On the intrinsic instability of the advective-diffusive transport

Abstract

[Abstract] A new theory for the advective–diffusive phenomenon is described in this study and the causes for the failure of the conventional numerical methods for this problem are investigated. It is shown that Fick’s law —the constitutive equation of the transport problem— is the cause of the appearance of oscillations in the numerical solutions of predominantly advective problems. Fick’s law leads to the unreasonable result that mass can propagate at an infinite speed. We propose a new formulation for the advective–diffusive problem by using a constitutive equation derived by M. Carlo Cattaneo in 1958 for thermodynamic and pure–diffusion problems. This new approach overcomes the problem of mass propagation at an infinite speed. It is also shown that the advective–diffusive problem is a wave–like problem. Hence, a pollutant diffuses like a wave in a fluid. A detailed analysis of the new equations shows an important conclusion: a critical fluid velocity exists for each advective–diffusive problem. When fluid velocity is greater or equal than this critical speed the steady state problem is not anymore a well–posed problem and the transient problem is as well ill–posed if it is stated as a bounduary value problem. In this case we should formulate the advective–diffusive problem as an initial value problem. Furthermore, we propose stability conditions for the steady state advective– diffusive problem. Several problems have been solved to check the good behaviour of the numerical solution of the new equations and the proposed stability conditions.