Hydrostatic concepts form the foundation of fluid mechanics, crucial for civil engineers engaged in designing and managing water-related infrastructure such as dams, water tanks, and pipelines. Understanding the behaviour of fluids at rest enables accurate prediction of pressure distributions, which is vital for safe and efficient structural design. In Kenya, with growing urbanization and increasing demand for water supply systems in cities like Nairobi and Mombasa, mastery of hydrostatics ensures engineers can tackle challenges related to water storage and distribution effectively.

1.1 Introduction to Hydrostatic Concepts

Hydrostatics deals with the study of fluids at rest and the forces and pressures associated with them. It is a cornerstone for civil engineering applications involving reservoirs, water towers, and underground water tanks where fluid pressure impacts structural integrity. Kenyan civil engineers frequently apply hydrostatic principles when designing County Government water storage facilities or irrigation systems for farms in the Rift Valley.

1.1.1 Definition of terms

Precise understanding of hydrostatic terminology is essential for clear communication and accurate calculations in engineering projects. The following key terms are foundational to hydrostatics:

1.1.1.1 Viscosity

Viscosity is the property of a fluid that resists flow or the sliding of one layer of fluid over another, and it can be thought of as the fluid’s internal friction or “thickness”. Honey and engine oil have high viscosity and flow slowly, while water and petrol have low viscosity and flow easily. Although viscosity has no effect on a fluid that is completely at rest, it becomes important as soon as the fluid moves, governing friction losses in pipes, flow in channels, settling of particles in water treatment and the behaviour of lubricants. Example: the engineer designing a long rising main for an irrigation scheme on a farm in Naivasha used the viscosity of water to calculate friction losses and size the pumps.

Types of Viscosity

• Dynamic (absolute) viscosity (μ): The ratio of shear stress to the rate of change of velocity across fluid layers, measured in pascal seconds (Pa·s) or N·s/m².
• Kinematic viscosity (ν): Dynamic viscosity divided by density, ν = μ ÷ ρ, measured in m²/s.
• Newtonian fluids: Fluids such as water, air and most oils whose viscosity stays constant at a given temperature regardless of how fast they are sheared.
• Non-Newtonian fluids: Fluids such as fresh concrete, mud, paint and blood whose apparent viscosity changes with the rate of shear.
• Effect of temperature: The viscosity of liquids decreases as temperature rises, while the viscosity of gases increases.

Newton’s law of viscosity: τ = μ × (du ÷ dy)

where τ is the shear stress, μ is the dynamic viscosity and du ÷ dy is the velocity gradient between fluid layers.

Typical Viscosities at About 20 °C

Worked Example 1: Kinematic Viscosity

An oil has a dynamic viscosity of 0.1 Pa·s and a density of 900 kg/m³. Find its kinematic viscosity.

ν = μ ÷ ρ

ν = 0.1 ÷ 900

ν = 1.11 × 10⁻⁴ m²/s

Worked Example 2: Shear Stress in a Fluid Film

A plate moves at 0.5 m/s over a fixed surface separated by an oil film 2 mm thick. The oil viscosity is 0.1 Pa·s. Find the shear stress on the plate.

τ = μ × (u ÷ y)

τ = 0.1 × (0.5 ÷ 0.002)

τ = 0.1 × 250

τ = 25 Pa

1.1.1.2 Density

Density is the mass of a fluid per unit volume, commonly in kg/m³. It affects hydrostatic pressure since heavier fluids exert greater pressure at a given depth. For instance, brackish water used in coastal aquaculture in Kilifi has a different density from freshwater, altering pressure calculations.

Density is given by ρ = m ÷ V, and for water it is taken as 1,000 kg/m³ in most engineering calculations, being highest at about 4 °C and decreasing slightly as temperature rises. Closely related quantities are the specific weight (γ), which is the weight per unit volume, and the specific volume, which is the volume per unit mass. Density appears directly in the hydrostatic pressure equation, p = ρ g h, so it controls the pressure on tank walls, dam faces and pipes. Example: the engineer designing a seawater intake tank for a hotel’s desalination plant in Diani used a density of 1,025 kg/m³ instead of 1,000 kg/m³, increasing the calculated wall pressures.

Related Properties

Typical Densities of Fluids

• Fresh water: About 1,000 kg/m³, the standard reference for liquids.
• Seawater: About 1,025 kg/m³ because of dissolved salts.
• Mercury: About 13,600 kg/m³, used in manometers and barometers.
• Light oils and diesel: About 800 kg/m³ to 900 kg/m³, so they float on water.
• Air at sea level: About 1.2 kg/m³, much lower than liquids.

Worked Example 3: Density and Specific Weight

A fluid with a volume of 0.25 m³ has a mass of 255 kg. Find its density and specific weight.

ρ = m ÷ V

ρ = 255 ÷ 0.25

ρ = 1,020 kg/m³

γ = ρ × g

γ = 1,020 × 9.81

γ = 10,006 N/m³ = 10.01 kN/m³

1.1.1.3 specific gravity

Specific gravity (SG), also called relative density, is the ratio of the density of a substance to the density of a standard reference substance, which for liquids and solids is water at about 4 °C (1,000 kg/m³). Because it is a ratio of two densities, specific gravity has no units, and it quickly shows whether a substance will sink (SG greater than 1) or float (SG less than 1) in water. It is widely used to describe liquids such as oils and fuels, soils and aggregates, and manometer fluids, and it is measured with a hydrometer or density bottle. Example: a county water company laboratory in Nyeri measured the specific gravity of a coagulant solution with a hydrometer to confirm its strength before dosing it into the treatment works.

SG = Density of substance ÷ Density of water

Equipment for Measuring Specific Gravity

Typical Specific Gravities

• Water: SG = 1.00 by definition.
• Diesel and light oils: SG about 0.80 to 0.90, so they float on water.
• Seawater: SG about 1.025.
• Mercury: SG about 13.6.
• Soil solids: SG about 2.60 to 2.80, used in soil mechanics calculations.

Worked Example 4: Specific Gravity of an Oil

An oil has a density of 850 kg/m³. Find its specific gravity.

SG = ρ_oil ÷ ρ_water

SG = 850 ÷ 1,000

SG = 0.85

Worked Example 5: Density from Specific Gravity

Mercury has a specific gravity of 13.6. Find its density and specific weight.

ρ = SG × ρ_water

ρ = 13.6 × 1,000

ρ = 13,600 kg/m³

γ = ρ × g

γ = 13,600 × 9.81

γ = 133,416 N/m³ = 133.4 kN/m³

1.1.1.4 surface tension

Surface tension is the property of a liquid surface that makes it behave like a thin stretched elastic film, caused by the attraction between liquid molecules (cohesion) at the surface. It is measured as force per unit length, in N/m, and for water at 20 °C it is about 0.073 N/m. Surface tension explains why water forms droplets, why small insects can stand on water and, together with adhesion between the liquid and a solid, why water rises in narrow tubes and pores by capillary action. Capillary rise is important in civil engineering because it draws groundwater up through soils and masonry, causing rising damp and affecting foundations. Example: the walls of an old classroom block at a primary school in Kisumu showed rising damp because no damp-proof course had been provided, allowing water to rise through the pores of the stone by capillary action.

Capillary rise: h = (4 × σ × cos θ) ÷ (ρ × g × d)

where h is the height of rise, σ is surface tension, θ is the contact angle (about 0° for clean water and glass), ρ is density, g is gravitational acceleration and d is the tube diameter.

Effects and Applications of Surface Tension

• Capillary rise in soils: Water rises above the water table through fine soil pores, which can weaken subgrades and affect foundations.
• Rising damp in walls: Water rises through porous bricks, blocks and mortar unless a damp-proof course is provided.
• Errors in measuring tubes: Narrow piezometer and manometer tubes give false readings due to capillary rise, so tubes should be wide enough.
• Formation of droplets and bubbles: Surface tension holds droplets and bubbles together.
• Wetting and waterproofing: Water repellent coatings change the contact angle so water beads off surfaces instead of soaking in.

Worked Example 6: Capillary Rise in a Glass Tube

Find the capillary rise of water in a clean glass tube of 2 mm diameter. Take σ = 0.073 N/m, θ = 0°, ρ = 1,000 kg/m³ and g = 9.81 m/s².

h = (4 × σ × cos θ) ÷ (ρ × g × d)

h = (4 × 0.073 × 1) ÷ (1,000 × 9.81 × 0.002)

h = 0.292 ÷ 19.62

h = 0.0149 m = 14.9 mm

1.1.1.5 Pressure

Pressure is the force exerted per unit area by a fluid at rest on a surface, expressed in Pascals (Pa) or kilopascals (kPa). It acts perpendicular to any surface in contact with the fluid. For example, in a water tank at KNH, pressure increases with depth, influencing wall thickness design.

1.1.1.6 Hydrostatic Pressure

Hydrostatic pressure refers specifically to the pressure exerted by a fluid due to its weight when the fluid is at rest. It increases linearly with depth because of the increasing weight of the fluid column above. This principle is critical in calculating forces on retaining walls holding back water in dams.

1.1.1.7 Pascal’s Law

Pascal’s Law states that pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid. This principle underpins hydraulic systems used in civil engineering equipment like hydraulic jacks for bridge construction.

1.1.1.8 Gauge Pressure and Atmospheric Pressure

Gauge pressure is the pressure measured relative to atmospheric pressure, which is the pressure exerted by the earth’s atmosphere at sea level (approximately 101.3 kPa). Civil engineers must distinguish between absolute and gauge pressures when designing pressurized water supply systems in Nairobi.

Chapter Summary

This chapter introduced the fundamental hydrostatic concepts essential for understanding fluid behavior at rest, beginning with key terms such as viscosity, density, specific gravity, and surface tension that characterize fluid properties. It then explored the intrinsic properties of fluids including compressibility, elasticity, and capillarity, which influence how fluids respond under various conditions. Different types of pressure were examined, highlighting gauge pressure, absolute pressure, and atmospheric pressure as critical measures in fluid mechanics. The chapter detailed the magnitude and position of pressures in fluids through hydrostatic pressure, pressure head, and pressure energy, explaining how these concepts determine force distribution in fluids. Pascal's law was presented with its statement and practical applications, demonstrating how pressure applied to a confined fluid is transmitted undiminished throughout. The law of floatation and buoyancy was discussed with reference to Archimedes' principle, outlining the conditions under which objects float, sink, or remain neutrally buoyant. Finally, the chapter covered the various applications of buoyancy in real-world scenarios, emphasizing its significance in engineering and fluid systems.