In this section, we will study fluids, which is any substance which has the ability to flow. Typically, liquids and gases have this property and take the form of containers containing them. We can also define a fluid as a substance that, subject to a tangential force (shear), continuously deforms.

Fluids

First, it is interesting to consider the properties of fluids standing (static), an area known as hydrostatics. In this case, the following parameters are fundamental:

Density ( \(\rho\) )
The specific mass or density \(\rho\) of a body is defined as the mass present in a unit volume, that is: $$\rho = \frac{m}{V},$$ where \(m\) is the mass of the body and \(V\) volume. Note that knowing the volume of a body and what material it is made, we can calculate the mass of that material as $$m = \rho V.$$
Densities of different substances. The figure shows the same substance in containers of the same size. Note that one container has greater mass (molecules) than the other in the same space, and therefore, a higher density.

Note that:
  • The density unit in the \(IS\) is \(\frac{kg}{m^3}\) .
  • Density is a scalar.
  • Each substance has a density, which is one of its characteristics.
  • The density of a gas changes considerably with the pressure, but the variation of the density of a liquid with the pressure is negligible and, for practical purposes, it is considered that it does not change. That is, gases are compressible, but liquids are not.
The table below shows the density of some substances.
Substance
Specific mass ( \(kg/m^3\) )
Air ( \(0^oC, 1 atm\) )
\(1.21\)
Water ( \(20^oC, 1 atm\) )
\(1.0 \times 10^3\)
Ice
\(0.92 \times 10^3\)
Concrete
\(2.3 \times 10^3\)
Aluminum
\(2.7 \times 10^3\)
Iron
\(7.85 \times 10^3\)
Lead
\(11.3 \times 10^3\)
Earth: crust
\(2.8 \times 10^3\)
Earth: Core
\(9.5 \times 10^3\)
Pressure ( \(P\) )
The pressure in an area is nothing more than a normal force per unit area and is a scalar (see figure below).
Pressure in a fluid. The figure illustrates a fluid under the action of a force \(\vec{F}\). We can decompose \(\vec{F}\) in the normal direction of the liquid surface, \(\vec{F_n}\) and in the tangent direction of the surface, \(\vec{F_t}\). To calculate the pressure, only the normal component is used, \(\vec{F_n}\).
The pressure due to a force \(\vec{F}\) into an area \(A\) may be calculated with the formula: $$P = \frac{F_n}{A},$$ where \(F_n\) is the modulus of the normal surface component of the force \(\vec{F}\).
Note that:
  • The pressure \(P\) at the point of an equilibrium fluid is the same in all directions.
  • The pressure in the \(IS\) unit is newton per square meter, \(\frac{N}{m^2} = Pascal (Pa)\) .
  • The manometer is the instrument used to measure pressures in general. In a car, for example, there is a manometer to measure the pressure of the oil that lubricates the engine.
    The table below lists the pressure of some systems.
    System
    Value (Pascal)
    Sun Center
    \(2 \times 10^{16}\)
    Earth's center
    \(4 \times 10^{11}\)
    Largest ocean depth
    \(1.1 \times 10^8\)
    Automobile tire
    \(2 \times 10^5\)
    Air at sea level
    \(1.0 \times 10^5\)
    Normal blood
    \(1.6 \times 10^4\)
    Maximum tolerable sound
    \(30\)
    Minimum detectable sound
    \(3 \times 10^{-5}\)

Atmospheric Pressure

The atmospheric pressure occurs due to the weight of the air column that presses the systems in the vicinity of the surface of the Earth. The higher the air column the greater the pressure, which is why we feel the pressure change when we travel from high places to low regions and vice versa. At sea level, we have that the atmospheric pressure has the following value: $$1 atm = 1.01 \times 10^5 Pa = 760 mmHg = \\ = 760 Torr = 14.7 Lib/in^2 (psi)$$ Atmospheric pressure is measured with an instrument called a barometer (see figure).

The mercury barometer is an instrument that measures atmospheric pressure. It consists only of a capillary tube where one end is closed. This tube is then filled with mercury and this tube is turned into a container, as shown. In this way, the mercury height inside the tube is proportional to the atmospheric pressure on the container surface.