Entropy solutions of a scalar conservation law modeling sedimentation in vessels with varying cross-sectional area
(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.
(Less)
- author
- Bürger, Raimund ; Careaga, Julio and Diehl, Stefan LU
- organization
- publishing date
- 2017
- 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
- Society for Industrial and Applied Mathematics
- 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
- 2025-01-07 14:48:06
@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}},
keywords = {{Conservation law; Method of characteristics; Shock wave; Variable cross section}},
language = {{eng}},
number = {{2}},
pages = {{789--811}},
publisher = {{Society for Industrial and Applied Mathematics}},
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}},
doi = {{10.1137/16M1083177}},
volume = {{77}},
year = {{2017}},
}