Fluids

A fluid is a substance that continuously deforms under the action of a shear stress, no matter how small. Unlike solids, fluids have no fixed shape and conform to the shape of their container. Both gases and liquids are fluids.

Pressure is a scalar quantity defined as the ratio of the normal force exerted on a surface to the area of that surface: $P = F/A$. In the International System it is measured in pascals (Pa), where $1\,\text{Pa} = 1\,\text{N/m}^2$. Other common units include the atmosphere ($1\,\text{atm} = 101325\,\text{Pa}$), the millimeter of mercury ($1\,\text{mmHg} \approx 133.3\,\text{Pa}$), and the bar ($1\,\text{bar} = 10^5\,\text{Pa}$).

In fluids at rest, pressure is isotropic: it acts with equal intensity in all directions. At any point in the fluid, the force on a small surface element is always perpendicular to the surface and has the same magnitude regardless of orientation.

Difference between liquids and gases:

• Incompressible ($\rho = \text{const}$): density is constant. Volume does not change appreciably with pressure. Typical of liquids (water, oil, mercury).
• Compressible ($\rho$ varies with $P$ and $T$): density depends on pressure and temperature according to an equation of state. Typical of gases (air, steam).

Everyday examples:

• Dams: water pressure increases with depth, which is why dams are built thicker at the base.
• Underwater pressure: at 10 m depth total pressure is about 2 atm (1 atm atmospheric + 1 atm hydrostatic).
• Drinking with a straw: by sucking we lower the pressure in our mouth, and atmospheric pressure pushes the liquid up.
• Atmospheric pressure: decreases with altitude; aircraft cabins are pressurized to maintain a comfortable pressure.

Pascal's principle states that a pressure change applied to an incompressible fluid confined in a container is transmitted undiminished and instantaneously to every point in the fluid and to the container walls.

Hydraulic press: two connected cylinders of area $A_1$ and $A_2$, filled with fluid (usually oil). Applying $F_1$ on the small piston generates $P = F_1/A_1$, which is transmitted unchanged to the large piston, producing $F_2 = P \cdot A_2 = F_1 \cdot (A_2/A_1)$. The ratio $A_2/A_1$ is called the mechanical advantage and can be very large (tens or hundreds).

Energy conservation: $W_1 = F_1 d_1 = W_2 = F_2 d_2$. Since $F_2 > F_1$, it follows that $d_1 > d_2$: what is gained in force is lost in displacement.

Real-world applications:

• Hydraulic brakes: the force of the foot on the pedal is multiplied to actuate the brake pads.
• Car lifts: a small piston lifts vehicles weighing tons.
• Excavators and cranes: hydraulic arms that lift heavy loads.
• Dentist chairs and barber chairs: height adjustment using hydraulic systems.
• Earthmoving machinery: shovels and buckets operated by high-pressure hydraulic circuits.

Stevin's law (the fundamental law of hydrostatics) describes pressure in a fluid at rest under gravity. Considering a fluid column of density $\rho$ and height $h$: $$P(h) = P_0 + \rho g h$$ where $P_0$ is the pressure at the free surface (typically atmospheric pressure), $\rho$ is the fluid density, $g = 9.81$ m/s² is the gravitational acceleration, and $h$ is the depth from the surface.

Key properties:

• Pressure increases linearly with depth: $\Delta P = \rho g \Delta h$.
• It depends only on depth and fluid density, not on container shape (hydrostatic paradox).
• Two points at the same depth in a homogeneous fluid have the same pressure.

Communicating vessels: containers connected at the base reach the same liquid level regardless of shape, because the pressure at the base must be the same in all branches: $\rho_1 g h_1 = \rho_2 g h_2$. If the fluids differ, the heights are inversely proportional to the densities.

Real-world applications and examples:

• Dams: pressure grows with depth; dams have a triangular cross-section, wider at the base to withstand the greater pressure.
• Submarines: at 100 m depth pressure is $\approx 10^6$ Pa (10 atm). The hull must be extremely strong.
• Water towers: in water supply systems, the tower height determines the pressure in the pipes.
• Mercury barometer: the mercury column is supported by atmospheric pressure: $P_{atm} = \rho_{Hg} g h$.
• Blood pressure: the heart pumps blood generating a pressure that varies with body position relative to the heart.

Archimedes' principle: a body immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid: $$F_A = \rho_{fluid}\,g\,V_{submerged}$$
Physical origin: the buoyant force arises from the pressure difference between the lower and upper parts of the body (Stevin's law). The greater pressure at the bottom generates a net upward force.

Floating conditions:

• $\rho_{body} < \rho_{fluid}$: the body floats (wood on water).
• $\rho_{body} = \rho_{fluid}$: neutral equilibrium (suspended at any depth).
• $\rho_{body} > \rho_{fluid}$: the body sinks (iron in water).

For a floating body: $V_{subm}/V_{tot} = \rho_{body}/\rho_{fluid}$.
Real-world applications and examples:

• Ships: a steel ship floats because its average density (including air in the holds) is lower than that of water.
• Submarines: control submersion by filling or emptying ballast tanks with water.
• Hot air balloons: hot air has lower density than the surrounding air, generating an upward buoyant force.
• Hydrometers: instruments that measure liquid density using the flotation of a graduated float.
• Icebergs: about 90% of the volume is submerged ($\rho_{ice}=917$ kg/m³, $\rho_{seawater}=1025$ kg/m³).
• Apparent weight: a submerged object feels lighter; the difference between real weight and apparent weight is the buoyant force.