The problem of removing an unyielded visco-plastic material from the walls of a duct is considered. This forms a prototype problem for the mechanical removal of soft solids from the duct walls in both oilfield well construction and food processing.

We consider two-layer axial flows for which both fluids may be characterized as Herschel–Bulkley fluids, a class of fluids including Newtonian, Bingham and power-law models. If the yield stress of the displaced fluid is sufficiently large, it is possible for static layers of fluid to persist at the walls of the duct, as has been shown in Allouche, Frigaard & Sona (2000). Following Prager (1954) two variational principles are derived for these flows. These may be loosely interpreted as rate-of-strain minimization and stress maximization principles. The rate-of-strain minimization principle leads directly to existence and uniqueness results for this class of flows. The stress maximization principle leads to a number of qualitative results. An adaptation of the stress maximization principle also allows us to define the concept of a maximal static wall layer in terms of minimization of a certain functional over the space of admissible fluid–fluid interfaces, i.e. a shape optimization problem. We present a number of geometrically simple examples that demonstrate the validity of this method.