Hydrostatic Equilibrium - Mathematical Consideration

Mathematical Consideration

Newton's laws of motion state that a volume of a fluid which is not in motion or which is in a state of constant velocity must have zero net force on it. This means the sum of the forces in a given direction must be opposed by an equal sum of forces in the opposite direction. This force balance is called the hydrostatic balance.

We can split the fluid into a large number of cuboid volume elements. By considering just one element, we can work out what happens to the fluid as a whole.

There are 3 forces: the force downwards onto the top of the cuboid from the pressure, P, of the fluid above it is, from the definition of pressure,

Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is

Finally, the weight of the volume element causes a force downwards. If the density is ρ, the volume is V and g the standard gravity, then:

The volume of this cuboid is equal to the area of the top or bottom, times the height — the formula for finding the volume of a cube.

By balancing these forces, the total force on the fluid is

This sum equals zero if the fluid's velocity is constant. Dividing by A,

Or,

P_top − P_bottom is a change in pressure, and h is the height of the volume element – a change in the distance above the ground. By saying these changes are infinitesimally small, the equation can be written in differential form.

Density changes with pressure, and gravity changes with height, so the equation would be:

Note finally that this last equation can be derived by solving the three-dimensional Navier-Stokes equations for the equilibrium situation where

Then the only non-trivial equation is the -equation, which now reads

Thus, hydrostatic balance can be regarded as a particularly simple equilibrium solution of the Navier-Stokes equations.