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Entropy solutions of a scalar conservation law modeling sedimentation in vessels with varying cross-sectional area

Bürger, Raimund; Careaga, Julio and Diehl, Stefan LU (2017) In SIAM Journal on Applied Mathematics 77(2). p.789-811
Abstract

The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a nonconvex flux function and a weight function that depends on spatial position. The sought unknown is the local solids' volume fraction. For the most important cases of vessels with downward-decreasing cross-sectional area and flux functions with at most one infection point, entropy solutions of this problem are constructed by the method of characteristics. Solutions exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the space-time plane. These trajectories are given by ordinary differential equations... (More)

The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a nonconvex flux function and a weight function that depends on spatial position. The sought unknown is the local solids' volume fraction. For the most important cases of vessels with downward-decreasing cross-sectional area and flux functions with at most one infection point, entropy solutions of this problem are constructed by the method of characteristics. Solutions exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the space-time plane. These trajectories are given by ordinary differential equations that arise from the jump condition. It is shown that three qualitatively different solutions may occur in dependence of the initial concentration. The potential application of the findings is a new method of flux identification via settling tests in a suitably shaped vessel. Related models also arise in flows of vehicular traffic, pedestrians, in pipes with varying cross-sectional area, and on curved surfaces.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Conservation law, Method of characteristics, Shock wave, Variable cross section
in
SIAM Journal on Applied Mathematics
volume
77
issue
2
pages
23 pages
publisher
SIAM Publications
external identifiers
  • scopus:85019113706
  • wos:000402388000011
ISSN
0036-1399
DOI
10.1137/16M1083177
language
English
LU publication?
yes
id
a306ae49-57ae-49c6-b340-13db551920b6
date added to LUP
2017-06-08 08:25:31
date last changed
2018-01-07 12:06:33
@article{a306ae49-57ae-49c6-b340-13db551920b6,
  abstract     = {<p>The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a nonconvex flux function and a weight function that depends on spatial position. The sought unknown is the local solids' volume fraction. For the most important cases of vessels with downward-decreasing cross-sectional area and flux functions with at most one infection point, entropy solutions of this problem are constructed by the method of characteristics. Solutions exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the space-time plane. These trajectories are given by ordinary differential equations that arise from the jump condition. It is shown that three qualitatively different solutions may occur in dependence of the initial concentration. The potential application of the findings is a new method of flux identification via settling tests in a suitably shaped vessel. Related models also arise in flows of vehicular traffic, pedestrians, in pipes with varying cross-sectional area, and on curved surfaces.</p>},
  author       = {Bürger, Raimund and Careaga, Julio and Diehl, Stefan},
  issn         = {0036-1399},
  keyword      = {Conservation law,Method of characteristics,Shock wave,Variable cross section},
  language     = {eng},
  number       = {2},
  pages        = {789--811},
  publisher    = {SIAM Publications},
  series       = {SIAM Journal on Applied Mathematics},
  title        = {Entropy solutions of a scalar conservation law modeling sedimentation in vessels with varying cross-sectional area},
  url          = {http://dx.doi.org/10.1137/16M1083177},
  volume       = {77},
  year         = {2017},
}