Effective rheology of Bingham fluids in a rough channel

We derive the volumetric flow rate versus pressure drop of a Bingham fluid in one-dimensional channels of variable apertures in the lubrication approximation. A characteristic length scale, $a^*$ characterizing the flow is introduced in order to distinguish between a high and a low flow rate regime. We illustrate the calculation for channels with periodically varying apertures. We then go on to consider apertures that are self affine. We determine how the scaling properties of the aperture field is reflected in the effective flow equations. Finally, a series expansion for high and low flow rates that works very well over the entire range of flow rates is proposed. This truncated expansion allows us to predict the domain of validity of the two expansions by comparing $a^*$ to the aperture distribution.