CELE Hydraulics & Fluid Mechanics — Properties of FluidsRevision Notes

Formulas

Exam Tips

• When a problem gives s, immediately write ρ = s × 1000 kg/m³ and γ = s × 9.81 kN/m³ as your first two lines.
• Board problems often chain properties: given s, find γ, then find pressure or buoyant force — practise the chain.
• If only γ is given, get ρ = γ/g before applying any viscosity or kinetic energy formula.

Key Points

• Density ρ is mass per unit volume (kg/m³). Standard water: ρ_w = 1000 kg/m³ at 4 °C.
• Specific weight γ is weight per unit volume (N/m³ or kN/m³): γ = ρg. Standard water: γ_w = 9.81 kN/m³.
• Specific gravity s is the dimensionless ratio of a fluid's density (or specific weight) to that of water at 4 °C: s = ρ/ρ_w = γ/γ_w.
• Specific volume v_s = 1/ρ (m³/kg) — the reciprocal of density, rarely asked alone but appears in thermodynamics-adjacent problems.
• For any fluid: ρ = s × 1000 kg/m³ and γ = s × 9.81 kN/m³.
• These three properties are interrelated through g = 9.81 m/s². Never mix g = 9.8 and g = 9.81 within the same problem.

Definitions

Section Title

1. Density, Specific Weight, and Specific Gravity

Common Mistakes

• Confusing γ (N/m³) with ρ (kg/m³): they differ by a factor of g = 9.81 m/s².
• Using γ_w = 9800 N/m³ instead of 9810 N/m³ — use 9810 unless the problem states otherwise.
• Forgetting that specific gravity is dimensionless — never attach units to s.
• Applying the formula γ = ρg but substituting ρ in g/cm³ (CGS) instead of converting to kg/m³ first.

Formulas

Exam Tips

• If a problem gives ν and ρ, always recover μ = ν × ρ before computing shear stress.
• For a linear velocity profile (two parallel plates), dv/dy = V/h. Confirm the profile is linear before using this shortcut.
• Kinematic viscosity of water at 20 °C ≈ 1 × 10⁻⁶ m²/s — memorize this standard value.
• Board problems on force: F = τ × A. Compute τ first, then multiply by plate area.

Key Points

• Viscosity is a fluid's internal resistance to shear (flow-layer sliding). It is NOT related to density alone.
• Newton's Law of Viscosity: τ = μ (dv/dy). The shear stress τ is proportional to the velocity gradient dv/dy.
• μ (mu) = dynamic (absolute) viscosity, SI unit: Pa·s (= N·s/m² = kg/m·s). Old unit: poise (1 P = 0.1 Pa·s).
• ν (nu) = kinematic viscosity = μ/ρ, SI unit: m²/s. Old unit: stoke (1 St = 10⁻⁴ m²/s).
• Newtonian fluids: μ is independent of shear rate dv/dy (water, air, most light oils). Non-Newtonian fluids (paint, blood, cement slurry) have μ that varies with shear rate.
• For liquids: μ decreases as temperature rises (molecules move apart, reducing intermolecular bonds). For gases: μ increases with temperature (molecular momentum exchange increases).
• On a board exam, unless stated otherwise, assume a linear (Couette) velocity profile: dv/dy = V/h where V is plate velocity and h is film thickness.

Definitions

Section Title

2. Viscosity

Common Mistakes

• Substituting ν where μ is required (or vice versa): check units — Pa·s vs m²/s.
• Using h in mm without converting to m in dv/dy = V/h.
• Assuming viscosity increases with temperature for liquids — it decreases (opposite of gases).
• Confusing the unit poise (P) with Pascal: 1 P = 0.1 Pa·s, not 0.1 Pa.

Formulas

Exam Tips

• The formula h = 4σcosθ/(γd) is the most-tested form on PRC boards — commit it to memory.
• Smaller tube diameter → larger capillary rise: h is inversely proportional to d.
• When the problem says 'clean glass tube' and the liquid is water, assume θ = 0° automatically.
• Capillary rise h for water in a 1 mm glass tube ≈ 29.7 mm — a useful sanity-check benchmark.

Key Points

• Surface tension σ (sigma) is the force per unit length (N/m) or equivalently energy per unit area (J/m²) at a liquid surface, caused by unbalanced intermolecular cohesive forces.
• Surface tension causes capillary rise (wetting liquids: water on glass, θ < 90°) or capillary depression (non-wetting: mercury on glass, θ > 90°).
• Capillary rise formula: h = 4σ cos θ / (γ d), where d is the tube diameter.
• Alternatively with radius r: h = 2σ cos θ / (γ r). Both forms are used on boards — know both.
• For water on clean glass: θ ≈ 0°, so cos θ = 1 (maximum rise). For mercury: θ ≈ 140°, cos θ is negative → depression.
• Surface tension decreases with increasing temperature.
• Practical significance: capillary action in fine soils (relevant to soil–water interaction, drainage design), drop formation, and wetting of concrete surfaces.

Definitions

Section Title

3. Surface Tension and Capillarity

Common Mistakes

• Using radius r in the formula h = 4σcosθ/(γd) — the factor of 4 goes with diameter d; use factor 2 with radius r.
• Forgetting to convert d from mm to m before substituting into the formula.
• Treating σ as N/m² (pressure units) — surface tension is N/m (force per unit length), not a stress.
• Using cos(0°) = 0 instead of cos(0°) = 1 for clean water-glass contact.

Formulas

Exam Tips

• Memorize: E_B of water = 2.2 GPa. This value is given in most board problems but sometimes it is the answer check.
• The fractional compression Δρ/ρ = Δp/E_B is small for water — if you get more than 1%, recheck your units.
• Board exam context: compressibility appears in Mach-number and water-hammer theory questions, not in pipe-flow or channel-flow calculations.

Key Points

• The bulk modulus of elasticity E_B (also written K) measures a fluid's resistance to compression: a large E_B means nearly incompressible.
• E_B = -dp / (dV/V) = dp / (dρ/ρ). Units: Pa or GPa.
• Water: E_B ≈ 2.2 GPa — very large, so water is treated as incompressible in most civil engineering hydraulics.
• Gases have very small E_B (highly compressible); liquids are nearly incompressible.
• The fractional change in density: Δρ/ρ = Δp / E_B.
• Water hammer (hydraulic transient) is the one situation where water's compressibility cannot be ignored — wave speed c = √(E_B/ρ).
• Compressibility κ = 1/E_B (the inverse of bulk modulus).

Definitions

Section Title

4. Compressibility and Bulk Modulus

Common Mistakes

• Confusing the sign convention: volume decreases when pressure increases, hence the negative sign in E_B = -dp/(dV/V).
• Expressing E_B in MPa for water instead of GPa — E_B ≈ 2200 MPa = 2.2 GPa.
• Applying incompressibility assumption (E_B → ∞) to a water-hammer problem where compressibility is the whole point.

Formulas

Exam Tips

• Board exam clue: whenever a problem mentions pump suction, high-speed flow, or low-pressure zones, think vapor pressure and cavitation.
• Standard p_v values to know: water at 20 °C = 2.34 kPa; at 100 °C = 101.3 kPa.
• NPSH problems appear in the fluid machinery portion — link vapor pressure to pump selection criteria.

Key Points

• Vapor pressure p_v is the absolute pressure at which a liquid boils at a given temperature. It increases with temperature.
• At 20 °C, p_v of water ≈ 2.34 kPa (absolute). At 100 °C, p_v = 101.325 kPa (atmospheric) — water boils.
• Cavitation occurs when the local absolute pressure in a flowing liquid drops to or below p_v. The liquid locally vaporizes, forming vapor bubbles.
• When bubbles collapse (implode) on solid surfaces, they cause high-pressure pulses leading to pitting, erosion, and noise — cavitation damage.
• Critical locations for cavitation: pump suction inlets, turbine runner blades, valves operating at high velocity, pipe bends at high elevation.
• Net Positive Suction Head (NPSH) design criterion in pump selection is a direct application of vapor pressure.
• Thoma's cavitation parameter σ_T = (NPSH_available) / H_pump connects vapor pressure to pump performance.

Definitions

Section Title

5. Vapor Pressure and Cavitation

Common Mistakes

• Using gauge pressure instead of absolute pressure when comparing local pressure to p_v — always use absolute.
• Thinking cavitation is caused by high temperature alone — it is caused by low local pressure (which can occur even at 20 °C in high-velocity zones).
• Confusing boiling (due to heating at constant pressure) with cavitation (due to pressure drop at roughly constant temperature) — both reach the saturation curve but by different paths.