Beschreibung
This is the second volume of a set of two devoted to the operator approach to linear problems in hydrodynamics. 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 second part of the present volume collects nonself-adjoint problems on small motions and normal oscillations of a viscous fluid filling a bounded region.
Inhaltsverzeichnis
InhaltsangabeTable 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
