2.6 Forces Imposed by Fluid Flow
67
Until now we have discussed the inertia force on a body far away from a
solid surface, under the assumption of a perfect fluid. Experiments as well as
theoretical analysis showed that the closer the body to a solid boundary, the
higher the inertia coefficient. For example, Wilson and Reid (1963) reported
that the Crn. coefficient for a pipeline seated on the sea floor reached a value
of 3.29. Massel and Done (1993) in their analysis of hard coral stability under
cyclone induced waves theoretically showed that the inertia coefficient, Cm, for
spherical coral on the substratum is 51/32 ; : : : : ; ; 1.59, which can be compared with
the value of 1.5, corresponding to a sphere in infinite fluid.
2.6.5 Bodies Falling in Fluid
Organic and inorganic particles suspended in sea water are subjected to acceleration due to gravity and settle through the water column. Their vertical
velocity increases, but there is some limit to the velocity growth due to increasing drag. The faster the particle moves, the greater the drag acting on
the particle. At some velocity, the upward drag force becomes equal to the
downward force of the particle's weight, reduced by the buoyancy according to
Archimedes law (see Sect. 2.2). At this velocity there is no net force acting.
Therefore according to Newton's first law, acceleration ceases and the body
moves with a constant velocity. This highest velocity which a body reaches in
free fall is called the terminal velocity.
To demonstrate the balance of forces controlling the terminal velocity, we
will determine this velocity for a small sphere of diameter D (Fig. 2.30). At
terminal velocity, the balance of forces can be written as follows:
weight - buoyancy = drag.
(2.91 )
As the weight and buoyancy of sphere are (4/3)7r Psg(D /2)3 and (4/3)7r Pwg(D /2)3,
respectively, the left-hand side of Eq. (2.91) becomes:
weight - buoyancy = ~7r Pwg (:: _ 1) (~) 3,
(2.92)
in which Ps is the density of sphere. The drag force for a small sphere, moving
slowly in calm water, is accurately predicted by Stokes' equation (C.56), i.e.:
drag = 37rpwvDw,
(2.93)
where w is the terminal velocity. Equating drag and reducing weight gives the
following expression for terminal velocity:
w = (E!.. _ 1) gD
2
Pw
18v'
(2.94)
67
Until now we have discussed the inertia force on a body far away from a
solid surface, under the assumption of a perfect fluid. Experiments as well as
theoretical analysis showed that the closer the body to a solid boundary, the
higher the inertia coefficient. For example, Wilson and Reid (1963) reported
that the Crn. coefficient for a pipeline seated on the sea floor reached a value
of 3.29. Massel and Done (1993) in their analysis of hard coral stability under
cyclone induced waves theoretically showed that the inertia coefficient, Cm, for
spherical coral on the substratum is 51/32 ; : : : : ; ; 1.59, which can be compared with
the value of 1.5, corresponding to a sphere in infinite fluid.
2.6.5 Bodies Falling in Fluid
Organic and inorganic particles suspended in sea water are subjected to acceleration due to gravity and settle through the water column. Their vertical
velocity increases, but there is some limit to the velocity growth due to increasing drag. The faster the particle moves, the greater the drag acting on
the particle. At some velocity, the upward drag force becomes equal to the
downward force of the particle's weight, reduced by the buoyancy according to
Archimedes law (see Sect. 2.2). At this velocity there is no net force acting.
Therefore according to Newton's first law, acceleration ceases and the body
moves with a constant velocity. This highest velocity which a body reaches in
free fall is called the terminal velocity.
To demonstrate the balance of forces controlling the terminal velocity, we
will determine this velocity for a small sphere of diameter D (Fig. 2.30). At
terminal velocity, the balance of forces can be written as follows:
weight - buoyancy = drag.
(2.91 )
As the weight and buoyancy of sphere are (4/3)7r Psg(D /2)3 and (4/3)7r Pwg(D /2)3,
respectively, the left-hand side of Eq. (2.91) becomes:
weight - buoyancy = ~7r Pwg (:: _ 1) (~) 3,
(2.92)
in which Ps is the density of sphere. The drag force for a small sphere, moving
slowly in calm water, is accurately predicted by Stokes' equation (C.56), i.e.:
drag = 37rpwvDw,
(2.93)
where w is the terminal velocity. Equating drag and reducing weight gives the
following expression for terminal velocity:
w = (E!.. _ 1) gD
2
Pw
18v'
(2.94)