Equations of Motion for Incompressible Viscous Fluids: With Mixed Boundary Conditions

Description

This monograph explores the motion of incompressible fluids by presenting and incorporating various boundary conditions possible for real phenomena. The authors’ approach carefully walks readers through the development of fluid equations at the cutting edge of research, and the applications of a variety of boundary conditions to real-world problems. Special attention is paid to the equivalence between partial differential equations with a mixture of various boundary conditions and their corresponding variational problems, especially variational inequalities with one unknown. A self-contained approach is maintained throughout by first covering introductory topics, and then moving on to mixtures of boundary conditions, a thorough outline of the Navier-Stokes equations, an analysis of both the steady and non-steady Boussinesq system, and more. Equations of Motion for Incompressible Viscous Fluids is ideal for postgraduate students and researchers in the fields of fluid equations, numerical analysis, and mathematical modelling. Preface Contents 1 Miscellanea of Analysis 1.1 Banach Space, Fixed Point and Basics of Mapping 1.1.1 Banach Space 1.1.2 Fixed-Point Theorems 1.1.3 Basics of Mappings 1.2 Lebesgue Space, Convergence 1.2.1 Lebesgue Space 1.2.2 Convergence of Sequences of Functions 1.3 Sobolev Space 1.3.1 Definition of Sobolev Space 1.3.2 Density and Continuation 1.3.3 Imbedding 1.3.4 Trace 1.3.5 Some Inequalities 1.4 Space of Abstract Functions 1.4.1 Abstract Functions and Its Derivatives 1.4.2 Compactness 1.5 Operator Equations and Operator-Differential Equations 1.5.1 Monotone Operator Equation 1.5.2 Pseudo-Monotone Operator Equation 1.5.3 Operator-Differential Equations 1.6 Convex Functional 1.7 Some Elementary Inequalities References 2 Fluid Equations 2.1 Derivation of Equations for Fluid Motion 2.1.1 Navier-Stokes Equations 2.1.2 Equations of Motion for Fluid Under Consideration of Heat 2.2 Boundary Conditions for the Navier-Stokes Equations 2.2.1 Boundary Conditions on the Walls 2.2.2 Boundary Conditions on Symmetric Planes 2.2.3 Boundary Conditions on Inlets and Outlets 2.2.4 Outflow Boundary Conditions on Imaginary Boundary 2.2.5 Boundary Conditions on Free Surfaces 2.3 Bilinear Forms for Hydrodynamics 2.3.1 Bilinear Forms 2.3.2 Variational Formulations for Mixed Boundary Value Problems of the Navier-Stokes Equations 2.4 Bibliographical Remarks 2.4.1 Fluid Equations 2.4.2 Boundary Conditions of the Navier-Stokes Equations 2.4.3 Bilinear Forms for Hydrodynamics References 3 The Steady Navier-Stokes System 3.1 Properties on the Boundary Surfaces of Vector Fields 3.1.1 The Second Fundamental Form and Shape Operator of Surface 3.1.2 Properties on the Boundary Surface of Vector Fields 3.2 Variational Formulations of the Steady Problems 3.3 Existence of Solutions to the Steady Problems 3.4 Bibliographical Remark References 4 The Non-steady Navier-Stokes System 4.1 Existence of a Solution: The Case of Total Pressure 4.1.1 Problem and Variational Formulation 4.1.2 An Auxiliary Problem by Elliptic Regularization 4.1.3 Proof of the Existence of a Solution 4.1.4 The Stokes Problem 4.2 Existence and Uniqueness of Solutions: The Case of Static Pressure 4.2.1 Existence and Uniqueness of Solutions to Problem I 4.2.2 Existence and Uniqueness of Solutions to Problem II 4.2.3 Existence and Uniqueness of Solutions for Perturbed Data 4.3 Bibliographical Remarks References 5 The Steady Navier-Stokes System with Friction Boundary Conditions 5.1 Variational Formulations of Problems 5.1.1 Variational Formulation: The Case of Static Pressure 5.1.2 Variational Formulation: The Case of Total Pressure 5.1.3 Variational Formulation: The Stokes Problem 5.2 Solutions to Variational Inequalities 5.3 Existence and Uniqueness of Solutions to the Steady Navier-Stokes Problems 5.4 Bibliographical Remarks References 6 The Non-steady Navier-Stokes System with Friction Boundary Conditions 6.1 Variational Formulations of Problems 6.1.1 Variational Formulation: The Case of Total Pressure 6.1.2 Variational Formulation: The Case of Static Pressure 6.1.3 Variational Formulation: The Stokes Problem 6.2 The Existence and Uniqueness of Solutions to Variational Inequalities 6.3 Solutions to the Non-steady Navier-Stokes Problems 6.3.1 Existence of a Solution: The Case of Total Pressure 6.3.2 Existence of a Unique Solution: The Case of Static Pressure 6.3.3 Existence of a Unique Solution: The Stokes Problem 6.4 Bibliographical Remarks References 7 The Steady Boussinesq System 7.1 Problems and Variational Formulations 7.1.1 Variational Formulation: The Case of Static Pressure 7.1.2 Variational Formulation: The Case of Total Pressure 7.2 Existence and Uniqueness of Solutions: The Case of Static Pressure 7.2.1 Existence of a Solution to an Auxiliary Problem 7.2.2 Existence and Estimates of Solutions to the Approximate Problem 7.2.3 Existence and Uniqueness of a Solution 7.3 Existence of a Solution: The Case of Total Pressure 7.4 Bibliographical Remarks References 8 The Non-steady Boussinesq System 8.1 Problems and Assumptions 8.2 Variational Formulations for Problems 8.2.1 Variational Formulations: The Case of Static Pressure 8.2.2 Variational Formulations: The Case of Total Pressure 8.3 Existence and Uniqueness of Solutions: The Case of Static Pressure 8.3.1 Existence and Estimation of Solutions to an Approximate Problem 8.3.2 Existence and Uniqueness of a Solution 8.4 Existence of a Solution: The Case of Total Pressure 8.4.1 Existence of a Solution to an Approximate Problem 8.4.2 Existence of a Solution 8.5 Bibliographical Remarks References 9 The Steady Equations for Heat-Conducting Fluids 9.1 Problems and Assumptions 9.2 Variational Formulations for Problems 9.2.1 Variational Formulation: The Case of Static Pressure 9.2.2 Variational Formulation: The Case of Total Pressure 9.3 Existence and Uniqueness of Solutions: The Case of Static Pressure 9.3.1 Existence of a Solution to an Auxiliary Problem 9.3.2 A Priori Estimates of Solutions to the Auxiliary Problem 9.3.3 Passing to Limits 9.4 Existence of a Solution: The Case of Total Pressure 9.5 Bibliographical Remarks References 10 The Non-steady Equations for Heat-Conducting Fluids 10.1 Problem and Variational Formulation 10.1.1 Problem and Assumption 10.1.2 Variational Formulation for Problem 10.2 Existence of a Solution 10.2.1 Existence of a Solution to an Approximate Problem 10.2.2 Estimates of Solutions to the Approximate Problem 10.2.3 Passing to the Limit 10.3 Bibliographical Remarks References Appendix Index Index