Low-Gravity Fluid Mechanics

1. Introduction.- 1.1 On Zero-Gravity.- 1.2 Surface or Capillary Forces.- 1.3 On the History of the Problem.- 1.4 Subject Matter of the Book.- I Statics.- 2. Equilibrium Shapes of a Liquid.- 2.1 Equilibrium Conditions.- 2.1.1 Basic Assumptions and Notations.- 2.1.2 Hydrostatic Conditions.- 2.1.3 Equilibrium of a Capillary Free Surface.- 2.1.4 Derivation of Equilibrium Conditions from the Variational Principle of Stationary Potential Energy.- 2.1.5 Vessel with a Nonsmooth Surface.- 2.1.6 Other Generalizations.- 2.2 The Equilibrium Surface Problem.- 2.2.1 Arbitrary Parametric Representation of a Surface.- 2.2.2 Equilibrium Surface with an Equation of the Type z=f(x,y).- 2.2.3 Axisymmetric Equilibrium Problem.- 2.2.4 Plane Equilibrium Problem.- 2.2.5 Equation for a Bundle of Equilibrium Surfaces.- 2.2.6 On Mass Force Potential and Similitude Numbers.- 2.3 The Construction of Simply Connected Axisymmetric Equilibrium Shapes.- 2.3.1 Family of Equilibrium Lines.- 2.3.2 Determination of an Equilibrium Line from Known Values of ? and ?.- 2.4 The Axisymmetric Problem for a Vessel at Rest.- 2.4.1 General Remarks.- 2.4.2 Positive Loading.- 2.4.3 Negative Loading.- 2.4.4 Zero-Gravity.- 2.4.5 Examples.- 2.4.6 Reference Commentaries.- 2.5 Axisymmetric Shapes of a Rotating Liquid Under Zero-Gravity.- 2.5.1 Properties of the Solutions of Equilibrium Differential Equations.- 2.5.2 Solution in Elliptic Integrals.- 2.5.3 Equilibrium Surfaces in a Vessel.- 2.5.4 Equilibrium Shapes of a Rotating Drop.- 2.5.5 Reference Commentaries.- 2.6 Axisymmetric Rotation Problem in a Gravitational Force Field.- 2.6.1 General Remarks.- 2.6.2 Cylinder.- 2.6.3 Cone and Sphere.- 2.6.4 Application of the First Integral of the Equilibrium Equation.- 2.7 Axisymmetric Problem for Large Bond Numbers.- 2.7.1 Formulation of the Problem.- 2.7.2 Construction of the Asymptotic Expansion.- 2.7.3 Boundary Layer Equation.- 2.7.4 Remarks.- 2.7.5 Reference Commentaries.- 2.8 An Axisymmetric Flat (Gently Sloping) Equilibrium Surface.- 2.8.1 General Remarks.- 2.8.2 Cone and Cylinder.- 2.8.3 Drop on a Plane.- 2.9 Doubly Connected Axisymmetric Equilibrium Surfaces.- 2.9.1 State of Rest Under Zero-Gravity.- 2.9.2 Gravitational Force Field.- 2.9.3 Rotation of a Liquid Under Zero-Gravity.- 2.9.4 Annular Equilibrium Shapes.- 2.10 The Plane Problem for a Gravitational Force Field.- 2.10.1 Properties of Integral Curves.- 2.10.2 Solution in Elliptic Integrals.- 2.10.3 Determination of the Shape of a Symmetric Equilibrium Line.- 2.10.4 Channel with a Semi-Infinite Cross Section.- 2.10.5 Linearized Problem.- 2.11 The Two-Dimensional Problem for Large Bond Numbers.- 2.11.1 Formulation of the Two-Dimensional Problem.- 2.11.2 Construction of the Asymptotic Expansion.- 2.12 Numerical Methods of Constructing Equilibrium Surfaces of a General Type.- 2.12.1 Variational Problem.- 2.12.2 Method of Local Variations.- 2.12.3 Optimal Discretization Method.- 2.12.4 Review of Other Methods and Results.- 2.13 Small Perturbations of the Equilibrium Surface.- 2.13.1 Formulation of the Problem.- 2.13.2 Linearization of the Problem.- 2.13.3 Axisymmetric Case.- 2.13.4 Small Bond Numbers.- 2.13.5 Gently Sloping (Flat) Surfaces.- 2.13.6 Nonaxisymmetric Shapes of a Drop on a Plane.- 2.13.7 On Numerical Construction of Equilibrium Surfaces.- 3. Stability of Equilibrium States of a Liquid.- 3.1 Introduction.- 3.2 The Second Variation of Potential Energy.- 3.2.1 Expression for the Second Variation.- 3.2.2 Spectral Stability Criterion.- 3.2.3 Axisymmetric Unperturbed Problem.- 3.2.4 Parameter-Dependent System.- 3.2.5 Stability of Cylindrical Equilibrium Surfaces in a Channel.- 3.3 Simply Connected Equilibrium Surfaces in the Axisymmetric Problem.- 3.3.1 Properties of Eigenvalues.- 3.3.2 Maximal Stability Regions.- 3.3.3 Critical Value of the Parameter X1.- 3.4 Critical Values of the Boundary Parameter for Main Types of Force Fields.- 3.4.1 Zero-Gravity.- 3.4.2 Horizontal Equilibrium Surface in a Uniform Gravitational Force Field.- 3.4.3 Uniform Field, Positive Loads.- 3.4.4 Uniform Field, Negative Loads.- 3.4.5 Liquid Rotating Under Zero-Gravity.- 3.4.6 Example.- 3.5 The Determination of Critical Loads.- 3.5.1 Cylindrical Vessel.- 3.5.2 Liquid in a Cone.- 3.5.3 Pendent Liquid Drop on a Horizontal Plane.- 3.5.4 Liquid with a Doubly Connected Free Surface.- 3.6 On the Stability of Axisymmetric Equilibrium Forms of a Rotating Liquid.- 3.6.1 Liquid in a Cylindrical Vessel.- 3.6.2 Cylindrical Free Surface.- 3.6.3 Liquid Column Between Parallel Plates.- 3.6.4 Stability of an Infinite Liquid Column.- 3.7 Two Classical Problems.- 3.7.1 Rotating Liquid Drop.- 3.7.2 Annular Equilibrium Configurations.- 3.8 The Stability of Cylindrical Equilibrium Surfaces in Channels for Specific Force Fields.- 3.8.1 General Case.- 3.8.2 Stability of Symmetric Equilibrium States.- 3.8.3 Zero-Gravity Conditions.- 3.8.4 Horizontal Equilibrium Surface in a Gravitational Force Field.- 3.8.5 Uniform Field, Positive Loads.- 3.8.6 Uniform Field, Negative Loads.- 3.8.7 Rectangular Channel.- 3.8.8 Plane Drop on a Horizontal Plate.- 3.8.9 Sectorial Channel.- 3.9 Stable Equilibrium of a Free Surface in Contact with the Edge of a Vessel.- 3.9.1 Formulation of the Problem.- 3.9.2 Conditions of Nonnegative ?U.- 3.9.3 Conditions of Stability in Terms of ?2U.- 3.9.4 Axisymmetric Case.- 3.9.5 Drop in Contact with an Edge.- 3.9.6 Limiting Case.- 3.10 Quasistatic Evolution and Rupture of an Equilibrium Shape.- 3.10.1 Introduction.- 3.10.2 Extrusion of a Liquid from a Circular Orifice.- 3.10.3 Transporting Water in a Sieve.- 3.11 Equilibrium Stability of a Liquid Zone.- 3.11.1 Zero-Gravity.- 3.11.2 Floating Zone Method.- 3.11.3 Gravitational Force Field.- 3.11.4 Problem I.- 3.11.5 Problems II and III.- 3.11.6 General Stability Criterion.- 3.12 The Stability of an Equilibrium State with an Unconnected Free Surface.- 3.12.1 Formulation of the Problem.- 3.12.2 Analogue of Maxwell's Problem.- 3.12.3 Vessels with Cylindrical Sections.- 3.12.4 Sectional Axisymmetric Problem.- 3.12.5 Properties of Eigenvalues.- 3.12.6 Method of Calculating Stability.- 3.12.7 Example.- 3.12.8 Closed Systems.- 3.12.9 Reference Commentaries.- 4. Bifurcation of Equilibrium States.- 4.1 Introduction.- 4.2 Bifurcation of Equilibrium Shapes. The General Case.- 4.2.1 Case of the Expansibility of a Solution in Integral Powers of the Parameter.- 4.2.2 Expansion of the Solution into Integral and Half-Integral Powers of the Parameter.- 4.2.3 Variants.- 4.3 Bifurcation in a Circular Cylinder for a Gravitational Force Field.- 4.3.1 Formulation of the Problem.- 4.3.2 The Case ? = ?/2.- 4.3.3 Arbitrary Wetting Angle.- 4.3.4 Stability Margin.- 4.4 Other Problems.- 4.4.1 Pendent Drop on a Horizontal Plane.- 4.4.2 Bifurcation of the Axisymmetric Shape at the Edge of a Vessel.- 4.4.3 Rotating Cylindrical Column of Liquid.- 4.4.4 Rotating Drop.- 4.4.5 Plane Problem for a Rectangular Channel.- 4.4.6 Plane Problem of a Pendent Drop.- 4.5 On the Concept of the Stability Margin in Problems of Fluid Mechanics.- 4.5.1 Introduction.- 4.5.2 Three Examples.- 4.5.3 On Continuous Chains of Stable States.- 4.5.4 Possible Approach to the Concept of the Stability Margin.- 4.5.5 Other Approaches.- 4.6 The Lyapunov-Schmidt Method.- 4.6.1 Fredholm Operators.- 4.6.2 Local Extensions of Solutions of Nonlinear Equations.- 4.6.3 Application to the Problem of Bifurcation of Equilibrium Shapes.- II Small Oscillations.- 5. Small Oscillations of an Ideal Liquid.- 5.1 Introduction.- 5.2 Formulation of the Small Oscillations Problem for an Ideal Liquid.- 5.2.1 Basic Equations.- 5.2.2 Dynamic Condition on an Equilibrium Surface.- 5.2.3 Transition to Dimensionless Variables. Normal Oscillations.- 5.3 Simple Problems Admitting Separation of Variables.- 5.3.1 Cylindrical Vessel.- 5.3.2 Rectangular Channel.- 5.3.3 Conic Vessel.- 5.3.4 Vessel in the Form of a Cylindrical Sector. Liquid Column.- 5.3.5 Spherical Self-Gravitating Liquid Layer.- 5.3.6 Supplement. Nonlinear Radial Oscillations of a Bubble.- 5.4 Transition to the Operator Equation.- 5.4.1 Derivation of the Equation.- 5.4.2 Orthogonal Expansion of the Space L2(?) of Vector-Functions.- 5.4.3 Projection of Euler's Equation.- 5.4.4 Refinement of the Properties of Operators A and B.- 5.4.5 Energy Spaces.- 5.5 Normal Oscillations and Variational Methods for Determining Their Frequencies.- 5.5.1 Spectrum Structure, Completeness of the System of Eigenfunctions.- 5.5.2 On the Stability of Equilibrium States.- 5.5.3 Extremal Properties of Eigenvalues.- 5.5.4 Ritz Method.- 5.5.5 Application of the Eigenfunctions of Operators A and B.- 5.6 Oscillations of a System of Immiscible Liquids.- 5.6.1 Formulation of the Problem.- 5.6.2 Transition to Operator Equation.- 5.6.3 Variational Methods.- 5.6.4 Some Alternative Versions.- 5.6.5 Examples.- 5.6.6 Operator Treatment of the Problem of Oscillations of a System of Immiscible Liquids.- 5.7 A Rotating Liquid. Application of the Function of State.- 5.7.1 General Case.- 5.7.2 Function of State.- 5.7.3 Normal Oscillations.- 5.7.4 Oscillations of a Liquid Column Under Zero-Gravity.- 5.7.5 Asymptotic Formulas.- 5.7.6 General Conclusions.- 5.7.7 Variational Approach.- 5.7.8 System of Immiscible Liquids.- 5.7.9 Oscillations of Two Liquids in a Cylindrical Vessel.- 5.8 Functional Approach to the Problem of Oscillations of an Ideal Rotating Liquid.- 5.8.1 Projection of Equations of Motion.- 5.8.2 Transition to the Operator Equation.- 5.8.3 Properties of Operators of the Problem.- 5.8.4 Normal Oscillations.- 5.8.5 On the Existence of Surface Waves.- 5.8.6 On the Completeness and Minimality of Surface Wave Modes.- 5.8.7 On the Existence of Internal Waves.- 6. Methods of Calculating Linear Oscillations of an Ideal Liquid.- 6.1 Plane Oscillations in a Rectangular Channel.- 6.1.1 Ritz Method.- 6.1.2 Computations.- 6.1.3 Results of Computations.- 6.2 Plane Oscillations in a Sectorial Channel.- 6.2.1 Method of Collocation.- 6.2.2 Results of Computations.- 6.3 Plane Problem on the Oscillations of a Weightless Drop Abutting a Plane.- 6.3.1 Introduction.- 6.3.2 Integral Equations Method.- 6.3.3 Results of Computations.- 6.3.4 Asymptotic Formulas for Small ?.- 6.3.5 Plane Problem on the Oscillations of a Bubble.- 6.4 Plane Oscillations in a Circular Channel.- 6.4.1 Application of the Ritz Method.- 6.4.2 Results of Computations.- 6.4.3 Asymptotic Formulas for Small and Large Fillings.- 6.5 Oscillations in a Circular Cylindrical Vessel.- 6.5.1 Introduction.- 6.5.2 Application of Method 1.- 6.5.3 Application of Method 2.- 6.5.4 Results of Computations; the Principal Mode.- 6.5.5 Results of Computations; Other Modes.- 6.5.6 On the Calculation of Frequencies and Oscillation Modes of a Rotating Liquid.- 6.6 Oscillations in a Spherical Vessel.- 6.6.1 Introduction.- 6.6.2 Method of Numerical Solution.- 6.6.3 Results of Computations.- 6.6.4 Asymptotic Formulas for Small and Large Fillings.- 6.6.5 Oscillations of a Weightless Drop Adjoining a Plane.- 6.6.6 Oscillations of a Rotating Liquid Drop.- 7. Linear Oscillations of a Viscous Liquid.- 7.1 Introduction.- 7.2 Formulation of the Problem. Properties of the Spectrum.- 7.2.1 Basic Equations.- 7.2.2 Normal Oscillations.- 7.2.3 Properties of the Spectrum.- 7.2.4 Stability Theorem.- 7.2.5 Application of Galerkin's Method.- 7.3 Free Oscillations of a Self-Gravitating Liquid Globe.- 7.3.1 Generalized Spherical Functions.- 7.3.2 Characteristic Equation of the Problem.- 7.3.3 Properties jof the Spectrum for any ?.- 7.3.4 Asymptotic Formula for the Minimum Eigenvalue for l??.- 7.3.5 Asymptotic Formula for Low Viscosity.- 7.3.6 Asymptotic Formula for High Viscosity.- 7.3.7 Other Cases.- 7.4 Oscillations of a Rotating Liquid Circle. Model Problem.- 7.4.1 Derivation of the Characteristic Equation.- 7.4.2 Properties of the Spectrum for ?0 = 0.- 7.4.3 General Properties of the Spectrum for ?0 ? 0.- 7.4.4 Limiting Cases.- 7.4.5 Effect of Gyroscopic Stabilization.- 7.4.6 Special Cases.- 7.5 Oscillations of a Low-Viscosity Rotating Liquid Ring.- 7.5.1 Basic Equations.- 7.5.2 Problem of Normal Oscillations of an Ideal Liquid.- 7.5.3 Low Viscosity.- 7.5.4 Oscillations of a Coaxial System of Liquids.- 7.6 Oscillations of a Spherical Layer of a Low-Viscosity, Self-Gravitating Liquid.- 7.6.1 Formulation of the Problem.- 7.6.2 Boundary Layer Method.- 7.6.3 Solution of the Characteristic Equation.- 7.6.4 Variants.- 7.6.5 Oscillations of a Concentric System of Liquids.- 7.7 Application of the Boundary Layer Method for an Axisymmetric Vessel.- 7.7.1 General Scheme of the Boundary Layer Method.- 7.7.2 Derivation of Boundary Layer Solutions.- 7.7.3 Determination of Correction to the Complex Damping Factor.- 7.7.4 Special Cases.- 7.8 Rotating Masses of Low-Viscosity Liquids.- 7.8.1 Peculiarities of the Problem.- 7.8.2 Determination of the Quantity ?2.- 7.8.3 Special and Limiting Cases; Results of Calculations.- 7.9 General Problem of the Oscillations of a Rotating Viscous Capillary Liquid.- 7.9.1 Formulation of the Problem.- 7.9.2 Two Auxiliary Problems.- 7.9.3 Transition to the Operator Equations.- 7.9.4 Transformation of System (7.9.17); Properties of the Operators of the Problem.- 7.9.5 The Basic Theorem.- 7.9.6 Asymptotic Form for High Viscosity.- 7.10 Reference Indications for Part II.- 7.10.1 Oscillations of an Ideal Liquid.- 7.10.2 Oscillations of a Viscous Liquid.- III Convection.- 8. Convection in a Self-Gravitating Liquid.- 8.1 Introduction.- 8.2 Formulation of the Problem.- 8.2.1 Equilibrium Conditions.- 8.2.2 Equations of Free Convection.- 8.2.3 Operator Equations.- 8.3 Stability Boundary for a Spherical Layer with Rigid Walls.- 8.3.1 Separation of Variables.- 8.3.2 Properties of Kernels Gk(r, s).- 8.3.3 Properties of Eigenvalues.- 8.3.4 Methods of Finding the Critical Rayleigh Numbers.- 8.3.5 Convection in a Spherical Vessel.- 8.3.6 Convection in a Thin Layer.- 8.4 Spherical Layer with a Free Outer Surface.- 8.4.1 Boundary-Value Problem.- 8.4.2 Reduction to an Integral Equation.- 8.4.3 Properties of Kernels.- 8.4.4 Properties of Eigenvalues.- 8.4.5 Thin Spherical Layer.- 8.4.6 Layer with an Inner Free Surface.- 8.5 Bifurcation of Solutions.- 8.5.1 Bifurcation Points.- 8.5.2 Bifurcation Equation in the Convection Problem.- 8.5.3 Application of the Method of Indeterminate Coefficients.- 8.5.4 Stability of Convective Motion.- 8.5.5 Calculation of Convection Flow in a Sphere.- 8.5.6 Direct Numerical Computation.- 8.5.7 Influence of Rotation.- 8.5.8 Some Additional Reference Commentaries to Chap. 8.- 9. Thermocapillary Convection.- 9.1 Introduction.- 9.1.1 Formulation of the Problem.- 9.1.2 Reference Commentaries.- 9.2 Stability Boundary.- 9.2.1 Spherical Layer.- 9.2.2 Thin Layer.- 9.2.3 A Bubble and a Drop.- 9.2.4 Rectangular Channel: Formulation of the Problem.- 9.2.5 Application of Galerkin's Method.- 9.2.6 Reference Commentaries.- 9.3 Convective Flows in a Spherical Layer After Loss of Stability.- 9.3.1 Formulation of the Problem.- 9.3.2 Difference Scheme.- 9.3.3 Flow Structure and Intensities.- 9.4 Forced Thermocapillary Flows in a Rectangular Channel.- 9.4.1 Formulation of the Problem; the Newton-Kantorovich Method.- 9.4.2 Structure of Convection.- 9.4.3 Effect of the Curvature of Free Surface.- 9.4.4 Reference Commentaries.- 9.5 Thermocapillary Motion in a Thin Liquid Layer.- 9.5.1 Formulation of the Problem.- 9.5.2 Derivation of the Equation for a Thin Layer.- 9.5.3 Steady-State Solution.- 9.5.4 Limiting Regimes of Spreading of a Drop.- 9.5.5 Stability of a Layer of Constant Thickness.- 9.6 Nonlinear Thermocapillary Convection in Processes of Space Technology (Reference Commentaries).- 9.6.1 Effect of Convection on Preparation of New Materials Under Zero-Gravity Conditions.- 9.6.2 Migration of a Bubble and a Drop.- References.- References to Preface and Introduction.- References to Part I (Chapters 2-4).- Supplementary References.- References to Part II (Chapters 5-7).- Supplementary References.- References to Part III (Chapters 8-9).- Supplementary References.