14
1. Basic Concepts of Fluid Flow
rotv = O .
(1.35)
From this condition it follows that there exists a velocity potential @, such that
the velocity vector can be defined as v = -grad@. The continuity equation
for an incompressible flow, div v = 0, then becomes a Laplace equation for
the potential @:
div (grad @) = 0 .
(1.36)
The momentum equation can then be integrated to give the Bernoulli equation, an algebraic equation that can be solved once the potential is known.
Potential flows are therefore described by the scalar Laplace equation. The
latter cannot be solved analytically for arbitrary geometries, although there
are simple analytical solutions (uniform flow, source, sink, vortex), which
can also be combined to create more complicated flows e.g. flow around a
cylinder.
For each velocity potential @ one can also define the corresponding streamfunction 9. The velocity vectors are tangential to streamlines (lines of constant streamfunction); the streamlines are orthogonal to lines of constant
potential, so these families of lines form an orthogonal flow net.
Potential flows are important but not very realistic. For example, the potential theory leads t o D'Alembert7s paradox, i.e. a body experiences neither
drag nor lift in a potential flow.
1.7.4 Creeping (Stokes) Flow
When the flow velocity is very small, the fluid is very viscous, or the geometric
dimensions are very small (i.e. when the Reynolds number is small), the
convective (inertial) terms in the Navier-Stokes equations play a minor role
and can be neglected (see the dimensionless form of the momentum equation,
Eq. (1.30)). The flow is then dominated by the viscous, pressure, and body
forces and is called creeping flow. If the fluid properties can be considered
constant, the momentum equations become linear; they are usually called
Stokes equations. Due to the low velocities the unsteady term can also be
neglected, a substantial simplification. The continuity equation is identical
to Eq. (1.32), while the momentum equations become:
1
div ( p grad ~ i )
- - div (p ii) + bi = 0 .
P
(1.37)
Creeping flows are found in porous media, coating technology, micro-devices
etc.
1.7.5 Boussinesq Approximation
In flows accompanied by heat transfer, the fluid properties are normally functions of temperature. The variations may be small and yet be the cause of the
1. Basic Concepts of Fluid Flow
rotv = O .
(1.35)
From this condition it follows that there exists a velocity potential @, such that
the velocity vector can be defined as v = -grad@. The continuity equation
for an incompressible flow, div v = 0, then becomes a Laplace equation for
the potential @:
div (grad @) = 0 .
(1.36)
The momentum equation can then be integrated to give the Bernoulli equation, an algebraic equation that can be solved once the potential is known.
Potential flows are therefore described by the scalar Laplace equation. The
latter cannot be solved analytically for arbitrary geometries, although there
are simple analytical solutions (uniform flow, source, sink, vortex), which
can also be combined to create more complicated flows e.g. flow around a
cylinder.
For each velocity potential @ one can also define the corresponding streamfunction 9. The velocity vectors are tangential to streamlines (lines of constant streamfunction); the streamlines are orthogonal to lines of constant
potential, so these families of lines form an orthogonal flow net.
Potential flows are important but not very realistic. For example, the potential theory leads t o D'Alembert7s paradox, i.e. a body experiences neither
drag nor lift in a potential flow.
1.7.4 Creeping (Stokes) Flow
When the flow velocity is very small, the fluid is very viscous, or the geometric
dimensions are very small (i.e. when the Reynolds number is small), the
convective (inertial) terms in the Navier-Stokes equations play a minor role
and can be neglected (see the dimensionless form of the momentum equation,
Eq. (1.30)). The flow is then dominated by the viscous, pressure, and body
forces and is called creeping flow. If the fluid properties can be considered
constant, the momentum equations become linear; they are usually called
Stokes equations. Due to the low velocities the unsteady term can also be
neglected, a substantial simplification. The continuity equation is identical
to Eq. (1.32), while the momentum equations become:
1
div ( p grad ~ i )
- - div (p ii) + bi = 0 .
P
(1.37)
Creeping flows are found in porous media, coating technology, micro-devices
etc.
1.7.5 Boussinesq Approximation
In flows accompanied by heat transfer, the fluid properties are normally functions of temperature. The variations may be small and yet be the cause of the