A model for peak formation in the two-phase equations
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- by Björn Sjögreen, Katarina Gustavsson and Reynir Levi Gudmundsson PDF
- Math. Comp. 76 (2007), 1925-1940 Request permission
Abstract:
We present a hyperbolic-elliptic model problem related to the equations of two-phase fluid flow. The model problem is solved numerically, and properties of its solution are presented. The model equation is well-posed when linearized around a constant state, but there is a strong focusing effect, and very large solutions exist at certain times. We prove that the model problem has a smooth solution for bounded times.References
- F.M. Auzerais, R.J. Jackson, and W.B. Russel, The resolution of shocks and the effect of compressible sediments in transient settling, J. of Fluid Mechanics 195 (1988), 437–462.
- R. Bürger, S. Evje, K. Hvistendahl Karlsen, and K.-A. Lie, Numerical methods for the simulation of the settling of flocculated suspensions, Chemical Engineering Journal 80 (2000), 91–104.
- María Cristina Bustos, Fernando Concha, Raimund Bürger, and Elmer M. Tory, Sedimentation and thickening, Mathematical Modelling: Theory and Applications, vol. 8, Kluwer Academic Publishers, Dordrecht, 1999. Phenomenological foundation and mathematical theory. MR 1747460, DOI 10.1007/978-94-015-9327-4
- M. Dorobantu, Numerical integration of 1d consolidation models, Trita-na-9503, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, 100 44 Stockholm, SWEDEN, 1995.
- Donald A. Drew and Stephen L. Passman, Theory of multicomponent fluids, Applied Mathematical Sciences, vol. 135, Springer-Verlag, New York, 1999. MR 1654261, DOI 10.1007/b97678
- Reynir Levi Gudmundsson, On the well-posedness of the two-fluid model for dispersed two-phase flow in 2D, Report TRITA-NA-0223, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 2002.
- Reynir Levi Gudmundsson and Björn Sjögreen, Numerical experiments with two-fluid equations for particle-gas flow ii: Granular temperature effects, Report TRITA-NA-0442, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 2004.
- Reynir Levi Gudmundsson and Jacob Yström, Numerical experiments with two-fluid equations for particle-gas flow, Part of report TRITA-NA-0222, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 2002.
- Katarina Gustavsson and Björn Sjögreen, Numerical study of a viscous consolidation model, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 569–578. MR 2053206
- Thomas Hillen, Christian Rohde, and Frithjof Lutscher, Existence of weak solutions for a hyperbolic model of chemosensitive movement, J. Math. Anal. Appl. 260 (2001), no. 1, 173–199. MR 1843975, DOI 10.1006/jmaa.2001.7447
- H.-O. Kreiss and J. Yström, Parabolic problems which are ill-posed in the zero dissipation limit, Math. Comput. Modelling 35 (2002), no. 11-12, 1271–1295. MR 1910453, DOI 10.1016/S0895-7177(02)00085-7
- Heinz-Otto Kreiss and Jens Lorenz, Initial-boundary value problems and the Navier-Stokes equations, Pure and Applied Mathematics, vol. 136, Academic Press, Inc., Boston, MA, 1989. MR 998379
- S. N. Kruzkov, First order quasilinear equations in several independent variables, Math. USSR. Sb. 10 (1970), 217–243.
- K. Mattson, Summation-by-parts operators for high order finite difference methods, Ph.D. thesis, Uppsala University, Information Technology, Department of Scientific Computing, 2003.
- Bo Strand, Summation by parts for finite difference approximations for $d/dx$, J. Comput. Phys. 110 (1994), no. 1, 47–67. MR 1259900, DOI 10.1006/jcph.1994.1005
- M. Ungarish, Hydrodynamics of suspensions, Springer-Verlag, 1993.
- B.G.M. van Wachem, J.C. Shouten, C.M. van den Bleck, and J.L. Sinclair, Comparative analysis of CFD models of dense gas-solid systems, American Institute of Chemical Engineering Journal 47 (2001), 1035–1051.
- Jacob Yström, On two-fluid equations for dispersed incompressible two-phase flow, Comput. Vis. Sci. 4 (2001), no. 2, 125–135. Second AMIF International Conference (Il Ciocco, 2000). MR 1946992, DOI 10.1007/s007910100064
Additional Information
- Björn Sjögreen
- Affiliation: Royal Institute of Technology, 100 44 Stockholm, Sweden
- Address at time of publication: Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-550, Livermore, California 94551
- Email: sjogreen2@llnl.gov
- Katarina Gustavsson
- Affiliation: Royal Institute of Technology, 100 44 Stockholm, Sweden
- Email: katarina@nada.kth.se
- Reynir Levi Gudmundsson
- Affiliation: Royal Institute of Technology, 100 44 Stockholm, Sweden
- Email: rlg@nada.kth.se
- Received by editor(s): June 20, 2005
- Received by editor(s) in revised form: June 4, 2006
- Published electronically: May 30, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1925-1940
- MSC (2000): Primary 76T25, 65M99, 35L60
- DOI: https://doi.org/10.1090/S0025-5718-07-01992-8
- MathSciNet review: 2336274