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Stability analysis and numerical simulation of non-Newtonian fluids of Oldroyd kind [Elektronische Ressource] / vorgelegt von Nicoleta Dana Scurtu

210 pages
Stability Analysis and NumericalSimulation of Non-Newtonian Fluidsof Oldroyd KindDen Naturwissenschaftlichen Fakult atender Friedrich-Alexander-Universit at Erlangen-Nurn bergzurErlangung des Doktorgradesvorgelegt vonNicoleta Dana Scurtuaus BrasovAls Dissertation genehmigt von den Naturwissen-schaftlichen Fakult aten der Universit at Erlangen-Nurn bergTag der mundlic hen Prufung: 24. Oktober 2005Vorsitzender derPromotionskommision: Prof. Dr. D.-P. H aderErstberichterstatter: Prof. Dr. Eberhard B anschZweitberichterstatter: Prof. Dr. Wolfgang BorchersTo the man who pleases him,God gives wisdom, knowledge and happiness...(the Bible)AcknowledgmentFirst of all, I am especially thankful to Prof. Dr. Eberhard B ansch who gave me the oppor-tunity to work in his research group in the fascinating eld of computational uid dynamicsand for the time he spent to supervise my research.I also want to thank Prof. Dr. Kunibert G. Siebert who kindly placed his program packageAlbert at my disposal which was the starting point and the major tool for my numericalcomputations.Finally, I want to acknowledge my colleagues from Weierstra Institute Berlin and fromBrandenburg Technical University of Technology Cottbus for their moral encouragement,for text improvements and for their hints concerning computational issues.Erlangen , July 2005 Nicoleta D. ScurtuContentsIntroduction 11 Description of Newtonian and non-Newtonian uids 51.
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Stability Analysis and Numerical
Simulation of Non-Newtonian Fluids
of Oldroyd Kind
Den Naturwissenschaftlichen Fakult aten
der Friedrich-Alexander-Universit at Erlangen-Nurn berg
zur
Erlangung des Doktorgrades
vorgelegt von
Nicoleta Dana Scurtu
aus BrasovAls Dissertation genehmigt von den Naturwissen-
schaftlichen Fakult aten der Universit at Erlangen-Nurn berg
Tag der mundlic hen Prufung: 24. Oktober 2005
Vorsitzender der
Promotionskommision: Prof. Dr. D.-P. H ader
Erstberichterstatter: Prof. Dr. Eberhard B ansch
Zweitberichterstatter: Prof. Dr. Wolfgang BorchersTo the man who pleases him,
God gives wisdom, knowledge and happiness...
(the Bible)Acknowledgment
First of all, I am especially thankful to Prof. Dr. Eberhard B ansch who gave me the oppor-
tunity to work in his research group in the fascinating eld of computational uid dynamics
and for the time he spent to supervise my research.
I also want to thank Prof. Dr. Kunibert G. Siebert who kindly placed his program package
Albert at my disposal which was the starting point and the major tool for my numerical
computations.
Finally, I want to acknowledge my colleagues from Weierstra Institute Berlin and from
Brandenburg Technical University of Technology Cottbus for their moral encouragement,
for text improvements and for their hints concerning computational issues.
Erlangen , July 2005 Nicoleta D. ScurtuContents
Introduction 1
1 Description of Newtonian and non-Newtonian uids 5
1.1 Kinematics, deformation and balance laws . . . . . . . . . . . . . . . . . . . . 5
1.2 Frame indi erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Simple incompressible uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Newton’s viscosity law . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Quasi-Newtonian models . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Di eren tial type uids (Rivlin-Ericksen) . . . . . . . . . . . . . . . . . 11
1.3.4 Rate type uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5 Integral type uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Mathematical formulation 19
2.1 The Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The elastic-viscous-split-stress method . . . . . . . . . . . . . . . . . . 19
2.1.2 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Dimensionless Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The high Wei en berg number problem . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Notes on the high Wei en berg number problem . . . . . . . . . . . . . . . . . 24
3 Existence results and nite element formulation 25
3.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The stationary Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The discontinuous Galerkin method . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Time approximation using the fractional -scheme 39
4.1 Application of the fractional step -scheme to the Oldroyd system . . . . . . 40
4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Well-posedness of the subproblems . . . . . . . . . . . . . . . . . . . . 43vi Contents
4.2.2 Fixed-point iteration scheme for the stationary Oldroyd system . . . . 44
5 Stability analysis 47
5.1 Spectral analysis of the linearized Oldroyd system . . . . . . . . . . . . . . . 47
5.1.1 Spectral analysis of the linearized continuous Oldroyd system . . . . . 47
5.1.2 Spectral of the -scheme for the linearized Oldroyd system . . 52
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Strong stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Choice of the splitting parameters k and ! . . . . . . . . . . . . . . . 66
Plots of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Contribution of the

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