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This book gives an introduction to the dynamics of inviscid,
incompressible fluids. By omitting effects inherent to viscosity
and compressibility, the book examines fluid flow at its most
fundamental level, while retaining the essential nonlinear
processes that makes fluid mechanics of enduring interest to
physicists, mathematicians and engineers. The book is intended
as a text for a beginning graduate-level course on fluid
mechanics, which in many institutions of higher learning deals
principally, and in some cases entirely, with inviscid flows.
Unlike more classical texts on inviscid fluids, the book also
contains extensive coverage of vorticity transport phenomena in
both two- and three-dimensional spaces and of a variety of
computational methods for inviscid flows. The later chapters of
the book might also be used as a text for more advanced fluid
dynamics courses, for instance on vortex dynamics, or as a
professional reference.
The
first chapter discusses the importance of inviscid flows and
examines modifications to ideal flow behavior in real fluids
caused by viscous forces in high Reynolds number laminar and
turbulent flows. Chapter 2 provides background on vector and
tensor analysis, which is used extensively in the remainder of
the text. Chapters 3-5 give a rigorous introduction to the
continuum mechanics of fluid flows, where at this stage both
viscosity and compressibility are included in the governing
equations. Aspects of kinematics that are important to the study
of fluid motion are developed in Chapter 3, including flow
lines, stretching and rotation of fluid elements, and the
transport theorem. Chapter 4 introduces mass and momentum
conservation and derives local and control-volume forms of the
conservation laws. Incompressibility is introduced as a
constraint form of the compressible flow theory. Chapter 5 deals
with discontinuity surfaces, which may occur either internal to
a flow or along boundaries of the flow domain. A modified form
of the transport theorem is developed and used to derive
discontinuity jump conditions, which are applied to obtain
boundary conditions for different flows. The effects of surface
tension at a material interface are also discussed.
Inviscid
flows are typically solved using solution methods, both
analytical and computational, that are not based directly on the
continuity and momentum conservation equations, but rather on
theorems that are derived from these equations. These general
theorems are developed in Chapters 6-8. Chapter 6 derives vector
representation theorems that, when applied to the velocity
field, yield a relationship between the vorticity and rate of
dilatation fields and the velocity that these fields generate.
Chapter 7 develops the vorticity transport equation and examines
the various laws governing vorticity transport in two- and
three-dimensional flows. A discussion of how these laws are
modified for viscous fluids is given at the end of Chapter 7.
Chapter 8 covers theorems associated with the pressure field,
both with and without vorticity present.
There exist a number of solution methods for two-dimensional
inviscid flows that cannot be extended to three-dimensional
spaces. These two-dimensional solution methods are described in
Chapters 9-11. Chapter 9 examines the analogy between analytic
functions of a complex variable and potential flow solutions,
and shows how complex variable theory can be used to simplify
flow solutions and to transform one solution into other flow
solutions. Chapter 10 applies the complex variable analogy to
develop theorems for forces and moments acting on a body
immersed in a potential flow field. The dynamics of vortex
patches and sheets in two-dimensional flows is examined in
Chapter 11, including development of different computational
methods for two-dimensional vorticity transport.
Solutions methods for three-dimensional flows are covered in
Chapters 12-14; many of these methods can be applied to
two-dimensional flows as a special case. Chapter 12 examines
three-dimensional potential flows, including analytic solutions
for axisymmetric flows, an approximate solution method for flow
past slender bodies, and computational methods of the
boundary-integral type for general potential flows. Chapter 13
provides an in-depth discussion of axisymmetric vorticity
dynamics, including analytic solutions for vortex rings,
computational methods for vorticity transport in axisymmetric
flows, and approximate methods for solution of axisymmetric wave
propagation on a vortex core. Chapter 14 describes different
approximations used to model the motion of thin-core vortex
tubes.
The final two chapters are devoted to perturbations
of equilibrium solutions and ensuing stability issues. Chapter
15 covers the dynamics of waves motions on an internal fluid
interface and at a free surface. Solutions of the linear wave
equations are examined, and a computational method is described
for nonlinear wave motions. Chapter 16 covers stability of
inviscid flows, progressing from vortex systems governed by
ordinary differential equations, to interfacial wave
instabilities, and finally to instabilities of rotating flows
and shear flows.
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