A solution with free boundary for non-Newtonian fluids with Drucker–Prager plasticity criterion

Ntovoris, E.; Regis, M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 46

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We study a free boundary problem which is motivated by a particular case of the flow of a non-Newtonian fluid, with a pressure depending yield stress given by a Drucker–Prager plasticity criterion. We focus on the steady case and reformulate the equation as a variational problem. The resulting energy has a term with linear growth while we study the problem in an unbounded domain. We derive an Euler–Lagrange equation and prove a comparison principle. We are then able to construct a subsolution and a supersolution which quantify the natural and expected properties of the solution; in particular, we show that the solution has in fact compact support, the boundary of which is the free boundary.

The model describes the flow of a non-Newtonian material on an inclined plane with walls, driven by gravity. We show that there is a critical angle for a non-zero solution to exist. Finally, using the sub/supersolutions we give estimates of the free boundary.

Classification : 76A05, 49J40, 35R35
Keywords: Non-Newtonian fluid, Drucker–Prager plasticity, variational inequality, free boundary

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