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

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 ∂ u( x,t )∂ t + ∂ /∂ x( F( u( x,t ),x ) ) = s( t )δ ( x ), where δ 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, · ) 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, · ) that have been used earlier can be omitted. The steady-state solutions are also presented.