AEE Mechanical Engineering Core · Chapter 4

Fluid Mechanics

What to remember

Pressure at depth is p = ρgh; the centre of pressure always lies below the centroid; a floating body is stable when the metacentre M is above the centre of gravity G.
Flow type is decided by Reynolds number (Re = ρVD/μ): laminar below about 2000 in a pipe, turbulent above about 4000. Head loss is major (friction, Darcy-Weisbach) or minor (bends, entry, enlargement).
Bernoulli's equation (p/ρg + V²/2g + z = constant) is energy conservation along a streamline for steady, incompressible, frictionless flow. Most flow meters (venturi, orifice, pitot, notch) are applications of it.

1. Fluid properties

A fluid deforms continuously under a shear stress. Standard properties:

Density ρ = mass/volume (water about 1000 kg/m³). Specific weight w = ρg (water about 9810 N/m³). Specific gravity = density of fluid / density of water. Mercury is about 13.6.
Dynamic viscosity μ: shear stress τ = μ (du/dy) (Newton's law of viscosity). Unit: Pa·s = N·s/m². (1 poise = 0.1 Pa·s.)
Kinematic viscosity ν = μ/ρ. Unit: m²/s. (1 stokes = 10⁻⁴ m²/s.)
A Newtonian fluid obeys the linear law above (water, air, oils). Non-Newtonian fluids do not (paints, blood, pastes).
Viscosity of a liquid decreases when temperature rises. Viscosity of a gas increases when temperature rises.
Surface tension σ (N/m) acts at a free surface. Capillary rise in a tube: h = 4σ cosθ / (ρ g d). Thinner tube means higher rise. Pressure inside a droplet: Δp = 4σ/d; inside a soap bubble: Δp = 8σ/d.
Bulk modulus K = −dp/(dV/V). Speed of sound in a fluid is c = √(K/ρ); in an ideal gas c = √(γRT).

2. Fluid statics

Pressure at a point is the same in all directions (Pascal's law). Variation with depth: p = p₀ + ρgh.
Absolute pressure = gauge pressure + atmospheric pressure. Vacuum pressure = atmospheric − absolute. Standard atmosphere is about 101.3 kPa, or about 10.33 m of water, or 760 mm of mercury.
Manometers: the U-tube, differential and inclined types measure pressure from the height of a liquid column. Equal pressure holds at the same level in the same continuous liquid.
Total force on a submerged plane surface: F = ρ g A h̄, where h̄ is the depth of the centroid.
Centre of pressure: h* = h̄ + I_G sin²θ / (A h̄). It is always below the centroid (except for a horizontal surface, where they coincide). For a vertical rectangle with its top edge at the free surface, h* = 2/3 of the depth.
Common I_G: rectangle bd³/12; circle πd⁴/64; triangle bh³/36.
Buoyancy (Archimedes): upward force = weight of fluid displaced, acting through the centre of buoyancy B (centroid of displaced volume).

Condition	Floating body
M above G (GM positive)	Stable
M coinciding with G	Neutral
M below G (GM negative)	Unstable

Metacentric height GM = BM − BG, with BM = I/V (I is the second moment of the waterline area about the axis of tilt, V is the displaced volume).

Worked example: a rectangular barge 10 m long, 6 m wide and 3 m draught has I = 10×6³/12 = 180 m⁴ and V = 180 m³, so BM = 1.0 m.

3. Fluid kinematics and dynamics

Steady flow: properties at a point do not change with time. Uniform flow: velocity does not change with position. In steady flow, streamline, pathline and streakline coincide.
Continuity (incompressible): A₁V₁ = A₂V₂ = Q. Example: a 20 cm pipe narrowing to 10 cm; area ratio 4, so velocity rises from 2 m/s to 8 m/s.
Irrotational flow has zero vorticity. A velocity potential exists only for irrotational flow. A stream function exists for any 2-D incompressible flow. Streamlines and equipotential lines cut at right angles.
Euler's equation along a streamline, integrated for incompressible flow, gives Bernoulli: p/ρg + V²/2g + z = constant (head in metres: pressure head + velocity head + elevation head).
Bernoulli assumptions: steady, incompressible, frictionless (inviscid), along one streamline. With real fluids add head loss h_L between the two sections.
Momentum equation: force = rate of change of momentum = ρQ(V₂ − V₁). It is used for jet force on plates, pipe bends and the loss at a sudden enlargement.

4. Flow measurement

Device	Principle / formula
Pitot tube	Stagnation pressure gives V = √(2Δp/ρ) = √(2gh); with Δp = 2 kPa in water, V = 2 m/s
Venturimeter	Q = Cd a₁a₂ √(2gh) / √(a₁² − a₂²); Cd about 0.95–0.99; low loss
Orifice meter	Same form as venturi; Cd about 0.6; larger permanent loss
Orifice (tank)	Torricelli V = √(2gH); Cd = Cv × Cc
Rectangular notch/weir	Q = (2/3) Cd L √(2g) H^{3/2}
Triangular (V) notch	Q = (8/15) Cd tan(θ/2) √(2g) H^{5/2}

Cv is velocity coefficient (about 0.97–0.99). Cc is contraction coefficient (about 0.62 for a sharp orifice). The vena contracta is the section of the jet where area is least.

5. Viscous flow in pipes

Reynolds number Re = ρVD/μ = VD/ν. It is the ratio of inertia force to viscous force. Pipe flow: laminar if Re < about 2000; transition 2000 to 4000; turbulent above about 4000.
Laminar flow in a circular pipe (Hagen-Poiseuille): parabolic profile; maximum velocity = 2 × mean velocity; friction factor f = 64/Re. Pressure drop Δp = 32 μ L V / D².
Darcy-Weisbach: h_f = f L V² / (2 g D). For turbulent flow with constant f, doubling velocity makes head loss 4 times. Hydraulic mean radius for a full pipe is D/4.
Moody chart gives f from Re and relative roughness. In fully rough flow f depends only on roughness.
Minor losses: sudden enlargement h = (V₁ − V₂)²/2g (example: V₁ = 4, V₂ = 2 m/s gives 4/19.62 = 0.204 m). Sudden contraction h ≈ 0.5 V₂²/2g. Entry loss about 0.5 V²/2g; exit loss V²/2g.
Pipes in series: same discharge, head losses add. Pipes in parallel: same head loss, discharges add.
Power transmission through a pipe is maximum when the friction head loss is one third of the supply head. Efficiency at that point is about 66.7%.
Water hammer: sudden valve closure produces a pressure surge. Surge tanks and slow closing reduce it. This matters in hydro-power penstocks.

6. Boundary layer, drag, compressible flow

Boundary layer: thin region near a wall where velocity rises from zero to nearly the free-stream value. Thickness δ is the distance where u = 0.99 U.
Separation occurs under an adverse pressure gradient (pressure rising along the flow). It gives a wake and high pressure drag.
Streamlined bodies: mostly skin-friction drag. Bluff bodies: mostly pressure (form) drag. Drag force F_D = ½ C_D ρ A V².
Mach number M = V/c. Compressibility can be neglected below about M = 0.3. M < 1 subsonic, M = 1 sonic, M > 1 supersonic. Example: for air at 300 K, c = √(1.4 × 287 × 300), about 347 m/s.

7. Dimensional analysis and similarity

Buckingham π theorem: number of dimensionless groups = n − m (n variables, m fundamental dimensions). Example: 7 variables and 3 dimensions give 4 groups.
Reynolds number: inertia/viscous. Froude Fr = V/√(gL): inertia/gravity. Mach: inertia/elasticity. Weber: inertia/surface tension. Euler: pressure/inertia.
Model laws: Reynolds law for closed pipe flow and bodies fully submerged; Froude law for free-surface flows (ships, spillways, open channels).

Model law	Used for
Reynolds	Pipe flow, submerged bodies, aircraft at low speed
Froude	Ship hulls, spillways, weirs, waves
Mach	High-speed compressible flow

Exam traps

Dynamic viscosity (Pa·s) and kinematic viscosity (m²/s) are not interchangeable.
Viscosity of liquids falls with temperature, but viscosity of gases rises.
Centre of pressure is below the centroid; centre of buoyancy is the centroid of displaced liquid, not of the body.
Stable floating needs M above G, not B above G.
Darcy friction factor f (= 4 × Fanning factor) is used in h_f = fLV²/2gD.
Laminar f = 64/Re does not depend on roughness. Turbulent f depends on Re and roughness.
Rectangular notch head power is 3/2; triangular notch is 5/2.
Bernoulli does not apply across a pump or turbine unless head added/removed is included.

One-liners

1. Specific gravity of mercury is about 13.6.
2. Water at standard conditions: ρ about 1000 kg/m³, w about 9810 N/m³.
3. Gauge pressure = absolute − atmospheric.
4. Pressure inside a soap bubble is 8σ/d above outside.
5. Hydrostatic force on a vertical surface = ρ g A h̄.
6. Venturimeter has a converging section, throat and diverging section.
7. Laminar pipe flow: V_max = 2 V_mean.
8. Re critical for pipe flow is about 2000 (lower critical).
9. Cd = Cv × Cc.
10. A pitot tube measures stagnation (total) pressure.
11. Froude number governs free-surface flow similarity.
12. Boundary layer separation needs an adverse pressure gradient.

Practice questions

What is the SI unit of dynamic viscosity?
1. N/m
2. Pa·s
3. m²/s
4. kg/m³
Answer
B. Pa·s
Dynamic viscosity μ = τ/(du/dy) has unit N·s/m² = Pa·s.
Kinematic viscosity of a fluid is defined as
1. dynamic viscosity multiplied by density
2. shear stress divided by pressure
3. density divided by dynamic viscosity
4. dynamic viscosity divided by density
Answer
D. dynamic viscosity divided by density
ν = μ/ρ, unit m²/s.
A Newtonian fluid is one in which shear stress is
1. directly proportional to the velocity gradient
2. proportional to the square of the velocity gradient
3. inversely proportional to the velocity gradient
4. independent of the velocity gradient
Answer
A. directly proportional to the velocity gradient
Newton's law of viscosity: τ = μ du/dy (linear).
Statements on viscosity: 1. The viscosity of a liquid decreases when its temperature rises. 2. The viscosity of a gas increases when its temperature rises. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Liquids: cohesion falls with heat; gases: molecular momentum exchange rises with heat.
A liquid has a specific gravity of 0.8. Its density is
1. 1250 kg/m³
2. 800 kg/m³
3. 8000 kg/m³
4. 80 kg/m³
Answer
B. 800 kg/m³
ρ = 0.8 × 1000 = 800 kg/m³.
The gauge pressure at a depth of 10 m in water (take ρ = 1000 kg/m³, g = 9.81 m/s²) is
1. 196.2 kPa
2. 101.3 kPa
3. 9.81 kPa
4. 98.1 kPa
Answer
D. 98.1 kPa
p = ρgh = 1000 × 9.81 × 10 = 98 100 Pa.
Absolute pressure is equal to
1. gauge pressure minus atmospheric pressure
2. gauge pressure plus atmospheric pressure
3. gauge pressure divided by atmospheric pressure
4. atmospheric pressure minus vacuum pressure plus gauge pressure
Answer
B. gauge pressure plus atmospheric pressure
p_abs = p_gauge + p_atm.
The pressure at the bottom of a 5 m deep tank of oil of specific gravity 0.8 (gauge, g = 9.81 m/s²) is
1. 39.24 kPa
2. 3.924 kPa
3. 78.48 kPa
4. 49.05 kPa
Answer
A. 39.24 kPa
p = 800 × 9.81 × 5 = 39 240 Pa.
A vertical rectangular gate 2 m wide and 3 m deep has its top edge at the free surface of water. The total hydrostatic force on it is (g = 9.81 m/s²)
1. 58.86 kN
2. 29.43 kN
3. 176.58 kN
4. 88.29 kN
Answer
D. 88.29 kN
F = ρ g A h̄ = 1000 × 9.81 × 6 × 1.5 = 88 290 N.
For a vertical rectangular plane with its top edge at the free surface of a liquid, the centre of pressure lies below the free surface at
1. one-third of the depth of the plane
2. two-thirds of the depth of the plane
3. three-fourths of the depth of the plane
4. half the depth of the plane
Answer
B. two-thirds of the depth of the plane
h* = h̄ + I_G/(A h̄) = d/2 + d/6 = 2d/3.
Statements on a submerged plane surface: 1. The centre of pressure of an inclined submerged plane lies below its centroid. 2. For a horizontal submerged plane the centre of pressure coincides with the centroid. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
h* = h̄ + I_G sin²θ/(A h̄); for θ = 0 the extra term vanishes.
The buoyant force on a body immersed in a fluid equals
1. the weight of the body
2. the volume of the body only
3. the weight of the body minus its volume
4. the weight of the fluid displaced by the body
Answer
D. the weight of the fluid displaced by the body
Archimedes' principle.
A floating body is in stable equilibrium when
1. the metacentre lies above the centre of gravity
2. the centre of buoyancy lies above the centre of gravity
3. the metacentre coincides with the centre of buoyancy
4. the metacentre lies below the centre of gravity
Answer
A. the metacentre lies above the centre of gravity
GM = BM − BG must be positive.
A rectangular barge 10 m long, 6 m wide floats with a draught of 3 m. The distance BM is
1. 1.5 m
2. 1.0 m
3. 0.5 m
4. 3.0 m
Answer
B. 1.0 m
I = 10 × 6³/12 = 180 m⁴; V = 10 × 6 × 3 = 180 m³; BM = I/V = 1.0 m.
Water flows at 2 m/s in a 20 cm diameter pipe which reduces to 10 cm diameter. The velocity in the smaller section is
1. 4 m/s
2. 6 m/s
3. 16 m/s
4. 8 m/s
Answer
D. 8 m/s
A₁/A₂ = (20/10)² = 4, so V₂ = 4 × 2 = 8 m/s.
Which of the following is NOT an assumption in the basic Bernoulli equation?
1. The fluid is incompressible
2. The equation is applied along a streamline
3. The fluid is viscous
4. The flow is steady
Answer
C. The fluid is viscous
Bernoulli's equation assumes inviscid (frictionless) flow.
The three terms of the Bernoulli equation, p/ρg, V²/2g and z, are each expressed in units of
1. metres of fluid column (head)
2. newtons
3. pascals
4. joules per kilogram
Answer
A. metres of fluid column (head)
Each term is a head, i.e. length.
A pitot tube in water measures a stagnation-to-static pressure difference of 2 kPa. The flow velocity is
1. 20 m/s
2. 4 m/s
3. 2 m/s
4. 1 m/s
Answer
C. 2 m/s
V = √(2Δp/ρ) = √(2 × 2000/1000) = 2 m/s.
For flow in a circular pipe, laminar flow is generally assured when the Reynolds number is below about
1. 2000
2. 4000
3. 10 000
4. 100 000
Answer
A. 2000
Lower critical Re for pipe flow is about 2000.
Water (ρ = 1000 kg/m³, μ = 0.001 Pa·s) flows at 1 m/s in a 50 mm pipe. The Reynolds number and flow type are
1. 50, laminar
2. 500 000, laminar
3. 5000, laminar
4. 50 000, turbulent
Answer
D. 50 000, turbulent
Re = ρVD/μ = 1000 × 1 × 0.05/0.001 = 50 000 > 4000.
In fully developed laminar flow in a circular pipe, the maximum velocity is
1. equal to the mean velocity
2. twice the mean velocity
3. 1.5 times the mean velocity
4. half the mean velocity
Answer
B. twice the mean velocity
Parabolic profile: u_max = 2 V_mean.
If velocity in a long pipe is doubled in turbulent flow with the friction factor unchanged, the Darcy-Weisbach friction head loss becomes
1. eight times
2. one-half
3. four times
4. two times
Answer
C. four times
h_f ∝ V².
A pipe has f = 0.02, L = 100 m, D = 0.1 m and V = 2 m/s. The friction head loss (g = 9.81 m/s²) is about
1. 4.08 m
2. 2.04 m
3. 8.15 m
4. 0.41 m
Answer
A. 4.08 m
h_f = 0.02 × 100 × 4/(2 × 9.81 × 0.1) = 8/1.962 = 4.08 m.
For laminar flow with Re = 1600 in a circular pipe, the Darcy friction factor is
1. 0.4
2. 0.04
3. 0.064
4. 0.016
Answer
B. 0.04
f = 64/Re = 64/1600 = 0.04.
The hydraulic mean radius of a circular pipe running full of diameter D is
1. D
2. D/2
3. D/8
4. D/4
Answer
D. D/4
R = A/P = (πD²/4)/(πD) = D/4.
Water flows through a sudden enlargement; velocity falls from 4 m/s to 2 m/s. The head loss is about (g = 9.81 m/s²)
1. 1.0 m
2. 0.204 m
3. 0.408 m
4. 0.102 m
Answer
B. 0.204 m
h = (V₁ − V₂)²/2g = 4/19.62 = 0.204 m.
Statements on pipe losses: 1. Loss at a pipe entrance is a minor loss. 2. Loss due to friction along a long straight pipe is a major loss. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Friction along length is major; fittings and entry/exit are minor.
When pipes are connected in parallel, which quantity is the same for all branches?
1. diameter
2. discharge
3. velocity
4. head loss
Answer
D. head loss
Branches share the same end points, so head loss is equal; discharges add.
Maximum power is transmitted through a pipe when the head lost in friction is equal to
1. one-third of the total supply head
2. one-fourth of the total head
3. two-thirds of the total head
4. one-half of the total head
Answer
A. one-third of the total supply head
Setting d(P)/dV = 0 gives h_f = H/3.
The boundary layer thickness is conventionally the distance from the wall at which the velocity reaches
1. 99% of the free-stream velocity
2. 10% of the free-stream velocity
3. 50% of the free-stream velocity
4. the free-stream velocity exactly
Answer
A. 99% of the free-stream velocity
Standard definition δ: u = 0.99 U.
Separation of the boundary layer from a surface occurs due to
1. a favourable pressure gradient
2. a constant pressure
3. an adverse pressure gradient
4. very high wall roughness only
Answer
C. an adverse pressure gradient
Rising pressure along the flow slows near-wall fluid until reversal.
Statements on drag: 1. A streamlined body mainly experiences skin-friction drag. 2. A bluff body mainly experiences pressure (form) drag. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Separation and wake dominate drag on bluff bodies.
Compressibility effects in a gas flow may be ignored when the Mach number is below about
1. 1.0
2. 0.3
3. 1.5
4. 0.8
Answer
B. 0.3
Density change is under about 5% for M < 0.3.
The speed of sound in air at 300 K (γ = 1.4, R = 287 J/kg·K) is about
1. 434 m/s
2. 300 m/s
3. 120 m/s
4. 347 m/s
Answer
D. 347 m/s
c = √(1.4 × 287 × 300) = √120 540 ≈ 347 m/s.
Discharge over a triangular notch is proportional to head H raised to the power
1. 3/2
2. 1/2
3. 5/2
4. 2
Answer
C. 5/2
Q = (8/15) Cd tan(θ/2) √(2g) H^{5/2}.
A sharp-edged orifice has Cv = 0.98 and Cc = 0.62. The coefficient of discharge is about
1. 1.58
2. 0.608
3. 0.632
4. 0.36
Answer
B. 0.608
Cd = Cv × Cc = 0.98 × 0.62 = 0.6076.
Water issues from a small orifice under a constant head of 5 m (g = 9.81 m/s²). The theoretical jet velocity is about
1. 9.9 m/s
2. 7.0 m/s
3. 24.5 m/s
4. 4.9 m/s
Answer
A. 9.9 m/s
V = √(2gH) = √98.1 = 9.9 m/s.
The Froude number is the ratio of
1. pressure force to inertia force
2. inertia force to surface tension force
3. inertia force to viscous force
4. inertia force to gravity force
Answer
D. inertia force to gravity force
Fr = V/√(gL).
A model of a ship hull is tested in a towing tank. Dynamic similarity requires equality of
1. Reynolds number
2. Froude number
3. Mach number
4. Weber number
Answer
B. Froude number
Free-surface wave making is gravity dominated.
A physical problem has 7 variables and 3 fundamental dimensions. By Buckingham's π theorem the number of dimensionless groups is
1. 3
2. 7
3. 10
4. 4
Answer
D. 4
n − m = 7 − 3 = 4.
In steady flow, which three lines coincide?
1. streamline, pathline and streakline
2. streamline, equipotential line and pathline
3. pathline, timeline and streakline
4. streakline, timeline and streamline
Answer
A. streamline, pathline and streakline
Under steady flow all fluid particles trace the same path.
Statements on 2-D flow: 1. A velocity potential function exists only for irrotational flow. 2. A stream function exists for any 2-D incompressible flow. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Stream function follows from continuity; potential needs zero vorticity.
If the diameter of a capillary tube is halved, the capillary rise of water becomes
1. unchanged
2. four times
3. half
4. double
Answer
D. double
h = 4σcosθ/(ρgd) ∝ 1/d.
A U-tube mercury manometer (specific gravity 13.6) shows a column difference of 0.1 m. The gauge pressure is about (g = 9.81 m/s²)
1. 9.81 kPa
2. 13.3 kPa
3. 133 kPa
4. 1.33 kPa
Answer
B. 13.3 kPa
p = 13 600 × 9.81 × 0.1 = 13 342 Pa.
Pressure inside a soap bubble of diameter d and surface tension σ, above the outside pressure, is
1. 8σ/d
2. 2σ/d
3. 4σ/d
4. σ/d
Answer
A. 8σ/d
A bubble has two surfaces: Δp = 2 × 4σ/d.

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1.The atmospheric pressure at sea level is _________ atm.
2.The force exerted by an object perpendicular to the surface is called:
3.The force employed on a unit area of the surface is called _____.
4.The effect of thrust on a surface depends on the area on which it acts. Force acting on a smaller area exerts a ___________ compared to a larger area.
5.The density of an ideal gas can be doubled by halving the ______________.
6.Solid CO 2 gets converted directly into gaseous state on decrease of pressure to:
7.The speed of sound depends primarily on the nature and ____________ of the medium.
8.The speed of light is ______.
9.The magnitude of the buoyant force depends on the _______ of the fluid.
10.The mass per unit volume is called

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