Effective rheology of Bingham fluids in a rough channel

Abstract

We derive the volumetric flow rate vs. 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.

The Effect of Microstructure on Models for the Flow of a Bingham Fluid in Porous Media: One-Dimensional Flows

“…for a porous medium consisting of identical channels; this value may also be found in Talon et al (2014) where it is termed the effective pressure threshold or apparent critical pressure. The equivalent analysis for Hagen-Poiseuille flow is summarised in the "Appendix" section where the analogous function g(σ ) is defined and which applies to a circular pore of radius, h. This curve is shown in Fig.…”

“…Muraleva and Muraleva (2011) considered steady flow in an undulating channel using finite difference methods to determine the unyielded regions. Talon et al (2014) considered more general boundary imperfections for the channel within the lubrication approximation, and they applied their method to periodic and self-affine undulations. The computation of pipe flow in more complicated geometries, such as square cross sections, is hindered by the need to find the yield surfaces numerically.…”

Generalization of Darcy's law for Bingham fluids in porous media: From flow-field statistics to the flow-rate regimes

“…Indeed, just above the critical pressure, since only one flowing path is remaining, the flow rate evolves linearly with the pressure Q oc (A P -A Pc). This linearity results from the fact that inside this channel, the effective viscosity can be considered as constant (see [14]). For a larger applied pressure, more and more paths start to open progressively.…”

“…Unlike Newtonian fluids, their flow characteristics in porous media are poorly known due to the interplay of the complex rheology and porous structure. Consequently, the determination of a macroscopic constitutive law to relate the flow rate to the applied pressure has been the subject of many investigations in the past [ 1,[5][6][7][8][9][10][11][12][13][14][15] but remains a challenging and controversial issue [16,17].…”

Flow of yield stress and Carreau fluids through rough‐walled rock fractures: Prediction and experiments

“…Many complex fluids used in industrial applications exhibit yield stress behaviour, e.g., polymer solutions, waxy crude oils, volcanic lavas, emulsions, colloid suspensions, foams, etc. [ Coussot , ; Dimitriou and McKinley , ; Roustaei et al ., ; Talon et al ., ; Lavrov , ; Coussot , ]. Common examples of yield stress shear‐thinning fluids are the slurries or cement grouts injected to reinforce soils, the heavy oils, or the drilling fluids injected into rocks for the reinforcement of wells [ Lavrov , ; Coussot , ].…”