Physics for Geologists, Second edition

Fluids and fluid flow 117
and the forces acting on the two parallel sides are clearly equal and opposite
from the second proposition. Since pa = pb = p, is independent of the angle
a, which may arbitrarily be assigned a value and orientation, the pressure
at a point in a static liquid is equal in all directions. Note carefully that the
pressure on a submerged object is not equal in all directions.
(4) The horizontal force acting on a surface in a static fluid is the product
of the pressure and the vertical projection of the area Consider the prism
again. The force acting on side b, of unit area, is pb, the horizontal com-
ponent of which is pb sin a. But sin a is the area of side a, which is the area
of the projection of b onto a vertical surface. This proposition can also be
shown to be true for curved surfaces by considering very small areas and
their tangential slope.
The propositions above apply equally to the water filling the pore space of
particulate solids, such as sand. If we now fill the containers with sand, it is
clear that the water pressure in the pore spaces between the sand grains will
follow the same relationship (bearing in mind that the grains will displace
water and so raise the free upper surface). What about the pendular water?
The short answer is that there is no water that behaves as pendular water
in a single-phase liquid. It is the interfacial tension that makes the pendular
water immobile, and this does not exist in a single-phase liquid.
What pressure does the solid-water mixture exert on the bottom of the
container? and what about the solid component? What follows must be
regarded as thought experiments because in small containers the pressure
of particulate solids on the bottom does not increase once the thickness is
greater than twice the diameter of the container.
Clearly the weight of the water-saturated sand exerts a pressure on the
bottom of the container, and that total force is made up of a force due to
the water and a force due to the solids. The fractional porosity f is the ratio
of pore-water volume to total volume. The bulk weight is the sum of the
weights of the water and the solids, pbgv; and the bulk weight density is the
bulk weight divided by the bulk volume, pbg (=yb).
The total weight of the water-saturated sediment is made up of the weight
of solids
and the weight of the water in the pores
the sum of which is
Buoyancy reduces the effective weight of the solids by the weight of water
displaced. Should we not take this into account? No. If we do that we find
Copyright 2002 by Richard E. Chapman