  Advective dispersion of solutes in long thin axisymmetric channels is
important to the analysis and design of a wide range of devices, including
chemical separation systems and microfluidic chips. Despite extensive analysis
of Taylor dispersion in various scenarios, most studies focused on long-term
dispersion behavior and cannot capture the transient evolution of solute zone
across the spatial variations in the channel. In the current study, we analyze
the Taylor-Aris dispersion for arbitrarily shaped axisymmetric channels. We
derive an expression for solute dynamics in terms of two coupled ordinary
differential equations (ODEs). These two ODEs allow prediction of the time
evolution of the mean location and axial (standard deviation) width of the
solute zone as a function of the channel geometry. We compare and benchmark our
predictions with Brownian dynamics simulations for a variety of cases including
linearly expanding/converging channels and periodic channels. We also present
an analytical description of the physical regimes of transient positive versus
negative axial growth of solute width. Finally, to further demonstrate the
utility of the analysis, we demonstrate a method to engineer channel geometries
to achieve desired solute width distributions over space and time. We apply the
latter analysis to generate a geometry that results in a constant axial width
and a second geometry that results in a sinusoidal axial variance in space.
