Transient Taylor-Aris dispersion for time-dependent flows in straight channels

Taylor–Aris dispersion, the shear-induced enhancement of solute diffusion in the flow direction of the solvent, has been studied intensely in the past half century for the case of steady flow and single-frequency pulsating flows. Here, combining Aris’s method of moments with Dirac’s bra–ket formalism, we derive an expression for the effective solute diffusivity valid for transient Taylor–Aris dispersion in any given time-dependent, multi-frequency solvent flow through straight channels. Our theory shows that the solute dispersion may be greatly enhanced by the time-dependent parts of the flow, and it explicitly reveals how the dispersion coefficient depends on the external driving frequencies of the velocity field and the internal relaxation rates for mass and momentum diffusion. Although applicable to any type of fluid, we restrict the examples of our theory to Newtonian fluids, for which we both recover the known results for steady and single-frequency pulsating flows, and find new, richer structure of the dispersion as function of system parameters in multi-frequency systems. We show that the effective diffusivity is enhanced significantly by those parts of the time-dependent velocity field that have frequencies smaller than the fluid momentum diffusion rate and the solute diffusion rate.