Pascal’s law, or the principle of transmission of fluid-pressure, is a principle in fluid mechanics that states that: pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure ratio (initial difference) remains the same. The law was established by French mathematician, Blaise Pascal.

Pascal’s principle is defined as:

A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid

This principle is stated mathematically as:

Pascal's Law

where:

Pascal's Law 2 is the hydrostatic pressure (given in pascals in the SI system), or the difference in pressure at two points within a fluid column, due to the weight of the fluid;

ρ is the fluid density (in kilograms per cubic meter in the SI system);

g is acceleration due to gravity (normally using the sea level acceleration due to Earth’s gravity in metres per second squared);

Pascal's Law 3 is the height of fluid above the point of measurement, or the difference in elevation between the two points within the fluid column (in metres in SI).

The explanation of this formula is that the change in pressure between two elevations is due to the weight of the fluid between the elevations. Note that the variation with height does not depend on any additional pressures. Therefore Pascal’s law can be interpreted as saying that any change in pressure applied at any given point of the fluid is transmitted undiminished throughout the fluid.

Pascal’s principle applies to all fluids, whether gases or liquids and underpins the operation of hydraulics, e.g. the hydraulic press.

A hydraulic press is a machine that uses a hydraulic cylinder to generate a compressive force. It uses the hydraulic equivalent of a mechanical lever, and was also known as a Bramah press after the English inventor, Joseph Bramah. He invented and was issued a patent on this press in 1795.

The hydraulic press depends on Pascal’s principle above: the pressure throughout a closed system is constant. One part of the system is a piston acting as a pump, with a modest mechanical force acting on a small cross-sectional area; the other part is a piston with a larger area which generates a correspondingly large mechanical force. Only small-diameter tubing (which resists pressure more easily) is needed if the pump is separated from the press cylinder.

Pascal’s law: Pressure on a confined fluid is transmitted undiminished and acts with equal force on equal areas and at 90 degrees to the container wall.

A fluid, such as oil, is displaced when either piston is pushed inward. The small piston, for a given distance of movement, displaces a smaller amount of volume than the large piston, which is proportional to the ratio of areas of the heads of the pistons. Therefore, the small piston must be moved a large distance to get the large piston to move significantly. The distance the large piston will move is the distance that the small piston is moved divided by the ratio of the areas of the heads of the pistons. This is how energy, in the form of work in this case, is conserved and the Law of Conservation of Energy is satisfied. Work is force times distance, and since the force is increased on the larger piston, the distance the force is applied over must be decreased.

The pressurized fluid used, if not generated locally by a hand or mechanically-powered pump, can be obtained by opening a valve which is connected to a hydraulic accumulator or a continuously-running pump whose pressure is regulated by a relief valve. When more force needs to be generated than the available pressure would allow, or there is a need to use smaller, higher-pressure cylinders to save size and weight, a hydraulic intensifier can be used to increase the pressure acting on the press cylinder.

When the pressure on the press cylinder is released (the fluid returning to a reservoir), the force created in the press is reduced to a low value (which depends on the friction of the cylinder’s seals. The main piston does not retract to its original position unless an additional mechanism is employed.