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Fluid Mechanics · Chapter 2 of 10 · Beginner

Properties of Fluids

Density, viscosity, vapor pressure, surface tension: a handful of properties set how a fluid resists flow, when it boils, and how it climbs a tube. Get them right and every later calculation follows.

Readiness check

This chapter quantifies the fluid properties. Tick only what you can do closed-notes.

Recall density ρ and its units.
Compute a gradient (rate of change with distance).
Use force = stress × area.
Recall that liquids boil when pressure meets vapor pressure.
Look up a property at a stated temperature.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsReview the Reynolds number and units in Chapter 1.

3 or more weak itemsRevisit basic mechanics and units first.

The core idea

Viscosity is the property that resists flow: the shear stress is proportional to the velocity gradient, τ = μ(du/dy). Density, vapor pressure, and surface tension complete the picture.

τ = μ (du/dy)ν = μ/ρ, SG = ρ/ρ_watercapillary: h = 2σcosθ/(ρgr)

A fluid's resistance to shear is its viscosity μ; Newton's law of viscosity says the shear stress equals μ times the velocity gradient across the flow. Dividing by density gives the kinematic viscosity ν that appears in the Reynolds number. Two more properties matter at boundaries: vapor pressure sets when a liquid cavitates (boils due to low pressure), and surface tension drives capillary rise and droplet behaviour. Liquids are nearly incompressible; gases are not.

The skill works when: you read the right property at the stated temperature and apply τ = μ(du/dy) for the shear.

The skill breaks down when: a property is used at the wrong temperature, or viscosity is confused with density.

The concept. Newton's law of viscosity: for a Newtonian fluid the shear stress is a straight line through the origin with slope μ. Non-Newtonian fluids curve, thinning or thickening as they are sheared harder.

The skills, taught in order

Five properties carry through the whole course: density, viscosity, vapor pressure, surface tension, and compressibility.

2.1 Density and specific gravity

Density ρ is mass per volume; specific weight is γ = ρg; specific gravity SG = ρ/ρ_water is the dimensionless ratio to water (1000 kg/m³). Liquid density barely changes with pressure, but gas density follows the ideal-gas law, ρ = P/(RT).

2.2 Viscosity

Viscosity measures resistance to shear. Newton's law of viscosity, τ = μ(du/dy), relates the shear stress to the velocity gradient, with μ the dynamic viscosity (Pa·s). The kinematic viscosity is ν = μ/ρ (m²/s). Newtonian fluids (water, air, oil) have constant μ; others thin or thicken with shear rate. Liquid viscosity falls with temperature; gas viscosity rises.

2.3 Vapor pressure and cavitation

The vapor pressure P_v is the pressure at which a liquid boils at a given temperature. If the local pressure in a flow drops to P_v (say at a pump inlet or propeller tip), vapor bubbles form and collapse violently, called cavitation, which erodes metal and must be avoided.

2.4 Surface tension and capillarity

Surface tension σ (N/m) is the energy of a liquid surface, responsible for droplets and bubbles. In a small tube it drives capillary rise h = 2σcosθ/(ρgr), where θ is the contact angle and r the tube radius. The effect grows as the tube narrows.

2.5 Compressibility

The bulk modulus E_v = −dP/(dV/V) measures resistance to volume change. For water it is about 2.2 GPa, so liquids are treated as incompressible. Gases compress easily, but a gas flow is still treated as incompressible when its Mach number is below about 0.3.

Property (at 20 °C)	Water	Air (1 atm)
Density ρ (kg/m³)	998	1.20
Dynamic viscosity μ (Pa·s)	1.00×10⁻³	1.82×10⁻⁵
Kinematic viscosity ν (m²/s)	1.00×10⁻⁶	1.52×10⁻⁵
Vapor pressure / surface tension	2.34 kPa / 0.0728 N/m	n/a (gas)

Engineering connection: viscosity sets pipe friction and drag, vapor pressure limits pump suction, and surface tension matters in small channels, sprays, and coatings.

Worked example 1: viscous drag on a plate

A 0.3 m × 0.3 m plate slides at 1 m/s over a 1 mm thick film of oil (μ = 0.8 Pa·s) on a fixed surface. Find the shear stress and the force needed to move the plate.

Figure 1. The oil sticks to both the plate and the fixed surface (no-slip), so a linear velocity profile develops. The shear stress is μ times that gradient, and the force is the stress over the plate area.

ProblemFind the shear stress and driving force for the plate in Figure 1.
Given / findA = 0.3 × 0.3 m², V = 1 m/s, h = 0.001 m, μ = 0.8 Pa·s. Find τ and F.
AssumptionsNewtonian oil, linear velocity profile across the thin film, no-slip at both surfaces.
ModelNewton's law with a linear gradient du/dy = V/h, then F = τA.
Equationsτ = μ(du/dy) = μV/h F = τA
Solveτ = 0.8 × (1/0.001) = 800 Pa. A = 0.09 m², so F = 800 × 0.09 = 72 N.
CheckUnits: (Pa·s)(1/s) = Pa for τ, and Pa·m² = N for F. A thinner film (smaller h) would raise both, which is why bearings run on very thin, low-viscosity films to limit drag.
ConclusionThe whole resistance comes from viscosity acting across the velocity gradient. This is the model behind lubrication, journal bearings, and viscometers.

Result. Shear stress 800 Pa; force to move the plate 72 N.

Worked example 2: capillary rise

A clean glass tube of 1 mm diameter stands in water at 20 °C (σ = 0.0728 N/m, ρ = 998 kg/m³, contact angle ≈ 0°). How high does the water rise in the tube?

Figure 2. Surface tension pulls the wetting water up the narrow tube until its weight balances the upward pull. The rise grows as the tube narrows (h ∝ 1/r).

ProblemFind the capillary rise of water in the tube in Figure 2.
Given / findd = 1 mm so r = 0.5 mm, σ = 0.0728 N/m, ρ = 998 kg/m³, θ ≈ 0°. Find h.
AssumptionsClean tube (θ = 0, fully wetting), water at 20 °C, static column.
ModelBalance the surface-tension pull around the rim against the weight of the raised column.
Equationsh = 2σcosθ/(ρgr)
Solveh = (2 × 0.0728 × cos 0°)/(998 × 9.81 × 0.0005) = 0.1456/4.895 = 0.0297 m = 29.7 mm.
CheckThe rise is inversely proportional to radius, so a 0.5 mm tube would lift water about 59 mm. This is why capillary effects dominate in thin tubes and porous media but are invisible in a wide beaker.
ConclusionSurface tension produces measurable rise only at small scales. It matters in micro-channels, wicks, and soils, and is negligible in ordinary pipes.

Result. The water rises 29.7 mm in the 1 mm tube.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Confusing μ and ρ	Reynolds number or drag wrong	"Is this resistance to shear or mass per volume?"	μ is viscosity (Pa·s); ρ is density (kg/m³); ν = μ/ρ links them.
Ignoring temperature	Wrong viscosity used	"At what temperature is this property?"	Always read properties at the stated temperature.
Forgetting cavitation	Pump or valve erodes	"Does local pressure drop to P_v?"	Keep the local pressure above the vapor pressure.
Capillarity at large scale	Surface tension included in a wide pipe	"How small is the length scale?"	Capillary rise scales as 1/r; negligible in ordinary pipes.

Practice ladder

Level 1 · Direct skill

Oil (μ = 0.25 Pa·s) fills a 2 mm gap; the upper plate moves at 0.5 m/s. Find the shear stress.

Show answer

τ = μV/h = 0.25 × (0.5/0.002) = 0.25 × 250 = 62.5 Pa.

Level 2 · Mixed concept

A fluid has SG = 0.85 and ν = 5×10⁻⁵ m²/s. Find its density and dynamic viscosity.

Show answer

ρ = SG × 1000 = 850 kg/m³. μ = ρν = 850 × 5×10⁻⁵ = 0.0425 Pa·s.

Level 3 · Independent problem

Water at 20 °C flows through a region where the pressure drops to 2 kPa absolute. Does it cavitate?

Show answer

The vapor pressure of water at 20 °C is 2.34 kPa. Since the local pressure (2 kPa) is below P_v, the water boils locally: yes, it cavitates. Keeping pressure above 2.34 kPa would prevent it.

Level 4 · Transfer to real engineering

Find a real situation where one of these properties dominates (oil viscosity in a cold engine, cavitation on a propeller, capillary action in a paper towel). Estimate the relevant property and its effect.

What good work looks like

The governing property identified and read at the right temperature, a quantitative estimate (shear stress, vapor-pressure margin, or capillary rise), and a clear physical consequence.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used viscosity, not density, in this shear-stress calculation."

"Give me five fluids; I will rank their viscosities and explain the temperature trend."

"Compute the drag force." Applying τ = μ(du/dy) yourself is the skill.

"Will it cavitate?" Comparing local pressure to vapor pressure is the point.

Portfolio task

Measure or estimate one fluid property in a real setting: a viscous drag, a cavitation margin, or a capillary rise, with the property read at the correct temperature.

Must include: the property with its temperature, the governing equation, and a verified result with a physical interpretation.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write Newton's law of viscosity.

τ = μ(du/dy): shear stress equals viscosity times the velocity gradient.

2. Relate dynamic and kinematic viscosity.

ν = μ/ρ (m²/s); ν appears in the Reynolds number.

3. What is cavitation?

Local boiling when the pressure drops to the vapor pressure; the bubbles collapse and erode surfaces.

4. Give the capillary-rise formula.

h = 2σcosθ/(ρgr); the rise grows as the tube narrows.

5. When can a gas flow be treated as incompressible?

When the Mach number is below about 0.3.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the drag force and capillary rise from a blank page.

+3 daysOne viscosity and one cavitation or capillary problem.

+7 daysCarry density and pressure into fluid statics, Chapter 3.

+30 daysReuse viscosity for pipe friction and drag later.

Textbook mapping

Item	Mapping
Primary source	Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, Chapter 2 (Properties of Fluids)
Cross-reference	White, Ch. 1 · Munson, Ch. 1
Core topics	2.1 Density · 2.2 Viscosity · 2.3 Vapor pressure and cavitation · 2.4 Surface tension · 2.5 Compressibility
Engineering connection	Lubrication, pump cavitation limits, sprays, and micro-channels.
Read next	Chapter 3: Pressure and Fluid Statics.