Taylor dispersion in long straight channels of arbitrary and gradually changing size and shape

Advective dispersion of neutral solutes in flow is pertinent to the design of a host of physical systems, including microfluidic devices and chemical separation systems. The cross-sectional shapes of the channels within these devices are often complex and may vary (in shape or area) in the flow direction. However, previous analyses of Taylor–Aris dispersion in channels of varying geometry typically focus on either periodic channels or channels with simple cross-sections. To our knowledge, no reduced-order theory exists to predict dispersion in channels of arbitrary cross-section whose shape or size may vary slowly in the axial direction. In the current study, we develop a theoretical framework for a reduced-order model applicable to the analysis of dispersion in channels with such shapes given steady or time-dependent flow conditions. We first derive a one-dimensional partial differential equation for the area-averaged solute concentration and associated expressions for the full, three-dimensional tracer distribution. Next, we formulate two coupled ordinary differential equations that govern the solute axial mean position and variance. Additionally, we derive an ordinary differential equation to engineer channels to produce a priori specified solute variance distributions. We apply our model to the example geometries of axisymmetric channels of varying radii, channels with a varying ‘dumbbell’ cross-section and rectangular channels of varying width. We benchmark our model’s performance against Brownian dynamics simulations for all tested geometries. Overall, our framework is straightforward to apply to different geometries and flow conditions, and may yield critical insights for the design and optimisation of microfluidic systems.