Scraping of a thin layer of viscoplastic fluid

We analyse the scraping of a thin layer of viscoplastic fluid residing on a horizontal surface by a translating rigid scraper. This motion generates a mound of fluid upstream of the scraper and a residual layer behind it, both of which are modelled using a shallow layer theory for a Bingham or Herschel-Bulkley fluid. The flows ahead of and behind the scraper are coupled by the motion in the gap under the scraper, which is driven both by the translation of the scraper and by the induced pressure gradient due to the difference in flow thickness upstream and downstream. When the gap between the scraper and the underlying surface is sufficiently small, we find that a steady-state emerges after a relatively long transient, and that en route to this state, the unsteady dynamics exhibit a variety of regimes that are self-similar to leading order. We construct these solutions explicitly and derive key scalings for the temporal development of the flowing viscoplastic layer, as well as identifying the time scales at which there are transitions between the regimes. These predictions are confirmed by comparison with results from the numerical integration of the full system. Finally, we report results from preliminary laboratory experiments, which are compared with predictions from the shallow-layer theory, obtaining reasonable agreement once a slip boundary condition is included in the model, as motivated by experimental observations.