💧Fluid Mechanics Unit 3 Review

Pressure is the force per unit area that a fluid exerts on surfaces. This single concept underpins the design of everything from dams and submarines to hydraulic brakes and water distribution systems. Here, you'll build from the basic definition of pressure through hydrostatic distributions and into the measurement conventions used in practice.

Pressure in Fluids

Pressure at a point in a fluid is defined as the force acting perpendicular to a surface divided by the area of that surface:

$P = \frac{F}{A}$

where $F$ is the normal force and $A$ is the area over which it acts. The SI unit is the Pascal (Pa), with $1 \text{ Pa} = 1 \text{ N/m}^2$ . For reference, standard atmospheric pressure at sea level is about 101,325 Pa.

A few properties worth remembering:

Pressure acts perpendicular to any surface in contact with the fluid. Water pushes straight into the wall of a swimming pool, not along it.
Pressure is a scalar quantity. It has magnitude but no inherent direction. The force that results from pressure does have a direction (always normal to the surface), but pressure itself is just a number at a point.
At any given point in a static fluid, pressure is the same in all directions. This is sometimes called Pascal's law and is the reason hydraulic systems work: pressure applied at one point transmits equally throughout the fluid.

Pressure in fluids, Pressure & Pascal’s Principle – TikZ.net

Hydrostatic Pressure Concept

When a fluid is at rest, the only thing generating pressure differences from point to point is the weight of the fluid above. This weight-driven pressure is called hydrostatic pressure.

In a static fluid, pressure increases linearly with depth. The governing relationship is:

$\frac{dP}{dz} = -\rho g$

where $\rho$ is the fluid density, $g$ is gravitational acceleration (9.81 m/s²), and $z$ is the vertical coordinate measured positive upward. The negative sign tells you that pressure decreases as you move upward (or equivalently, increases as you go deeper).

The key takeaway: pressure at a given depth depends on two things, the fluid density and the height of the fluid column above that point. Mercury ( $\rho \approx 13{,}546 \text{ kg/m}^3$ ) is about 13.6 times denser than water, so a mercury column only ~0.76 m tall exerts the same pressure as a water column about 10.3 m tall. This is exactly why mercury barometers are compact enough to sit on a desk.

Pressure in fluids, Pascal’s Principle and Hydraulics – University Physics Volume 1

Hydrostatic Pressure Applications

Using the Hydrostatic Equation

Integrating the differential relation for a fluid of constant density gives the equation you'll use most often:

$P = P_0 + \rho g h$

$P_0$ is the pressure at the reference level (often atmospheric pressure at the free surface).
$h$ is the depth below that reference level (positive downward into the fluid).
$\rho$ is the fluid density (assumed constant for incompressible fluids like water).

To find the pressure difference between two points at different depths:

$\Delta P = \rho g \Delta h$

where $\Delta h$ is the vertical distance between the two points.

Solving hydrostatic problems step-by-step:

Pick a reference level. The free surface of the liquid is usually the easiest choice because you know the pressure there (typically atmospheric).
Define your coordinate direction. If $z$ points upward, depth $h$ is the distance measured downward from the reference, and pressure increases with increasing $h$ .
Identify knowns. List $P_0$ , $\rho$ , $g$ , and the depth or depths of interest.
Apply the equation. Plug into $P = P_0 + \rho g h$ for absolute pressure at a point, or use $\Delta P = \rho g \Delta h$ for the difference between two points.
Check units. Make sure density is in kg/m³, $g$ in m/s², and $h$ in meters so your answer comes out in Pascals.

Quick example: What's the gauge pressure 5 m below the surface of a freshwater lake?

$\Delta P = \rho g h = (1000 \text{ kg/m}^3)(9.81 \text{ m/s}^2)(5 \text{ m}) = 49{,}050 \text{ Pa} \approx 49.1 \text{ kPa}$

That's roughly half an atmosphere of additional pressure, which is why you already feel it in your ears at the bottom of a deep pool.

Atmospheric Pressure Effects

Atmospheric pressure is the pressure the Earth's atmosphere exerts on everything at the surface. At sea level, its standard value is:

101,325 Pa (1 atm)
14.696 psi (pounds per square inch)

Atmospheric pressure matters because it serves as the baseline for two different pressure scales:

Gauge pressure is pressure measured relative to atmospheric pressure. A tire pressure gauge reading 32 psi means the air inside is 32 psi above atmospheric. If the gauge reads zero, the pressure inside equals atmospheric.

Absolute pressure is the total pressure, including the atmosphere. It equals gauge pressure plus atmospheric pressure: $P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}$ Absolute pressure can never be negative (you can't have less than zero molecules pushing on a surface), but gauge pressure can be negative when the absolute pressure is below atmospheric (a partial vacuum).

Atmospheric pressure also changes with altitude. At higher elevations the air column above you is shorter, so atmospheric pressure drops. One practical consequence: water boils when its vapor pressure equals the surrounding atmospheric pressure, so at high altitude (lower $P_{\text{atm}}$ ), water boils below 100 °C. In Denver, CO (elevation ~1,600 m), water boils at roughly 95 °C.

💧Fluid Mechanics Unit 3 Review