A delay-diffusion description for contaminant dispersion

The rate of longitudinal dispersion of a contaminant in a current depends upon concentration variations across the flow. These variations are generated by the combination of a longitudinal concentration gradient and a cross-stream velocity shear. However, the response is not instantaneous and so not only is the dispersion coefficient D(τ) time-dependent, but also there is a memory of the concentration distribution at earlier times. In this paper it is shown that these features are accurately reproduced by the delay-diffusion equation \[ \partial_t\bar{c} \overline{u}\partial_x\bar{c} - \overline{\kappa}\partial^2_x\bar{c} - \int_{0}^{\infty}\partial_{\tau} D\partial^2_x\bar{c}\left(x-\int_0^{\tau}\tilde{u}(\tau^{\prime})\,d\tau^{\prime},t-\tau\right)\,d\tau = \bar{q}(x, t), \] where a transport velocity for the free decay of cross-stream concentration variations.