← Fluid Mechanics & Hydraulics · GATE Civil

Chapter 1 of 10

Fluid Properties & Statics

In the GATE Civil syllabus under Fluid Mechanics & Hydraulics · 2 parts

Properties of Fluids · Fluid Statics & Pressure Measurement

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (15 sections)

Part 1 of 2

Properties of Fluids

Last reviewed 16 Sept 2026 · 5 min read

What a fluid is

A fluid is a substance that deforms continuously under the action of a shear stress, however small. Liquids and gases are fluids. A solid resists shear by a finite deformation; a fluid at rest cannot sustain any shear stress.

Liquids have a definite volume and a free surface; nearly incompressible.
Gases fill their container; highly compressible.

Continuum assumption: fluid properties are treated as continuous functions of position, ignoring molecular structure — valid when the dimensions of interest are much larger than the mean free path of molecules.

Mass and weight properties

Property	Definition	Water at 4 °C (standard)
Mass density $ρ$	Mass per unit volume (kg/m³)	1000 kg/m³
Specific weight $w$ or $γ$	Weight per unit volume $= ρ g$ (N/m³)	9810 N/m³
Specific volume $v$	Volume per unit mass $= 1/ ρ$ (m³/kg)	0.001 m³/kg
Specific gravity $S$ (relative density)	$ρ_{f l u i d} / ρ_{w a t er}$ (liquids)	1.0 (mercury 13.6)

Viscosity

Viscosity is the property by which a fluid resists relative motion (shear) between adjacent layers.

FormulaNewton's law of viscosity

τ = μ \frac{d u}{d y}

$τ$ = shear stress; $d u / d y$ = velocity gradient (rate of shear strain); $μ$ = dynamic (absolute) viscosity.

Kinematic viscosity: $ν = \frac{μ}{ρ}$

Quantity	SI unit	CGS unit
Dynamic viscosity $μ$	N·s/m² = Pa·s	poise (1 poise = 0.1 N·s/m²; 1 centipoise = 0.001 N·s/m²)
Kinematic viscosity $ν$	m²/s	stoke (1 stoke = 1 cm²/s = $1 0^{- 4}$ m²/s)

Water at 20 °C: $μ$ ≈ 1.0 × 10⁻³ N·s/m² (1 centipoise), $ν$ ≈ 1.0 × 10⁻⁶ m²/s.

Effect of temperature

Liquids: viscosity decreases with temperature (cohesive forces between molecules weaken).
Gases: viscosity increases with temperature (molecular momentum exchange increases).

Newtonian and non-Newtonian fluids

Fluid	Shear stress–rate relation	Examples
Newtonian	Linear through the origin (constant $μ$ )	Water, air, most oils
Non-Newtonian — shear thinning (pseudoplastic)	Apparent viscosity decreases with shear rate	Paints, blood, polymer solutions
Shear thickening (dilatant)	Apparent viscosity increases	Starch–water, some suspensions
Bingham plastic	Needs a yield stress before flowing, then linear	Toothpaste, sewage sludge, drilling mud
Thixotropic / rheopectic	Viscosity changes with time of shearing	Some gels, printer ink
Ideal fluid	No viscosity, incompressible (theoretical)	—

Compressibility and bulk modulus

K = - \frac{d p}{d V / V} = \frac{d p}{d ρ / ρ}, compressibility = \frac{1}{K}

Water: $K$ ≈ 2.1–2.2 × 10⁹ N/m² — liquids are treated as incompressible except in problems such as water hammer. Gases: for isothermal compression $K = p$ ; for adiabatic compression $K = k p$ ( $k$ = ratio of specific heats).

Surface tension

At the free surface of a liquid, molecules are pulled inward, so the surface behaves like a stretched membrane. Surface tension $σ$ is the tensile force per unit length of a line on the surface (N/m). Water–air at 20 °C: about 0.073 N/m; mercury about 0.48 N/m.

FormulaPressure due to surface tension

Liquid droplet (diameter $d$ ): $Δ p = \frac{4 σ}{d}$
Soap bubble (two surfaces): $Δ p = \frac{8 σ}{d}$
Liquid jet (cylinder): $Δ p = \frac{2 σ}{d}$

Capillarity

Rise or fall of a liquid in a small tube due to adhesion and cohesion:

h = \frac{4 σ cos θ}{ρ g d}

Water in glass ( $θ \approx 0°$ ): adhesion > cohesion → liquid rises, concave meniscus.
Mercury in glass ( $θ \approx 130°$ –140°): cohesion > adhesion → liquid is depressed, convex meniscus.

Capillary errors are significant in small piezometer tubes; tubes should be at least about 6 mm in diameter to limit error.

Vapour pressure and cavitation

Vapour pressure — partial pressure of vapour in equilibrium with its liquid; rises with temperature (water ≈ 2.34 kPa at 20 °C; equals atmospheric pressure at 100 °C).
When local absolute pressure falls to the vapour pressure, the liquid boils at that temperature, forming vapour bubbles. When these bubbles move into higher-pressure regions they collapse violently — cavitation — causing noise, vibration, loss of efficiency and pitting erosion of pump impellers, turbine runners, spillway surfaces and siphons.

Worked examples

Worked ExampleExample 1 — viscous shear

A plate 0.025 mm from a fixed plate moves at 60 cm/s and needs a force of 2 N per m² to maintain the speed. Find the dynamic viscosity of the fluid between them.

Solution. $\frac{d u}{d y} = \frac{0.6}{0.025 \times 1 0 ^{- 3}} = 24000$ s⁻¹ $μ = \frac{τ}{d u / d y} = \frac{2}{24 000} = 8.33 \times 1 0^{- 5}$ N·s/m² $= 8.33 \times 1 0^{- 4} poise$

Worked ExampleExample 2 — kinematic viscosity

An oil has $μ$ = 0.05 poise and specific gravity 0.9. Find $ν$ in stokes and m²/s.

Solution. $μ = 0.005$ N·s/m²; $ρ = 900$ kg/m³. $ν = \frac{0.005}{900} = 5.56 \times 1 0^{- 6}$ m²/s $= 0.0556 stoke$

Worked ExampleExample 3 — pressure inside a droplet

Find the excess pressure inside a water droplet of 0.04 mm diameter ( $σ$ = 0.0725 N/m).

Solution. $Δ p = \frac{4 \times 0.0725}{0.04 \times 1 0 ^{- 3}} = 7250 N/ m^{2}$

Worked ExampleExample 4 — capillary rise

Find the capillary rise of water in a 3 mm glass tube ( $σ$ = 0.0735 N/m, $θ$ = 0).

Solution. $h = \frac{4 \times 0.0735}{1000 \times 9.81 \times 0.003} = 0.00999$ m $= 10 mm$

Worked ExampleExample 5 — bulk modulus

Water's volume decreases by 0.15% when pressure rises from 70 N/cm² to 130 N/cm². Find $K$ .

Solution. $d p = 60$ N/cm² $= 6 \times 1 0^{5}$ N/m²; $d V / V = 0.0015$ . $K = \frac{6 \times 1 0 ^{5}}{0.0015} = 4 \times 1 0^{8} N/ m^{2}$

Frequently tested points

A fluid deforms continuously under any shear stress.
$τ = μ d u / d y$ ; $ν = μ / ρ$ ; 1 poise = 0.1 N·s/m²; 1 stoke = 10⁻⁴ m²/s.
Liquid viscosity falls with temperature; gas viscosity rises.
Bingham plastic needs a yield stress; ideal fluid has zero viscosity.
Droplet $4 σ / d$ ; bubble $8 σ / d$ ; jet $2 σ / d$ .
Capillary rise $4 σ cos θ / (ρ g d)$ ; mercury depressed.
Cavitation when pressure drops to vapour pressure.

Common MistakeCommon mistakes

Using $8 σ / d$ for a liquid droplet (that is a soap bubble with two surfaces).
Mixing poise with N·s/m² (factor of 10).
Stating that gas viscosity decreases with temperature.

Revision SummaryChapter summary

Fluids cannot resist shear at rest; the continuum model describes them.
Density, specific weight and specific gravity describe mass properties.
Viscosity follows Newton's law for Newtonian fluids and varies oppositely with temperature for liquids and gases.
Compressibility, surface tension, capillarity and vapour pressure explain water hammer, droplets, capillary rise and cavitation.

Part 2 of 2

Fluid Statics & Pressure Measurement

Last reviewed 16 Sept 2026 · 6 min read

Pressure

Pressure (intensity of pressure) is the normal force per unit area exerted by a fluid: $p = F / A$ (N/m² = pascal; 1 bar = 10⁵ Pa).

Pascal's law

At a point in a fluid at rest, the pressure is the same in all directions: $p_{x} = p_{y} = p_{z}$ . (Pressure acts normal to any surface; there is no shear in a static fluid.)

Applications: hydraulic press, hydraulic jack, hydraulic brakes — a small force on a small piston produces a large force on a large piston: $\frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}}$ .

Hydrostatic law

In a static fluid, pressure increases with depth:

\frac{d p}{d z} = - ρ g \Rightarrow p = ρ g h (gauge, below a free surface)

Pressure is the same at all points on a horizontal plane within the same continuous fluid.
The shape of the container does not matter (hydrostatic paradox) — only the vertical depth.
Pressure head $h = \frac{p}{ρ g}$ : the height of a column of the fluid that produces the pressure. Converting between liquids: $h_{1} S_{1} = h_{2} S_{2}$ .

Absolute, gauge and vacuum pressure

p_{ab s} = p_{a t m} + p_{g a ug e}, p_{v a c uu m} = p_{a t m} - p_{ab s}

Atmospheric pressure at sea level ≈ 101.325 kPa = 10.33 m of water = 760 mm of mercury.
Gauge pressure is measured relative to local atmosphere (positive or negative).
Vacuum (negative gauge) pressure — below atmospheric; absolute zero is a perfect vacuum.
Atmospheric pressure is measured by a barometer (mercury or aneroid).

Pressure measuring devices

Piezometer

A simple vertical tube connected to the pipe: $p = ρ g h$ . Limitations: cannot measure negative (vacuum) pressures (air enters), impractical for high pressures (very tall tube), unsuitable for gases.

Simple U-tube manometer

A U-tube containing a heavier manometric liquid (usually mercury, $S_{2}$ ) connected to the pipe carrying fluid of $S_{1}$ . Write pressures along the tube, equating pressures at the same level in the same liquid.

FormulaSimple U-tube manometer (pipe on the left, right limb open)

Positive pressure at A, fluid rising $h_{1}$ above the datum in the left limb, manometric liquid difference $h_{2}$ :

p_{A} + ρ_{1} g h_{1} = ρ_{2} g h_{2} \Rightarrow p_{A} = ρ_{2} g h_{2} - ρ_{1} g h_{1}

Negative (vacuum) pressure at A (manometric liquid higher in the left limb):

p_{A} = - ρ_{1} g h_{1} - ρ_{2} g h_{2}

Differential U-tube manometer

Measures the difference in pressure between two points A and B. For two pipes at the same level carrying the same fluid ( $S_{1}$ ) with mercury ( $S_{2}$ ) reading $x$ :

p_{A} - p_{B} = x g (ρ_{2} - ρ_{1}) \Rightarrow head difference = x (\frac{S _{2}}{S _{1}} - 1)

For water with mercury: $h = 12.6 x$ .

Inverted U-tube manometer

A lighter manometric fluid (air or oil) in an inverted U: used for small pressure differences in liquids: $p_{A} - p_{B} = x g (ρ_{1} - ρ_{l i g h t})$ (same-level points).

Micromanometers and inclined manometers

Inclined manometer — the limb is inclined at $θ$ , so a small vertical difference $h = L sin θ$ produces a longer reading $L$ — higher sensitivity.
Micromanometer — enlarged reservoir (well) on one limb so that only one limb is read; used for very small differences.

Mechanical gauges

Bourdon tube gauge — a curved elliptical tube tends to straighten under pressure; pointer motion; used for high pressures and vacuum.
Diaphragm gauge, bellows gauge, dead-weight gauge (used for calibration), and electronic pressure transducers.

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