Low-Dimensional Models for Taylor Dispersion

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We use the Liapunov-Schmidt technique of classical bifurcation theory to derive a low-dimensional model for describing the time evolution of a non-reactive tracer in laminar flow through a tube. Unlike the other averaging techniques, the Liapunov Schmidt formalism leads to an exact averaged model which is valid for all times and converges for all values of parameters with any arbitrary initial or inlet conditions, including points sources in space and time, respectively. This model is consistent with the physics of system as the single mode combined model is parabolic in terms of cross-sectional average concentration and hyperbolic in terms of cup-mixing concentration after truncating up to the second order spatial and temporal derivatives. We also analyze the temporal evolution of spatial moments for the general initial release of a tracer and show that it does not have the centroid displacement or variance deficits predicted by other methods. The moment analysis suggests the existence of a critical time τ_CD named as convection-diffusion time which characterizes the dominance of diffusion over convection processes.