Entropy solutions and flux identification for a scalar conservation law modelling sedimentation in cones

Raimund Bürger

e-mail: rburger@ing-mat.udec.cl∗

CI2MA & Departamento de Ingeniería Matemática

Universidad de Concepción, Chile


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. Related models also arise in flows of
vehicular traffic, pedestrians, in pipes with varying cross-sectional area, and on curved surfaces.
In the first part of the talk, which is based on [3], entropy solutions of this problem are
constructed by the method of characteristics (see, e.g., [8]) for the most important cases of vessels
with downward-decreasing cross-sectional area and flux functions with at most one inflection
point. 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. These findings generalize an earlier treatment [1].

The potential application of the findings is a new method of flux identification via settling
tests in a suitably shaped vessel, which is considered in the second part of the talk related
to [4]. Precisely, a method is presented for the identification of a non-convex flux function of a
hyperbolic scalar conservation law that models sedimentation of solid particles in a liquid. While
all previous identification methods are based on data obtained from settling tests in cylindrical
vessels [2, 5, 6, 7], the novel approach is based on the richer solution behaviour produced
in a vessel with downward-decreasing cross-sectional area, where we limit the discussion to a
segment of a cone. Except for the initial homogeneous concentration, the data given for the
present inverse problem are the location of the decline of the supernatant-suspension interface
as a function of time. The inverse problem is solved by utilizing the construction of solutions of
the direct problem by the method of characteristics. In theory, the entire flux function can be
estimated from only one batch-settling experiment, and the solution is given by parametric and
explicit formulas for the flux function. The method is tested on synthetic data (for example,
generated by numerical simulations with a known flux) and on published experimental data.
Joint work with:
Julio Careaga, CI2MA & Departamento de Ingeniería Matemática, Universidad de Concepción,
Concepción, Chile
e-mail: juliocareaga@udec.cl

Stefan Diehl, Centre for Mathematical Sciences, Lund University, Lund, Sweden
e-mail: diehl@maths.lth.se

∗Partially funded by Fondecyt project 1130154; BASAL project CMM, Universidad de Chile and Centro
de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; CRHIAM, project CONI-
CYT/FONDAP/15130015; and Fondef project ID15I10291.


[1] Anestis, G., Eine eindimensionale Theorie der Sedimentation in Absetzbehältern veränder-
lichen Querschnitts und in Zentrifugen, PhD thesis, TU Vienna, Austria, 1981.

[2] Betancourt, F., Bürger, R., Diehl, S. and Mejías, C., Advanced methods of flux iden-
tification for clarifier-thickener simulation models, Minerals Eng., 63:2–15, 2014.

[3] Bürger, R., Careaga, J. and Diehl, S., Entropy solutions of a scalar conservation law

modelling sedimentation in vessels with varying cross-sectional area, SIAM J. Appl. Math., in


[4] Bürger, R., Careaga, J. and Diehl, S., Flux identification for scalar conservation laws

modelling sedimentation in vessels with varying cross-sectional area. Preprint 2016-40, Centro

de Investigación en Ingeniería Matemática, Universidad de Concepción; submitted.

[5] Bürger, R. and Diehl, S., Convexity-preserving flux identification for scalar conservation

laws modelling sedimentation, Inverse Problems, 29(4):045008, 2013.

[6] Coronel, A., James, F. and Sepúlveda, M., Numerical identification of parameters for a

model of sedimentation processes, Inverse Problems, 19(4):951–972, 2003.

[7] Diehl, S., Numerical identification of constitutive functions in scalar nonlinear convection-
diffusion equations with application to batch sedimentation., Appl. Numer. Math., 95:154–172,


[8] Holden, H. and Risebro, N.H., Front Tracking for Hyperbolic Conservation Laws. Second

Edition, Springer Verlag, Berlin, 2015.