# Advanced Fluid Mechanics

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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 of Advanced Fluid Mechanics compiles pertinent information that are introduced in the more advanced classes at the senior level and at the graduate level. “Advanced Fluid Mechanics” 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 (“Fundamentals”) 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.

## Key Features

- Offers detailed derivation of fundamental equations for better comprehension of more advanced mathematical analysis
- Provides groundwork for more advanced topics on boundary layer analysis, unsteady flow, turbulent modeling, and computational fluid dynamics
- Includes worked-out examples and end-of-chapter problems as well as a companion web site with sample computational programs and Solutions Manual

## Readership

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

## Table of Contents

- Chapter 1 - Fundamentals

1.1 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. Lagrange’s stream function for two-dimensional flows

b. Stream functions for three-dimensional flows,including Stokes stream function

1.5 Newton’s 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 flows

2.1 Inviscid flows

2.2 Irrotational flows and the velocity potential

a. Intersection of velocity potential lines

and streamlines in two dimensions

b. Basic two-dimensional irrotational flows

c. Hele-Shaw flows

d. Basic three-dimensional irrotational flows

e. Superposition and the method of images

f. Vortices near walls

g. Rankine half body

h. Rankine oval

i. Circular cylinder or sphere in a uniform stream

2.3 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 theorem

b. Maximum-minimum potential theorem

c. Maximum-minimum speed theorem

d. Kelvin’s minimum kinetic energy theorem

e. Maximum kinetic energy theorem

f. Uniqueness theorem

g. Kelvin’s persistence of circulation theorem

h. Weiss and Butler sphere theorems

Problems

Chapter 3 - Irrotational Two-Dimensional Flows

3.1 Complex variable theory applied to

two-dimensional irrotational flows

3.2 Flow past a circular cylinder with circulation

3.3 Flow past an elliptical cylinder with circulation

3.4 The Joukowski airfoil

3.5 Kármán-Trefftz and Jones-McWilliams airfoils 3.6 NACA airfoils

3.7 Lifting line theory

3.8 Kármán 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 waves

4.1 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 fluids

4.4 Waves in an accelerating container

4.5 Stability of a round jet

4.6 Local surface disturbance on a large body of fluid - Kelvin’s 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 absent

a. 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. Stokes’ first problem-impulsive motion of a plate

b. Stokes’ 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 Approximation

6.1 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 transformation

d. Crocco’s transformation

e. Mangler’s transformation for bodies of revolution

6.8 Boundary layers in rotating flows

Problems

Chapter 7 - Thermal Effects

7.1 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 jet

Problems

Chapter 8 - Low Reynolds number Flows

8.1 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 sphere

8.3 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 Oseen’s approximation for slow viscous flow

8.7 Resolution of the Stokes/Whitehead paradoxes

Problems

Chapter 9 - Flow stability

9.1 Linear stability theory of fluid flows

9.2 Thermal instability in a viscous fluid - Rayleigh-Bénard 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 turbulence

10.1 The why and the how of turbulence

10.2 Statistical approach - one point averaging

10.3 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 Dynamics

11.1 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 equations

Problems

Chapter 12 - Numerical solution of partial differential equations

12.1 Introduction

12.2 Relaxation methods

12.3 Surface singularities

12.4 One step methods

a. Forward time, centered space - explicit

b. Dufort-Frankel method - explicit

c. Crank-Nicholson method - implicit

d. Boundary layer equations - Crank-Nicholson

e. Boundary layer equations - hybrid method

f. Richardson extrapolation

g. Further choices for dealing with nonlinearities

h. Upwind differencing for convective acceleration terms

12.5 Multi-step, or alternating direction, methods

a. Alternating direction explicit (ADE) method

b.Alternating direction implicit (ADI) method

12.6 Method of characteristics

12.7 Leapfrog method - explicit

12.8 Lax-Wendroff method - explicit

12,9 MacCormack’s methods

a. MacCormack’s explicit method

b. MacCormack’s 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 coordinates

a. Cylindrical polar coordinates

b. Spherical polar coordinates

A8. Tensors in general coordinates

References

Index

## Product details

- No. of pages: 368
- Language: English
- Copyright: © Academic Press 2007
- Published: June 21, 2007
- Imprint: Academic Press
- Hardcover ISBN: 9780123708854
- eBook ISBN: 9780080549088

## About the Author

### William Graebel

Professor Emeritus William Graebel works in the Department of Mechanical Engineering at University of Michigan, USA.

#### Affiliations and Expertise

Professor Emeritus, Department of Mechanical Engineering, University of Michigan, USA