Taylor–Aris dispersion in shear-rate-dependent fluid flows

The spread of a pulse of solute in a pressure-driven channel flow is well described for a wide range of Newtonian flows for which the viscosity and diffusivity are constants. Over many decades, various extensions have been suggested for the dispersion in pressure-driven non-Newtonian channel flows. While many theoretical studies have examined the effect of shear-rate-dependent viscosity on dispersion for a variety of non-Newtonian constitutive models, the solute diffusivity has invariably been treated as a constant. This assumption, however, is in contrast to the expectation that the diffusivity of a colloidal particle is inversely related to the viscosity, e.g. recall the Stokes–Einstein relation. We account for this coupling of transport coefficients – viscosity and diffusivity – by assuming a generalised form of the Stokes–Einstein equation, inspired by the recognition that the viscosity is now a field, although only transport transverse to the main flow direction is relevant because of the common assumptions of Taylor–Aris dispersion. Thus, we derive a general formula for axial dispersion in steady, pressure-driven shear-rate-dependent flows in uniform channels. In particular, we apply our general relation to calculate the Taylor–Aris dispersion coefficient for steady flows of a shear-thinning Carreau fluid and a viscoelastic Phan-Thien–Tanner fluid. Finally, we highlight new theoretical questions raised by this transport situation, where the underlying diffusivity is also a (tensorial) field related to variations in viscosity.