Operator Approach to Linear Problems of Hydrodynamics

Table of Contents Volume II.- III: Motion of Bodies with Cavities ContainingViscous Incompressible Fluids.- 7: Motion of Bodies with Cavities Completely Filled with Viscous Incompressible Fluids.- 7.1 Motion of Fluids Completely Filling a Cavity in aStationary Body.- 7.1.1 Statement of the Problem and the Basic Equations.- 7.1.2 Reducing the Problem to a Differential Equation in aHilbert Space. Existence of Solutions.- 7.1.3 Structure of the Spectrum of the Problem.- 7.1.4 Perturbation of the Stationary Motion of a Fluid.- 7.1.5 Small Movements of a Fluid in a Rotating Container.- 7.2 Small Movements of a Gyrostate Around a Fixed Mass Center.- 7.2.1 Statement of the Problem and the Basic Equations.- 7.2.2 Transition to a Differential Equation in a Hilbert Space.- 7.2.3 Properties of the Translation Operator.- 7.2.4 Existence of Solutions of the Evolution Problem.- 7.2.5 Normal Oscillations.- 7.3 Rotating Motion of a Gyrostate.- 7.3.1 Statement of the Problem and the Basic Equations.- 7.3.2 Transition to a Differential Equation in a Hilbert Space.- 7.3.3 Properties of the Operators of the Problem.- 7.3.4 Normal Oscillations.- 7.3.5 Solvability of the Nonstationary Problem.- 7.4 Asymptotic Solutions for High Viscosity.- 7.4.1 Solving the Hydrodynamics Problem.- 7.4.2 Asymptotic Equations of the Motion of a Rigid Body.- 7.4.3 An Example.- 7.5 Oscillations of a Pendulum With a Cavity Completely Filled With aViscous Fluid.- 7.5.1 Towards the Statement of the Problem.- 7.5.2 Transition to a System of Operator Equations.- 7.5.3 The Indefinite Metric Approach.- 7.5.4 Other Properties of Solutions of the Spectral Problem.- 7.5.5 On Riesz and p-Basicity of Modes of Dissipative Waves.- 7.6 Problems on Fluids Flowing Through a Given Cavity.- 7.6.1 The Basic Equations.- 7.6.2 Application of the Abstract Scheme.- 7.6.3 Transition to Operator Equations inOrthogonal Subspaces.- 7.6.4 Theorem on Existence of a Generalized Solution.- 7.7 Convective Movements of Fluids in a Closed Cavity.- 7.7.1 Equations of Thermal Convection.- 7.7.2 Conditions of Mechanical Equilibrium.- 7.7.3 Final Statement of the Problem.- 7.7.4 Transition to an Operator Equation.- 7.7.5 Solvability of the Initial Boundary Value Problem.- 7.7.6 Normal Movements of a System Heated from Below.- 7.7.7 Normal Oscillations for Heating from Above.- 7.7.8 On Transition of Eigenvalues to the Left Half-Plane in theCase of Heating from Below.- 8: Motion of Viscous Fluids in Open Containers.- 8.1 Small Movements of Viscous Fluids in an OpenImmovable Container.- 8.1.1 Classical Statement of the Problem.- 8.1.2 Auxiliary Boundary Value Problems.- 8.1.3 Generalized Solutions of the Homogeneous NonstationaryProblem.- 8.1.4 Motions with Small Mass Forces.- 8.1.5 Equation of Energy Balance.- 8.1.6 Equation of Normal Oscillations.- 8.2 The Main Operator Pencil.- 8.2.1 Structure of the Spectrum of the Problem.- 8.2.2 Linearization of the Pencil.- 8.2.3 Mutual Relationships between Eigen-andAssociated Elements of the Two Pencils.- 8.2.4 Transformation to a Nondegenerate Pencil.- 8.2.5 The Property of Two-Multiple Basicity.- 8.2.6 Spectral Factorization of the Overdamped Pencil.Separate Basicity.- 8.2.7 Double-Sided Inequalities for the Two Branches ofEigenvalues.- 8.2.8 The General Case.- 8.3 Normal Oscillations and the Spectrum of the HydrodynamicsProblem.- 8.3.1 General Properties of the Spectrum.- 8.3.2 Influence of Fluid Viscosity on the Structure of theSpectrum of the Problem.- 8.3.3 Properties of Surface and Internal Waves.- 8.3.4 On the Basicity of Modes of Normal Oscillations.- 8.4 Oscillations of a Heavy Rotating Fluid.- 8.4.1 Statement of the Problem.- 8.4.2 Transition to a System of Operator Equations.- 8.4.3 Solvability of the Nonstationary Problem.- 8.4.4 Equations of Normal Oscillations.- 8.4.5 Investigation of the Spectral Problem.- 8.4.6 On the Completeness of Systems of Modes of NormalOscillations of the Initial Problem.- 8.5 Asymptotic Solutions for High Viscosity.- 8.5.1 The Cauchy Problem.- 8.5.2 Normal Oscillations.- 8.5.3 Motions under Mass Forces.- 8.5.4 The Case of Rotating Viscous Fluids.- 8.6 Oscillations of a System of Nonmixing Fluids.- 8.6.1 Statement of the Problem on Small Oscillations.- 8.6.2 Transition to a System of Operator Equations.- 8.6.3 The Theorem on Existence of a Generalized Solution..- 8.6.4 Normal Oscillations of a System of Fluids.- 8.6.5 On the Stability of the Relative Equilibrium State.- 8.7 Small Motions Around a Fixed Point of a Body with aCavity Partially Filled with Fluid.- 8.7.1 Basic Equations.- 8.7.2 Transition to a System of Operator Equations.- 8.7.3 Auxiliary Results.- 8.7.4 Existence of Solution of the Boundary Value Problem.- 8.8 Normal Oscillations of a Pendulum Partially Filled with aFluid (The Plane Problem).- 8.8.1 Statement of the Problem.- 8.8.2 Transition to a Differential Equation in aHilbert Space.- 8.8.3 Properties of Operator Coefficients of the EvolutionEquation.- 8.8.4 Normal Oscillations. Properties of the SpectralProblem.- 8.8.5 Theorem on Instability.- 8.8.6 On the Solvability of the Initial Boundary Value Problem.- 8.9 Convection in a Partially Filled Container.- 8.9.1 Statement of the Problem.- 8.9.2 Transition to a System of Operator Equations.- 8.9.3 Solvability of the Evolution Problem.- 8.9.4 Normal Convective Movements. Reduction to anOperator Pencil.- 8.9.5 Dissipatively Thermal and Surface Waves under theGeneral Law of Heat Transfer.- 8.9.6 Surface and Internal Waves for Heating From Above.- 8.9.7 Normal Oscillations for Heating from Below and for aGiven Heat Flow on the Free Surface.- 8.10 Sufficient Conditions of Instability for ConvectiveMovements of a Fluid.- 8.10.1 Transition to a Two-Parameter Pencil.- 8.10.2 On the Structure of the Kernels of OperatorCoefficients.- 8.10.3 On the Existence of Eigenvalues in the Left ComplexHalf-Plane.- 8.10.4 Double-Sided Estimates for Eigenvalues.- 8.10.5 Derivation of a Sufficient Condition for Instability.- 8.10.6 Remarks.- 9: Oscillations of Capillary Viscous Fluids.- 9.1 Statement of the Problem.- 9.1.1 Basic Equations and Boundary Conditions.- 9.1.2 Some Properties of Solutions to the NormalOscillation Problem.- 9.1.3 On the Spectrum Structure of Normal Oscillations.- 9.2 Oscilations of Capillary Fluids in Arbitrary Containers.- 9.2.1 Transition to a System of Operator Equations.- 9.2.2 Normal Oscillations. Properties of the Operators of theOscillations Problem.- 9.2.3 Normal Oscillations of a Rotating Fluid.- 9.2.4 Normal Oscillations of a Nonrotating Fluid.- 9.2.5 The Matrix Structure of the Main Operator.- 9.2.6 On the Finiteness of the Number of Nonreal Eigenvalues.- 9.2.7 Heuristic Considerations. The AbstractSpectral Problem.- 9.2.8 Heuristic Considerations. Physical Conclusions.- 9.2.9 The Solvability of the Evolution Problem.- 9.3 The Inverse of the Lagrange Theorem on Stability.- 9.3.1 Formulating the Theorem.- 9.3.2 Auxiliary Propositions.- 9.3.3 The Principle of Changing Stability.- 9.3.4 Transition to an Equation with a Compact Operator.- 9.3.5 Application of Perturbation Theory.- 9.3.6 The Existence of Eigenvalues in the Left ComplexHalf-Plane for an Arbitrary Viscosity Value.- 9.4 Motions of a Rigid Body Containing a Cavity Filled with aCapillary Fluid under Conditions of Complete Low Gravity.- 9.4.1 Statement of the Problem.- 9.4.2 Transition to a System of Operator Equations.- 9.4.3 Normal Oscillations. Transition to an Equation with aDissipative Operator.- 9.4.4 Properties of Normal Movements.- 9.4.5 The Existence of a Generalized Solution to theNonstationary Problem.- Appendix C: Remarks and Reference Comments to Part.- C.1 Chapter 7.- C.2 Chapter 8.- C.3 Chapter 9.- IV: Small Oscillations of Complex Hydrodynamic Systems.- 10: Oscillations of Partially Dissipative Hydrosystems.- 10.1 Statement of the Problem.- 10.1.1 The Classical Statement of the Problem.- 10.1.2 The Law of Full Energy Balance. Definition of aGeneralized Solution.- 10.1.3 Normal Oscillations. Statement of the Problem.- 10.2 Studying an Initial Boundary Value Problem.- 10.2.1 Projections of Euler and Navier-Stokes Equationson Orthogonal Subspaces.- 10.2.2 Auxiliary Boundary Value Problems.- 10.2.3 Properties of Matrix Blocks and TheirPhysical Meanings.- 10.2.4 Theorem on Correct Solvability of the InitialBoundary Value Problem.- 10.3 Model Problem on Normal Oscilations of PartiallyDissipative Hydrosystems.- 10.3.1 Statement of the Model Problem.- 10.3.2 Obtaining the Characteristic Equation.- 10.3.3 Studying the Characteristic Equation.- 10.3.4 General Conclusions and Hypotheses on the Structureof the Spectrum of the Hydrodynamics Problem.- 10.4 Normal Oscillations of a Partially DissipativeHydrosystem in an Arbitrary Domain.- 10.4.1 Transition to an Operator Pencil with BoundedOperator Coefficients.- 10.4.2 General Properties of the Spectrum.- 10.4.3 The Theorem on Spectrum Location.- 10.5 On the Completeness of the System of Modes of NormalOscillations.- 10.5.1 Auxiliary Results.- 10.5.2 Theorem on Completeness. Keldysh SchemeRealization.- 10.5.3 On the Existence of Branches of Eigenvalues with aLimit Point at Infinity. Heuristic Arguments.- 10.5.4 Concluding Remarks.- 11: Oscillations of Visco-Elastic and Relaxing Media.- 11.1 Visco-Elastic Fluids in Completely Filled Containers.- 11.1.1 A Model of a Visco-Elastic Fluid.- 11.1.2 Statement of the Initial Boundary Value Problem. 318 11.1.3 On the Solvability of the Initial BoundaryValue Problem.- 11.1.4 Normal Oscillations.- 11.2 Abstract Evolution and Spectral Problems Generated bySmall Motions of a Visco-Elastic Fluid.- 11.2.1 Statement of the Problem. Transition to anEquation with a Dissipative Operator.- 11.2.2 On the Solvability of the Cauchy Problem forIntegro-Differential Equations.- 11.2.3 Spectral Problem. Transition to an Equation with aBounded Operator.- 11.2.4 Properties of Operator Coefficients of the SpectralProblem.- 11.2.5 Properties of Solutions of the Spectral Problem.- 11.3 Small Motions and Normal Oscillations of a Visco-ElasticFluid in an Open Container.- 11.3.1 Mathematical Statement of the Problem.- 11.3.2 Transition to a System of Operator Equations.- 11.3.3 On the Solvability of the Initial Boundary ValueProblem.- 11.3.4 Normal Oscillations. Main Operator Pencil.- 11.4 Multiple Basicity of the System of Eigen-and Associated Elements for the Problem on Normal Oscillations of aVisco-Elastic Fluid in an Open Container.- 11.4.1 The Linearization of the Pencil.- 11.4.2 The Theorem on Basicity of the System of Root Elementsof the Linear Pencil.- 11.4.3 Connection between the System of Root Elementsof the Two Problems.- 11.4.4 The Theorem on Basicity of Special Form Elements.- 11.5 Additional Properties of Solutions of the Spectral Problem..- 11.5.1 On the Existence of Different Branches of Eigenvalues.- 11.5.2 On the Location of Nonreal Eigenvalues in theComplex Plane.- 11.5.3 Multiple Basicity and p-Basicity. An IndefiniteMetric Approach Using Krein Space Theory.- 11.6 Oscillations of Relaxing Fluids.- 11.6.1 Classical Statement of the Problem on Small Motionsof a Relaxing Fluid.- 11.6.2 Transition to an Initial Boundary Value Problem forOne Scalar Function.- 11.6.3 The Simplest Problem on Oscilations of a RelaxingFluid.- 11.6.4 On the Solvability of the Cauchy Problem for anAbstract Integro-Differential Equation Connected withSmall Motions of a Relaxing Fluid.- 11.6.5 Normal Oscilations of a Relaxing Fluid withVariable Medium Characteristics.- 11.6.6 Physical Conclusions.- Appendix D: Remarks and Reference Comments to Part.- D.1 Chapter 10.- D.2 Chapter 11.- Standard Reference Texts.- List of Symbols.