Operator Approach to Linear Problems of Hydrodynamics

, by Kopachevsky, Nikolay D.; Krein, S. G.

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ISBN: 9783764321901 | 3764321903
Cover: Hardcover
Copyright: 10/1/2003

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This is the second volume of a set of two devoted to the operator approach to linear problems in hydrodynamics (Volume 1: 3-7643-5406-2). It presents functional analytical methods applied to the study of small movements and normal oscillations of hydromechanical systems having cavities filled with either ideal or viscous fluids. The work relies on the authors' and their students' works of the last 30-40 years. The readers are not supposed to be familiar with the methods of functional analysis. The second part of the present volume collects nonself-adjoint problems on small motions and normal oscillations of a viscous fluid filling a bounded region.The book is particularly useful for researchers, engineers and students in fluid mechanics and mathematics interested in operator theoretical methods for the analysis of hydrodynamical problems.

Volume II

Preface

Table of Contents

Volume II

vii

Volume I xvi

Introduction

(4)

Part III: Motion of Bodies with Cavities Containing Viscous Incompressible Fluids

(244)

Chapter 7: Motion of Bodies with Cavities Completely Filled with Viscous Incompressible Fluids

7.1 Motion of Fluids Completely Filling a Cavity in a Stationary Body

(7)

7.1.1 Statement of the Problem and the Basic Equations

(1)

7.1.2 Reducing the Problem to a Differential Equation in a Hilbert Space. Existence of Solutions

(1)

7.1.3 Structure of the Spectrum of the Problem

(1)

7.1.4 Perturbation of the Stationary Motion of a Fluid

(2)

7.1.5 Small Movements of a Fluid in a Rotating Container

(3)

7.2 Small Movements of a Gyrostate Around a Fixed Mass Center

(6)

7.2.1 Statement of the Problem and the Basic Equations

(1)

7.2.2 Transition to a Differential Equation in a Hilbert Space

(1)

7.2.3 Properties of the Translation Operator

(2)

7.2.4 Existence of Solutions of the Evolution Problem

(1)

7.2.5 Normal Oscillations

(1)

7.3 Rotating Motion of a Gyrostate

7.3.1 Statement of the Problem and the Basic Equations

(1)

7.3.2 Transition to a Differential Equation in Hilbert Space

(1)

7.3.3 Properties of the Translation Operator

(1)

7.3.4 Normal Oscillations

(4)

7.3.5 Solvability of the Nonstationary Problem

7.4 Asymptotic Solution for High Viscosity

(16)

7.4.1 Solving the Hydrodynamics Problem

(1)

7.4.2 Asymptotic Equations of the Motion of a Rigid Body

(3)

7.4.3 An Example

(3)

7.5 Oscillations of a Pendulum With a Cavity Completely Filled With a Viscous Fluid

(22)

7.5.1 Towards the Statement of the Problem

(1)

7.5.2 Transition to a System of Operator Equations

(2)

7.5.3 The Indefinite Metric Approach

(2)

7.5.4 Other Properties of Solutions of the Spectral Problem

(6)

7.5.5 On Riesz and p-Basicity of Modes of Dissipative Waves

(11)

7.6 Problems on Fluids Flowing Through a Given Cavity

7.6.1 The Basic Equations

(1)

7.6.2 Application of the Abstract Scheme

(2)

7.6.3 Transition to Operator Equations in Orthogonal Subspaces

(1)

7.6.4 Theorem on Existence of a Generalized Solution

(1)

7.7 Convective Movements of Fluids in a Closed Cavity

(13)

7.7.1 Equations of Thermal Convection

(1)

7.7.2 Conditions of Mechanical Equilibrium

(1)

7.7.3 Final Statement of the Problem

(1)

7.7.4 Transition to an Operator Equation

(2)

7.7.5 Solvability of the Initial Boundary Value Problem

(1)

7.7.6 Normal Movements of a System Heated from Below

(2)

7.7.7 Normal Oscillations for Heating from Above

(2)

7.7.8 On Transition of Eigenvalues to the Left Half-Plane in the Case of Heating from Below

(5)

Chapter 8: Motion of Viscous Fluids in Open Containers

8.1 Small Movements of Viscous Fluids in an Open Immovable Container

(3)

8.1.1 Classical Statement of the Problem

(1)

8.1.2 Auxiliary Boundary Value Problems

(3)

8.1.3 Generalized Solutions of the Homogeneous Nonstationary Problem

(4)

8.1.4 Motions with Small Mass Forces

(1)

8.1.5 Equation of Energy Balance

8.1.6 Equation of Normal Oscillations

(1)

8.2 The Main Operator Pencil

(27)

8.2.1 Structure of the Spectrum of the Problem

(2)

8.2.2 Linearization of the Pencil

(2)

8.2.3 Mutual Relationships between Eigen- and Associated Elements of the Two Pencils

(2)

8.2.4 Transformation to a Nondegenerate Pencil

(1)

8.2.5 The Property of Two-Multiple Basicity

(2)

8.2.8 Spectral Factorization of the Overdamped Pencil. Separate Basicity

8.2.7 Double-Sided Inequalities for the Two Branches of Eigenvalues

(93)

8.2.8 The General Case

(2)

8.3 Normal Oscillations and the Spectrum of the Hydrodynamics Problem

(10)

8.3.1 General Properties of the Spectrum

(3)

8.3.2 Influence of Fluid Viscosity on the Structure of the Spectrum of the Problem

100

(1)

8.3.3 Properties of Surface and Internal Waves

101

(3)

8.3.4 On the Basicity of Modes of Normal Oscillations

104

(3)

8.4 Oscillations of a Heavy Rotating Fluid

107

(14)

8.4.1 Statement of the Problem

107

(2)

8.4.2 Transition to a System of Operator Equations

109

(1)

8.4.3 Solvability of the Nonstationary Problem

109

(3)

8.4.4 Equations of Normal Oscillations

112

(1)

8.4.5 Investigation of the Spectral Problem

113

(7)

8.4.6 On the Completeness of Systems of Modes of Normal Oscillations of the Initial Problem

120

(1)

8.5 Asymptotic Solutions for High Viscosity

121

(12)

8.5.1 The Cauchy Problem

121

(3)

8.5.2 Normal Oscillations

124

(3)

8.5.3 Motions under Mass Forces

127

(3)

8.5.4 The Case of Rotating Viscous Fluids

130

(3)

8.6 Oscillations of a System of Nonmixing Fluids

133

(7)

8.6.1 Statement of the Problem on Small Oscillations

134

(1)

8.6.2 Transition to a System of Operator Equations

135

(2)

8.6.3 The Theorem on Existence of a Generalized Solution

137

(1)

8.6.4 Normal Oscillations of a System of Fluids

138

(1)

8.6.5 On the Stability of the Relative Equilibrium Stat

139

(1)

8.7 Small Motions Around a Fixed Point of a Body with a Cavity Partially Filled with Fluid

140

(8)

8.7.1 Basic Equations

141

(1)

8.7.2 Transition to a System of Operator Equations

141

(2)

8.7.3 Auxiliary Results

143

(2)

8.7.4 Existence of Solution of the Boundary Value Problem

145

(3)

8.8 Normal Oscillations of a Pendulum Partially Filled with a Fluid (The Plane Problem)

148

(14)

8.8.1 Statement of the Problem

148

(1)

8.8.2 Transition to a Differential Equation in a Hilbert Space

149

(2)

8.8.3 Properties of Operator Coefficients of the Evolution Equation

151

(3)

8.8.4 Normal Oscillations. Properties of the Spectral Problem

154

(5)

8.8.5 Theorem on Instability

159

(2)

8.8.6 On the Solvability of the Initial Boundary Value Problem

161

(1)

8.9 Convection in a Partially Filled Container

162

(14)

8.9.1 Statement of the Problem

162

(2)

8.9.2 Transition to a System of Operator Equations

164

(1)

8.9.3 Solvability of the Evolution Problem

165

(22)

8.9.4 Normal Convective Movements. Reduction to an Operator Pencil

187

8.9.5 Dissipatively Thermal and Surface Waves under the General Law of Heat Transfer

169

(4)

8.9.6 Surface and Internal Waves for Heating From Above

173

(1)

8.9.7 Normal Oscillations for Heating from Below and for a Given Heat Flow on the Free Surface

174

(2)

8.10 Sufficient Conditions of Instability for Convective Movements of a Fluid

176

(11)

8.10.1 Transition to a Two-Parameter Pencil

176

(4)

8.10.2 On the Structure of the Kernels of Operator Coefficients

180

(1)

8.10.3 On the Existence of Eigenvalues in the Left Complex Half-Plane

181

(1)

8.10.4 Double-Sided Estimates for Eigenvalues

182

(1)

8.10.5 Derivation of a Sufficient Condition for Instability

183

(2)

8.10.6 Remarks

185

(2)

Chapter 9: Oscillations of Capillary Viscous Fluids

9.1 Statement of the Problem

187

(5)

9.1.1 Basic Equations and Boundary Conditions

187

(2)

9.1.2 Some Properties of Solutions to the Normal Oscillation Problem

189

(2)

9.1.3 On the Spectrum Structure of Normal Oscillations

191

(1)

9.2 Oscilations of Capillary Fluids in Arbitrary Containers

192

(25)

9.2.1 Transition to a System of Operator Equations

192

(2)

9.2.2 Normal Oscillations. Properties of the Operators of the Oscillations Problem

194

(6)

9.2.3 Normal Oscillations of a Rotating Fluid

200

(2)

9.2.4 Normal Oscillations of a Nonrotating Fluid

202

(3)

9.2.5 The Matrix Structure of the Main Operator

205

(1)

9.2.6 On the Finiteness of the Number of Nonreal Eigenvalues

206

(4)

9.2.7 Heuristic Considerations. The Abstract Spectral Problem

210

(2)

9.2.8 Heuristic Considerations. Physical Conclusions

212

(3)

9.2.9 The Solvability of the Evolution Problem

215

(2)

9.3 The Inverse of the Lagrange Theorem on Stability

217

(12)

9.3.1 Formulating the Theorem

218

(1)

9.3.2 Auxiliary Propositions

219

(3)

9.3.3 The Principle of Changing Stability

222

(1)

9.3.4 Transition to an Equation with a Compact Operator

223

(2)

9.3.5 Application of Perturbation Theory

225

(3)

9.3.6 The Existence of Eigenvalues in the Left Complex Half-Plane for an Arbitrary Viscosity Value

228

(1)

9.4 Motions of a Rigid Body Containing a Cavity Filled with a Capillary Fluid under Conditions of Complete Low Gravity

229

(8)

9.4.1 Statement of the Problem

229

(1)

9.4.2 Transition to a System of Operator Equations

230

(1)

9.4.3 Normal Oscillations. Transition to an Equation with a Dissipative Operator

231

(2)

9.4.4 Properties of Normal Movements

233

(1)

9.4.5 The Existence of a Generalized Solution to the Nonstationary Problem

233

(4)

Appendix C: Remarks and Reference Comments to Part III

C.1 Chapter 7

237

(3)

C.2 Chapter 8

240

(3)

C.3 Chapter 9

243

(6)

Part IV: Small Oscillations of Complex Hydrodynamic Systems

249

(166)

Chapter 10: Oscillations of Partially Dissipative Hydrosystems

10.1 Statement of the Problem

252

(8)

10.1.1 The Classical Statement of the Problem

252

(2)

10.1.2 The Law of Full Energy Balance. Definition of a Generalized Solution

254

(3)

10.1.3 Normal Oscillations. Statement of the Problem

257

(3)

10.2 Studying an Initial Boundary Value Problem

260

(11)

10.2.1 Projections of Euler and Navier-Stokes Equations on Orthogonal Subspaces

260

(3)

10.2.2 Auxiliary Boundary Value Problems

263

(3)

10.2.3 Properties of Matrix Blocks and Their Physical Meanings

266

(3)

10.2.4 Theorem on Correct Solvability of the Initial Boundary Value Problem

269

(2)

10.3 Model Problem on Normal Oscilations of Partially Dissipative Hydrosystems

271

(12)

10.3.1 Statement of the Model Problem

271

(2)

10.3.2 Obtaining the Characteristic Equation

273

(2)

10.3.3 Studying the Characteristic Equation

275

(6)

10.3.4 General Conclusions and Hypotheses on the Structure of the Spectrum of the Hydrodynamics Problem

281

(2)

10.4 Normal Oscillations of a Partially Dissipative Hydrosystem in an Arbitrary Domain

283

(17)

10.4.1 Transition to an Operator Pencil with Bounded Operator Coefficients

283

(3)

10.4.2 General Properties of the Spectrum

286

(7)

10.4.3 The Theorem on Spectrum Location

293

(7)

10.5 On the Completeness of the System of Modes of Normal Oscillations

300

(16)

10.5.1 Auxiliary Results

300

(2)

10.5.2 Theorem on Completeness. Keldysh Scheme Realization

302

(5)

10.5.3 On the Existence of Branches of Eigenvalues with a Limit Point at Infinity. Heuristic Arguments

307

(7)

10.5.4 Concluding Remarks

314

(2)

Chapter 11: Oscillations of Visco-Elastic and Relaxing Media

11.1 Visco-Elastic Fluids in Completely Filled Containers

316

(7)

11.1.1 A Model of a Visco-Elastic Fluid

316

(2)

11.1.2 Statement of the Initial Boundary Value Problem

318

(1)

11.1.3 On the Solvability of the Initial Boundary Value Problem

319

(2)

11.1.4 Normal Oscillations

321

(2)

11.2 Abstract Evolution and Spectral Problems Generated by Small Motions of a Visco-Elastic Fluid

323

(19)

11.2.1 Statement of the Problem. Transition to an Equation with a Dissipative Operator

324

(1)

11.2.2 On the Solvability of the Cauchy Problem for Integro-Differential Equations

325

(1)

11.2.3 Spectral Problem. Transition to an Equation with a Bounded Operator

326

(4)

11.2.4 Properties of Operator Coefficients of the Spectral Problem

330

(3)

11.2.5 Properties of Solutions of the Spectral Problem

333

(9)

11.3 Small Motions and Normal Oscillations of a Visco-Elastic Fluid in an Open Container

342

(7)

11.3.1 Mathematical Statement of the Problem

342

(1)

11.3.2 Transition to a System of Operator Equations

343

(2)

11.3.3 On the Solvability of the Initial Boundary Value Problem

345

(1)

11.3.4 Normal Oscillations. Main Operator Pencil

346

(3)

11.4 Multiple Basicity of the System of Eigen- and Associated Elements for the Problem on Normal Oscillations of a Visco-Elastic Fluid in an Open Container

349

(25)

11.4.1 The Linearization of the Pencil

349

(5)

11.4.2 The Theorem on Basicity of the System of Root Elements of the Linear Pencil

354

(1)

11.4.3 Connection between the System of Root Elements of the Two Problems

355

(8)

11.4.4 The Theorem on Basicity of Special Form Elements

363

(11)

11.5 Additional Properties of Solutions of the Spectral Problem

374

(16)

11.5.1 On the Existence of Different Branches of Eigenvalues

374

(3)

11.5.2 On the Location of Nonreal Eigenvalues in the Complex Plane

377

(7)

11.5.3 Multiple Basicity and p-Basicity. An Indefinite Metric Approach Using Krein Space Theory

384

(6)

11.6 Oscillations of Relaxing Fluids

390

(21)

11.6.1 Classical Statement of the Problem on Small Motions of a Relaxing Fluid

390

(2)

11.6.2 Transition to an Initial Boundary Value Problem for One Scalar Function

392

(2)

11.6.3 The Simplest Problem on Oscilations of a Relaxing Fluid

394

(3)

11.6.4 On the Solvability of the Cauchy Problem for an Abstract Integro-Differential Equation Connected with Small Motions of a Relaxing Fluid

397

(3)

11.6.5 Normal Oscilations of a Relaxing Fluid with Variable Medium Characteristics

400

(10)

11.6.6 Physical Conclusions

410

Appendix D: Remarks and Reference Comments to Part IV

D.1 Chapter 10

411

(1)

D.2 Chapter 11

412

(3)

Standard Reference Texts

415

(2)

Bibliography

417

(16)

List of Symbols

433

(4)

Subject Index

437

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Operator Approach to Linear Problems of Hydrodynamics

Summary

Table of Contents

Supplemental Materials