Fluid Mechanics: Fluid Statics

Fluid statics is the study of fluids at rest. Despite the lack of motion, static fluids generate forces that govern everything from dam design and ship stability to blood pressure measurement and hydraulic lifts. The key idea is simple: in a stationary fluid, shear stresses vanish and only normal stresses remain. Those normal stresses are what we call pressure, and pressure in a fluid has a predictable structure that makes engineering calculations possible.

Pressure in a Fluid at Rest

Pressure is defined as the normal force per unit area. In a static fluid, pressure at a point acts equally in all directions (an outcome often summarized as Pascal’s principle). That directional equality is what allows pressure to be treated as a scalar field rather than a vector field.

Hydrostatic Pressure Variation with Depth

The most important relationship in fluid statics is the hydrostatic equation. Consider a small fluid element of cross-sectional area $A$ and height $dz$ in a fluid of density $\rho$. Force balance in the vertical direction yields:

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

Integrating from a reference elevation gives a linear pressure increase with depth for incompressible fluids:

$$p=p_0+\rho gh$$

where:

• $p_0$ is the pressure at the reference level (often the free surface where $p_0$ equals atmospheric pressure),
• $h$ is the vertical depth below that level,
• $g$ is gravitational acceleration.

This relationship explains why divers experience higher pressure at depth, why deep tanks require thicker walls near the bottom, and why pressure gauges in pipelines must account for elevation differences.

Absolute Pressure, Gauge Pressure, and Atmospheric Effects

Pressure can be reported as:

• Absolute pressure, measured relative to a perfect vacuum.
• Gauge pressure, measured relative to the local atmospheric pressure.

They are related by $p_{\text{abs}}=p_{\text{atm}}+p_{\text{gauge}}$. In many practical problems involving open tanks or reservoirs, atmospheric pressure cancels out because it acts on all exposed free surfaces.

Pressure Distribution and Forces on Submerged Surfaces

Because pressure increases with depth, a submerged surface experiences a nonuniform load. Engineers often need two results: the resultant hydrostatic force and the location where it acts, called the center of pressure.

Resultant Force on a Plane Surface

For a plane area $A$ submerged in a liquid of constant density, the resultant hydrostatic force magnitude equals the pressure at the area’s centroid times the area:

$$F=\rho g{h}_{c}A$$

where $h_c$ is the depth of the centroid below the free surface. This compact formula is widely used for gates, viewing windows in aquariums, and access panels in tanks.

A practical implication is that two surfaces of the same area at the same centroid depth experience the same resultant force even if their shapes differ. Shape matters, however, for where the force acts.

Center of Pressure

The center of pressure lies below the centroid for vertical or inclined surfaces because pressure increases with depth. For a plane surface inclined at an angle, the location depends on the second moment of area. In general terms, the deeper portions contribute disproportionately to the moment, shifting the line of action downward.

For design, this matters as much as the force magnitude. A sluice gate might withstand the net load but still fail at hinges or supports if the center of pressure creates an unexpected moment.

Curved Surfaces: Horizontal and Vertical Components

Curved surfaces are common in pipes, tanks, and dams. The resultant hydrostatic force on a curved surface is typically found by resolving into components:

• The horizontal component equals the hydrostatic force on the vertical projection of the curved surface.
• The vertical component equals the weight of the imaginary column of fluid directly above the curved surface up to the free surface (with sign depending on whether the fluid pushes upward or downward).

This decomposition turns a complicated pressure field into manageable geometry and weight calculations.

Manometers and Pressure Measurement

Manometers measure pressure using fluid columns. They are valued for accuracy and simplicity, especially for calibration and laboratory measurements.

Piezometer and Simple Manometer Concepts

A piezometer is a vertical tube connected to a point in a liquid. The liquid rises to a height that represents the local gauge pressure:

$$p_{\text{gauge}}=\rho gh$$

Piezometers are limited to liquids and moderate pressures because very high pressures would require impractically tall tubes.

U-Tube Manometers and Differential Pressure

A U-tube manometer uses a manometric fluid (often mercury or another dense liquid) to compare pressures between two points. The governing principle is that, in a static connected fluid system, pressures at the same elevation in the same continuous fluid are equal. By stepping through the column and accounting for elevation changes and densities, you can determine pressure differences.

In practice, U-tube and inclined manometers are used to measure small pressure differences in airflow systems, filters, and ducts. An inclined tube increases resolution because a small vertical height change corresponds to a longer movement along the tube.

Buoyancy and Archimedes’ Principle

Buoyancy is one of the most recognizable outcomes of fluid statics. Any object immersed in a fluid experiences an upward force because pressure is higher at its lower surfaces than at its upper surfaces.

Buoyant Force

Archimedes’ principle states:

• The buoyant force equals the weight of the displaced fluid.

Mathematically:

$$F_B={\rho }_{f}g{V}_{\text{disp}}$$

where ${\rho }_f$ is the fluid density and $V_{\text{disp}}$ is the volume of fluid displaced by the body.

An object floats if its weight is less than or equal to the buoyant force. For a floating body, equilibrium requires:

$$W=F_B$$

So the submerged volume adjusts until the displaced fluid weight matches the object’s weight. This explains why a steel ship can float: it displaces a large volume of water due to its overall shape and enclosed air, reducing average density.

Apparent Weight and Submerged Objects

For an object fully submerged and held by a support, the support force is reduced by buoyancy. The apparent weight is:

$$W_{\text{apparent}}=W-F_B$$

This matters in underwater lifting, where buoyancy reduces crane loads, and in density measurement methods that compare weights in air and in a fluid.

Stability of Floating Bodies

Floating is not just about whether an object rises or sinks. It is also about whether it remains upright or capsizes when disturbed. Stability analysis in fluid statics focuses on the relationship between the center of gravity, center of buoyancy, and a geometric point called the metacenter.

Centers of Gravity and Buoyancy

• The center of gravity (G) is the point through which the body’s weight acts downward.
• The center of buoyancy (B) is the centroid of the displaced fluid volume, where the buoyant force effectively acts upward.

When a floating body is upright, $B$ lies along the vertical line through the displaced volume. If the body tilts, the submerged shape changes and $B$ shifts laterally.

Metacentric Height and Stability

For small angles of tilt, the buoyant force line of action intersects the original vertical centerline at the metacenter (M). The relative positions determine stability:

• Stable equilibrium if $M$ is above $G$ (positive metacentric height).
• Unstable equilibrium if $M$ is below $G$.
• Neutral equilibrium if $M$ coincides with $G$.

The metacentric height is $GM=M-G$ (a distance, not a force). Designers increase stability by lowering $G$ (adding ballast low in a ship) or by shaping the hull to increase the shift of $B$ when tilted.

A practical example is cargo loading. Stacking heavy cargo high raises $G$, reducing $GM$, and can make a vessel sluggish or dangerously prone to capsizing. Stability calculations are therefore central to marine operations, not just shipbuilding.

Why Fluid Statics Still Matters

Fluid statics underpins many systems that appear dynamic. Pumps and turbines rely on pressure baselines defined by hydrostatics. Sensor readings must be corrected for elevation and fluid density. Structural safety for dams, tanks, and submerged walls depends on accurate hydrostatic force and moment calculations. Even everyday measurements like blood pressure are rooted in manometric principles.

By mastering pressure distribution, manometers, buoyancy, and floating stability, you gain a framework that translates directly into reliable engineering design and clear physical intuition about how fluids push, support, and stabilize the world around us.