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

Juan Carlos De Los Reyes, Georg Stadler

Research output: Contribution to journalArticle

Abstract

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 Hölder 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.

Original languageEnglish (US)
Pages (from-to)575-602
Number of pages28
JournalInterfaces and Free Boundaries
Volume16
Issue number4
DOIs
StatePublished - 2014

Fingerprint

free boundaries
shear
shear flow
fluids
viscosity
formulations
Newton methods
regularity
infinity
strain rate
continuums
optimization
energy

Keywords

  • Additional regularity
  • Fictitious domain method
  • Mixed discretization
  • Non-Newtonian fluid mechanics
  • Semismooth Newton method
  • Shear thickening
  • Variational inequality

ASJC Scopus subject areas

  • Surfaces and Interfaces

Cite this

A nonsmooth model for discontinuous shear thickening fluids : Analysis and numerical solution. / De Los Reyes, Juan Carlos; Stadler, Georg.

In: Interfaces and Free Boundaries, Vol. 16, No. 4, 2014, p. 575-602.

Research output: Contribution to journalArticle

@article{8c722765b27c423fb16ab2b344ca71c7,
title = "A nonsmooth model for discontinuous shear thickening fluids: Analysis and numerical solution",
abstract = "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{\"o}lder 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 = "Additional regularity, Fictitious domain method, Mixed discretization, Non-Newtonian fluid mechanics, Semismooth Newton method, Shear thickening, Variational inequality",
author = "{De Los Reyes}, {Juan Carlos} and Georg Stadler",
year = "2014",
doi = "10.4171/IFB/330",
language = "English (US)",
volume = "16",
pages = "575--602",
journal = "Interfaces and Free Boundaries",
issn = "1463-9963",
publisher = "European Mathematical Society Publishing House",
number = "4",

}

TY - JOUR

T1 - A nonsmooth model for discontinuous shear thickening fluids

T2 - Analysis and numerical solution

AU - De Los Reyes, Juan Carlos

AU - Stadler, Georg

PY - 2014

Y1 - 2014

N2 - 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 Hölder 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.

AB - 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 Hölder 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.

KW - Additional regularity

KW - Fictitious domain method

KW - Mixed discretization

KW - Non-Newtonian fluid mechanics

KW - Semismooth Newton method

KW - Shear thickening

KW - Variational inequality

UR - http://www.scopus.com/inward/record.url?scp=84916934264&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84916934264&partnerID=8YFLogxK

U2 - 10.4171/IFB/330

DO - 10.4171/IFB/330

M3 - Article

VL - 16

SP - 575

EP - 602

JO - Interfaces and Free Boundaries

JF - Interfaces and Free Boundaries

SN - 1463-9963

IS - 4

ER -