Taylor dispersion in arbitrarily shaped axisymmetric channels

Advective dispersion of solutes in long thin channels is important to the analysis and design of a wide range of devices, including chemical separations and microfluidic chips. Despite extensive analysis of Taylor dispersion in various scenarios, most studies have focused on the long-term dispersion behavior of solute through channels of uniform cross-section. In the current study, we analyze the Taylor-Aris dispersion for straight, axisymmetric channels with arbitrary (axial) distributions of diameter. We derive an expression for solute dynamics in terms of two coupled ordinary differential equations (ODEs). These ODEs enable predictions of the time evolution of the mean location and axial dimension (standard deviation) of the solute zone as a function of time and/or the channel geometry. We compare and benchmark our predictions with Brownian dynamics simulations for a variety of cases including linearly expanding and converging channels and channels with periodic diameter distributions. We present an analytical description of the physical regimes of transient positive versus negative axial growth of the solute dimension. Lastly, we demonstrate a method to engineer channel geometries to achieve desired solute width distributions over space and time. We apply this analysis to the design of a geometry that yields a constant axial width and a second geometry that yields a sinusoidal axial variance in space.