On the lubrication paradox and the use of regularisation methods for lubrication flows

Advances in the simulation of viscoplastic fluid flows using interior-point methods

“…Therefore, no problem regarding the lack of point-wise convergence for stresses seem to appear with the IPM contrary to what has been observed for regularized models [37,38]. The location of the yielded regions are reported in Figure 9 for different Bingham numbers and are consistent with results from the literature (see for instance [10,39]).…”

Effective rheology of Bingham fluids in a rough channel

“…Lubrication approximations have been investigated extensively for Newtonian fluids in confined geometries (e.g., [32,33]). For visco-plastic fluids, the limit of lubrication has also been investigated by various authors in different geometries (see [34] for film flows, [22][23][24]35] for confined geometries and [36,37] for experiments). In this case, the lubrication hypothesis becomes very restrictive essentially due to rigidity of the plug regions.…”

“…The present work relates to earlier work considering channel flow with periodically varying aperture [22][23][24]. In those articles, the authors investigated the validity of the lubrication approximation [25] which can become inconsistent with the presence of a unyielded plug along the center of the channel.…”

Issues in the flow of yield-stress liquids

“…It is straightforward to demonstrate from strictly kinematical arguments that continuity of the velocity and stress at the yield surface cannot be satisfied within the context of conventional lubrication theory, and asymptotic methods must be used with delicacy (Lipscomb and Denn 1984); the issues are addressed in recent work by Putz et al (2009). Variational methods can be used, but these are best for bounding macroscopic quantities (the drag coefficient, for example) and less satisfactory for establishing the details of velocity and stress fields.…”

“…The "gold standard" for checking the validity of calculations of flow around a sphere in a Bingham fluid is that of Beris et al (1985). Some new results demonstrating convergence problems when the yield surface is discontinuous are described in Putz et al (2009). Overall, the regularization methodology appears to be satisfactory in most instances and is incorporated into commercial CFD codes, although some authors (e.g., Putz and Frigaard 2010) have recently employed an augmented Lagrangian approach that allows more accurate determination of the unyielded regions.…”