# Advanced Fluid Mechanics

**By**

- William Graebel

Fluid mechanics is the study of how fluids behave and interact under various forces and in various applied situations, whether in liquid or gas state or both. The author compiles pertinent information that are introduced in the more advanced classes at the senior level and at the graduate level. âAdvanced Fluid Mechanicsâ courses typically cover a variety of topics involving fluids in various multiple states (phases), with both elastic and non-elastic qualities, and flowing in complex ways. This new text will integrate both the simple stages of fluid mechanics (âFundamentalsâ) with those involving more complex parameters, including Inviscid Flow in multi-dimensions, Viscous Flow and Turbulence, and a succinct introduction to Computational Fluid Dynamics. It will offer exceptional pedagogy, for both classroom use and self-instruction, including many worked-out examples, end-of-chapter problems, and actual computer programs that can be used to reinforce theory with real-world applications.Professional engineers as well as Physicists and Chemists working in the analysis of fluid behavior in complex systems will find the contents of this book useful.All manufacturing companies involved in any sort of systems that encompass fluids and fluid flow analysis (e.g., heat exchangers, air conditioning and refrigeration, chemical processes, etc.) or energy generation (steam boilers, turbines and internal combustion engines, jet propulsion systems, etc.), or fluid systems and fluid power (e.g., hydraulics, piping systems, and so on)will reap the benefits of this text.

**Audience**

Graduate-level students in Mechanical, Aerospace & Aeronautical, Chemical, Environmental and Biomechanical Engineering; Graduate-level students in Chemistry and Physics ; Professional engineers in mechanical, chemical, materials, environmental, and biomedical engineering; Physicists and Chemists working in the analysis of fluid behavior in complex systems

Hardbound, 368 Pages

Published: June 2007

Imprint: Academic Press

ISBN: 978-0-12-370885-4

## Contents

- Chapter 1 - Fundamentals1.1 Introduction 1.2 Velocity, acceleration and the material derivative 1.3 The local continuity equation 1.4 Path lines, stream lines and the stream function a. Lagrangeâs stream function for two-dimensional flows b. Stream functions for three-dimensional flows,including Stokes stream function 1.5 Newtonâs momentum equation 1.6 Stress 1.7 Rates of deformation 1.8 Constitutive relations for Newtonian fluids 1.9 Equations for Newtonian fluids 1.10 Boundary conditions 1.11 Vorticity and circulation 1.12 The vorticity equation 1.13 The work-energy equation 1.14 The first law of thermodynamics 1.15 Dimensionless parameters 1.16.Non-Newtonian fluids 1.17 Moving coordinate systems Problems Chapter 2 - Inviscid irrotational flows2.1 Inviscid flows 2.2 Irrotational flows and the velocity potential a. Intersection of velocity potential lines and streamlines in two dimensionsb. Basic two-dimensional irrotational flows c. Hele-Shaw flows d. Basic three-dimensional irrotational flows e. Superposition and the method of imagesf. Vortices near walls g. Rankine half body h. Rankine oval i. Circular cylinder or sphere in a uniform stream2.3 Singularity distribution methods a. Two and three-dimensional slender body theory b. Panel methods 2.4 Forces acting on a translating sphere 2.5 Added mass and the Lagally theorem 2.6 Theorems for irrotational flow a. Mean value and maximum modulus theoremb. Maximum-minimum potential theorem c. Maximum-minimum speed theorem d. Kelvinâs minimum kinetic energy theoreme. Maximum kinetic energy theorem f. Uniqueness theorem g. Kelvinâs persistence of circulation theorem h. Weiss and Butler sphere theoremsProblems Chapter 3 - Irrotational Two-Dimensional Flows3.1 Complex variable theory applied to two-dimensional irrotational flows 3.2 Flow past a circular cylinder with circulation3.3 Flow past an elliptical cylinder with circulation 3.4 The Joukowski airfoil 3.5 KÃ¡rmÃ¡n-Trefftz and Jones-McWilliams airfoils 3.6 NACA airfoils 3.7 Lifting line theory 3.8 KÃ¡rmÃ¡n vortex street 3.9 Conformal mapping and the Schwarz-Christoffel transformation 3.10 Cavity flows 3.11 Added mass and forces and moments for two-dimensional bodies Problems Chapter 4 - Surface and interfacial waves4.1 Linearized free surface wave theory a. Infinitely long channel b. Waves in a container of finite size 4.2 Group velocity 4.3 Waves at the interface of two dissimilar fluids4.4 Waves in an accelerating container 4.5 Stability of a round jet 4.6 Local surface disturbance on a large body of fluid - Kelvinâs ship wave 4.7 Shallow depth free surface waves - cnoidal and solitary waves 4.8 Ray theory of gravity waves for non-uniform depths Problems Chapter 5 - Exact solutions of the Navier-Stokes equations 5.1 Solutions to the steady-state Navier-Stokes equations when convective acceleration is absenta. Two-dimensional flow between parallel planes b. Poiseuille flow in a rectangular conduit c. Poiseuille flow in a round tube d. Poiseuille flow in tubes of arbitrarily shaped cross-section e. Couette flow between circular cylinders 5.2 Unsteady flows when convective acceleration is absent a. Stokesâ first problem-impulsive motion of a plate b. Stokesâ second problem-oscillation of a plate 5.3 Other unsteady flows when convective acceleration is absent a. Impulsive plane Poiseuille and Couette flows b. Impulsive circular Couette flow 5.4 Steady flows when convective acceleration is present. a. Plane stagnation point flow b. Three-dimensional stagnation point flow c. Flow into convergent or divergent channels d. Flow in a spiral channel e. Flow due to a round laminar jet f. Flow due to a rotating disk Problems Chapter 6 - The Boundary Layer Approximation6.1 Introduction to boundary layers 6.2 The boundary layer equations 6.3 Boundary layer thickness 6.4 Falkner-Skan solutions for flow past a wedge a. Boundary layer on a flat plate b. Stagnation point boundary layer flow c. General case 6.5 The integral form of the boundary layer equation 6.6 Axisymmetric laminar jet 6.7 Flow separation 6.8 Transformations for non-similar boundary layer solutions a. Falkner transformation b. von Mises transformation c. Combined Mises-Falkner transformationd. Croccoâs transformation e. Manglerâs transformation for bodies of revolution6.8 Boundary layers in rotating flows Problems Chapter 7 - Thermal Effects7.1 Thermal boundary layers 7.2 Forced convection on a horizontal flat plate a. Falkner-Skan wedge thermal boundary layer b. Isothermal flat plate c. Flat plate with constant heat flux 7.3 The integral method for thermal convection a. Flat plate with a constant temperature region b. Flat plate with constant heat flux 7.4 Heat transfer near the stagnation point of an isothermal cylinder 7.5 Natural convection on an isothermal vertical plate 7.6 Natural convection on a vertical plate with uniform heat flux 7.7 Thermal boundary layer on inclined flat plates 7.8 Integral method for natural convection on an isothermal vertical plate 7.9 Temperature distribution in an axisymmetric jetProblems Chapter 8 - Low Reynolds number Flows8.1 Stokes approximation 1. Doublet 2a. Stokeslet for steady flows 2b. Stokeslet for unsteady flows 3a. Rotlet for steady flows 3b. Rotlet for unsteady flows 8.2 Slow steady flow past a solid sphere8.3 Slow steady flow past a liquid sphere 8.4 Flow due to a sphere undergoing simple harmonic motion 8.5 General translation of a sphere 8.6 Oseenâs approximation for slow viscous flow 8.7 Resolution of the Stokes/Whitehead paradoxesProblems Chapter 9 - Flow stability9.1 Linear stability theory of fluid flows 9.2 Thermal instability in a viscous fluid - Rayleigh-BÃ©nard convection 9.3 Stability if flow between rotating circular cylinders - Couette-Taylor instability 9.4 Stability of plane flows Problems Chapter 10 - Turbulence and transition to turbulence10.1 The why and the how of turbulence 10.2 Statistical approach - one point averaging10.3 Zero-equation turbulent models 10.4 One-equation turbulent models 10.5 Two-equation turbulent models 10.6 Stress-equation models 10.7 Equations of motion in Fourier space 10.8 Quantum theory models 10.9 Large eddy models 10.10 Phenomenologic observations 10.11 Conclusions Chapter 11 - An Introduction To Computational Fluid Dynamics11.1 Introduction 11.2 Numerical calculus 11.3 Numerical integration of ordinary differential equations 11.4 The finite element method 11.5 Linear stability problems - invariant imbedding and Riccati methods 11.6 Errors, accuracy, and stiff equationsProblems Chapter 12 - Numerical solution of partial differential equations12.1 Introduction 12.2 Relaxation methods 12.3 Surface singularities 12.4 One step methods a. Forward time, centered space - explicitb. Dufort-Frankel method - explicit c. Crank-Nicholson method - implicit d. Boundary layer equations - Crank-Nicholson e. Boundary layer equations - hybrid methodf. Richardson extrapolation g. Further choices for dealing with nonlinearitiesh. Upwind differencing for convective acceleration terms 12.5 Multi-step, or alternating direction, methodsa. Alternating direction explicit (ADE) methodb.Alternating direction implicit (ADI) method12.6 Method of characteristics 12.7 Leapfrog method - explicit 12.8 Lax-Wendroff method - explicit 12,9 MacCormackâs methods a. MacCormackâs explicit method b. MacCormackâs implicit method 12.10 Discrete vortex methods (DVM) 12.11 Cloud in cell method (CIC) Problems Appendix - Mathematical aids A1. Vector differential calculus A2. Vector integral calculus A3. Fourier series and integrals A4. Solution of ordinary differential equations a. Method of Frobenius b. Mathieu equation c. Finding eigenvalues - the Riccatti method A5. Index notation A6. Tensors in Cartesian coordinates A7. Tensors in orthogonal curvilinear coordinatesa. Cylindrical polar coordinates b. Spherical polar coordinates A8. Tensors in general coordinates References Index