Longitudinal dispersion in flows that are homogeneous in the streamwise direction

The longitudinal dispersion of a pulse of passive scalar contaminant within a flow that is unchanging in the downstream direction is established to be a function of the mean-velocity-history of molecules from a uniform distribution on the flow cross-section. The specific case of laminar flow between parallel plates is investigated in detail, wherein it is found that $$\frac{1}{2}\frac{{\overline {d\eta (t)^2 } }}{{dt}} = \frac{9}{{2\pi ^6 }}\frac{{\bar U^2 d^2 }}{k}\sum\limits_{n = 1}^\infty {\frac{1}{{n^6 }}} \left( {1 - \exp \left( {\frac{{ - (2n\pi )^2 tK}}{{d^2 }}} \right)} \right),$$ where η(t) is the displacement relative to an axis located byūt whenū is the discharge velocity,K is the molecular diffusivity andd is the plate separation. This flow is used to show that the molecule mean velocity-histories can be approximated by a function that does not directly depend on the release position of the molecule on the flow cross-section. The results of a numerical simulation of the dispersion in this flow are presented for comparison.