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Interfaces and Free Boundaries

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Volume 16, Issue 4, 2014, pp. 575–602
DOI: 10.4171/IFB/330

Published online: 2014-12-10

A nonsmooth model for discontinuous shear thickening fluids: Analysis and numerical solution

Juan Carlos De los Reyes[1] and Georg Stadler[2]

(1) Escuela Politécnica Nacional, Quito, Ecuador
(2) New York University, USA

We propose a nonsmooth continuum mechanical model for discontinuous shear thickening flow. The model obeys a formulation as energy minimization problem and its solution satisfies a Stokes type system with a nonsmooth constitute relation. Solutions have a free boundary at which the behavior of the fluid changes. We present Sobolev as well as H¨older regularity results and study the limit of the model as the viscosity in the shear thickened volume tends to infinity. A mixed problem formulation is discretized using finite elements and a semismooth Newton method is proposed for the solution of the resulting discrete system. Numerical problems for steady and unsteady shear thickening flows are presented and used to study the solution algorithm, properties of the flow and the free boundary. These numerical problems are motivated by recently reported experimental studies of dispersions with high particle-to-fluid volume fractions, which often show a sudden increase of viscosity at certain strain rates.

Keywords: Shear thickening, non-Newtonian fluid mechanics, variational inequality, additional regularity, mixed discretization, semismooth Newton method, fictitious domain method

De los Reyes Juan Carlos, Stadler Georg: A nonsmooth model for discontinuous shear thickening fluids: Analysis and numerical solution. Interfaces Free Bound. 16 (2014), 575-602. doi: 10.4171/IFB/330