Taylor diffusion in time-dependent flow

In this paper, we consider the problem of calculating the axial concentration profile of a solute transported by a time-dependent flow in a rigid straight pipe. This generalizes the result that Taylor derived in 1953 for calculating the axial concentration profile of a solute in a steady flow. Using asymptotic analysis, we derive a time-dependent diffusion equation for the mean concentration profile along the axial-direction in a pipe. In the special case of time-independent flow, our result reduces to that of Taylor.