Properties of Fluids - Revision Notes
This chapter lays the quantitative foundation for all of Hydraulics and Fluid Mechanics. Every downstream topic — hydrostatic pressure, pipe flow, open-channel flow, pump selection — depends on knowing exactly what a fluid weighs, how it resists shear, how it clings to surfaces, and when it is about to vaporize. Board examinees are expected to compute specific weight, specific gravity, viscous shear stress, capillary rise, bulk modulus compression, and to identify cavitation conditions — all in SI units. Master the six core formulas here and the rest of Hydraulics becomes arithmetic.
1. Density, Specific Weight, and Specific Gravity
Formulas
γ = ρ g
Example
Oil with ρ = 850 kg/m³: γ = 850 × 9.81 = 8338.5 N/m³ = 8.34 kN/m³
Variables
γ = specific weight (N/m³); ρ = density (kg/m³); g = 9.81 m/s²
Application
Convert between mass-based and weight-based descriptions of a fluid
s = ρ / ρ_w = γ / γ_w
Example
Gasoline s = 0.72 → ρ = 720 kg/m³; γ = 0.72 × 9810 = 7063 N/m³
Variables
s = specific gravity (dimensionless); ρ_w = 1000 kg/m³; γ_w = 9810 N/m³
Application
Identify fluid type; quickly compute ρ or γ from a given s
v_s = 1 / ρ
Example
Water: v_s = 1/1000 = 0.001 m³/kg
Variables
v_s = specific volume (m³/kg); ρ = density (kg/m³)
Application
Used in compressibility and thermodynamic fluid problems
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
Density (ρ)
Mass of fluid per unit volume, in kg/m³.
Importance
Base property from which γ and s are derived; appears in every momentum and energy equation.
Specific Weight (γ)
Weight (gravitational force) of fluid per unit volume, in N/m³ or kN/m³.
Importance
Direct link to hydrostatic pressure: p = γh.
Specific Gravity (s)
Dimensionless ratio of fluid density to water density at 4 °C.
Importance
Allows rapid comparison of fluids; used in manometer and buoyancy problems.
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.
2. Viscosity
Formulas
τ = μ (dv/dy)
Example
Oil film h = 2 mm, plate velocity V = 3 m/s, μ = 0.005 Pa·s: τ = 0.005 × (3/0.002) = 7.5 Pa
Variables
τ = shear stress (Pa = N/m²); μ = dynamic viscosity (Pa·s); dv/dy = velocity gradient (s⁻¹ or m/s per m)
Application
Compute shear stress in a fluid film between a moving plate and a fixed surface
ν = μ / ρ
Example
Water at 20 °C: μ = 0.001 Pa·s, ρ = 998 kg/m³ → ν = 0.001/998 = 1.002 × 10⁻⁶ m²/s
Variables
ν = kinematic viscosity (m²/s); μ = dynamic viscosity (Pa·s); ρ = density (kg/m³)
Application
Reynolds number Re = ρVD/μ = VD/ν uses kinematic viscosity directly
F = μ A (V/h)
Example
A = 0.3 m², V = 1.5 m/s, h = 0.5 mm = 0.0005 m, μ = 0.1 Pa·s: F = 0.1 × 0.3 × (1.5/0.0005) = 90 N
Variables
F = shear force (N); A = plate area (m²); V = velocity (m/s); h = film thickness (m)
Application
Force required to slide a plate over a viscous film at constant speed
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
Dynamic (Absolute) Viscosity (μ)
The proportionality constant between shear stress and velocity gradient in Newton's Law of Viscosity. Units: Pa·s.
Importance
Appears directly in shear force and pipe-flow head-loss formulas (Hagen-Poiseuille).
Kinematic Viscosity (ν)
Dynamic viscosity divided by density: ν = μ/ρ. Units: m²/s.
Importance
Used in Reynolds number, Darcy-Weisbach friction factor charts, and dimensional analysis.
Newtonian Fluid
A fluid for which μ is constant regardless of shear rate — the shear stress vs. velocity gradient plot is a straight line through the origin.
Importance
All standard hydraulics formulas assume Newtonian behavior; board exams use water and oil unless stated.
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.
3. Surface Tension and Capillarity
Formulas
h = 4σ cos θ / (γ d)
Example
Water, d = 2 mm, σ = 0.0728 N/m, θ = 0°: h = 4(0.0728)(1)/(9810 × 0.002) = 0.2912/19.62 = 0.01484 m = 14.84 mm
Variables
h = capillary rise or depression (m); σ = surface tension (N/m); θ = contact angle (degrees); γ = specific weight of liquid (N/m³); d = tube inside diameter (m)
Application
Height to which a liquid rises (or is depressed) in a capillary tube
h = 2σ cos θ / (γ r)
Example
Same problem with r = 1 mm = 0.001 m: h = 2(0.0728)(1)/(9810 × 0.001) = 0.1456/9.81 = 0.01484 m ✓
Variables
r = tube inside radius (m) = d/2
Application
Alternative form using radius — verify which form the problem uses by checking given data
p_inside - p_outside = 4σ / d (spherical droplet)
Example
Water droplet d = 0.5 mm: Δp = 4(0.0728)/0.0005 = 582 Pa
Variables
Pressure difference across a spherical liquid droplet or bubble surface
Application
Explains excess pressure inside a droplet — occasionally tested in theory questions
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
Surface Tension (σ)
The tensile force per unit length acting along the surface of a liquid due to unequal cohesive forces on surface molecules. Units: N/m.
Importance
Governs capillary action; standard value for water at 20 °C: σ ≈ 0.0728 N/m.
Contact Angle (θ)
The angle between the liquid–solid interface and the liquid–air interface at the line of contact. θ < 90° → wetting (rise); θ > 90° → non-wetting (depression).
Importance
Determines the sign and magnitude of capillary action.
Capillarity
The phenomenon of liquid rising or falling in a narrow tube due to the interplay of adhesion (liquid to solid), cohesion (liquid to liquid), and surface tension.
Importance
Explains moisture movement in soils and building materials — relevant to seepage and waterproofing design.
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.
4. Compressibility and Bulk Modulus
Formulas
E_B = dp / (dρ/ρ) = -dp / (dV/V)
Example
Water under Δp = 2 MPa, E_B = 2.2 GPa: Δρ/ρ = 2×10⁶ / 2.2×10⁹ = 9.09 × 10⁻⁴ (≈ 0.091% compression)
Variables
E_B = bulk modulus (Pa or GPa); dp = pressure increment (Pa); dρ/ρ = fractional density change; dV/V = fractional volume change (negative because volume decreases with pressure)
Application
Compute the fractional compression of a liquid under a given pressure increase
c = √(E_B / ρ)
Example
Water: c = √(2.2×10⁹ / 1000) = √(2.2×10⁶) = 1483 m/s
Variables
c = speed of sound / pressure wave in the fluid (m/s); E_B in Pa; ρ in kg/m³
Application
Water-hammer wave speed in rigid pipes
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
Bulk Modulus of Elasticity (E_B)
The ratio of an applied pressure increment to the resulting fractional change in volume (or density). A measure of a fluid's stiffness against compression. Units: Pa (GPa for liquids).
Importance
Confirms water is effectively incompressible for steady-flow problems; critical for water-hammer calculations.
Compressibility (κ)
The reciprocal of the bulk modulus: κ = 1/E_B. Units: Pa⁻¹.
Importance
A high κ means easily compressible (gases); a low κ means nearly incompressible (liquids).
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.
5. Vapor Pressure and Cavitation
Formulas
p_v increases with T (lookup table)
Example
Water at 30 °C: p_v ≈ 4.24 kPa abs. A pump suction line at that temperature must maintain p_suction > 4.24 kPa abs.
Variables
p_v = vapor pressure (kPa abs); T = fluid temperature (°C)
Application
Determine cavitation risk: if p_local ≤ p_v, cavitation occurs
NPSH_available = (p_s / γ) + (V_s² / 2g) - (p_v / γ)
Example
If p_s/γ = 8 m, V_s²/2g = 0.5 m, p_v/γ = 0.24 m: NPSH_a = 8 + 0.5 - 0.24 = 8.26 m
Variables
p_s = suction pressure; V_s = suction velocity; p_v = vapor pressure; γ = specific weight
Application
Ensures the pump does not cavitate; NPSH_available must exceed NPSH_required from pump manufacturer
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
Vapor Pressure (p_v)
The saturation pressure of a liquid at a given temperature — the pressure at which it transitions from liquid to vapor. Always expressed as absolute pressure.
Importance
Sets the lower bound on permissible pressure in any hydraulic system; governs cavitation onset.
Cavitation
The formation of vapor cavities (bubbles) in a liquid when local pressure drops to the vapor pressure, followed by violent collapse when pressure recovers.
Importance
Causes structural damage to pumps, turbines, and pipe fittings; avoided through proper NPSH design.
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.
Connections
- Specific weight γ directly feeds into hydrostatic pressure: p = γh — Chapter 2 (Pressure and Pressure Measurement).
- Density ρ appears in the continuity equation (ρAV = const) and Bernoulli's equation — Chapter 3 (Fluid Kinematics).
- Dynamic viscosity μ is the core parameter of Hagen-Poiseuille pipe-flow theory and the Darcy-Weisbach friction factor — Chapter 5 (Pipe Flow).
- Kinematic viscosity ν appears in the Reynolds number Re = VD/ν, which determines laminar vs turbulent flow regimes — Chapter 5.
- Vapor pressure p_v is critical for Net Positive Suction Head (NPSH) calculations in pump selection — Chapter 8 (Fluid Machinery).
- Bulk modulus E_B governs water-hammer wave speed c = √(E_B/ρ) — Chapter 9 (Hydraulic Transients).
- Surface tension and capillarity connect to soil capillarity and seepage in Geotechnical Engineering — relevant cross-subject link.
- Specific gravity s is used in manometer equations involving multiple fluids — Chapter 2 (Pressure Measurement).
- The concept of Newtonian vs non-Newtonian fluid behavior sets the scope of standard hydraulics formulas — all subsequent chapters implicitly assume Newtonian fluids.
Quick review questions
An unknown liquid has a specific gravity of 0.92. What are its density and specific weight?
Answer
ρ = 920 kg/m³; γ = 9025.2 N/m³ ≈ 9.03 kN/m³
ρ = s × ρ_w = 0.92 × 1000 = 920 kg/m³. γ = ρg = 920 × 9.81 = 9025.2 N/m³. Alternatively, γ = s × γ_w = 0.92 × 9810 = 9025.2 N/m³.
A flat plate of area 0.4 m² slides over a 0.8 mm oil film at a velocity of 2 m/s. If μ = 0.08 Pa·s, what force is required?
Answer
F = 80 N
τ = μ(V/h) = 0.08 × (2/0.0008) = 0.08 × 2500 = 200 Pa. F = τ × A = 200 × 0.4 = 80 N.
Water (σ = 0.0728 N/m, θ = 0°, γ = 9810 N/m³) is placed in a glass tube of diameter 1 mm. How high does it rise?
Answer
h ≈ 29.7 mm
h = 4σcosθ/(γd) = 4(0.0728)(cos 0°)/(9810 × 0.001) = 0.2912/9.81 = 0.02969 m ≈ 29.7 mm. Note: smaller tube → higher rise.
A pressure of 4 MPa is applied to water with E_B = 2.2 GPa. What is the fractional change in density?
Answer
Δρ/ρ = 1.818 × 10⁻³ (≈ 0.182%)
Δρ/ρ = Δp/E_B = 4 × 10⁶ / 2.2 × 10⁹ = 1.818 × 10⁻³. Water is nearly incompressible — less than 0.2% compression under 4 MPa.
Distinguish dynamic viscosity from kinematic viscosity and give the SI unit for each.
Answer
μ (dynamic) = Pa·s; ν (kinematic) = m²/s; related by ν = μ/ρ
Dynamic viscosity μ measures absolute resistance to shear (Pa·s). Kinematic viscosity ν = μ/ρ factors out density (m²/s). Kinematic viscosity is used in Reynolds number and Moody-chart calculations.
A Newtonian oil has μ = 0.3 Pa·s and ρ = 900 kg/m³. Find its kinematic viscosity in m²/s and in stokes.
Answer
ν = 3.33 × 10⁻⁴ m²/s = 3.33 stokes
ν = μ/ρ = 0.3/900 = 3.33 × 10⁻⁴ m²/s. Since 1 stoke = 10⁻⁴ m²/s: ν = 3.33 St.
What is cavitation and at what pressure does it initiate for water at 20 °C?
Answer
Cavitation initiates when local absolute pressure drops to the vapor pressure: p_v ≈ 2.34 kPa (abs) at 20 °C.
Cavitation is the vaporization of liquid due to local pressure falling to the vapor pressure, forming vapor bubbles that collapse violently. At 20 °C, water's vapor pressure is approximately 2.34 kPa absolute.
Mercury (θ = 140°, σ = 0.48 N/m, s = 13.6) is in a glass tube of d = 3 mm. Does it rise or fall, and by how much?
Answer
Mercury falls (is depressed) by approximately 9.8 mm.
γ_Hg = 13.6 × 9810 = 133,416 N/m³. cos(140°) = −0.766. h = 4(0.48)(−0.766)/(133416 × 0.003) = −1.4707/400.25 = −0.003674 m ≈ −3.7 mm. Recheck: h = 4(0.48)(0.766)/(13.6 × 9810 × 0.003) = 1.4707/400.25 = 0.00367 m = 3.67 mm depression. Magnitude ≈ 3.7 mm depression (sign indicates fall).
Diagrams
Properties of Fluids — Chapter Overview Mind Map
A hierarchical mind map showing all five major property groups and their key sub-concepts. Use this as a navigation guide when reviewing the chapter and as a final-night summary tool before the board exam.
Step-by-Step: Solving a Viscous Shear Force Problem
A decision-tree flowchart guiding the solver through the viscous shear force calculation: identify the velocity profile, compute the gradient, apply Newton's Law of Viscosity, and obtain the force. Includes a critical unit-check branch to avoid common exam errors.
Capillary Rise Calculation Procedure
A procedural flowchart for solving capillary rise/depression problems. Shows how to handle missing contact angle, unit conversion of diameter, and the sign convention distinguishing rise from depression.
Phase Transition and Cavitation: Pressure-Temperature States of Water
A state diagram illustrating the phase transition pathway from liquid water to vapor and the cavitation cycle. Clarifies that cavitation is a pressure-driven (not only temperature-driven) phenomenon, helping reviewees distinguish boiling from cavitation.
Relative Bulk Modulus: Water vs Common Engineering Fluids
A proportional pie chart comparing the bulk moduli of common engineering fluids. Mercury is extremely stiff (25 GPa), glycerin is stiffer than water, and hydraulic oil is somewhat more compressible. All liquids have far larger E_B than gases, confirming the incompressibility assumption.
Vapor Pressure of Water vs Temperature — Key Reference Points
A timeline showing how vapor pressure of water rises rapidly with temperature. At 20 °C (typical Philippine ambient temperature), p_v = 2.34 kPa absolute — this is the critical lower-bound pressure for cavitation-free pump operation in Philippine tropical conditions.
Exam strategy
On PRC Civil Engineer board examinations, Properties of Fluids questions account for approximately 10–15% of the Hydraulics section. The highest-yield items are: (1) computing shear force from viscosity given plate area and film thickness — always draw the diagram, label h and V, confirm linear profile; (2) specific gravity/density/specific weight chain calculations — write γ = ρg and s = ρ/1000 immediately; (3) capillary rise using h = 4σcosθ/(γd) — convert d to meters, and check whether rise or depression (sign of cosθ); (4) bulk modulus compression — divide Δp by E_B in the same units. Time management: these problems are largely formula-plug items; if you cannot get an answer in 90 seconds, flag and move on. Common time-wasters are unit conversions (mm to m, poise to Pa·s, stokes to m²/s) — prepare a quick unit-conversion mental checklist: h in m, d in m, μ in Pa·s, σ in N/m, E_B in Pa or GPa consistent with Δp. Always sanity-check: capillary rise for water in a 1 mm tube ≈ 30 mm; viscous shear stress in typical oil films is on the order of Pa to kPa; water compression under MPa-level pressures is less than 0.1%.