2
1. Basic Concepts of Fluid Flow
eventually produces a more random type of flow that is called turbulent; the
process of laminar-turbulent transition is an important area in its own right.
Finally, the ratio of the flow speed to the speed of sound in the fluid (the
Mach number) determines whether exchange between kinetic energy of the
motion and internal degrees of freedom needs to be considered. For small
Mach numbers, Ma < 0.3, the flow may be considered incompressible; otherwise, it is compressible. If Ma < 1, the flow is called subsonic; when Ma > 1,
the flow is supersonic and shock waves are possible. Finally, for Ma > 5 , the
compression may create high enough temperatures to change the chemical
nature of the fluid; such flows are called hypersonic. These distinctions affect
the mathematical nature of the problem and therefore the solution method.
Note that we call the flow compressible or incompressible depending on the
Mach number, even though compressibility is a property of the fluid. This
is common terminology since the flow of a compressible fluid at low Mach
number is essentially incompressible.
In many flows, the effects of viscosity are important only near walls, so
that the flow in the largest part of the domain can be considered as inviscid.
In the fluids we treat in this book, Newton's law of viscosity is a good approximation and it will be used exclusively. Fluids obeying Newton's law are
called Newtonian; non-Newtonian fluids are important for some engineering
applications but are not treated here.
Many other phenomena affect fluid flow. These include temperature differences which lead to heat transfer and density differences which give rise to
buoyancy. They, and differences in concentration of solutes, may affect flows
significantly or, even be the sole cause of the flow. Phase changes (boiling,
condensation, melting and freezing), when they occur, always lead to important modifications of the flow and give rise to multi-phase flow. Variation of
other properties such as viscosity, surface tension etc. may also play important role in determining the nature of the flow. With only a few exceptions,
these effects will not be considered in this book.
In this chapter the basic equations governing fluid flow and associated
phenomena will be presented in several forms: (i) a coordinate-free form,
which can be specialized to various coordinate systems, (ii) an integral form
for a finite control volume, which serves as starting point for an important
class of numerical methods, and (iii) a differential (tensor) form in a Cartesian
reference frame, which is the basis for another important approach. The basic
conservation principles and laws used to derive these equations will only
be briefly summarized here; more detailed derivations can be found in a
number of standard texts on fluid mechanics (e.g. Bird et al., 1962; Slattery,
1972; White, 1986). It is assumed that the reader is somewhat familiar with
the physics of fluid flow and related phenomena, so we shall concentrate on
techniques for the numerical solution of the governing equations.
1. Basic Concepts of Fluid Flow
eventually produces a more random type of flow that is called turbulent; the
process of laminar-turbulent transition is an important area in its own right.
Finally, the ratio of the flow speed to the speed of sound in the fluid (the
Mach number) determines whether exchange between kinetic energy of the
motion and internal degrees of freedom needs to be considered. For small
Mach numbers, Ma < 0.3, the flow may be considered incompressible; otherwise, it is compressible. If Ma < 1, the flow is called subsonic; when Ma > 1,
the flow is supersonic and shock waves are possible. Finally, for Ma > 5 , the
compression may create high enough temperatures to change the chemical
nature of the fluid; such flows are called hypersonic. These distinctions affect
the mathematical nature of the problem and therefore the solution method.
Note that we call the flow compressible or incompressible depending on the
Mach number, even though compressibility is a property of the fluid. This
is common terminology since the flow of a compressible fluid at low Mach
number is essentially incompressible.
In many flows, the effects of viscosity are important only near walls, so
that the flow in the largest part of the domain can be considered as inviscid.
In the fluids we treat in this book, Newton's law of viscosity is a good approximation and it will be used exclusively. Fluids obeying Newton's law are
called Newtonian; non-Newtonian fluids are important for some engineering
applications but are not treated here.
Many other phenomena affect fluid flow. These include temperature differences which lead to heat transfer and density differences which give rise to
buoyancy. They, and differences in concentration of solutes, may affect flows
significantly or, even be the sole cause of the flow. Phase changes (boiling,
condensation, melting and freezing), when they occur, always lead to important modifications of the flow and give rise to multi-phase flow. Variation of
other properties such as viscosity, surface tension etc. may also play important role in determining the nature of the flow. With only a few exceptions,
these effects will not be considered in this book.
In this chapter the basic equations governing fluid flow and associated
phenomena will be presented in several forms: (i) a coordinate-free form,
which can be specialized to various coordinate systems, (ii) an integral form
for a finite control volume, which serves as starting point for an important
class of numerical methods, and (iii) a differential (tensor) form in a Cartesian
reference frame, which is the basis for another important approach. The basic
conservation principles and laws used to derive these equations will only
be briefly summarized here; more detailed derivations can be found in a
number of standard texts on fluid mechanics (e.g. Bird et al., 1962; Slattery,
1972; White, 1986). It is assumed that the reader is somewhat familiar with
the physics of fluid flow and related phenomena, so we shall concentrate on
techniques for the numerical solution of the governing equations.