservation Laws (Oxford University Press, Oxford, UK, 2000).
9. R. Bürger, F. Concha, K.-K. Fjelde, and K. H. Karlsen, Numerical simulation of the settling of polydisperse suspensions of spheres, Powder Technol. 113, 30-54 (2000).
10. R. Bürger, K.-K. Fjelde, K. Höfler, and K. H. Karlsen, Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations, J. Eng. Math. 41, 167-187 (2001).
11. R. Bürger, K. H. Karlsen, E. M. Tory, and W. L. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, ZAMM Z. Angew. Math. Mech. 82, 699-722 (2002).
12. R. Bürger and A. Kozakevicius, Adaptive multiresolution WENO schemes for multi-species kinematic flow models, J. Comput. Phys. 224, 1190-1222 (2007).
13. M. C. Bustos and F. Concha, Settling velocities of particulate systems: 10. A numerical method for solving Kynch sedimentation processes, Int. J. Mineral Process. 57, 185-203 (1999).
14. M. C. Bustos, F. Concha, R. Bürger, and E. M. Tory, Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999).
15. C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38, 33-41 (1972).
16. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Second Ed. (Springer-Verlag, Berlin, 2005).
17. A. H. Falls and W. M. Schulte, Theory of three component, three phase displacement in porous media, SPE Reserv. Eng. 7, 377-384 (1992).
18. A. H. Falls and W. M. Schulte, Features of three component, three phase displacement in porous media, SPE Reserv. Eng. 7, 426-432 (1992).
19. E. Fried and B.C. Roy, Gravity-induced segregation of cohesionless granular mixture. In: K. Hutter and N. Kirchner (eds.), Dynamic Response of Granular and Porous Materials under Large and Catastrophic Deformations (Springer-Verlag, Berlin 2003, pp. 393-421).
20. V. Haugse, K. H. Karlsen, K.-A. Lie, and J. R. Natvig, Numerical solution of the polymer system by front tracking, Transp. Porous Media 44, 63-83 (2001).
21. H. Holden, L. Holden, and R. A. Høegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one dimension, Comput. Math. Applic. 15, 595-602 (1988).
22. H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws (Springer-Verlag, New York, 2002).
23. E. Isaacson, D. Marchesin, B. Plohr, and B. Temple, The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems I., SIAM J. Appl. Math. 48, 1009-1032 (1988).
24. E. Isaacson and B. Temple, The Riemann problem near a hyperbolic singularity. II, III, SIAM J. Appl. Math. 48 1287-1301, 1302-1318 (1988).
25. R. Juanes and K.-A. Lie, Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver, Transp. Porous Media 67, 375-393 (2007).
26. R. Juanes and K.-A. Lie, Numerical modeling of multiphase first-contact miscible flows. Part 2. Front-tracking/streamline simulation, Transp. Porous Media, to appear.
27. R. Juanes and T. W. Patzek, Three-phase displacement theory: An improved description of relative permeabilities, SPE J. 9, 302-313 (2004).
28. R. Juanes and T. W. Patzek, Analytical solution to the Riemann problem of three-phase flow in porous media, Transp. Porous Media 55, 47-70 (2004).
29. B. L. Keyfitz and H. C. Kranzer, A viscosity approximation to a system of conservation laws with no classical Riemann solution. In: Nonlinear Hyperbolic Problems (Bordeaux, 1988), Lecture Notes in Math. 1402 (Springer-Verlag, Berlin 1989), pp. 185-197.
30. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160, 214-282 (2000).
31. G.J. Kynch, A theory of sedimentation, Trans. Farad. Soc. 48, 166-176 (1952).
32. P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, 537-566 (1957).
33. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, PA, 1973).
34. R. J. LeVeque, Numerical Methods for Conservation Laws, Second Ed. (Birkhäuser Verlag, Basel, 1992).
35. K.-A. Lie and R. Juanes, A front-tracking method for the simulation of three-phase flow in porous media, Comput. Geosci. 9, 29-59 (2005).
36. T. P. Liu, The Riemann problem for general 2×2 conservation laws, Trans. Amer. Math. Soc. 199, 89-112 (1974).
37. T P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53, 78-88 (1976).
38. M. J. Lockett and K. S. Bassoon, Sedimentation of binary particle mixtures, Powder Technol. 24, 1-7 (1979).
39. A. Majda and R. L. Pego, Stable viscosity matrices for systems of conservation laws, J. Diff. Eqns. 56, 229-262 (1985).
40. D. Marchesin and B. Plohr, Wave structure in WAG recovery, SPE paper 56480.
41. J. H. Masliyah, Hindered settling in a multiple-species particle system, Chem. Eng. Sci. 34, 1166-1168 (1979).
42. R. Menikoff and B. J. Plohr, The Riemann problem for fluid flow of real materials, Rev. Mod. Phys. 61, 75-130 (1989).
43. S. Müller and A. Voß, The Riemann Problem for the Euler equations with nonconvex and nonsmooth equation of state: Construction of wave curves, SIAM J. Sci. Comp. 28, 651-681 (2006).
44. N. H. Risebro, A front tracking alternative to the random choice method, Proc. Amer. Math. Soc. 117, 1125-1139 (1993).
45. N. H. Risebro and A. Tveito, Front tracking applied to a nonstrictly hyperbolic system of conservation laws, SIAM J. Sci. Stat. Comput. 12, 1401-1419 (1991).
46. F. Rosso and G. Sona, Gravity-driven separation of oil-water dispersions, Adv. Math. Sci. Appl. 11, 127-151 (2001).
47. W. Schneider, G. Anestis, and U. Schaflinger, Sediment composition due to settling of particles of different sizes, Int. J. Multiphase Flow 11, 419-323 (1985).
48. P. T. Shannon, E. Stroupe, and E. M. Tory, Batch and continuous thickening, Ind. Eng. Chem. Fund. 2, 203-211 (1963).
49. M. Torrilhon, Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics, J. Plasma Phys. 69, 253-276 (2003).
50. B. Wendroff, An analysis of front tracking for chromatography, Acta Appl. Math. 30, 265-285 (1993).
51. G. C. K. Wong and S. C. Wong, A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers, Transp. Res. A 36, 827-841 (2002).
52. A. Zeidan, A. Rohani, and A. Bassi, Dynamic and steady-state sedimentation of poly disperse suspension and prediction of outlets particle-size distribution, Chem. Eng. Sci. 59, 2619-2632 (2004).
53. A. Zeidan, S. Rohani, A. Bassi, and P. Whiting, Review and comparison of solids settling velocity models, Rev. Chem. Eng. 19, 473-530 (2003).