An Introduction to the Mechanics of Incompressible Fluids

Description

This open access book allows the reader to grasp the main bulk of fluid flow problems at a brisk pace. Starting with the basic concepts of conservation laws developed using continuum mechanics, the incompressibility of a fluid is explained and modeled, leading to the famous Navier-Stokes equation that governs the dynamics of fluids. Some exact solutions for transient and steady-state cases in Cartesian and axisymmetric coordinates are proposed. A particular set of examples is associated with creeping or Stokes flows, where viscosity is the dominant physical phenomenon. Irrotational flows are treated by introducing complex variables. The use of the conformal mapping and the Joukowski transformation allows the treatment of the flow around an airfoil.  The boundary layer theory corrects the earlier approach with the Prandtl equations, their solution for the case of a flat plate, and the von Karman integral equation. The instability of fluid flows is studied for parallel flows using the Orr-Sommerfeld equation. The stability of a circular Couette flow is also described. The book ends with the modeling of turbulence by the Reynolds-averaged Navier-Stokes equations and large-eddy simulations. Each chapter includes useful practice problems and their solutions.   The book is useful for engineers, physicists, and scientists interested in the fascinating field of fluid mechanics. Foreword Preface Acknowledgments Contents 1 Incompressible Newtonian Fluid Mechanics 1.1 Introduction 1.1.1 Circular Couette Flow 1.1.2 Flow Around a Cylinder 1.2 Fluid Kinematics 1.2.1 Material and Spatial Descriptions 1.2.2 Velocity, Material Derivative and Acceleration 1.2.3 Jacobian 1.2.4 Reynolds Transport Theorem 1.3 Velocity Gradient and Associated Tensors 1.4 Mass Conservation 1.5 Equation of Motion 1.6 Equation of Energy 1.7 Constitutive and State Equations 1.8 Incompressible Navier–Stokes Equations 1.9 Boundary and Initial Conditions 1.9.1 No Slip Wall 1.9.2 Interface 1.9.3 Laminar Free Surface 1.9.4 Perfect Fluid 1.10 Thermodynamics Considerations and Incompressibility 1.10.1 Compressible Fluid and Compressible Navier–Stokes Equations 1.10.2 Incompressibility 1.10.3 Boussinesq Approximation for Weakly Dilatable Fluids 1.11 The Method of Control Volume Exercises 3 Exact Solutions of the Navier–Stokes Equations 3.1 Plane Stationary Flows 3.1.1 Plane Couette Flow 3.1.2 Plane Poiseuille Flow 3.1.3 Flow of an Incompressible Fluid on an Inclined Plane 3.2 Axisymmetric Stationary Flows 3.2.1 Circular Couette Flow 3.2.2 Circular Poiseuille Flow in a Cylindrical Pipe 3.2.3 Helical Flow Between Two Circular Cylinders in Relative Motion 3.3 Plane Transient Flows 3.3.1 Transient Flow in a Semi-infinite Space 3.3.2 Flow on an Oscillating Plane 3.3.3 Channel Flow with a Pulsatile Pressure Gradient 3.4 Axisymmetric Transient Flows 3.4.1 Starting Transient Poiseuille Flow 3.4.2 Pulsating Flow in a Circular Pipe 3.5 Plane Periodic Solutions 3.6 Pipe Flow 3.6.1 Polynomial solutions 3.6.2 The Rectangular Pipe Exercises 4 Vorticity and Vortex Kinematics 4.1 Kinematic Considerations 4.2 Dynamic Vorticity Equation 4.2.1 General Equation 4.2.2 Physical Interpretation of Vorticity Dynamics for the Incompressible Perfect Fluid 4.2.3 The Vorticity Number 4.3 Vorticity Equation for a Viscous Newtonian Fluid 4.4 Circulation Equation 4.5 Vorticity Equation for a Perfect Fluid 4.6 Bernoulli's Equation 4.7 Vorticity Production on a Solid Wall 4.8 Flow Behind a Grid 4.9 Taylor–Green Vortex Exercises 5 Stokes Flow 5.1 Plane Creeping Flows 5.1.1 Flow in a Corner 5.2 Two-Dimensional Corner Moffatt Eddies 5.2.1 Real Solutions for lamdaλ (alpha greater than 73.15 Superscript degreesα>73.15°) 5.2.2 Complex Solutions for lamdaλ (alpha less than 73.15 Superscript degreesα