Mathematical Fluid Dynamics, Present and Future

Name: Mathematical Fluid Dynamics, Present and Future | Tokyo, Japan, November 2014
Brand: Springer
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Tokyo, Japan, November 2014

Yoshihiro Shibata Yukihito Suzuki(Editor)

Springer (Publisher)

Published on 1. December 2016

XIX, 613 pages

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Content

Intro
Preface
Acknowledgements
Contents
Contributors
Part I Multiphase Flows
1 Nonconvergence of the Capillary Stress Functional for Solutions of the Convective Cahn-Hilliard Equation
1.1 Introduction
1.2 Notation and Basic Assumptions
1.3 Nonconvergence Result
References
2 On the Interface Formation Model for Dynamic Triple Lines
2.1 Introduction
2.2 Integral Balances
2.3 Transport Theorems
2.4 Local Balances
2.5 Entropy Production and Closure Relations
2.6 Isothermal Case with Vanishing Triple Line Mass
2.7 Thermodynamical Consistency and Equilibria
References
3 Global Solvability of the Problem on Two-Phase Capillary Fluid Motion in the Oberbeck--Boussinesq Approximation
3.1 Statement of the Problem and the Main Result
3.2 An Energy Estimate of the Solution
3.3 Linearized Problems
3.4 Global Solvability of the Problem (3.1), (3.4), (3.3)
3.4.1 Conclusions
References
4 Stability of Steady Flow Past a Rotating Body
4.1 Motivation and Introduction
4.2 Auxiliary Results
4.3 The Main Theorem on Stability
References
5 Asymptotic Structure of Steady Stokes Flow Around a Rotating Obstacle in Two Dimensions
5.1 Introduction
5.2 Results
5.3 Fundamental Solution
5.4 Proof of Theorem 5.2.1
5.5 Proof of Theorem 5.2.2
References
6 Toward Understanding Global Flow Structure
6.1 Introduction
6.2 Phenomena
6.2.1 Localized Convection Patterns in Binary Fluid Convection
6.2.2 Localized Convection Patterns in Bioconvection
6.2.3 Surface Switching
6.3 Analysis Methods
6.3.1 Orbit Analysis Applying Covariant Lyapunov Analysis
6.3.2 Generating Cellular Automata Rule from Measurement Data Alone
6.4 Concluding Remarks
References
7 Mathematical and Numerical Analysis of the Rayleigh-Plesset and the Keller Equations
7.1 Introduction
7.2 Mathematical Models for Motion of a Spherical Bubble
7.2.1 The Rayleigh-Plesset Equation
7.2.2 The Rayleigh-Plesset-Keller Equation
7.3 Mathematical Analysis
7.4 A Hamiltonian Formulation of the Rayleigh-Plesset-Keller Equation
7.4.1 A Hamiltonian Formulation of the Rayleigh-Plesset Equation
7.4.2 A Hamiltonian Formulation of the Keller-Herring Equation
7.5 Discrete Gradient Schemes for the Rayleigh-Plesset and Keller Equations
7.6 Numerical Results
7.6.1 The Inviscid Rayleigh-Plesset Equation
7.6.2 The Keller Equation
7.7 Concluding Remarks
References
8 On the Amplitude Equation of Approximate Surface Waves on the Plasma-Vacuum Interface
8.1 Introduction
8.2 The Plasma-Vacuum Interface Problem
8.3 The Asymptotic Expansion
8.4 The First Order Equations
8.5 The Second Order Equations
8.5.1 The Second Order Equations in the Plasma Region
8.5.2 The Second Order Equations in Vacuum
8.5.3 The Second Order Jump Conditions
8.5.4 The Kernel
8.6 Noncanonical Variables and Well-Posedness
8.6.1 Well-Posedness of the Amplitude Equation
8.6.2 Regularity of the First Order Terms U(1),V(1)
References
9 On the mathcalR-Bounded Solution Operator and the Maximal Lp-Lq Regularity of the Stokes Equations With Free Boundary Condition
9.1 Introduction
9.2 Reduced Stokes Problem
9.2.1 Equivalence Between Stokes and Reduced Stokes Equations
9.2.2 On the mathcalR Bounded Solution Operators for the Reduced Stokes Problem
9.2.3 Time Dependent Reduced Stokes Equations
9.3 Reduced Stokes Equations in mathbbRN and mathbbRN+
9.3.1 Reduced Stokes Equations in mathbbRN
9.3.2 Reduced Stokes Equations in mathbbRN+ with Free Boundary Condition
9.3.3 Reduced Stokes Equations in mathbbRN+ with Non-slip Boundary Condition
9.4 On the mathcalR Bounded Solution Operators in a Bent Half-Space
9.4.1 Unit Outer Normal and Laplace-Beltrami Operator in a Bent-Half Space
9.4.2 Reduced Stokes Equations with Free Boundary Condition in a Bent-Half Space
9.4.3 Reduced Stokes Equations with Non-slip Boundary Condition in a Bent-Half Space
9.5 Proof of Theorem 9.2.1
9.5.1 Some Preparation for the Proof of Theorem 9.2.1
9.5.2 Local Solutions
9.5.3 Construction of a Parametrix
9.5.4 Representation of the Remainder Terms Vi(?)(f, f, fb)
9.5.5 Proof of Theorem 9.2.1
9.6 Proof of Theorem 9.2.2
9.6.1 Existence Part
9.6.2 The Weak Laplace Problem with Dirichlet Condition
9.6.3 Uniqueness Part
9.7 Proofs of Theorems 9.1.4 and 9.1.5
References
10 On the Solvability of Free Boundary Problem for Viscous Compressible Fluids in an Infinite Time Interval
10.1 Introduction
10.2 Linear Problem
10.3 Nonlinear Problem
References
11 Classical Solvability of the Two-Phase Radial Viscous Fingering Problem in a Hele-Shaw Cell
11.1 Introduction
11.2 Formulation of the Problem
11.3 Reformulation of the Problem
11.4 Linear Problem
11.5 Nonlinear Problem: Proof of Theorem 11.2.2
11.6 Passing to the Limit eto0: Proof of Theorem 11.2.1
References
12 Investigation of Bubble Clouds in a Cavitating Jet
12.1 Introduction
12.2 Experimental Observations
12.2.1 Experimental Apparatus
12.2.2 Instantaneous Photograph of Cavitating Jet and Visualization of Flow Around the Jet
12.2.3 Observation of Unsteady Behavior of Bubble Cloud Using Shadowgraph Technique and Schlieren Method with High-Speed Video Camera
12.2.4 Incidence Frequencies of Pressure Pulses Generated by Cavitating Jet
12.2.5 Erosion Test of Aluminum Specimen Using Cavitating Jet
12.3 Numerical Analysis
12.3.1 Continuum Model of Homogeneous Two-Phase Flow of Cavitating Jet
12.3.2 Computational Results of Cavitating Jet
12.3.3 Spherical Cloud Model of Cavitation Bubbles
12.3.4 Using Parameters for Flow Simulation of Cloud Cavitation
12.3.5 Results and Discussion on Pressure Pulse in Cavitation Cloud
12.3.6 Influence of Cloud Size and Downstream Pressure on the Impulsive Pressure
12.3.7 Concluding Remarks and Further Outlook
References
Part II Other Related Topics
13 Weak Solutions to Problems Involving Inviscid Fluids
13.1 Introduction
13.2 Abstract Problem, Main Result
13.2.1 Abstract Problem
13.2.2 Subsolutions
13.2.3 Main Result
13.3 Convex Integration
13.4 Oscillatory Lemma, Infinitely Many Solutions
13.5 Examples
13.5.1 Euler-Fourier System
13.5.2 Quantum Fluids
13.5.3 Binary Mixtures of Compressible Fluids
13.6 Continuity at the Initial Time, Admissible Solutions
13.6.1 Example, Dissipative Solutions to the Euler-Fourier System
References
14 Enstrophy Variations in the Incompressible 2D Euler Flows and a Point Vortex System
14.1 Introduction
14.2 The 2D Euler-a Equations and the aPV System
14.3 Enstrophy Variations in the aPV System
14.3.1 The aPV System
14.3.2 Collapse of a-Point Vortices and the Enstrophy Dissipation
14.4 Summary and Discussion
References
15 Thermodynamical Consistent Modeling and Analysis of Nematic Liquid Crystal Flows
15.1 Introduction
15.2 Thermodynamical Consistent Modeling
15.3 Thermodynamical Consistency and Stability
15.4 Analysis of the Non-isothermal Simplified Model
References
16 Statistical Mechanics of Quasi-geostrophic Vortices
16.1 Introduction
16.2 Quasi-geostrophic Approximation and the Equations of Motion of Point Vortices
16.3 Maximum Entropy Theory and the Three-Dimensional Mean Field Equation
16.4 Direct Numerical Simulation of the Point Vortex System
16.5 Maximum Entropy States
16.5.1 Three-Dimensional Maximum Entropy States in a Cubic Box
16.5.2 Influence of the Aspect Ratio of the Periodic Domain
16.6 Stability of the Maximum Entropy States
16.6.1 Arnold's Method
16.6.2 Numerical Simulation of Quasi-geostrophic Equation
16.7 Summary
References
17 Heat Convection of Compressible Viscous Fluids. III
17.1 Introduction
17.2 Formulation
17.3 Linearized System
References
18 Error Estimates of a Stabilized Lagrange--Galerkin Scheme of Second-Order in Time for the Navier--Stokes Equations
18.1 Introduction
18.2 A Lagrange--Galerkin Scheme of Second-Order in Time
18.3 Main Results
18.4 Proofs of Theorems1 and 2
18.4.1 Preparations
18.4.2 An Estimate at Each Time Step
18.4.3 Proof of Theorem1
18.4.4 Proof of Theorem2
18.5 Numerical Results
18.6 Conclusions
References
19 Chaotic Dynamics in an Integro-Differential Reaction-Diffusion System in the Presence of 0:1:2 Resonance
19.1 Introduction
19.1.1 Numerical Results
19.1.2 Main Results
19.2 Bifurcation Structure Around a 1-Mode Stationary Solutions
19.2.1 The Case When a4c4&0
19.2.2 The Case When a4c4&0
19.2.3 Case Study
19.3 Heteroclinic Cycles on a Blow-Up Vector Field
19.3.1 Transformation to the Dumortier-Kokubu's Normal Form
19.3.2 Heteroclinic Cycles in the Scaling Family
References
20 Linfty-Stability of Discontinuous Traveling Waves in a Radiating Gas Model
20.1 Introduction
20.2 Local Solvability
20.3 Asymptotic Stability
20.3.1 Energy Estimates Away from the Discontinuity
20.3.2 Energy Estimates over the Entire Domain
References
21 Mathematical Analysis and Numerical Simulations for a Model of Atherosclerosis
21.1 Introduction
21.2 Atherosclerosis Mathematical Modeling
21.3 Existence, Uniqueness and Boundedness of Solutions
21.3.1 Notations
21.3.2 Parabolic Problem with Nonlinear Boundary Conditions
21.3.3 Monotone Iterative Method
21.3.4 Existence of Upper and Lower Solutions
21.3.5 Existence-comparison Theorem
21.3.6 The Case of Linear Boundary Conditions
21.4 Numerical Simulations
21.5 Conclusions
References
22 Regularity for the Solution of a Stochastic Partial Differential Equation with the Fractional Laplacian
22.1 Introduction
22.2 Formulation of Our Problem
22.2.1 Notations
22.2.2 Definition of Solutions
22.2.3 Existence and Uniqueness of Solutions of (22.1)
22.3 Regularity of the Solution
References

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Mathematical Fluid Dynamics, Present and Future

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