The convection-diffusion equation for a finite domain with time varying boundaries

A solution is developed for a convection-diffusion equation describing chem ical transport with sorption, decay, and production. The problem is formula ted in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the prob lem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusi on equation which satisfies a Dirichlet condition at the left end of the ha lf line. This problem does not appear to have been solved in the literature , and the resulting representation should be useful for problems of practic al interest. Authors of previous works on problems of this type have eliminated the unkn own exit concentration by assuming a continuous concentration at the outflo w boundary. This yields a well-posed problem by forcing a homogeneous Neuma nn exit, widely known as Danckwerts condition. We provide a solution to tha t problem and use it to produce an estimate which demonstrates that Danckwe rts condition implies a zero concentration at the outflow boundary, even fo r a long flow domain and a large time. (C) 2001 Elsevier Science Ltd. All r ights reserved.