Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 52

Pipes tapped into
main pipe to measure
pressure

Discharge,
Q in

Reservoir

p A

p B

p A > p B

Q in = Q out

dx

Discharge,
Q out

Pressure force gradient = -( p B - p A )/ dx
= -d p /d x (net force per unit volume)

Fig. 3.23 Pressure gradient in moving fluid.

3.5.4 Horizontal gradients of static pressure in
moving fluids

particular flow depth due to energy used up in overcoming
viscous and turbulent frictional resistance. Using the defi-
nition diagram we can see that because of the downstream
decrease of pressure there is a net positive force acting in
the x -direction of

Rather confusingly, the static pressure condition also refers
to moving fluid when the pressure is measured normal to
the flow direction (Fig. 3.23); a downstream pressure gra-
dient always exists. Pressure decreases downstream at a

d p /d x per unit fluid volume. We
return to the energy consequences of pressure changes in
moving fluids in Section 3.12.

3.6 Buoyancy force

We have seen that any mass, m , in a gravity field is acted
upon by gravity equal to the force, m g . This is not quite
the most general formulation of the situation for we can-
not always ignore the density of the ambient medium. We
must consider the magnitude of weight force acting upon
a mass when the mass is immersed, that is, we must meas-
ure the weight force of a pear underwater (Fig. 3.24). In
such cases both the mass of substance and gravity are con-
stant at a point in space and the weight force must only
depend upon the contrast in density between that of the
mass and of the ambient medium. In situations arising in
meteorology or oceanography, neighboring air or water
masses may have densities that vary only slightly
(Fig. 3.25) and the buoyancy must be taken into account.
On the other hand, it is usual to neglect the tiny density
differences between solid Earth material and air where the
ratio between density and typical silicate minerals making
up rocks is only 4

Archimedes principle tells us that A is acted upon by an
upthrust equal to the weight of ambient fluid displaced. This
is because of the vertical gradient in hydrostatic pressure
between the bottom and top of A . If the upthrust is less than
the weight of the immersed or partially immersed substance,
that is, density of A is greater than that of B , then descent
will occur; vice versa for ascent. Generally when we mix two
substances of contrasting density,

B , any motion, or
the lack of it, depends only upon the sign of the density con-
trast,

A and

, and not in any way upon the magnitude
of the masses involved. Three conditions are possible: neu-
tral buoyancy (

A B

gives rise to a net downward force causing descent of A ; pos-
itive buoyancy when negative

0); negative buoyancy when positive

gives rise to a net upward
force causing ascent of A . In each case the buoyant force per
unit volume of substance, F B , is given by the expression g .

The speed of any resultant motion due to this force depends
upon other properties of the fluids involved such as absolute
mass and viscosity. Examples of buoyancy in the oceans and
ocean lithosphere are given in Fig. 3.27.

10 4 . In problems involving sedimenta-
tion of such particle through air we may ignore buoyancy,
but not through water (Fig. 3.26).

3.6.1 Archimedes and the buoyant force 3.6.2 Reduced gravity

Generally, for any solid or fluid mass, A , that rests within
or partly within another ambient solid or liquid mass, B ,

From the above discussion, you can appreciate that any
mass partly or wholly immersed in an ambient medium of