zbMATH — the first resource for mathematics

A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. (English) Zbl 1013.35052
Under some idealizing assumptions the authors reduce a model of continuous sedimentation to a conservation law where the flux function depends continuously on the spatial variable. A weak solution to this equation is constructed by means of the front tracking method.

35L65 Hyperbolic conservation laws
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Barton, N.G.; Li, C.-H.; Spencer, S.J., Control of a surface of discontinuity in continuous thickeners, J. austral. math. soc. ser. B, 33, 269-289, (1992) · Zbl 0758.35009
[2] Bressan, A., Global solutions to systems of conservation laws by wave-front tracking, J. math. anal. appl., 170, 414-432, (1992) · Zbl 0779.35067
[3] Bressan, A.; Liu, T.-P.; Yang, T., L1 stability estimates for n×n conservation laws, Arch. rat. mech. anal., 149, 1-22, (1999)
[4] Bürger, R.; Tory, E.M., On upper rarefaction waves in batch settling, Powder technol., 108, 74-87, (2000)
[5] Bustos, M.C.; Concha, F., Settling velocities of particulate systems10. A numerical method for solving kynch sedimentation processes, Int. J. mineral process., 57, 185-203, (1999)
[6] Bustos, M.C.; Concha, F.; Bürger, R.; Tory, E.M., Sedimentation and thickening, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0936.76001
[7] Bustos, M.C.; Concha, F.; Wendland, W.L., Global weak solutions to the problem of continuous sedimentation of an ideal suspension, Math. meth. appl. sci., 13, 1-22, (1990) · Zbl 0722.35054
[8] Bustos, M.C.; Paiva, F.; Wendland, W.L., Entropy boundary conditions in the theory of sedimentation of ideal suspensions, Math. meth. appl. sci., 19, 679-697, (1996) · Zbl 0855.35077
[9] Chancelier, J.P.; Cohen de Lara, M.; Pacard, F., Analysis of a conservation PDE with discontinuous fluxa model of settler, SIAM J. math. appl., 54, 954-995, (1994) · Zbl 0811.35077
[10] F. Concha, A. Barrientos, M.C. Bustos, Phenomenological model of high capacity thickening, in: Proceedings of the 19th International Mineral Processing Congress (XIX IMPC), San Francisco, 1995, pp. 75-79 (Chapter 14).
[11] Dafermos, C.M., Polygonal approximations of solutions to the initial value problem for a conservation law, J. math. anal. appl., 38, 33-41, (1972) · Zbl 0233.35014
[12] Diehl, S., On scalar conservation laws with point source and discontinuous flux function, SIAM J. math. anal., 26, 1425-1451, (1995) · Zbl 0852.35094
[13] Diehl, S., A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. appl. math., 56, 388-419, (1996) · Zbl 0849.35142
[14] Diehl, S., Dynamic and steady-state behavior of continuous sedimentation, SIAM J. appl. math., 57, 991-1018, (1997) · Zbl 0889.35062
[15] Diehl, S., On boundary conditions and solutions for ideal thickener-clarifier units, Chem. eng. J., 80, 119-133, (2000)
[16] DiPerna, R.J., Global existence of solutions to nonlinear systems of conservation laws, J. diff. eqns., 20, 187-212, (1976) · Zbl 0314.58010
[17] T. Gimse, N.H. Risebro, Riemann problems with a discontinuous flux function, in: Proceedings of the Third Conf. Hyp. Problems, Uppsala, Sweden, 1990. · Zbl 0789.35102
[18] Gimse, T.; Risebro, N.H., Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. math. anal., 23, 635-648, (1992) · Zbl 0776.35034
[19] Holden, H.; Holden, L.; Høegh-Krohn, R., A numerical method for first order nonlinear scalar conservation laws in one dimension, Comp. math. appl., 15, 595-602, (1988) · Zbl 0658.65085
[20] H. Holden, N.H. Risebro, Front Tracking for Conservation Laws, Springer-Verlag, New York, 2002. · Zbl 1006.35002
[21] Klingenberg, C.; Risebro, N.H., Convex conservation laws with discontinuous coefficients, Comm. PDE, 20, 1959-1990, (1995) · Zbl 0836.35090
[22] Klingenberg, C.; Risebro, N.H., Stability of a resonant system of conservation laws modeling polymer flow with gravitation, J. diff. eqns., 170, 344-380, (2001) · Zbl 0977.35083
[23] Kunik, M., A numerical method for some initial-value problems of one scalar hyperbolic conservation law, Math. meth. appl. sci., 15, 495-509, (1992) · Zbl 0761.65075
[24] Kunik, M., A solution formula for a non-convex scalar hyperbolic conservation law with monotone initial data, Math. meth. appl. sci., 16, 895-902, (1993) · Zbl 0823.35116
[25] Kynch, G.J., A theory of sedimentation, Trans. Faraday soc., 48, 166-176, (1952)
[26] Lev, O.; Rubin, E.; Sheintuch, M., Steady state analysis of a continuous clarifier – thickener system, Aiche j., 32, 1516-1525, (1986)
[27] Petty, C.A., Continuous sedimentation of a suspension with a nonconvex flux law, Chem. eng. sci., 30, 1451-1458, (1975)
[28] Risebro, N.H., A front-tracking alternative to the random choice method, Proc. amer. math. soc., 117, 1125-1139, (1993) · Zbl 0799.35153
[29] Risebro, N.H.; Tveito, A., Front tracking applied to a nonstrictly hyperbolic system of conservation laws, SIAM J. sci. stat. comput., 12, 1401-1419, (1991) · Zbl 0736.65075
[30] Risebro, N.H.; Tveito, A., A front tracking method for conservation laws in one dimension, J. comp. phys., 101, 130-139, (1992) · Zbl 0756.65120
[31] Temple, B., Global solution of the Cauchy problem for a 2×2 non-strictly hyperbolic system of conservation laws, Adv. appl. math., 3, 335-375, (1982) · Zbl 0508.76107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.