Introduction to Hydrostatics
- The simplified Navier Stokes equation for hydrostatics is
a vector equation, which can be split into three components. The
convention will be adopted that gravity always acts in the negative
z direction. Thus,
and the three components of the hydrostatics equation reduce to
Since pressure is now only a function of z, total derivatives
can be used for the z-component instead of partial derivatives.
In fact, this equation can be integrated directly from some point
1 to some point 2. Assuming both density and gravity remain nearly
constant from 1 to 2 (a reasonable approximation unless there
is a huge elevation difference between points 1 and 2), the z-component
Another form of this equation, which is
much easier to remember is
This is the only hydrostatics equation needed. It is easily remembered
by thinking about scuba diving. As a diver goes down, the pressure
on his ears increases. So, the pressure "below" is greater
than the pressure "above."
Some "rules" to remember about hydrostatics
- Recall, for hydrostatics, pressure can be found from the simple
- There are several "rules" or comments which directly
result from the above equation:
- If you can draw a continuous line through the same fluid
from point 1 to point 2, then p1 = p2 if
z1 = z2.
For example, consider the oddly shaped container below:
By this rule, p1 = p2 and p4 =
p5 since these points are at the same elevation in
the same fluid. However, p2 does not equal p3
even though they are at the same elevation, because one cannot
draw a line connecting these points through the same fluid.
In fact, p2 is less than p3 since
mercury is denser than water.
- Any free surface open to the atmosphere has atmospheric
(This rule holds not only for hydrostatics, by the way, but for
any free surface exposed to the atmosphere, whether that
surface is moving, stationary, flat, or curved.) Consider the
hydrostatics example of a container of water:
The little upside-down triangle indicates a free surface, and
means that the pressure there is atmospheric pressure, pa.
In other words, in this example, p1 = pa.
To find the pressure at point 2, our hydrostatics equation is
- In most practical problems, atmospheric pressure is
assumed to be constant at all elevations (unless the change
in elevation is extremely large).
Consider the example below, in which water is pumped from one
large reservoir to another, as indicated:
Again, the little upside-down triangle indicates a free surface,
and means that the pressure there is atmospheric pressure, pa.
In other words, in this example, p1 = pa
and p2 = pa. But since point 2 is higher
in elevation than point 1, the local atmospheric pressure at 2
is a little lower than that at point 1. To be precise, our hydrostatics
equation must be used to account for the difference in elevation
between points 1 and 2:
However, since the density of the water in the problem is so much
greater than that of the air, it is common to ignore the difference
between p1 and p2, and call them both the
same value of atmospheric pressure, pa.
- The shape of a container does not matter in hydrostatics.
(Except of course for very small diameter tubes, where surface
tension becomes important.)
Consider the three containers in the figure below:
At first glance, it may seem that the pressure at point 3 would
be greater than that at point 1 or 2, since the weight of the
water is more "concentrated" on the small area at the
bottom, but in reality, all three pressures are identical. Use
of our hydrostatics equation confirms this conclusion, i.e.
In all three cases, the thin column of water above the point in
question at the bottom is identical. Pressure is a force per unit
area, and over a small area at the bottom, that force is due to
the weight of the water above it, which is the same in all three
cases, regardless of the container shape.
- Pressure is constant across a flat fluid-fluid interface.
For example, consider the container in the figure below, which
is partially filled with mercury, and partially with water:
In this case, our hydrostatics equation must be used twice, once
in each of the liquids.
Note that if the interface is not flat, but curved, there will
be a pressure difference across that interface. For example, consider
the junction of an air-water interface with a vertical wall. Due
to surface tension, the water creeps up the wall, causing the
interface to be curved. The pressure at the top of the interface
is atmospheric, but the pressure just below the curved portion
of the interface is less than atmospheric, due to surface tension
in the interface.