5.2 Fluid statics

Hydrostatic Pressure and Depth

Fluid statics deals with how liquids and gases behave when they're not moving. Understanding pressure in stationary fluids is essential for designing tanks, dams, pipelines, and any system where fluid sits at rest. This section covers hydrostatic pressure, forces on submerged surfaces, buoyancy, and pressure measurement.

Hydrostatic Pressure Principles

Hydrostatic pressure is the pressure a fluid at rest exerts due to the weight of fluid above a given point. A few key properties define how it works:

• It acts equally in all directions at a given depth (this is called isotropic behavior). A pressure sensor at 5 m depth reads the same value whether it faces up, down, or sideways.
• It depends only on depth, not on the shape or size of the container. A narrow tube and a wide tank filled to the same height with the same fluid produce identical pressures at the bottom.
• It's a scalar quantity, meaning it has magnitude but no inherent direction.

Relationship Between Hydrostatic Pressure and Depth

Hydrostatic pressure increases linearly with depth. The governing equation is:

$P = \rho g h$

where:

• $P$ = hydrostatic pressure (Pa)
• $\rho$ = fluid density (kg/m³)
• $g$ = acceleration due to gravity (9.81 m/s²)
• $h$ = depth below the fluid surface (m)

This means the pressure difference between any two points in a static fluid depends only on the vertical distance between them and the fluid's density.

Quick example: For a swimming pool filled with water ($\rho = 1000$ kg/m³), the gauge pressure at 3 m depth is:

$P = (1000)(9.81)(3) = 29{,}430 \text{ Pa} \approx 29.4 \text{ kPa}$

At 1 m depth, it's only about 9.8 kPa. The pressure scales directly with depth.

Force and Center of Pressure

Hydrostatic Force on Submerged Surfaces

When a fluid pushes against a submerged surface (like a dam wall or a tank panel), the total force depends on the pressure at the surface's centroid (geometric center) and the surface area:

$F = P_c A = \rho g h_c A$

where:

• $F$ = resultant hydrostatic force (N)
• $P_c$ = pressure at the centroid of the surface
• $h_c$ = depth of the centroid below the fluid surface
• $A$ = area of the submerged surface

The direction of this force is always perpendicular to the surface, pointing from the high-pressure side toward the low-pressure side. For a vertical dam wall, that means the force acts horizontally, pushing outward. And because pressure increases with depth, the lower portions of the wall experience more force than the upper portions.

Center of Pressure

The center of pressure is the single point where the total hydrostatic force effectively acts on a submerged surface. Because pressure increases with depth, the center of pressure is always below the centroid for a vertically submerged surface.

Its location is given by:

$y_p = \frac{I_c}{A h_c} + h_c$

where:

• $y_p$ = depth of the center of pressure below the fluid surface
• $I_c$ = second moment of area (moment of inertia) about the centroidal axis of the surface
• $A$ = area of the surface
• $h_c$ = depth of the centroid below the surface

Why does this matter? If you're designing a gate or hatch that pivots, you need to know where the resultant force acts, not just how large it is. Placing the pivot at the center of pressure means zero net moment from the fluid, which is critical for engineering design.

For a vertically submerged rectangle of height $b$ and width $w$, the second moment of area is $I_c = \frac{wb^3}{12}$. As the surface gets deeper (larger $h_c$), the term $\frac{I_c}{Ah_c}$ shrinks, and the center of pressure approaches the centroid.

Buoyancy and Archimedes' Principle

Buoyancy Concept

Buoyancy is the net upward force a fluid exerts on an immersed object. It arises because pressure increases with depth, so the fluid pushes harder on the bottom of an object than on the top.

The buoyant force equals:

$F_b = \rho_f g V_d$

where:

• $F_b$ = buoyant force (N)
• $\rho_f$ = density of the fluid (not the object)
• $g$ = acceleration due to gravity
• $V_d$ = volume of fluid displaced by the object

Three outcomes are possible:

• Object floats: Buoyant force equals the object's weight. The object displaces only enough fluid to balance its weight, so it's partially submerged.
• Object sinks: The object's weight exceeds the maximum buoyant force (when fully submerged). This happens when the object is denser than the fluid.
• Neutral buoyancy: Weight exactly equals the buoyant force while fully submerged. The object neither rises nor sinks. Submarines achieve this by adjusting ballast water.

The stability of a floating object depends on the relative positions of its center of gravity and its center of buoyancy (the centroid of the displaced fluid volume). If the center of buoyancy shifts to create a restoring moment when the object tilts, the object is stable.

Archimedes' Principle and Applications

Archimedes' principle states that the buoyant force on an immersed object equals the weight of the fluid displaced by that object. This is the physical basis for the equation above.

A few practical applications:

• Ship design: A steel ship floats because its hull encloses a large air-filled volume, displacing enough water so that the buoyant force matches the ship's total weight. Even though steel is denser than water, the average density of the ship (steel + air) is less than water.
• Hot air balloons: Heated air inside the balloon is less dense than the surrounding cooler air, so the weight of displaced air exceeds the balloon's weight, producing a net upward force.
• Hydrostatic weighing: By weighing an object in air and then submerged in water, you can calculate its volume and density. This technique is used in materials testing and even body composition analysis.

Manometers and Pressure Measurement

Manometers

Manometers measure pressure differences using columns of liquid. The most common type is the U-tube manometer: a U-shaped tube partially filled with a dense liquid (often mercury, $\rho = 13{,}600$ kg/m³, or water).

How to read a U-tube manometer:

1. One end connects to the point where you want to measure pressure. The other end is open to the atmosphere (or to a second pressure point).
2. The pressure difference pushes the manometer fluid to different heights in each arm of the U-tube.
3. Measure the height difference $h$ between the two liquid levels.
4. Calculate the pressure difference:

$P_1 - P_2 = \rho_m g h$

where $\rho_m$ is the density of the manometer fluid and $h$ is the height difference.

Example: A U-tube mercury manometer connected to a gas pipeline shows a height difference of 15 cm (0.15 m). The gauge pressure in the pipeline is:

$P_{gauge} = (13{,}600)(9.81)(0.15) = 20{,}012 \text{ Pa} \approx 20.0 \text{ kPa}$

Other manometer types include:

• Inclined manometers: The tube is tilted at an angle, which stretches the liquid movement along a longer scale. This makes them more sensitive for measuring small pressure differences.
• Differential manometers: Both ends connect to different points in a system, directly measuring the pressure difference between them without referencing the atmosphere.

Pressure Measurement Devices

Beyond manometers, mechanical gauges are widely used:

• Bourdon tube gauges use a curved metal tube that straightens under pressure, moving a needle on a dial. These are the round gauges you'll see on compressed air tanks and boilers.
• Diaphragm gauges use a flexible membrane that deflects under pressure, suitable for lower pressure ranges or corrosive fluids.

When solving pressure measurement problems, always clarify whether you're working with gauge pressure (relative to atmospheric) or absolute pressure (relative to a perfect vacuum). Most manometers and mechanical gauges read gauge pressure. To convert: $P_{abs} = P_{gauge} + P_{atm}$, where standard atmospheric pressure is approximately 101.325 kPa.