Fluid Statics

Since fluids are almost always in motion, there is not all that much to fluid statics. Yay!

Really, the only bits are the hydrostatic equation (which is used as the basis of any fluid statics problem), pressure variation with depth, and forces on submerged surfaces – though these can be tricky.

The Hydrostatic Equation

The derivation of this equation is simple.

• Take a fluid particle, where the area at the top and bottom is A and the height is δz

• Taking the pressure at the bottom as P, we can work out the pressure at the top as P + the difference in pressure:

• Using F = PA, calculate the force on the bottom and top:

• The gravitational force of the fluid particle (the weight) is given as:

• Now, we can balance the forces on the fluid particle:

This is the Hydrostatic Equation, and integrating both sides gives the equation for change in pressure with change in height that you are probably familiar with:

Note that you are probably familiar with ΔP = ρgh. This is the same, h is just defined as (z₁-z₂), hence the negative sign disappears. This form is sometimes known as the integrated form.

It is important to note that in this derivation, density has been assumed to be constant. In reality, this is often not the case.

Gauge Pressure

In the derivation above, we have ignored atmospheric pressure. This is because we assume it acts the same on each surface, so cancels out.

Pressure that ignores atmospheric pressure is called gauge pressure, as it is the pressure a gauge will read (for example on a bicycle pump).

Manometry

A manometer is a device used to measure a pressure difference:

If we know the difference in heights of the fluid interfaces, l, we can calculate the pressure difference of P₁ and P₂, because the pressure at A (a point in height that we chose to define) must be the same in each side:

Combining these equations gives:

As you can see, the height l’ is irrelevant.

Hydrostatic Forces on Flat Surfaces

Since pressure varies with depth, the magnitude of the force on a submerged surface also varies with depth: it is known as a distributed force.

When combining fluid mechanics with solid mechanics, however, this is not helpful. Instead, we want to know the resultant force of the pressure on the surface. There are three things we need to know to fully define the resultant force:

1. The magnitude

2. The point of application

3. The direction

The direction is easy: pressure force always acts perpendicular to the surface. To find the magnitude and direction we use a particular form of two-dimensional integration: surface integrals.

We know that the pressure force on an infinitesimally small area is given by:

Integrating this on a surface gives the sum of the forces on the infinitesimally small areas: