# Problem of Second Grade Fluids in Convex Polyhedrons

@article{Bernard2012ProblemOS, title={Problem of Second Grade Fluids in Convex Polyhedrons}, author={Jean-marie Bernard}, journal={SIAM J. Math. Anal.}, year={2012}, volume={44}, pages={2018-2038} }

This paper studies the solutions of a three-dimensional grade-two fluid model with a tangential boundary condition in a polyhedron. We begin to split the problem into a system with a generalized Stokes problem and a transport equation, as Girault and Scott have done in the two-dimensional case. But, compared to the two-dimensional problem, we have an additional term that is difficult to bound which requires regularity of the solutions and we have to prove that the solutions of the transport…

## Topics from this paper

## 3 Citations

Fully nonhomogeneous problem of two-dimensional second grade fluids

- MathematicsMathematical Methods in the Applied Sciences
- 2018

This article studies the solutions of a two-dimensional grade-two fluid model with a fully non-homogeneous boundary condition for velocity u. Compared to problems with a homogeneous or tangential…

An asymptotic duality between the Oldroyd-Maxwell and grade-two fluid models

- 2021

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upperconvected and lower-convected Maxwell…

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

- Comptes Rendus. Mathématique
- 2021

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell…

## References

SHOWING 1-10 OF 26 REFERENCES

Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition

- Mathematics
- 1999

This article studies the solutions in H1 of a two-dimensional grade-two fluid model with a non-homogeneous Dirichlet tangential boundary condition, on a Lipschitz-continuous domain. Existence is…

Finite-element discretizations of a two-dimensional grade-two fluid model

- Mathematics
- 2001

We propose and analyze several finite-element schemes for solving a grade-two fluid model, with a tangential boundary condition, in a two-dimensional polygon. The exact problem is split into a…

Stationary problem of second-grade fluids in three dimensions: existence, uniqueness and regularity

- Mathematics
- 1999

This paper is devotcd to the stationary problem of second-grade fluids, in the case where α 1 + α 2 = 0, in three dimensions. In relation to the problem in two dimensions, studied by E. H. Ouazar,…

Elliptic Equations in Polyhedral Domains

- Mathematics
- 2010

This is the first monograph which systematically treats elliptic boundary value problems in domains of polyhedral type. The authors mainly describe their own recent results focusing on the Dirichlet…

Stationary Stokes and Navier-Stokes systems on two-or three-dimensional domains with corners

- Mathematics
- 1989

The $H^s $-regularity (s being real and nonnegative) of solutions of the Stokes system in domains with corners is studied. In particular, a $H^2 $-regularity result on a convex polyhedron that…

Regularized finite element discretizations of a grade‐two fluid model

- Mathematics
- 2005

UMMARY
We consider a system with three unknowns in a two-dimensional bounded domain which models the flow of a grade-two non-Newtonian fluid. We propose to compute an approximation of the solution…

Thermodynamics and stability of fluids of third grade

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1980

Today, even though the Clausius-Duhem inequality is widely considered to be of central importance in the subject of continuum thermomechanics, it is also believed to be a somewhat special…

Weak and classical solutions of a family of second grade fluids

- Mathematics
- 1997

Abstract This paper shows that the decomposition method introduced by Cioranescu and Ouazar [Non-linear Partial Differential Equations 109, 178 (1984)] allows the proof of global existence in time of…

The Mathematical Theory of Finite Element Methods

- Mathematics
- 1994

Contents: Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element Space.- Polynomial Approximation Theory in Sobolev…

Finite Element Approximation of the Navier-Stokes Equations

- Mathematics
- 1979

Mathematical foundation of the stokes problem.- Numerical solution of the stokes problem a classical method.- A mixed finite element method for solving the stokes problem.- The stationary…