(37c) Low-Dimensional Models for Taylor Dispersion

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
Ï?_CD named as convection-diffusion time which characterizes
the dominance of diffusion over convection processes.