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Buoyancy
Consider the air/water system described in the previous section.
Let
be some volume, bounded by a closed surface
, that straddles the plane
, and is thus partially occupied by water, and partially
by air. The
component of the net force acting on the fluid (i.e., either water or air) contained within
is written (see Section 1.3)

(2.5) 
where

(2.6) 
is the stress tensor for a static fluid (see Section 1.5), and

(2.7) 
the gravitational force density. (Recall that the indices
,
, and
refer to the
,
, and
axes, respectively. Thus,
, et cetera.)
The first term on the righthand side of Equation (2.5) represents the net surface force acting across
, whereas the second term represents the net volume force distributed throughout
.
Making use of the tensor divergence theorem (see Section B.4), Equations (2.5)(2.7)
yield the following expression for the net force:

(2.8) 
where

(2.9) 
and
Here,
is the net surface force, and
the net volume force.
It follows from Equations (2.4) and (2.9) that

(2.12) 
where
.
Here,
is the volume of that part of
which lies below the waterline, and
the
total mass of water contained within
.
Moreover, from Equations (2.2), (2.10), and (2.11),

(2.13) 
It can be seen that the net surface force,
, is directed vertically upward, and exactly balances the
net volume force,
, which is directed vertically downward. Of course,
is the weight of the water contained
within
. On the other hand,
, which is generally known as the buoyancy force,
is the resultant pressure of the water immediately surrounding
. We conclude that, in equilibrium, the net buoyancy force acting across
exactly balances the weight of the water inside
, so that the total force acting on the contents of
is zero, as
must be the case for a system in mechanical equilibrium. We can also deduce that
the line of action of
(which is vertical) passes through the center of gravity of the water inside
. Otherwise, a net torque would act on the contents of
, which would contradict our assumption that the system is in
mechanical equilibrium.
Next: Equilibrium of Floating Bodies
Up: Hydrostatics
Previous: Hydrostatic Pressure
Richard Fitzpatrick
20160331