A finite element approximation for the steady solution of a second-grade fluid model

Sequeira, Adélia; Baía, Margarida

doi:10.1016/s0377-0427(99)00149-1

Cited by 8 publications

(6 citation statements)

References 24 publications

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“…Secondly, due to the presence of convective terms in both momentum and constitutive equations, adequate discretizations procedure must be used such as discontinuous finite elements, Galerkin Least Square (GLS) stabilization [3], or the characteristics method. Thirdly, the presence of nonlinear terms prevents numerical methods to converge at high Deborah numbers, this being consistent with theoretical [2,13,19,24,27] and experimental [23] studies.…”

Section: Numerical Analysis Of a Simplified Problemsupporting

confidence: 80%

A finite element/Monte-Carlo method for polymer dilute solutions

Bonvin

Picasso

2001

Computing and Visualization in Science

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A finite element/Monte-Carlo method is proposed for solving the flow of a polymer dilute solution. The mass and momentum equations are supplemented with stochastic differential equations which model the dynamics of the polymer chains. Finite elements and a Monte-Carlo method are used to solve the problem. A theoretical analysis is performed on a simplified, deterministic, Oldroyd-B problem. Numerical results are compared to experimental ones.

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Section: Numerical Analysis Of a Simplified Problemsupporting

confidence: 80%

A finite element/Monte-Carlo method for polymer dilute solutions

Bonvin

Picasso

2001

Computing and Visualization in Science

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“…The statement of the above existence result is similar (although not the same) to those of [4,21], in which the convective term in the extra-stress constitutive equation are considered (see also [26] for analogous results on a second-grade fluid). However, a different technique is used in this paper, allowing a priori and a posteriori error estimates to be obtained more easily, with other assumptions.…”

Section: Moreover the Mapping λ ∈ [0 λ] → U H (λ) ∈ X H Is Continuosupporting

confidence: 77%

“…Secondly, due to the presence of convective terms in both momentum and constitutive equations, adequate discretizations procedure must be used such as discontinuous finite elements, GLS stabilization procedures [5], or the characteristics method. Thirdly, the presence of nonlinear terms prevents numerical methods to converge at high Deborah numbers, this being consistent with theoretical [4,19,21,23,26] and experimental [22] studies.…”

Section: Introductionsupporting

confidence: 80%

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Existence,a priorianda posteriorierror estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Picasso

Rappaz

2001

ESAIM: M2AN

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Abstract.In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived. An Elastic Viscous Split Stress (EVSS) scheme related to the GLS method is introduced. Numerical results confirm the theoretical predictions.Mathematics Subject Classification. 65N30, 65N12, 76A10.

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“…We finally observe that this "splitting method" has also been successfully applied to numerical studies of viscoelastic fluids by A. Sequeira and her associates; see, e.g., [100], [101].…”

Section: Lemma 23 a Compact Mapping M Of A Closed Bounded Convex Smentioning

confidence: 71%

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

Galdi

Oberwolfach Seminars

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IntroductionBlood flow per se is a very complicated subject. Thus, it is not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and challenging.The role of mathematics in the investigation of blood flow properties -as in the most part of applied sciences -is twofold and is directed toward the accomplishment of the following objectives. The first one, of a more theoretical nature, is the validation of the models proposed by engineers, and consists in securing conditions under which the governing equations possess the fundamental requirements of well-posedness, such as existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more applied character, is to prove that these models give a satisfactory interpretation of the observed phenomena. In general, both tasks present serious difficulties in that they require the study of several different, and frequently combined, topics that include, among others, Navier-Stokes equations, non-Newtonian fluid models, nonlinear elasticity, fluid-structure interaction and multi-phase flow. It must be added that some of these topics are still at the beginning of a systematic mathematical research, whereas some others are in a continuous growth.As a matter of fact, the initiation or the methodical investigation of several of the above research areas was just motivated by problems arising in blood flow. Moreover, blood flow can also pose challenging questions in "classical" topics, questions that, in the past, happened to receive little or no attention at all. A typical example is provided by the problem of the flow of a Navier-Stokes liquid in an unbounded piping system, under a given time-periodic flow-rate, that has been "discovered" only in 2005, thanks to the work of H. Beirão da Veiga; see [7]. G.P. GaldiIt seemed to me hopeless to present and to describe in this article all relevant aspects of the mathematical analysis related to blood flow, even at an introductory level. Therefore, I preferred to concentrate on some, of the many, topics which are at the foundation of this analysis, and to point out directions for future research. More specifically, I focused on three different subjects which are the content of as many separate chapters.The first chapter deals with the study of some fundamental properties of the flow of a Navier-Stokes liquid in a piping system, which can be either unbounded or bounded. Here, I have concentrated the analysis mostly on steady-state and time-periodic motions, and on their attainability. There are several reasons for this choice. On the one hand, because these types of motions are the most "elementary" to occur in the arterial and venuous system, and, on the other hand, because the initial-boundary value problem in an unbounded piping system has been investigated in full detail in the most recent article [86], to which I refer the interested reader.The second chapter is dedicated to the mathematical analysis of certain non-Newtonian...

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A finite element approximation for the steady solution of a second-grade fluid model

Cited by 8 publications

References 24 publications

A finite element/Monte-Carlo method for polymer dilute solutions

A finite element/Monte-Carlo method for polymer dilute solutions

Existence,a priorianda posteriorierror estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

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