Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions (Oberwolfach Seminars, 48)

✍ Scribed by Barbara Kaltenbacher

• Publisher: Birkhäuser
• Year: 2018
• Tongue: English
• Leaves: 320
• Category: Library

✦ Synopsis

This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions. The first chapter provides a general description of a fluid-structure interaction, which is formulated within a realistic framework, where the structure subject to a frictional damping moves within the fluid. The second chapter then offers a multifaceted description, with often surprising results, of the case of the static interface; a case that is argued in the literature to be a good model for small, rapid oscillations of the structure. The third chapter describes flow-structure interaction where the compressible Navier-Stokes equations are replaced by the linearized Euler equation, while the solid is taken as a nonlinear plate, which oscillates in the surrounding gas flow. The final chapter focuses on a the equations of nonlinear acoustics coupled with linear acousticsor elasticity, as they arise in the context of high intensity ultrasound applications.

✦ Table of Contents

Contents
Preface
Chapter 1 An introduction to a fluid-structure model
1.1 Introduction
1.2 The model in Eulerian coordinates
1.3 The Lagrangian formulation
1.4 The nonhomogeneous Stokes problem
1.5 The variable coefficient nonhomogeneous Stokes problem
1.6 Static fluid-structure system with constant coefficients
1.7 Variable fluid-structure system with given coefficients
1.8 Equipartition of energy estimate
Acknowledgments
Bibliography
Chapter 2 Linear parabolic-hyperbolic fluid-structure interaction models. The case of static interface
2.0 Orientation
2.1 Physical model
Part I: The undamped case χ=0, α=0
2.2 The model
2.3 The operator A is dissipative on H
2.4 The point spectrum σp(A) of A in C+: σp(A) ∩ C+ is (i) either only the origin, or else (ii) a countable set on the imaginary axis (depending on the geometry of Ωs)
2.5 The adjoint operator A: dissipativity and point spectrum σp(A) of A: A+: σp(A) ∩ C+ = σp(A) ∩ C+)
2.6 The operators A and A generate s.c. contraction semigroups on H
2.6.1 The subspace H ≡ [Null (A)] of co-dimension invariant under the semigroup dAT. The restriction operator A = A│H is boundedly invertible
2.6.2 Generation of s.c. contraction semigroups: by A and A∗ onH; and by A and A∗ on the original space H
2.7 For 0 ≠ ∈ R in case (a); for 0 ≠ r ≠ r+rnj+,–, j = , ..., J, ind case (b) of proposition 2.1.4, then (irI – A)–1 ∈L(H)
2.8 The main result of higher regularity
2.9 High-level initial conditions. Regularity in the tangential direction [3]
2.9.1 Slashing the u and w variables by a first-order operator B on Ω, tangential to the boundary ΓfUΓs
2.9.2 Analysis of the commutator terms [B, Δ]u and [B, Δ]ω in thehalf-space
2.10 Proof of theorem 2.8.1
2.10.1 Boosting the regularity for the structural component ω: proof of theorem 2.8.1(a)
2.10.2 Boosting the regularity for fluid components {u, p}: proof of theorem 2.8.1(b)–(c)
2.11 Extension of theorem 2.8.1 to a forcing term in equation (1.1a)
2.12 Backward uniqueness [11]
Part II: The interior damped case χ=1, α=0
2.13 The model, main results
2.14 Proof of the uniform stabilization theorem 2.13.2 on H
2.15 Proof of the spectral properties of theorem 2.13.3 on H
Part III: The case of dissipation at the interface χ=0, α=1
2.16 The model. Statement of main results
2.17 Proof of theorem 2.16.3: basic estimate modulo l.o.t
2.18 Proof of theorem 2.17.2: absorption of l.o.t. in (1.17.3): from (2.17.3) to (2.17.1)
2.19 Additional complementary results
2.19.1 Problem (2.16.1) with partial damping at the interface
2.19.2 Problem (2.1.1) with α=0 and full interior damping
Appendix A. Some regularity results of a non-homogeneous Stokes problem
Appendix B. From the B.C. (2.1.1f ) to the B.C. (2.9.3f )
Appendix C. The gradient, divergence and Laplace operators in the M-S system of coordinates
Appendix D: A special overdetermined elliptic problem (SOEP): geometries which satisfy it, or do not satisfy it. [12]
Acknowledgement
Bibliography
Chapter 3 Flow-plate interactions: well-posedness and long time behavior
3.1 Dynamical systems and long-time behavior of solutions
3.1.1 Definitions and notions
3.2 Panel flutter and the flow-plate interaction modeling
3.2.1 Applications
3.2.2 Modeling
3.3 Panel flutter nonlinear dynamical system
3.3.1 Functional setup
3.3.2 Definition of solutions
3.3.3 Preliminary remarks
3.3.4 Dynamical system in the subsonic case
3.3.5 Dynamical system in the supersonic case
3.4 Long-time behavior of flow-plate interactions
3.4.1 Reduced, delay dynamics
3.4.2 Compact global attractor for delay structural dynamics
3.4.3 Strong stability to equilibrium for full flow-plate interaction
3.4.4 Basics
3.4.5 Smooth data result
3.4.6 Large static and viscous damping
3.5 Relevant work on related models
3.5.1 The Berger nonlinearity
3.5.2 Rotational inertia and thermal effects—velocity smoothing
3.5.3 Kutta–Joukowsky condition
3.5.4 Piston-theoretic models
3.6 Open problems and model extensions
3.6.1 Viscous flows
3.6.2 The transonic regime
3.6.3 The free boundary condition
3.6.4 Full von Karman model
3.6.5 Axial flow
Acknowledgments
Bibliography
Chapter 4 Some aspects in nonlinear acoustics: structure-acoustic coupling and shape optimization
4.1 Introduction
4.2 Models
4.2.1 Derivation of models
4.2.2 Analysis
4.3 Coupling
4.3.1 Motivation
4.3.2 Nonlinear damping
4.3.3 Acoustic-acoustic coupling
4.3.4 Acoustic-elastic coupling
4.4 Shape optimization
4.4.1 Optimization problem
Gradient computation
4.5 Outlook
Acknowledgment
Bibliography
Correction to: Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions