Definition of Pressure: Force, Work Done, & Mechanical Energy

[Greg Bernhardt]

Definition/Summary

Pressure is normal force per area, or work done per volume, or mechanical energy per volume (mechanical energy density).

Static pressure, $P$, in a fluid (a liquid or gas or plasma), is measured across a surface which moves with the flow. It is the same in all directions at any point (unless viscosity is significant at that point). It is usually simply called "pressure".

Dynamic pressure in a fluid is the macroscopic kinetic energy density, $\frac{1}{2}\,\rho\,v^2$.
Total pressure in a fluid is pressure (static pressure) plus dynamic pressure, $P\ +\ \frac{1}{2}\,\rho\,v^2$. It is the pressure measured across a stationary surface.

At any point in a mixture of gases, the pressure is equal to the sum of the partial pressures of the individual gases.

The SI unit of pressure is the pascal (Pa), equal to one joule per cubic metre (J/m³), or Newton per square metre (N/m²), or kilogram per metre per second squared (kg/m.s²).

Equations

Force = pressure times area:
$$\boldsymbol{F}\,=\,\int_SP\,\hat{\boldsymbol{n}}\,dA\ \ \ \ \ \ (F = PA\ \ \text{for constant pressure on a flat surface})$$
where $\hat{\boldsymbol{n}}$ is the unit vector normal (perpendicular) to the surface S

Pressure in a stationary liquid of density $\rho$ at depth $d$ below a surface exposed to atmospheric pressure $P_a$:
$$P\ =\ P_a\,+\,\rho g d$$
Bernoulli's equation along any streamline of a steady incompressible non-viscous flow:
$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ constant$$
Bernoulli's equation along any streamline of a steady non-viscous flow:
$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \rho\,\epsilon\ =\ constant$$
or:
$$\frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ +\ \text{enthalpy per unit mass}\ =\ constant$$

Extended explanation

If a pipe narrows, the fluid must flow faster, because of conservation of mass.

Since the energy is greater, the (static) pressure must be less, ultimately because of conservation of energy.

Dynamic pressure and Bernoulli's equation:

In fluid flow, we use measurements per volume or per mass. Density $\rho$ is mass per volume; energy density is energy per volume; and so on. So any ordinary dynamic equation should be convertible into a fluid dynamic equation by dividing everything by volume .

In particular, since work done per displaced volume is pressure, and since in steady non-viscous flow, energy minus work done per displaced volume is constant along any streamline, the ordinary equation for conservation of energy in a gravitational field,
$$\frac{1}{2} mv^2 + mgh + U = W + \mathrm{constant}$$
becomes Bernoulli's equation for steady non-viscous flow:
$$P + \frac{1}{2}\rho v^2 + \rho gh + \rho\epsilon = \mathrm{constant\ along\ any\ streamline}$$
In this equation, all four terms have dimensions of pressure. The first term is ordinary pressure (sometimes called static pressure); the second is kinetic energy density, usually called dynamic pressure; the third is gravitational potential energy density; and the fourth is internal energy density.

Atmospheric pressure:

For calculations involving a fluid, such as water, which is much denser than air, atmospheric pressure can be ignored, since it appears on both sides of the equation and can be taken to be constant, even at different heights. This is because the difference in pressure at different heights is $\Delta P = \rho_{\mathrm{fluid}}g\Delta h + \rho_{\mathrm{air}}g\Delta h$, so if the density of air is negligible compared with the density of the fluid, the difference in atmospheric pressure can be taken to be zero. This applies, for example, when calculating forces on the wall of a container and when calculating the speed of water exiting a hole.

Absolute pressure and gauge pressure:

Absolute pressure is another name for pressure, sometimes used to distinguish it from gauge pressure.

Gauge pressure is pressure minus atmospheric pressure. For example, the devices usually used for measuring tyre pressure measure gauge pressure.

Force on a surface:

Force = pressure times area, so for example:

The net force F_net on a flat vertical wall of a container of water — that is, the force resulting from water pressure inside minus atmospheric pressure outside — is the integral of the net force on each horizontal strip of width W and height dD at a depth of D below the surface:
$$F_\mathrm{net} = \int PW\,dD = \rho g \int WD\,dD$$

Speed of water exiting a hole:

If a hole is made in the side or bottom of a container of water at depth $D$ below the stationary top surface of the water, then the exit speed $v$ may be calculated by applying Bernoulli's equation along a streamline from the top surface (where the pressure is atmospheric pressure) to a point just outside the hole (where the pressure is also atmospheric pressure):
$$\frac{1}{2}\rho v^2 - \rho gD = 0$$

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[APhysicist]

To summarize, pressure is a measure of force per area or work done per volume. It can be static, dynamic, or total depending on the situation. It is measured in pascals and is subject to Bernoulli's equation for fluid flow involving non-viscous steady flows. Atmospheric pressure can be ignored for denser fluids such as water as it appears on both sides of the equation and can be taken to be constant. Force on a surface is pressure times area, and the speed of water exiting a hole can be calculated using Bernoulli's equation.