Advances in Mathematical Fluid Mechanics (eBook)

Foreword 8
Short Biography 10
Paolo Galdi – The Man and the Mathematician 12
Contents 18
Contributors 22
Isotropically and Anisotropically Weighted Sobolev Spaces for the Oseen Equation 27
Chérif Amrouche and Ulrich Razafison 27
1 Introduction 27
2 Scalar Oseen Potential in Three Dimensional Space 29
3 Weighted Hardy Inequalities 38
References 49
A New Model of Diphasic Fluids in Thin Films 51
Guy Bayada, Laurent Chupin, and Bérénice Grec 51
1 Introduction 51
2 Governing Equations 52
2.1 Modelling One Fluid in a Thin Domain 52
2.2 Modelling a Mixture in a Thin Domain 54
2.2.1 Modelling a Mixture and Taking the Surface Tension into Account 54
2.2.2 Modelling Diphasic Flows in a Thin Domain 56
3 Theoretical Results 57
4 Numerical Simulations 58
4.1 The Numerical Scheme 58
4.1.1 Time Discretization 58
4.1.2 Space Discretization 59
4.2 The Numerical Results 59
References 60
On the Global Integrability for Any Finite Power of the Full Gradient for a Class of Generalized Power Law Models bold0mu mumu p< 2p<
Hugo Beirão da Veiga 62
1 Main Result 62
2 Proof of Theorem 1 65
References 67
Steady Flow Around a Floating Body: The Rotationally Symmetric Case 68
Josef Bemelmans and Mads Kyed 68
1 Introduction 68
2 Approximation Scheme 71
3 A Solution to the Free-Boundary Problem for a Cylinder 74
4 A Solution to the Free-Boundary Problem for a Rotationally Symmetric Body 76
References 79
On a Stochastic Approach to Eddy Viscosity Models for Turbulent Flows 80
Luigi C. Berselli and Franco Flandoli 80
1 Introduction 80
2 Two Approaches to Turbulent Flows: LES and Stochastic Partial Differential Equations 82
2.1 The Paradigm of Large/Small Scales 82
2.2 Preliminaries on Stochastic Navier-Stokes Equations 84
2.2.1 On White Noise 84
2.2.2 On a Stochastic LES Model 85
2.2.3 Existence and Balance Laws for Stochastic Navier-Stokes Equations 86
2.3 Brownian Scaling 88
3 Scaling Transformations and Derivation of Generalized Smagorinsky Models 89
3.1 Derivation of a Family of Eddy Viscosity Models 90
3.2 Comparison with Alternate Scaling Previously Proposed 92
3.3 Is u a Large Scale Approximation of u? 94
3.4 Over-Damping and the Size of |2su|p-2 96
3.5 Modification of Lilly's Argument for General p 98
3.6 On a Wall Bounded Problem 100
4 Conclusions 104
References 104
Numerical Study of the Significance of the Non-Newtonian Nature of Blood in Steady Flow Through a Stenosed Vessel 107
Tomáš Bodnár and Adélia Sequeira 107
1 Introduction 107
2 Constitutive Models for Blood 109
2.1 Generalized Newtonian Models 110
2.2 Viscoelastic Models 112
2.2.1 Convected Derivatives and Oldroyd-B Model 113
2.3 Models Summary 115
3 Numerical Methods 116
3.1 Space Discretization 117
3.2 Time Advancing Scheme 118
3.3 Numerical Stabilization 119
4 Numerical Simulations and Results 120
4.1 Computational Domain and Model Parameters 120
4.2 Numerical Results 121
5 Conclusions and Remarks 126
References 127
A Priori Convergence Estimates for a Rough Poisson-Dirichlet Problem with Natural Vertical Boundary Conditions 129
Eric Bonnetier, Didier Bresch, and Vuk Milišic 129
1 Introduction 129
2 The Framework 131
3 Summary of the Results Obtained in the Periodic Case 133
3.1 The Cell Problems 133
3.1.1 The First Order Cell Problem 133
3.1.2 The Second Order Cell Problem 134
3.2 Standard Averaged Wall Laws 134
3.2.1 A First Order Approximation 134
3.2.2 A Second Order Approximation 134
3.3 Compact Form of the Full Boundary Layer Ansatz 135
4 The Non Periodic Case: A Vertical Corrector 136
4.1 Microscopic Decay Estimates 136
4.2 Test Functions: From Macro to Micro and Vice-Versa 150
5 A New Proof of Convergence for Standard Averaged Wall Laws 151
5.1 The Full First Order Boundary Layer Approximation: Error Estimates 151
5.1.1 The Neumann Part 153
5.1.2 The Dirichlet Part 154
5.2 The Standard Averaged Wall Law: New Error Estimates 156
6 Conclusion 157
References 157
Vortex Induced Oscillations of Cylinders at Low and Intermediate Reynolds Numbers 159
Roberto Camassa, Bong Jae Chung, Philip Howard, Richard McLaughlin, and Ashwin Vaidya 159
1 Introduction 159
2 Experimental Setup 161
3 Results 162
4 Discussion 167
References 168
One-dimensional Modelling of Venous Pathologies: Finite Volume and WENO Schemes 170
Nicola Cavallini and Vincenzo Coscia 170
1 Introduction 170
2 Mathematical Modelling 172
3 Numerical Modelling 175
3.1 Numerical Treatment of Boundary Conditions 180
4 Applications 184
4.1 Accuracy Analysis 184
4.2 Valve Test Case 185
4.3 Leg Veins Network 186
5 Conclusions 190
References 192
On the Energy Equality for Weak Solutions of the 3D Navier-Stokes Equations 194
Alexey Cheskidov, Susan Friedlander, and Roman Shvydkoy 194
1 Introduction 194
2 Preliminaries 195
3 The Proof of Theorem 1 196
References 198
The bold0mu mumu (p-q)(p-q)dest(p-q)(p-q)(p-q)(p-q) Coupled Fluid-Energy Systems 199
Luisa Consiglieri 199
1 The Formulation of the Problem 199
2 Existence Results 203
2.1 The Differentiable Flux 204
2.2 The Nondifferentiable Flux 205
3 The Differentiable Flux 206
3.1 Proof of Theorem 1 207
3.2 Proof of Theorem 2 208
3.3 Proof of Theorem 3 209
4 The Nondifferentiable Flux 209
References 211
A Potential-Theoretic Approach to the Time-Dependent Oseen System 213
Paul Deuring 213
1 Introduction 213
2 Notations, Definitions, Auxiliary Results 216
3 Solving Integral Equation (11) 221
References 235
Regularity of Weak Solutions for the Navier-Stokes Equations Via Energy Criteria 237
Reinhard Farwig, Hideo Kozono, and Hermann Sohr 237
1 Introduction and Main Results 237
2 Proof 240
References 248
Looking for the Lost Memory in Diffusion-Reaction Equations 250
José Augusto Ferreira and Paula de Oliveira 250
1 Introduction 250
2 The Simplest Non Brownian Model 252
2.1 Introduction 252
2.2 Well-Posedness in Hadamard's Sense 253
2.3 Asymptotic Behaviour of the Model 254
3 An Hybrid Method for Diffusion: Brownian and Non Brownian Character 257
3.1 Introduction 257
3.2 An Energy Estimate for the Jeffrey's Equation 260
3.3 A Splitting Method 261
4 A Non Brownian Model for a Quasi-linear Reaction-Diffusion Problem 265
4.1 Introduction 265
4.2 Energy Estimates for the PDE 266
4.3 Energy Estimates for the Fully Discrete Approximation 267
4.4 Convergence 269
References 271
Maximum Principle and Gradient Estimates for Stationary Solutions of the Navier-Stokes Equations: A Partly Numerical Investigation 273
Robert Finn, Abderrahim Ouazzi, and Stefan Turek 273
1 Introduction 273
2 Test Configurations 275
3 Numerical Methods 276
4 Results and Analysis 276
5 Some Conclusions 289
References 289
A Study of Shark Skin and Its Drag Reducing Mechanism 290
Elfriede Friedmann, Julia Portl, and Thomas Richter 290
1 Introduction 290
2 Modeling of Shark Skin 291
3 Direct Simulations 295
4 Scientific Visualization of our Flow Data 298
4.1 The Visualization Pipeline 298
4.2 A Small Album on the Visualization of Flow over Shark Skin 300
5 Drag Reduction Mechanism of Shark Skin 302
References 303
Stability of Poiseuille Flow in a Porous Medium 305
Antony A. Hill and Brian Straughan 305
1 Introduction 305
2 Linear Instability 306
3 Numerical Results 307
4 Nonlinear Energy Stability Theory 309
References 311
Towards a Geometrical Multiscale Approach to Non-Newtonian Blood Flow Simulations 312
João Janela, Alexandra Moura, and Adélia Sequeira 312
1 Introduction 312
2 The Mathematical Models 313
2.1 The Geometrical Multiscale Approach 315
2.1.1 The Lumped Parameters Model 316
2.1.2 Coupling the 3D and the 0D Model 317
3 The Numerical Methods 318
3.1 Solution of a Reduced Problem 320
4 Numerical Results 323
5 Conclusions 325
References 326
The Role of Potential Flow in the Theory of the Navier-Stokes Equations 327
Daniel D. Joseph 327
1 Helmholtz Decomposition 327
2 Irrotational Viscous Stress and the Irrotational Viscous Dissipation 328
3 Navier-Stokes Equations and Viscous Dissipation in Decomposed Form 329
4 Computation of Drag from Dissipation 330
5 Stokes Flow Around a Sphere of Radius a in a Uniform Stream U 330
6 Biot-Savart Law 331
References 332
Small Perturbations of Initial Conditions of Solutions of the Navier-Stokes Equations in the L3-Norm and Applications 334
Petr Kucera 334
1 Introduction 334
2 Small Perturbations of the Initial Velocity in the Norm "026B30D ."026B30D 2+ "026B30D ."026B30D 3 336
3 Strong Solution of the Navier-Stokes Equations with Large Initial Conditions 341
References 343
Streaming Flow Effects in the Nearly Inviscid Faraday Instability 344
Elena Martín and José M. Vega 344
1 Introduction 344
2 Coupled Amplitude-Mean Flow Equations 345
3 Results and Conclusions 350
References 352
The Dirichlet Problems for Steady Navier-Stokes Equations in Domains with Thin Channels 354
Yuliya V. Namlyeyeva, Šarka Necasová, and Igor Igorievich Skrypnik 354
1 Introduction and Formulation of the Problem 354
2 Assumptions on the Sequence of Perforated Domains {s}s=1 and Formulation of the Main Result 357
3 Pointwise and Integral Estimates of the Auxiliary Functions 360
4 Auxiliary Statements and Cut-off Functions 369
5 Asymptotic Expansion of Solutions and Proof of the Main Result 371
6 Appendix 379
References 380
Existence of Weak Solutions to the Equations of Natural Convection with Dissipative Heating 382
Joachim Naumann and Jörg Wolf 382
1 Introduction 382
2 Notations. Statement of the Main Result 384
3 Proof of the Main Theorem 386
3.1 Approximate Solutions 386
3.2 A-priori Estimates 389
3.3 Passage to the Limit 394
References 398
A Weak Solvability of the Navier-Stokes Equation with Navier's Boundary Condition Around a Ball Striking the Wall 400
Jirí Neustupa and Patrick Penel 400
1 Motivation, Introduction and Notation 400
2 A Formal Study of the Initial-Boundary Value Problem (1)--(5) and the Main Theorem 404
3 Construction of the Auxiliary Function at and its Properties 407
4 The Time Discretized Boundary Value Problems 411
5 The Non-stationary Approximations, Their Estimates and Weak Convergence 413
6 The Limit Function u: A Solution of the Weak Problem (17) 414
7 Concluding Remarks 421
References 421
On the Influence of an Absorption Term in Incompressible Fluid Flows 423
Hermenegildo B. de Oliveira 423
1 Introduction 423
2 The Modified Ellis Problem 425
3 Notation and Auxiliary Results 426
4 Weak Formulation 428
5 Extinction in Time 430
6 Discussion and Extensions 434
References 437
Adaptive FE Eigenvalue Computation with Applications to Hydrodynamic Stability 439
Rolf Rannacher 439
1 Introduction 439
2 An Example from Flow Control 441
3 The Navier-Stokes Problem and its Stability Analysis 444
4 The Galerkin Finite Element Approximation 446
5 The Perturbed Stability Eigenvalue Problem 448
6 The Effect of Non-normality 451
7 Practical Stability Analysis of the Navier-Stokes Equations 456
7.1 The Deficiency Test 457
7.2 Evaluation of the Error Representation 457
7.3 Strategies for Mesh Refinement 459
8 Numerical Examples 460
8.1 Burgers Equation 460
8.2 Plane Shear Flow (Couette Flow) 460
8.3 Stability of Drag-Minimal Flow 461
References 463
Numerical Simulation of Laminar Incompressible Fluid-Structure Interaction for Elastic Material with Point Constraints 465
Mudassar Razzaq, Jaroslav Hron, and Stefan Turek 465
1 Introduction 465
2 Fluid-Structure Interaction Problem Formulation 466
2.1 Fluid 466
2.2 Structure 467
2.3 Interaction Condition 468
3 Discretization and Solution Techniques 468
3.1 Time Discretization 468
3.1.1 Basic -Scheme 469
3.1.2 Backward Euler-Scheme 469
3.1.3 Crank-Nicholson-Scheme 470
3.1.4 Fractional-Step--Scheme 470
3.1.5 A Modified Fractional-Step--Scheme 470
3.2 Space Discretization 472
3.2.1 The Conforming Element Q2 P1 472
3.3 Solution Algorithm 473
3.4 Multigrid Solver 475
4 Objectives and Problem Configuration 475
4.1 Geometry of the Problem 476
4.2 Boundary and Initial Conditions 477
5 Experimental Results 478
6 Numerical Investigations 478
6.1 Results for 1mm Thick Beam 479
6.2 Results for 0.04mm Thick Beam 482
7 Summary and Future Developments 482
References 485
On Stokes' Problem 487
Remigio Russo 487
1 Introduction 487
2 The Stokes Formulae 490
3 The Classical Potential Theory 497
4 A Variant of G. Verchota's Approach to Potential Theory for Lipschitz Domains 505
5 Existence, Uniqueness and Regularity of Very Weak Solutions in Bounded Lipschitz Domains 509
6 Existence and Uniqueness of Very Weak Solutions in Lipschitz Exterior Domains 511
7 The Stokes Paradox for the Robin Problem 514
8 An Existence Theorem to the Steady-State Navier-Stokes Problem 518
References 521
On a C0 Semigroup Associated with a Modified Oseen Equation with Rotating Effect 526
Yoshihiro Shibata 526
1 Introduction and Main Results 527
2 Analysis of the Resolvent Problem in R3 531
3 Interior Problem 540
4 A Preparation for Construction of Parametrix in an Exterior Domain 543
5 On a Unique Existence Theorem to the Nonstationary Problem for the Initial Data in Lq, R+2() 548
6 Proofs of Main Results 556
References 563
A New Approach to the Regularity of Weak bold0mu mumu LqLqdestLqLqLqLq-Solutions of Stokes and Similar Equations via the Cosserat Operator 565
Christian G. Simader 565
1 Introduction 565
2 Notations 566
3 Regularity Theorems for , 2 and the Cosserat Operator 567
4 The Solution of the Equation div v = with Prescribed Boundary Values: Applications to Stokes' Equation and Similar Equations 574
References 584
Large Time Behavior of Energy in Some Slowly Decreasing Solutions of the Navier-Stokes Equations 585
Zdenek Skalák 585
1 Introduction 585
2 Presentation and Proof of Main Results 588
References 592
A Selected Survey of the Mathematical Theory of 1D Flows 593
Ivan Straškraba 593
1 Introduction and Main Results 593
2 Further Cases 596
References 599
A Numerical Method for Nonstationary Stokes Flow 600
Werner Varnhorn 600
1 Introduction and Notation 600
2 An Implicit Euler Scheme 603
3 Hydrodynamical Potential Theory 609
4 A Boundary Element Method 617
References 622
A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations 623
Jörg Wolf 623
1 Introduction and Main Result 623
2 Preliminary Lemmas 627
3 Fundamental Estimates for Caloric Functions 629
4 Caccioppoli-Type Inequality 631
5 Proof of Theorem1 637
References 639
An In Vitro Device for Evaluation of Cellular Response to Flows Found at the Apex of Arterial Bifurcations 641
Zijing Zeng, Bong Jae Chung, Michael Durka, and Anne M. Robertson 641
1 Introduction 642
2 Methods 644
2.1 Governing Equations 644
2.1.1 Flow Parameters: Wall Shear Stress and Wall Shear Stress Gradient 645
2.2 Flow in Idealized Bifurcation Models 647
2.3 Rationale Behind Design of Fluid Domain in the Bifurcation Chamber 649
3 T-Chamber Design: Analysis and Results 651
3.1 Fluid Domain 651
3.1.1 Bifurcation Region 652
3.1.2 Test Regions II and III 655
3.2 Reservoir Design 656
3.2.1 Methods of Decreasing Fluid Volume 658
3.3 Validation of T-Chamber Design 658
4 Assembly Design and Manufacture 661
5 Discussion 662
References 664