Entropy Boundary Conditions in the Theory of Sedimentation of Ideal Suspensions

Bustos, María Cristina; Paiva, Freddy; Wendland, Wolfgang L.

doi:10.1002/(sici)1099-1476(199606)19:9<679::aid-mma784>3.0.co;2-l

Cited by 27 publications

(20 citation statements)

References 18 publications

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“…We give the definition of the weak entropy solution to the initial boundary problems (1.1) (also see [3,6,11,15]). …”

Section: Construction Of Piecewise Smooth Solutionmentioning

confidence: 99%

“…For the initial boundary value problem (1.1) whose initial data and boundary data are general bounded variation functions, the existence and uniqueness of the global weak entropy solution in the sense of (2.1) have been obtained, and the global weak entropy solution satisfies the following boundary entropy condition (2.2) (see also [11,3,15]):…”

Section: Satisfies the Following Inequalitymentioning

confidence: 99%

“…The initial boundary value problem of scalar conservation laws plays an important roles in the mathematical modelling and simulation of the practical problem of the one-dimensional sedimentation processes and traffic flow on highways [1][2][3][4][5]. Bardos et al [6] established the existence and uniqueness of the weak entropy solution in the BV-setting for the initial boundary value problems of scalar conservation laws, respectively, by the vanishing viscosity method and by Kruzkov's method [7] (about proving the existence of the solution of the initial value problem for conservation laws by the vanishing viscosity method, see also [8][9][10] etc.).…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty for scalar conservation laws with a boundary effect is to have a good formation of the boundary condition. Namely, for a fixed initial value as (1.1) 2 , we really cannot impose such a condition on the boundary as (1.1) 3 , and the boundary condition is necessarily linked to the entropy condition (The approach for dealing with the boundary in [6] has been followed by many authors [3,[11][12][13][14][15], who extended the previous results in different directions). In other words, the weak entropy solution u(x, t) for (1.1) does not admit a trace at the boundary, namely, u(0, t) does not always equal u b (t), whereas, as a viscosity approximation of the weak entropy solution for (1.1), the solution of the initial boundary value problems of parabolic Equation (1.2) does admit a fixed trace at the boundary.…”

Section: Introductionmentioning

confidence: 99%

“…However, the geometric structure of the solution is much more difficult due to the presence of the boundary. The authors in paper [3] constructed the global weak entropy solution to the initial boundary problem on a bounded interval for some special initial boundary data with three pieces of constant data corresponding to the practical problem of continuous sedimentation of an ideal suspension, but they have not obtained the general result for all cases of three pieces of constant data. In our previous work [42], we considered the initial boundary value problem (1.1) with three pieces of constant data, i.e., u b (t)(t > 0) is a constant function and u 0 (x)(x > 0) is a function with two pieces of constant.…”

Section: Introductionmentioning

confidence: 99%

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Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Liu

Pan

2005

Acta Math Sinica

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This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L 1 -error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(ε 1/2 ) in L 1 -norm; otherwise, as in the initial value problem, the L 1 -error bound is O(ε| ln ε|).

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“…We give the definition of the weak entropy solution to the initial boundary problems (1.1) (also see [3,6,11,15]). …”

Section: Construction Of Piecewise Smooth Solutionmentioning

confidence: 99%

Section: Satisfies the Following Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Liu

Pan

2005

Acta Math Sinica

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Entropy boundary and jump conditions in the theory of sedimentation with compression

Bürger

Wendland

1998

Math. Meth. Appl. Sci.

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We present an initial–boundary value problem of a quasilinear degenerate parabolic equation for the settling and consolidation of a flocculated suspension. The corresponding definition of generalized solutions is formulated. It is based on an entropy integral inequality in the sense of Kružkov. From this definition, jump and entropy conditions that have to be satisfied at discontinuities, and an entropy condition valid on one boundary of the computational domain are derived. The latter implies a set‐valued reformulation of the original boundary condition. It is interpreted geometrically and characterized by the solution of an auxiliary hyperbolic Riemann problem. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons, Ltd.

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A critical look at the kinematic‐wave theory for sedimentation–consolidation processes in closed vessels

Bürger

Kunik

2001

Math Methods in App Sciences

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SUMMARYThe two-phase ow of a occulated suspension in a closed settling vessel with inclined walls is investigated within a consistent extension of the kinematic wave theory to sedimentation processes with compression. Wall boundary conditions are used to spatially derive one-dimensional ÿeld equations for planar ows and ows which are symmetric with respect to the vertical axis. We analyse the special cases of a conical vessel and a roof-shaped vessel. The case of a small initial time and a large time for the ÿnal consolidation state leads to explicit expressions for the ow ÿelds, which constitute an important test of the theory. The resulting initial-boundary value problems are well posed and can be solved numerically by a simple adaptation of one of the newly developed numerical schemes for strongly degenerate convection-di usion problems. However, from a physical point of view, both the analytical and numerical results reveal a deÿciency of the general ÿeld equations. In particular, the strongly reduced form of the linear momentum balance turns out to be an oversimpliÿcation. Included in our discussion as a special case are the Kynch theory and the well-known analyses of sedimentation in vessels with inclined walls within the framework of kinematic waves, which exhibit the same shortcomings. In order to formulate consistent boundary conditions for both phases in a closed vessel and in order to predict boundary layers in the presence of inclined walls, viscosity terms should be taken into account.

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Entropy Boundary Conditions in the Theory of Sedimentation of Ideal Suspensions

Cited by 27 publications

References 18 publications

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Entropy boundary and jump conditions in the theory of sedimentation with compression

A critical look at the kinematic‐wave theory for sedimentation–consolidation processes in closed vessels

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