Fluid Statics, Manometry, and Buoyancy

Introduction and Context

Fluid statics concerns fluids at rest and the forces that arise because of pressure. Although a stationary fluid has no shear stress, it can exert substantial normal forces on containers, gates, dams, tanks, pipes, and machine components. Hydrostatic principles are therefore fundamental in the design of hydraulic equipment, storage vessels, pressure-measuring devices, ships, submersible vehicles, and industrial process systems.

This chapter develops the pressure field in a fluid at rest, explains manometers, determines hydrostatic forces and centers of pressure on submerged surfaces, and establishes the principles governing buoyancy and floating-body stability. The treatment assumes an incompressible fluid unless compressibility is explicitly discussed.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Distinguish absolute, gauge, and vacuum pressure.
2. Derive and apply the hydrostatic pressure relation.
3. Analyze piezometers, U-tube manometers, differential manometers, and inclined manometers.
4. Calculate resultant hydrostatic forces and centers of pressure on plane and curved surfaces.
5. Apply Archimedes’ principle to immersed and floating bodies.
6. Determine metacentric height and assess the stability of floating bodies.
7. Identify assumptions, measurement errors, and practical limitations in hydrostatic analysis.

1. Fundamental Concepts of Fluid Statics

A fluid at rest cannot sustain a tangential or shear stress. It can, however, exert pressure normal to every surface. Pressure is the normal force per unit area:

where p is pressure in pascals (Pa), F_n is normal force in newtons (N), and A is area in square metres (m²). Pressure is a scalar property at a point in a fluid at rest; it has the same magnitude in every direction.

Pressure may be reported in three common forms:

• Absolute pressure: measured relative to a perfect vacuum.
• Gauge pressure: measured relative to local atmospheric pressure.
• Vacuum pressure: the amount by which pressure is below atmospheric pressure.

The relationships are

and, when a pressure is below atmospheric,

Here p_atm is atmospheric pressure, p_g is gauge pressure, and p_vac is vacuum pressure. Gauge pressure can be negative, but absolute pressure cannot be negative.

Density, specific weight, and specific gravity

Density is mass per unit volume:

Specific weight is weight per unit volume:

where ρ is density (kg/m³), m is mass (kg), V is volume (m³), γ is specific weight (N/m³), and g is gravitational acceleration (approximately 9.81 m/s²). Specific gravity is the ratio of a fluid’s density to the density of water at a stated reference temperature:

Specific gravity is dimensionless. Density and viscosity vary with temperature, so engineering calculations should use properties at the actual operating temperature whenever accuracy matters.

2. Hydrostatic Pressure Distribution

Consider a small rectangular fluid element of thickness dz, with the vertical coordinate z increasing upward. Since the fluid is stationary, the vertical forces must balance. The pressure force on the lower face exceeds that on the upper face by the weight of the element. This gives

For a constant-density fluid, integration between points 1 and 2 gives

or, if point 2 is at a vertical depth h below point 1,

This is the fundamental hydrostatic equation. It applies to a continuous fluid at rest with constant density and uniform gravitational acceleration. Pressure increases linearly with depth, and all points at the same elevation in the same connected fluid have equal pressure.

For a free surface exposed to the atmosphere, the gauge pressure at depth h is

The pressure distribution is triangular when measured from a free surface. If the surface pressure is nonzero, the absolute pressure diagram is a rectangle due to the surface pressure plus a triangle due to the fluid weight.

A useful pressure head is

It represents the height of a column of the same fluid that produces pressure p. Pressure-head methods are particularly convenient in manometer calculations.

Worked Example 1: Pressure at depth

Water has density 998 kg/m³ at the operating temperature. Determine the gauge pressure 4.5 m below an open tank surface.

Using p_g = ρgh:

p_g = 998 × 9.81 × 4.5 = 44,060 Pa

Thus, the gauge pressure is approximately 44.1 kPa. If atmospheric pressure is 101.3 kPa absolute, the absolute pressure is approximately 145.4 kPa.

3. Pressure Measurement and Manometry

Manometers determine pressure by balancing it against the hydrostatic pressure of one or more liquid columns. They are simple, reliable, and useful for calibration, laboratory measurements, and low-pressure industrial applications.

Piezometer

A piezometer is a transparent vertical tube connected to a liquid system. The liquid rises until the column weight balances the pressure. For a liquid of density ρ:

A piezometer is unsuitable for gases, very high pressures, and liquids that would rise above practical tube heights.

U-tube manometer

A U-tube manometer contains a manometric liquid, often mercury, connected to the system under investigation. To analyze it, begin at a known pressure and move through the connected columns. Moving downward through a stationary liquid adds ρg∆z to pressure; moving upward subtracts it. At the same elevation in a continuous manometric liquid, pressures are equal.

For a simple U-tube measuring the gauge pressure of a liquid lighter than the manometric liquid, the result depends on the elevations and densities. The safest method is not to memorize a single arrangement-specific equation but to write a pressure balance along the actual path.

Differential manometer

A differential manometer measures the pressure difference between two points. If both tapping points contain the same fluid and a heavier manometric fluid separates the columns, a common form is

for a suitable geometry in which h is the vertical difference between interfaces, ρ_f is the process-fluid density, and ρ_m is the manometric-fluid density. The sign and elevation terms must be adapted to the actual arrangement. This equation assumes immiscible fluids, static columns, and negligible capillary effects.

Inclined manometer

An inclined tube magnifies the visible displacement for small pressure differences. If the column moves a length L along a tube inclined at angle θ to the horizontal, the vertical height difference is

The pressure difference is then calculated from the relevant hydrostatic balance. Inclined manometers improve readability but require accurate angle measurement and careful leveling.

Manometer best practices

Use a manometric liquid that is chemically compatible, sufficiently dense, and immiscible with the process fluid. Remove trapped air, ensure that connections are leak-free, read menisci consistently, and correct for temperature-dependent density when precision is required. Mercury offers high density and compact columns but presents toxicity and disposal concerns; safer alternatives may be preferable.

4. Hydrostatic Forces on Plane Surfaces

A submerged surface experiences a distributed pressure force. The resultant force on a plane surface is

when the surface is exposed to the atmosphere and gauge pressure is used. Here p_c is pressure at the area centroid, A is surface area, and h_c is the vertical depth of the centroid below the free surface. If the free surface pressure is not atmospheric, use the absolute or gauge pressure at the centroid consistently.

The equation follows because pressure varies linearly with depth and the average pressure over a plane surface equals the pressure at its centroid. The force acts normal to the surface.

Center of pressure

The center of pressure is the point through which the resultant force acts. It lies below the centroid for a vertical or inclined submerged plane because deeper portions experience greater pressure. For an inclined plane whose centroid is at vertical depth h_c,

where h_R is the vertical depth of the center of pressure, I_G is the second moment of area about a centroidal axis parallel to the free surface, and θ is the angle between the plane and the free surface. For a vertical plane, θ = 90° and therefore sin² θ = 1. For a horizontal plane, pressure is uniform and the center of pressure coincides with the centroid.

The formula assumes a plane surface, a linear hydrostatic pressure field, constant density, and a free surface with negligible curvature.

Common second moments of area include:

Worked Example 2: Vertical rectangular gate

A vertical rectangular gate is 2 m wide and 3 m high. Its top edge is 1 m below the water surface. Determine the resultant force and center of pressure.

The area is A = 2 × 3 = 6 m². The centroid is at

h_c = 1 + 3/2 = 2.5 m.

For water, take ρ = 1000 kg/m³:

F_R = 1000 × 9.81 × 2.5 × 6 = 147,150 N.

Thus, F_R ≈ 147 kN. For the rectangle,

I_G = bd³/12 = 2 × 3³/12 = 4.5 m⁴.

Since the gate is vertical,

h_R = 2.5 + 4.5/(2.5 × 6) = 2.8 m.

The force acts 2.80 m below the free surface, or 0.20 m below the gate centroid.

5. Hydrostatic Forces on Curved Surfaces

For a curved surface, pressure forces have different directions, so the resultant is determined through components.

The horizontal component equals the hydrostatic force on the vertical projection of the curved surface:

where A_v is the area of the vertical projection and h_c is its centroid depth.

The vertical component equals the weight of the imaginary fluid directly above the curved surface up to the free surface:

Its line of action passes through the centroid of the imaginary volume, subject to the direction determined by whether the fluid lies above or below the surface. The resultant magnitude is

and its inclination to the horizontal is

These relations assume a static fluid and a clearly defined free surface. Atmospheric forces may cancel when they act on both sides; otherwise, they must be included.

6. Buoyancy and Archimedes’ Principle

A body immersed in a fluid experiences a net upward force because pressure increases with depth. Archimedes’ principle states that the buoyant force equals the weight of the fluid displaced:

where F_B is buoyant force, ρ_f is fluid density, and V_d is displaced volume. The buoyant force acts through the center of buoyancy, which is the centroid of the displaced fluid volume.

For a fully immersed body, V_d equals the body volume. For a floating body, only the submerged volume is displaced. The vertical equilibrium condition for a freely floating body is

where W is the body weight.

If a homogeneous body floats in a liquid, its submerged-volume fraction is

provided the body is in static equilibrium and the liquid density is uniform.

Stability of floating bodies

A floating body is in equilibrium when its weight through the center of gravity balances buoyancy through the center of buoyancy. A small angular displacement changes the submerged geometry and shifts the center of buoyancy. The new buoyancy line intersects the original centerline at the metacenter, denoted M.

• Stable equilibrium: metacenter above center of gravity, GM > 0.
• Neutral equilibrium: metacenter coincides with center of gravity, GM = 0.
• Unstable equilibrium: metacenter below center of gravity, GM < 0.

The metacentric height is

where BM is the metacentric radius and BG is the distance from the center of buoyancy to the center of gravity. The metacentric radius is

where I_W is the second moment of area of the waterplane about the axis of heel and V_d is displaced volume. This small-angle stability criterion assumes a modest heel angle and a body whose geometry changes smoothly.

A wide waterplane generally increases I_W and therefore improves initial stability. A high center of gravity reduces GM and may make a vessel unsafe. Initial stability does not guarantee stability at large angles, where free-surface shape, deck immersion, cargo movement, and flooding become important.

Free-surface effect

Liquid moving in a partially filled tank shifts its center of gravity when a vessel heels. This reduces effective metacentric height. The reduction is commonly expressed as

where GG_v is the virtual rise of the center of gravity, ρ_l is liquid density, I_t is the tank free-surface second moment about the heel axis, and W is vessel weight. The effective stability becomes approximately GM_effective = GM − GG_v. Tanks should therefore be subdivided or kept either full or empty where operationally feasible.

7. Applied Case: Hydrostatic Design of a Water-Storage Tank Outlet Gate

Illustrative engineering case. A municipal water tank has a vertical rectangular outlet gate 1.5 m wide and 2.0 m high. The top of the gate is 0.8 m below the water surface. The gate is hinged at its top and restrained by a linkage at its bottom. Determine the resultant hydrostatic force and the force required at the bottom linkage, neglecting gate weight and friction.

Analysis

The centroid depth is

h_c = 0.8 + 2.0/2 = 1.8 m.

The gate area is A = 1.5 × 2.0 = 3.0 m². Therefore,

F_R = 1000 × 9.81 × 1.8 × 3.0 = 52,974 N.

The second moment about the horizontal centroidal axis is

I_G = 1.5 × 2.0³/12 = 1.0 m⁴.

The center of pressure is

h_R = 1.8 + 1.0/(1.8 × 3.0) = 1.985 m.

Thus, the force acts 1.185 m below the top hinge. Taking moments about the hinge, the bottom linkage force F_L acts at 2.0 m from the hinge:

F_L × 2.0 = 52,974 × 1.185.

Hence,

F_L ≈ 31,400 N = 31.4 kN.

Outcome and learning points

The gate must safely transmit approximately 53 kN of resultant load, while the idealized bottom linkage carries approximately 31 kN. In design, dynamic effects, corrosion, seal friction, impact during opening, nonuniform geometry, and a suitable safety factor must also be considered. The example demonstrates that force magnitude alone is insufficient: the center of pressure determines the moment and therefore the actuator or linkage requirement.

8. Limitations, Safety, and Professional Practice

Hydrostatic equations are powerful but depend on assumptions. Errors may arise from fluid motion, density stratification, temperature variation, trapped gas, capillary action, inaccurate elevation readings, corrosion, and leakage. A manometer connected to a vibrating or pulsating system may not display a meaningful steady pressure. For such systems, a pressure transducer with suitable dynamic response is preferable.

Engineers should distinguish design pressure from normal operating pressure and account for transient conditions such as water hammer, rapid valve closure, tank filling, and emergency flooding. Pressure vessels and storage tanks must comply with applicable national codes and occupational safety requirements. High-pressure systems require relief devices, inspection, material compatibility checks, and documented calibration.

Modern systems increasingly combine hydrostatic calculations with digital pressure sensors, data acquisition, computational fluid dynamics, and digital twins. Artificial intelligence can help detect sensor drift or abnormal pressure patterns, but it does not replace conservation laws, calibration, or professional engineering judgment. Sensor data should be traceable, secure, and validated against independent measurements.

Best-Practice Procedure for Hydrostatic Problems

1. Draw the geometry and mark the free surface and all elevations.
2. Select a pressure reference: absolute or gauge.
3. Identify fluid densities and verify units.
4. For manometers, trace a continuous path and add pressure when moving downward, subtracting when moving upward.
5. For plane surfaces, find area and centroid depth before calculating force.
6. Locate the center of pressure using the correct second moment of area.
7. For curved surfaces, resolve forces into horizontal and vertical components.
8. For floating bodies, check both force equilibrium and metacentric stability.
9. Check dimensions, significant figures, signs, and physical plausibility.
10. Include safety factors and operating transients in real designs.

Chapter Summary

Fluid statics describes pressure and force behavior in fluids at rest. The hydrostatic relation dp/dz = −ρg shows that pressure increases with depth and leads, for constant density, to p = p₀ + ρgh. Pressure may be expressed as absolute, gauge, or vacuum pressure. Manometers measure pressure through balanced liquid columns; their analysis requires careful elevation tracking and consistent density data.

The resultant force on a plane submerged surface equals the pressure at its centroid multiplied by area, while the center of pressure lies below the centroid for nonhorizontal surfaces. Curved-surface forces are resolved using the vertical projection and the imaginary-fluid-volume method. Archimedes’ principle states that buoyancy equals the weight of displaced fluid. Floating-body stability depends on the relative locations of the center of gravity, center of buoyancy, and metacenter, with positive metacentric height indicating initial stability. Practical engineering requires attention to free-surface effects, measurement limitations, safety codes, transient loading, and calibration.

Key Terms

Pressure: Normal force exerted per unit area.

Absolute pressure: Pressure measured from a perfect vacuum.

Gauge pressure: Pressure measured relative to atmospheric pressure.

Vacuum pressure: Amount by which pressure is below atmospheric pressure.

Density: Mass per unit volume of a fluid.

Specific weight: Weight per unit volume, equal to ρg.

Specific gravity: Ratio of a substance’s density to a reference density.

Hydrostatic pressure: Pressure in a fluid at rest.

Pressure head: Pressure divided by specific weight.

Piezometer: Vertical liquid tube used to measure pressure head.

Manometer: Device that measures pressure using liquid-column differences.

Resultant force: Single force equivalent to a distributed pressure loading.

Center of pressure: Point through which the resultant hydrostatic force acts.

Second moment of area: Geometric property used to locate centers of pressure and assess stability.

Buoyancy: Upward force exerted by a fluid on an immersed body.

Center of buoyancy: Centroid of the displaced fluid volume.

Metacenter: Point governing the initial stability of a floating body.

Metacentric height: Distance between the metacenter and center of gravity.

Free-surface effect: Reduction in floating stability caused by liquid movement in a partially filled tank.

Review Questions

Short Answer Questions

1. Why does pressure increase with depth in a stationary fluid?
2. Distinguish between absolute pressure and gauge pressure.
3. State Archimedes’ principle.
4. Why is the center of pressure below the centroid of a vertical submerged plate?
5. What does a positive metacentric height indicate?

Long Answer Questions

1. Derive the hydrostatic pressure equation for an incompressible fluid and state its assumptions.
2. Explain the operation and limitations of piezometers, U-tube manometers, and inclined manometers.
3. Derive the resultant-force and center-of-pressure relations for an inclined plane surface.
4. Explain how horizontal and vertical components are used to determine force on a curved surface.
5. Discuss the effect of a partially filled tank on the stability of a floating vessel.

Analytical / Application Questions

1. A vertical rectangular gate is 2.4 m wide and 1.8 m high, with its top 1.2 m below a water surface. Calculate the resultant force and center-of-pressure depth.
2. A vessel displaces 120 m³ of seawater with density 1025 kg/m³. Determine its supported weight and explain how a rise in the center of gravity affects initial stability.
3. Design a pressure-measurement approach for a hot, vibrating pipeline. Compare a differential manometer with a pressure transducer, considering accuracy, safety, maintenance, and dynamic response.

Multiple Choice Questions with Answers

Question 1

For an open tank containing water, what is the gauge pressure 3 m below the free surface? Take ρ = 1000 kg/m³ and g = 9.81 m/s².

A. 9.81 kPa

B. 29.4 kPa

C. 98.1 kPa

D. 294 kPa

Correct Answer: B. 29.4 kPa

Question 2

In a connected stationary fluid of constant density, points at the same elevation have:

A. Equal pressure

B. Equal velocity

C. Equal absolute temperature only

D. Zero pressure

Correct Answer: A. Equal pressure

Question 3

A piezometer is most suitable for measuring:

A. Very high gas pressure

B. Pressure head in a liquid

C. Vacuum in a sealed gas vessel

D. Rapid pressure fluctuations

Correct Answer: B. Pressure head in a liquid

Question 4

The resultant hydrostatic force on a plane submerged surface is calculated using:

A. Pressure at the lowest point multiplied by area

B. Pressure at the centroid multiplied by area

C. Atmospheric pressure multiplied by perimeter

D. Specific gravity divided by area

Correct Answer: B. Pressure at the centroid multiplied by area

Question 5

For a vertical submerged plane surface, the center of pressure is generally:

A. Above the free surface

B. At the top edge

C. At the centroid exactly

D. Below the centroid

Correct Answer: D. Below the centroid

Question 6

The buoyant force on a fully immersed body equals:

A. The weight of the body only

B. The pressure at the centroid times area

C. The weight of fluid displaced

D. The mass of the body divided by volume

Correct Answer: C. The weight of fluid displaced

Question 7

A floating body with positive metacentric height has:

A. Initial stable equilibrium

B. Initial unstable equilibrium

C. Neutral equilibrium necessarily

D. No buoyant force

Correct Answer: A. Initial stable equilibrium

Question 8

If a manometric liquid column is inclined at angle θ to the horizontal and moves a length L, its vertical height difference is:

A. L cos θ

B. L tan θ

C. L sin θ

D. L/ sin θ

Correct Answer: C. L sin θ

Question 9

For a curved submerged surface, the horizontal hydrostatic-force component equals the force on the:

A. Horizontal projection

B. Vertical projection

C. Entire wetted perimeter

D. Center of buoyancy

Correct Answer: B. Vertical projection

Question 10

A water gate has area 4 m² and centroid depth 2 m. Using ρ = 1000 kg/m³ and g = 9.81 m/s², its resultant gauge force is:

A. 19.6 kN

B. 39.2 kN

C. 78.5 kN

D. 98.1 kN

Correct Answer: C. 78.5 kN