Variational inequality solutions and finite stopping time for a class of shear-thinning flows

The aim of this paper is to study the existence of variational inequality weak solutions and of a finite stopping time for a large class of generalized Newtonian fluids shear-thinning flows. The existence of dissipative solutions for such flows is known since \cite{abbatiello-feireisl-20}. We submit here an alternative approach using variational inequality solutions as presented in \cite{duvaut-lions} in the two-dimensional Bingham flow. In order to prove the existence of such solutions we regularize the non-linear term and then we apply a Galerkin method for finally passing to the limit with respect to both regularization and Galerkin discretization parameters. In a second time, we prove the existence of a finite stopping time for Ostwald-De Waele and Bingham flows in dimension \(N \in \{2, 3\}\).

[1]  Michael Ruzicka,et al.  Natural second-order regularity for parabolic systems with operators having (p,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{ , 2021, Calculus of Variations and Partial Differential Equations.

[2]  P. Coussot.  Rhéophysique, la matière dans tous ses états , 2020 .

[3]  Zhong Tan,et al.  Regularity of weak solutions to a class of nonlinear problem with non-standard growth conditions , 2020 .

[4]  E. Feireisl,et al.  On a class of generalized solutions to equations describing incompressible viscous fluids , 2019, Annali di Matematica Pura ed Applicata (1923 -).

[5]  H. B. Veiga,et al.  On the energy equality for solutions to Newtonian and non-Newtonian fluids , 2019, Nonlinear Analysis.

[6]  P. Maremonti,et al.  Existence of Regular Time-Periodic Solutions to Shear-Thinning Fluids , 2018, Journal of Mathematical Fluid Mechanics.

[7]  Viorel Barbu,et al.  Controllability and Stabilization of Parabolic Equations , 2018 .

[8]  L. Berselli,et al.  Global regularity properties of steady shear thinning flows , 2016, 1607.06297.

[9]  Franck Boyer,et al.  Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models , 2012 .

[10]  M. Ruzicka,et al.  Existence of Weak Solutions for Unsteady Motions of Herschel–Bulkley Fluids , 2012 .

[11]  Mohammed Kbiri Alaoui,et al.  On Degenerate Parabolic Equations , 2011, Int. J. Math. Math. Sci..

[12]  Kazufumi Ito,et al.  Optimal control of parabolic variational inequalities , 2010 .

[13]  L. Berselli,et al.  Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities , 2010 .

[14]  J. Frehse,et al.  Non-homogeneous generalized Newtonian fluids , 2008 .

[15]  Jörg Wolf,et al.  Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity , 2007 .

[16]  L. Diening,et al.  Strong Solutions for Generalized Newtonian Fluids , 2005 .

[17]  K. Rajagopal,et al.  Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities , 2002 .

[18]  Josef Málek,et al.  On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case $p\geq2$ , 2001, Advances in Differential Equations.

[19]  J. Málek.  Weak and Measure-valued Solutions to Evolutionary PDEs , 1996 .

[20]  H. Amann.  Stability of the Rest State of a viscous incompressible fluid , 1994 .

[21]  C. F. Curtiss,et al.  Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics , 1987 .

[22]  F. Patrone,et al.  On the Optimal Control of Variational Inequalities , 1975, Optimization Techniques.

[23]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[24]  Shan Suthaharan,et al.  Modeling and Algorithms , 2016 .

[25]  Robinson Rodrigo Sadowski.  Three-Dimensional Navier–Stokes Equations , 2016 .

[26]  R. Glowinski,et al.  Qualitative properties and approximation of solutions of Bingham flows: On the stabilization for large time and the geometry of the support , 2010 .

[27]  L. Diening,et al.  Existence of weak solutions for unsteady motions of generalized Newtonian fluids , 2010 .

[28]  Anne M. Robertson,et al.  Rheological models for blood , 2009 .

[29]  Josef Málek,et al.  On Analysis of Steady Flows of Fluids with Shear-Dependent Viscosity Based on the Lipschitz Truncation Method , 2003, SIAM J. Math. Anal..

[30]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[31]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[32]  E. Becker.  Simple Non-Newtonian Fluid Flows , 1980 .

[33]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[34]  L. Nirenberg,et al.  On elliptic partial differential equations , 1959 .

[35]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.