Hydrostatic pressure

Hydrostatic pressure is the pressure exerted by a fluid in equilibrium within the fluid due to its own weight. As we will see below, pressure increases with depth because there will be more fluid above and therefore the weight will increase too.

When a fluid is in hydrostatic equilibrium, the forces exerted by the fluid on the walls of its container must act perpendicular to the walls. If these forces had a component tangent to the walls, the fluid couldn’t be at rest (it would flow).

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Hydrostatics has many applications both in science and engineering. It allows studying many phenomena related to the storage, transport and usage of fluids; for instance, plate tectonics, or arterial pressure can be understood in terms of a fluid in hydrostatic equilibrium.

In order to simplify the calculations we will assume that the fluid is incompressible: its density does not depend on pressure.

Let’s consider a fluid contained in a vessel as shown in the figure below. For the fluid element shown in the figure to be at rest, the forces within the fluid must act normal to the imaginary surfaces isolating it from its surroundings:

Moreover, for the fluid element to be at rest the following equalities must be met:

In the figure below the fluid element is shown but this time forces are expressed in terms of pressure. The weight of the fluid element is shown too:

In hydrostatic equilibrium the net force acting on the fluid element along the vertical axis must be zero:

Where A is the area of both the top and bottom faces of the fluid element.

Now, isolating dp and using the definition of density:

where dy is the height of the fluid element.

The above expression can be used to calculate the hydrostatic pressure as a function of depth as well as the atmospheric pressure as a function of height. See related pages below to see how both expressions are obtained.

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