Smooth thin-layer asymptotic expansions for free-surface yield-stress flows

We derive two asymptotic expansions with a smooth velocity field for free-surface viscoplastic flows down an inclined plane in the shallow-flow approximation. The first expansion is based on the classical Herschel–Bulkley constitutive law by using asymptotic matching at the interface between the pseudo-plug and the sheared layer. In contrast to previous works, where authors considered only one term in the transition layer, we compute two extra terms to guarantee a smooth transition of the inertial contribution from the sheared layer to the pseudo-plug. However, the terms associated to the transition layer are solutions of nonintegrable equations, thus preventing the potential use of this expansion for deriving a shallow-flow model. The second asymptotic expansion is based on an alternative tensorial extension of the Herschel–Bulkley law, for which the alignment between the yield-stress tensor and the strain-rate tensor is relaxed, while the von Mises criterion is kept. In this case, smooth asymptotic expansions of the velocity field are given by fully analytical expressions. Comparison of these two expansions with experiments shows that both give essentially equivalent and relatively good agreement.