Variational methods in the theory of the fluidity of a viscous-plastic medium

The blocking of an inhomogeneous Bingham fluid. Applications to landslides

“…We can use now Proposition 4.2 to deduce that the Bingham fluid is blocked, a contradiction. The expression of u follows from (16), keeping in mind that σ = −F − C 0 .…”

“…More precisely, we study the link between the yield limit distribution and the external forces distribution (or the mass density distribution) for which the flow of the Bingham fluid is blocked or exhibits rigid zones. In opposition to the previous works dealing only with homogeneous Bingham fluids [8,9,11,12,[16][17][18], we are interested in a fluid whose yield limit is inhomogeneous.…”

A duality based semismooth Newton framework for solving variational inequalities of the second kind

“…From the graphics it can be observed that the Bingham fluid behaves like a solid in the center of the cross section. This phenomenon is described in detail in [32], where the so-called nucleus of the Bingham inequality is analyzed, and it is verified numerically in, e.g., [6,12]. Further, a complementarity relation between the velocity and the Euclidean norm of the dual variable may be observed.…”

“…The flow of a visco-plastic fluid in a pipe was studied theoretically in [13,32] and numerically in [12]. It fits into our framework by choosing Y = H 1 0 (Ω), where Ω ⊂ R 2 denotes a bounded domain with a Lipschitz boundary.…”

“…From a practical point of view, applications of VIs of the second kind involve many important phenomena such as friction, visco-plastic fluid flow or problems in elasto/visco-plasticity. Some analytical results for such problems can be found, e.g., in [9,12,28,32,37].…”