A conservation Law with Point Source and Discontinuous Flux Function Modelling Continuous Sedimentation

Abstract

Continuous sedimentation of solid particles in a liquid takes place in a clarifier-thickener unit, which has one feed inlet and two outlets. The aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions. The conservation of mass yields the scalar conservation law $\frac{\partial u( {x,t} )}{\partial t} + \frac{\partial }{\partial x}( {F( {u( {x,t} ),x} )} ) = s( t )\delta ( x )$, where $\delta $ is the Dirac measure, s is a source, and F is a flux function, which is discontinuous at three points in the one-dimensional space coordinate x. Under certain regularity assumptions a procedure for constructing a solution, locally in time, is presented. The nonlinear phenomena are complicated, and so is the general construction of a solution. The problem of nonuniqueness due to the discontinuities of $F( {u, \cdot } )$ is handled by a generalized entropy condition. An advantage of this approach is that the a priori boundary conditions (at the discontinuities of $ F( {u, \cdot } ) $ that have been used earlier can be omitted. The steady-state solutions are also presented.