This textbook is a pedagogic introduction to a number of phenomena employing fluid mechanics. Beginning with basic concepts and conservation laws for neutral and charged fluids, the authors apply and develop them to understand aerodynamics, locomotion of micro-organisms, waves in air and water, shock waves, hydrodynamic and hydromagnetic instabilities, stars and black holes, blood flow in humans, and superfluids. The approach is to consider various striking topics on fluid mechanics, without losing necessary mathematical rigor. The book balances the qualitative explanations with formal treatment, in a compact manner. A special focus is given to the important and difficult subject of turbulence and the book ends with a discussion on turbulence in quantum fluids. The textbook is dotted by a number of illustrative examples, mostly from real life, and exercises.
The textbook is designed for a one semester course and addresses students at undergraduate and graduate level in physics or engineering, who want to research in the fields as diverse as aeronautics, meteorology, cosmology, biomechanics, and mathematical physics. It is requested knowledge of an undergraduate level course on mathematical methods to better understand the topics presented here.
Author(s): Sudhir Ranjan Jain, Bhooshan S. Paradkar, Shashikumar M. Chitre
Publisher: Springer
Year: 2023
Language: English
Pages: 283
City: Cham
Foreword
Preface
Acknowledgments
Contents
1 Introduction
1.1 Hydrodynamic Description: Validity
1.1.1 Streamlines, Streaklines, Pathlines
1.1.2 Torricelli's Law
1.2 Conservation Laws and Governing Equations
1.2.1 Continuity Equation
1.2.2 Boussinesq Approximation
1.2.3 Anelastic Approximation
1.2.4 Validity of the Approximations
1.3 Stress-Strain Relationship
1.4 Conservation of Momentum
1.5 Energy
2 Fluid Equations from Kinetic Theory
2.1 Continuum Approximation in Fluid Mechanics
2.2 Derivation of Conservation Laws of Fluid Mechanics
2.2.1 Mass Conservation Equation
2.2.2 Momentum Conservation Equation
2.2.3 Energy Conservation Equation
2.3 Closure of the Conservation Laws
2.3.1 Constitutive Relations for a Newtonian Fluid
2.3.2 Equations of State
2.4 Governing Equations for the Newtonian Fluid
2.4.1 Governing Equations for the Compressible Fluid
2.4.2 Governing Equations for an Incompressible Fluid
3 Vorticity
3.1 Vorticity Equation
3.1.1 Vorticity Equation for the Incompressible Fluid
3.2 Kelvin's Circulation Theorem
3.3 Charged Fluids and Magnetovorticity
4 Potential Flows in Two Dimensions
4.1 Use of Complex Analysis in Potential Flow Theory
4.1.1 Complex Potentials of the Form W(z) = C Zn
4.1.1.1 Construction of Uniform Flows
4.1.1.2 Flow Inside a Wedge
4.1.1.3 Flow over a Sharp Edge
4.1.1.4 Flow due to a Doublet
4.1.2 Complex Potentials of the Form W(z) = C logz
4.1.2.1 Line Source/Sink Flows
4.1.2.2 Line Vortex Flows
4.1.3 Potential Flow Past a Cylinder
4.1.4 Flow Past a Rotating Cylinder
4.2 Conformal Transformations
4.2.1 Joukowski Transformation
4.2.2 Joukowski Transformation for Flow Past an Ellipse
4.2.3 Joukowski Transformation for Flow over an Airfoil
4.3 Force on a Body Immerse in a Potential Flow
4.3.1 Force on Cylinder with Circulation Around It
4.3.2 D'Alembert's Paradox and Its Resolution
5 Viscous Flow
5.1 Poiseuille Flow
5.2 Flow of Tar Down an Inclined Plane
5.3 Stokes Problems
5.3.1 Stokes First Problem
5.3.2 Stokes Second Problem
6 Low Reynolds Number Flows
6.1 Reynolds Number and Its Significance
6.2 Stokes Flow (Re 1)
6.2.1 Axisymetric Stokes Flow
6.2.2 Stokes Drag on a Sphere
6.3 General Properties of Stokes Flow
6.3.1 Kinematic Reversibility
6.3.2 Reciprocal Theorem
6.3.3 Force and Torque on a Body of Arbitrary Shape Inside a Low Re Flow
6.4 Scallop Theorem for Locomotion in Micro-Organisms
6.4.1 Flagellar Locomotion
6.5 Low Reynolds Number Flows in Lubrication
7 Physiological Hydrodynamics
7.1 Basics: Blood Flow Along Arteries
7.2 Blood Flow, Pumped by Human Heart
7.2.1 Response of Arterial Walls to Pressure
7.2.2 Blood Flow in an Artery
8 Water Waves
8.1 Small-Amplitude Surface Gravity Waves
8.1.1 Linear Shallow-Water Waves
8.1.2 Deep-Water Waves
8.1.3 Effect of Surface Tension
8.2 Waves from a Local Pulsed Source
8.3 Cerenkov Emission and Kelvin Wakes
8.4 Analogy Between Shallow Water Waves and Ion-Acoustic Waves in Plasmas
8.5 Korteweg-de Vries Equation and Solitons
8.5.1 Periodic Waves, Solitons
8.5.2 Analytical Solution of KdV Equation
9 Magnetohydrodynamics
9.1 Equations of Motion
9.1.1 Ohm's Law for Completely Ionized Gas
9.1.2 Generalization of Ohm's Law for Weakly (Partially) Ionized Gas
9.2 Role of Maxwell Stresses in Magnetohydrodynamics
9.2.1 Lorentz Force
9.2.2 Hydromagnetic Waves
9.3 Force-Free Magnetic Fields and Beltrami Flows
9.3.1 Magnetic Field Configurations
9.3.2 Hydromagnetic Waves
9.4 Ferraro's Law of Isorotation
9.5 Magnetic Reconnections and Formation of Current Sheets
10 Tensor Virial Theorem and Applications
10.1 Neutron Stars
11 Stability Problems in Hydrodynamics and Hydromagnetics
11.1 Linear Stability Analysis
11.2 Normal Mode Analysis
11.3 Jeans Instability
11.3.1 Case of Uniform Rotation
11.4 Rayleigh-Taylor Instability
11.5 Kelvin-Helmholtz Instability
11.6 Rayleigh-Bènard Convection
11.7 Parker Instability and Magnetic Buoyancy
11.8 Various Wave Modes for an Ideal Atmosphere
12 Shock Waves
12.1 Conservation Laws and Rankine-Hugoniot Relations
12.2 Blast Waves
12.3 Structure of a Shock
12.3.1 Structure of Weak Shocks
12.3.2 Estimation of Shock Thickness
12.3.3 Velocity Profile Through a Weak Shock
13 Astrophysical Fluid Mechanics
13.1 Accretion onto Compact Objects
13.2 Magnetic Fields as Agents of Transporting Angular Momentum
13.3 Origin and Maintenance of Cosmic Magnetic Fields
13.4 Dynamo Theory
13.5 Understanding the Black Hole Evaporation Using Classical Fluid Flow
13.5.1 Fluid Dynamics and the Geometric Metric for Massless Scalar Field
14 An Introduction to Classical Turbulence
14.1 Taylor-Couette Flow: From Newton to Chandrasekhar
14.1.1 Stability of Couette Flow
14.1.1.1 Rayleigh's Criterion
14.2 Chaos and Turbulence
14.2.1 Lorenz Equations
14.3 Nonlinear Wave Interaction
14.4 Weak Wave Turbulence
14.5 Turbulent Heat Flow: Structures and Scaling
15 Superfluid Hydrodynamics and Quantum Turbulence
15.1 The Superfluidity of Helium II
15.2 Quantum Fluid Mechanics
15.2.1 Quantized Vortices
15.2.2 Dynamics of Quantized Vortex Rings
15.2.3 Uniformly Rotating He II
15.3 Quantum Turbulence
15.3.1 Landau Two-Fluid Model for a One-Component Fluid
15.3.2 Kinetic Theory
15.4 Comparing Classical and Quantum Turbulence
15.5 A Very Brief Summary of Some Experimental Methods
15.6 A Few Questions
Solutions
Problems of Chapter 1
Problems of Chapter 2
Problems of Chapter 3
Problems of Chapter 4
Problems of Chapter 5
Problems of Chapter 6
Problems of Chapter 7
Problems of Chapter 8
Problems of Chapter 9
Problems of Chapter 12
Problems of Chapter 14
References
Index
