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Hydraulics & Fluid Mechanics – Complete Study Notes

GATE ESE / IES SSC JE State PSC RRB JE

Comprehensive chapter-wise notes covering all 16 topics of Hydraulics and Fluid Mechanics — fluid properties, manometry, hydrostatics, buoyancy, kinematics, Bernoulli's equation, flow measurement, viscous flow, pipe flow, drag and lift, boundary layer, turbulent flow, dimensional analysis, weirs, hydraulic turbines, pumps, and open channel flow. All formulae, diagrams, derivations, worked examples and exam-focused tables included.

Ch 1 · Fluid Properties Ch 2 · Manometry Ch 3 · Hydrostatic Forces Ch 4 · Buoyancy & Floatation Ch 5 · Fluid Kinematics Ch 6 · Fluid Dynamics & Flow Measurement Ch 7 · Viscous Flow Ch 8 · Flow Through Pipes Ch 9 · Drag & Lift Ch 10 · Boundary Layer Ch 11 · Turbulent Flow Ch 12 · Dimensional Analysis Ch 13 · Notches & Weirs Ch 14 · Impact of Jets & Turbines Ch 15 · Pumps Ch 16 · Open Channel Flow ★ Quick Revision

1Fluid Properties►

1.1 Definition and Classification

A fluid is a substance that deforms continuously under the application of any shear stress, however small. Unlike a solid (which deforms by a fixed amount), a fluid flows — velocity continues to increase as long as shear stress is applied.

Property	Symbol	Definition	SI Unit	Water (20°C)	Air (20°C)
Density	ρ	Mass per unit volume = m/V	kg/m³	998	1.204
Specific weight	γ	Weight per unit volume = ρg	N/m³	9789	11.8
Specific gravity	S or G	ρ_fluid / ρ_water	dimensionless	1.0	0.0012
Dynamic viscosity	μ	τ = μ(du/dy); resistance to shear	Pa·s (N·s/m²)	1.002×10⁻³	1.81×10⁻⁵
Kinematic viscosity	ν	μ/ρ; momentum diffusivity	m²/s (Stoke = 10⁻⁴ m²/s)	1.004×10⁻⁶	1.51×10⁻⁵
Bulk modulus	K	K = −V(dP/dV) = ρ(dP/dρ)	N/m² (Pa)	2.18×10⁹	—
Surface tension	σ	Force per unit length at free surface	N/m	0.0728	—
Vapour pressure	p_v	Pressure at which liquid boils	Pa (kPa)	2.34 kPa	—

1.2 Newton's Law of Viscosity

τ = μ × (du/dy)
τ = shear stress (Pa); μ = dynamic viscosity; du/dy = velocity gradient (rate of shear deformation)

Newtonian fluids: τ ∝ (du/dy) — linear relationship; μ = constant
Examples: Water, air, most oils

Non-Newtonian fluids: τ ≠ μ(du/dy)
Power law: τ = K(du/dy)^n
n < 1: Pseudoplastic (shear thinning) — paint, blood, polymer solutions
n > 1: Dilatant (shear thickening) — cornstarch in water, wet sand
n = 1: Newtonian
Bingham plastic: τ = τ₀ + μ_B(du/dy) — needs yield stress before flowing (toothpaste, sewage sludge)

1.3 Viscosity — Temperature Dependence

Liquids: viscosity DECREASES with temperature (intermolecular cohesion weakens)
μ(T) ≈ A × e^(B/T) [Andrade's equation; T in Kelvin]

Gases: viscosity INCREASES with temperature (molecular momentum transfer increases)
μ(T) ∝ T^0.5 to T^0.75 [Sutherland's law for gases]

1.4 Compressibility and Bulk Modulus

Bulk modulus: K = −dP / (dV/V) = ρ × dP/dρ
Compressibility: β = 1/K
Speed of sound in fluid: c = √(K/ρ)
For water: c ≈ 1480 m/s; for air: c ≈ 343 m/s (γ = 1.4, R = 287 J/kg·K)
Mach number: Ma = V/c; Ma < 0.3 → incompressible assumption valid

1.5 Surface Tension and Capillarity

Pressure inside a droplet: ΔP = 4σ/d (d = diameter of droplet; two surfaces)
Pressure inside a bubble: ΔP = 8σ/d (soap bubble; two films)
Pressure inside a jet: ΔP = 2σ/d (liquid jet; one surface)

Capillary rise: h = 4σ cos θ / (ρgd)
σ = surface tension; θ = contact angle (0° for wetting); d = tube diameter
For water-glass: θ ≈ 0° → h ≈ 4σ/(ρgd); capillary RISE (wetting)
For mercury-glass: θ ≈ 140° → h negative; capillary DEPRESSION (non-wetting)

1.6 Cavitation

Cavitation occurs when local pressure in a flowing liquid falls below the vapour pressure, causing vapour bubbles to form. When these bubbles collapse (near high-pressure zones), they cause shock waves — leading to erosion, noise, and vibration in hydraulic machinery.

Cavitation number: σ_c = (p_atm − p_v) / (½ρV²)
Cavitation occurs when σ_c falls below a critical value (σ_c,crit) for the geometry
Prevention: maintain sufficient NPSH (Net Positive Suction Head) at pump inlet

📝 GATE Tip: ΔP inside droplet = 4σ/d; bubble = 8σ/d; jet = 2σ/d. Capillary rise h = 4σcosθ/(ρgd). Liquids: μ decreases with T; gases: μ increases with T. Newton's law τ = μ(du/dy) — the most tested definition in this chapter.

2Manometry►

2.1 Pressure and Pascal's Law

Pressure: p = F/A (Pa = N/m²); 1 bar = 10⁵ Pa; 1 atm = 101.325 kPa
Hydrostatic pressure equation: dp/dz = −ρg
Pressure at depth h: p = p₀ + ρgh (p₀ = surface pressure; ρgh = gauge pressure contribution)

Pascal's Law: pressure applied to an enclosed fluid is transmitted equally in all directions
Hydraulic jack: F₁/A₁ = F₂/A₂ (mechanical advantage = A₂/A₁)

2.2 Types of Manometers

Type	Principle	Range / Use
Simple U-tube	Heavy gauge fluid (mercury) balances pressure; read height difference	Moderate pressures; most common
Inverted U-tube	Light gauge fluid (air or oil) at top; for measuring small differential pressures in liquids	Small pressure differences in water/oil pipes
Inclined manometer	Limb inclined at angle θ; amplifies reading by 1/sinθ	Very small pressures; wind tunnel tests
Micro-manometer	Large reservoir + narrow tube; small pressure differences	Extremely small ΔP; laboratory
Differential manometer	Connects two pressure points; measures difference directly	Across orifices, valves, Venturi meters
Bourdon gauge	Coiled metal tube deforms under pressure; mechanical indicator	Industrial; wide pressure range

2.3 Manometer Calculation Procedure

Start from one point; add ρgh going DOWN; subtract ρgh going UP
The algebraic sum between two points = pressure difference (p₁ − p₂)

Simple U-tube connecting points A and B (gauge fluid density ρ_m, pipe fluid ρ_f):
p_A − p_B = ρ_m × g × h_m − ρ_f × g × (z_A − z_B)
where h_m = difference in gauge fluid meniscus levels

For gas pressure measurement (gas in pipe, gauge fluid = mercury):
p_A = ρ_Hg × g × h (gas weight negligible)
p_gauge = ρgh_mercury × g × h_mm

2.4 Pressure Prism and Hydrostatic Paradox

The pressure at any point depends only on depth and fluid density — NOT on the shape of the container
(Pascal / hydrostatic paradox: tall narrow tube and wide flat container at same depth have same pressure)

Atmospheric pressure = 101.325 kPa = 10.33 m of water = 760 mm of mercury
γ_water = 9.81 kN/m³ ≈ 9.79 kN/m³ at 20°C
γ_mercury = 133.4 kN/m³ (S_Hg = 13.6)

📝 GATE Tip: p = ρgh (gauge pressure at depth h). S_Hg = 13.6. 1 atm = 101.325 kPa = 760 mm Hg = 10.33 m water. Manometer rule: add pressure going down, subtract going up. For differential manometer: p_A − p_B = (ρ_gauge − ρ_fluid)gh.

3Hydrostatic Forces on Surfaces►

3.1 Force on a Plane Submerged Surface

Total (resultant) hydrostatic force:
F = ρg × ȳ × A
ȳ = depth of centroid of the area below free surface
A = area of the submerged surface

Centre of pressure (depth of action of F):
y_cp = ȳ + I_G / (ȳ × A)
I_G = second moment of area about centroidal axis parallel to free surface

Always: y_cp > ȳ (CP is always BELOW centroid; moves toward centroid as depth increases)

For inclined plane surface (angle θ to horizontal):
F = ρg × ȳ × A (ȳ = vertical depth of centroid; same formula)
y_cp (along incline) = ȳ_incline + I_G / (ȳ_incline × A)

Fig. 3.1 — Hydrostatic force F = ρgȳA acts at centre of pressure (CP) below centroid G

Fig. 3.2 — Second moment of area I_G about centroidal axis for common shapes; used in y_cp formula

3.2 Force on a Curved Submerged Surface

Horizontal component F_H = force on the vertical projection of the curved surface
F_H = ρg × ȳ_proj × A_proj (acts at CP of projected area)

Vertical component F_V = weight of real or imaginary fluid above the curved surface
F_V = ρg × Volume of fluid above the curved surface
F_V acts at centroid of this volume

Resultant force: F_R = √(F_H² + F_V²)
Direction: tan θ = F_V / F_H

3.3 Pressure on Dams and Lock Gates

For a rectangular dam (width b, depth H of water):
Total force: F = ½ × ρg × H² × b
Acts at H/3 from base (centroid of triangular pressure distribution)

Overturning moment about toe = F × H/3
Resisting moment = Weight of dam × distance from toe to CG
FOS against overturning = Resisting / Overturning ≥ 1.5

📝 GATE Tip: F = ρgȳA. CP: y_cp = ȳ + I_G/(ȳA). For rectangle: I_G = bd³/12. Curved surface: F_H = force on projection; F_V = weight of fluid above. CP always below centroid — and moves toward centroid as submergence depth increases.

4Buoyancy & Floatation►

4.1 Archimedes' Principle

Buoyant force F_B = weight of fluid displaced by the body
F_B = ρ_f × g × V_submerged
ρ_f = density of fluid; V_sub = submerged volume

For a FLOATING body: F_B = W (weight of body)
ρ_f × g × V_sub = ρ_b × g × V_total
Fraction submerged = ρ_body / ρ_fluid = S_body / S_fluid

For SINKING body: ρ_body > ρ_fluid → not in equilibrium; sinks
For FLOATING body: ρ_body < ρ_fluid → floats with fraction ρ_b/ρ_f submerged

4.2 Metacentre and Stability of Floating Bodies

Fig. 4.1 — Metacentre M: when body tilts, buoyancy line through new B' intersects original vertical at M; if M is above G → stable; if below → unstable

4.3 Metacentric Height

Metacentric height: GM = BM − BG [M above G: GM positive → stable]

BM = I_WL / V_sub
I_WL = second moment of waterplane area about longitudinal axis through centroid
V_sub = displaced volume

For a rectangular pontoon (L × B at waterline, draught d):
I_WL = L × B³ / 12
V_sub = L × B × d
BM = B² / (12d)
OB = d/2 (centroid of displaced volume from keel)
OG = height of CG from keel
GM = OB + BM − OG = d/2 + B²/(12d) − OG

Experimental GM: GM = W_s × x / (W × tanθ)
W_s = shifted weight, x = shift distance, θ = angle of tilt measured

📝 GATE Tip: F_B = ρgV_sub. Fraction submerged = ρ_body/ρ_fluid. BM = I_WL/V_sub. GM = BM − BG. For rectangle: BM = B²/(12d). Stable if GM > 0 (M above G). Experimental: GM = W_s·x/(W·tanθ). These five facts cover 95% of GATE questions on this topic.

5Fluid Kinematics►

5.1 Types of Flow

Classification	Steady / Unsteady	Uniform / Non-uniform	Laminar / Turbulent	Compressible
Criterion	∂V/∂t = 0 or ≠ 0	∂V/∂s = 0 or ≠ 0	Re < 2000 or > 4000	ρ varies or constant
Steady	V, p, ρ constant at each point with time	—	—	—
Uniform	—	V constant along streamline at a given time	—	—
Laminar	—	—	Re < 2000; layers slide; viscous dominates	—
Turbulent	—	—	Re > 4000; chaotic; inertia dominates	—

5.2 Flow Lines

Streamline: tangent at every point = velocity direction at that instant; no flow across
Pathline: actual path traced by a fluid particle over time
Streakline: locus of all particles that have passed through a point (visible in dye injection)
Streamtube: a bundle of streamlines; no flow through the walls

For STEADY flow: streamline = pathline = streakline (all three coincide)
For UNSTEADY flow: all three are generally different

5.3 Continuity Equation

General (3D, compressible):
∂ρ/∂t + ∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z = 0

Steady, incompressible (ρ = const):
∂u/∂x + ∂v/∂y + ∂w/∂z = 0 (Laplacian ∇·V = 0)

1D steady flow (stream tube):
A₁V₁ = A₂V₂ = Q (volume flow rate; constant for incompressible steady flow)

For compressible: ρ₁A₁V₁ = ρ₂A₂V₂ = ṁ (mass flow rate; constant)

5.4 Stream Function and Velocity Potential

Stream function ψ (for 2D, incompressible):
u = ∂ψ/∂y; v = −∂ψ/∂x
ψ = constant along a streamline; difference in ψ between two streamlines = flow rate per unit depth
ψ exists for all 2D flows; satisfies continuity automatically

Velocity potential φ (for irrotational flow):
u = −∂φ/∂x; v = −∂φ/∂y; w = −∂φ/∂z
φ = constant on equipotential lines; satisfies Laplace equation ∇²φ = 0
φ exists only for irrotational (potential) flow

Relationship: streamlines and equipotential lines are mutually perpendicular
Rotation: ω_z = ½(∂v/∂x − ∂u/∂y); Irrotational if ω = 0 everywhere

5.5 Vorticity and Circulation

Vorticity: ω⃗ = ∇ × V⃗ (2× angular velocity of fluid element)
Circulation: Γ = ∮ V⃗ · ds⃗ (line integral around a closed contour)
Stokes' theorem: Γ = ∬(∇ × V⃗) · dA⃗ = ∬ ω⃗ · dA⃗

Irrotational flow: ω = 0 everywhere, Γ = 0 for any contour not enclosing a vortex

📝 GATE Tip: Continuity: A₁V₁ = A₂V₂. Steady flow: streamline = pathline = streakline. u = ∂ψ/∂y, v = −∂ψ/∂x (stream function). u = −∂φ/∂x (potential). Streamlines ⊥ equipotentials. Irrotational: ∂v/∂x = ∂u/∂y.

6Fluid Dynamics & Flow Measurements►

6.1 Bernoulli's Equation

Euler's equation (along a streamline, inviscid flow):
dp/(ρg) + V·dV/g + dz = 0

Bernoulli's equation (steady, incompressible, inviscid, irrotational flow):
p/(ρg) + V²/(2g) + z = H = constant [total head]
OR: p/ρ + V²/2 + gz = constant [in J/kg]
OR: p + ½ρV² + ρgz = constant [in Pa]

Pressure head: p/(ρg)
Velocity head: V²/(2g)
Datum head: z
Total head H = sum of all three = constant along a streamline (Bernoulli)

Modified Bernoulli (with losses h_L and pump head H_p):
H₁ + H_p = H₂ + h_L

6.2 Flow Measuring Devices

Device	Type	Formula	C_d typical
Venturimeter	Differential pressure (pipe)	Q = C_d × A₁A₂/√(A₁²−A₂²) × √(2gh); h = differential head	0.96–0.99
Orifice meter	Differential pressure (pipe); simple	Q = C_d × A_o × √(2gh); h = pressure diff/(ρg)	0.60–0.65
Pitot tube	Point velocity measurement	V = C_v √(2g×Δh); Δh = stagnation − static head	C_v ≈ 0.98–1.0
Rotameter	Variable area (float in tapered tube)	Direct reading; float position ∝ flow rate	—
Notch / Weir	Open channel (see Ch 13)	Q ∝ H^(n) where n depends on shape	—

6.3 Orifice and Vena Contracta

Theoretical velocity (Torricelli): V_t = √(2gh) [h = head above orifice centre]

Actual velocity: V_a = C_v × V_t = C_v × √(2gh)
C_v = coefficient of velocity ≈ 0.98 (accounts for friction losses)

Coefficient of contraction: C_c = A_vena/A_orifice ≈ 0.64 for sharp-edged orifice

Actual flow: Q = C_d × A × √(2gh) C_d = C_v × C_c
C_d ≈ 0.61 (sharp-edged); 0.82 (bell-mouth); 0.51 (re-entrant)

For submerged orifice: Q = C_d × A × √(2g(h₁−h₂)) (h₁ = upstream head, h₂ = downstream)

6.4 Momentum Equation (Linear Momentum)

Reynolds Transport Theorem applied to momentum:
ΣF⃗ = d(mV⃗)/dt = ρQ(V⃗₂ − V⃗₁) [steady flow in a control volume]

Force on a pipe bend (x-direction):
F_x = ρQ(V₂cosθ₂ − V₁cosθ₁) + (p₂A₂cosθ₂ − p₁A₁cosθ₁)
F_y = ρQ(V₂sinθ₂ − V₁sinθ₁) + (p₂A₂sinθ₂ − p₁A₁sinθ₁)

Resultant force on fluid from pipe: F_R = √(F_x² + F_y²)
Force on pipe from fluid (Newton 3rd): equal and opposite to F_R on fluid

📝 GATE Tip: Bernoulli: p/ρg + V²/2g + z = H. Venturimeter C_d = 0.96–0.99; Orifice C_d ≈ 0.61; C_c ≈ 0.64; C_v ≈ 0.98. Torricelli: V = √(2gh). These three devices and their C_d values are tested every year.

7Viscous Flow of Incompressible Fluid►

7.1 Hagen-Poiseuille Flow (Laminar Pipe Flow)

For fully developed laminar flow in a circular pipe:
Velocity profile: u(r) = (R² − r²) × (−dp/dx) / (4μ) [parabolic]
Max velocity (at centreline): u_max = R² × (−dp/dx) / (4μ)
Mean velocity: V_mean = u_max / 2 = R²×(−dp/dx)/(8μ)

Hagen-Poiseuille law: Q = πR⁴ × (−dp/dx) / (8μ) = πD⁴ΔP / (128μL)

Head loss (H-P): h_f = 32μLV / (ρgD²) = 128μLQ / (πρgD⁴)
This is the Darcy-Weisbach with f = 64/Re for laminar flow.

Kinetic energy correction factor (α): α = 2.0 for laminar parabolic profile
Momentum correction factor (β): β = 4/3 for laminar flow

7.2 Laminar Flow Between Parallel Plates

Velocity profile (Couette-Poiseuille): u(y) = Uy/h + (1/2μ)(−dp/dx)y(h−y)
Pure Couette (pressure gradient zero): u = U × y/h [linear; top plate moving at U]
Pure Poiseuille (both plates fixed): u = (h²−y²)/(2μ) × (−dp/dx)/2 [parabolic]

Mean velocity for Poiseuille between plates: V = h² × (−dp/dx) / (12μ)

7.3 Reynolds Number and Transition

Re = ρVD/μ = VD/ν
Re < 2000: Laminar (viscous forces dominate)
2000 < Re < 4000: Transition (unstable; may be laminar or turbulent)
Re > 4000: Turbulent (inertia forces dominate)

Physical meaning: Re = (Inertia force) / (Viscous force) = ρV²L² / (μVL) = ρVL/μ
Critical Re in pipe: 2300 (theoretical); 2000 (engineering; conservative)

📝 GATE Tip: H-P: Q = πD⁴ΔP/(128μL). Velocity profile: u_max = 2V_mean (parabolic). α = 2.0 (KE correction), β = 4/3 (momentum correction) for laminar. f = 64/Re for laminar. These are the core Hagen-Poiseuille facts.

8Flow Through Pipes►

8.1 Darcy-Weisbach Equation

Major (friction) head loss: h_f = f × L × V² / (D × 2g) = f × L × Q² / (3.03D⁵)
f = Darcy friction factor (dimensionless)
D = pipe diameter; L = pipe length; V = mean velocity

Chezy formula: V = C√(RS); C = Chezy coefficient; R = D/4 (hydraulic radius for full pipe); S = h_f/L
Relation: f = 8g/C²

Darcy friction factor f:
Laminar: f = 64/Re
Turbulent smooth pipe (Blasius): f = 0.316/Re^0.25 [4000 < Re < 10⁵]
Turbulent (Colebrook-White, implicit): 1/√f = −2 log(ε/(3.7D) + 2.51/(Re√f))
ε = absolute pipe roughness (mm): Glass = 0.0015; Commercial steel = 0.046; Cast iron = 0.26; Concrete = 0.3–3.0

8.2 Minor Losses

Minor loss: h_m = K × V²/(2g) [K = loss coefficient]

Entry loss (sharp): K = 0.5
Entry loss (bell-mouth): K ≈ 0.04
Exit loss (pipe to reservoir): K = 1.0 (full velocity head lost)
Sudden expansion (Borda-Carnot): h_e = (V₁ − V₂)²/(2g); K = (1 − A₁/A₂)²
Sudden contraction: K = 0.5(1 − A₂/A₁)
Bend (90°): K = 0.3–1.5 depending on R/D
Gate valve (fully open): K = 0.2; Globe valve: K = 6–10

8.3 Pipes in Series and Parallel

Pipes in SERIES (same flow Q through each):
Q₁ = Q₂ = Q₃ = Q
h_f(total) = h_f1 + h_f2 + h_f3

Pipes in PARALLEL (same head loss across each):
h_f1 = h_f2 = h_f3
Q = Q₁ + Q₂ + Q₃

Equivalent pipe (same head loss and flow as the system):
Series: 1/C²eq = Σ(L_i / C²_i R_i A²_i) ... simplifies to:
L_eq/D_eq^5 = Σ(L_i/D_i^5) [same f, from h_f = fLQ²/(3.03D⁵)]
Parallel: L_eq/D_eq^5 = L / (ΣD_i^5/2)² ... from continuity and equal head loss

8.4 Pipe Network Analysis (Hardy-Cross)

For pipe networks: Kirchhoff's laws applied:
Node law: ΣQ = 0 at each junction (continuity)
Loop law: Σh_f = 0 around each loop (energy)

Hardy-Cross correction per loop:
ΔQ = −Σ(h_f) / (n × Σ|h_f/Q|)
n = exponent in h_f = kQ^n (n=2 for Darcy-Weisbach; n=1.85 for Hazen-Williams)
Iterate until ΔQ < tolerance

8.5 Water Hammer

Pressure rise due to sudden valve closure (Joukowsky equation):
ΔP = ρ × c × ΔV [ΔV = change in velocity = V₀ for complete closure]
c = celerity (wave speed) = 1 / √[ρ(1/K + D/(eE))]
K = bulk modulus of water; E = elastic modulus of pipe; e = pipe wall thickness

For rigid pipe: c = √(K/ρ) ≈ 1480 m/s (water)
For elastic pipe: c < 1480 m/s

Critical closure time: t_c = 2L/c (time for wave to travel to reservoir and return)
Slow closure (t > t_c): partial water hammer; less severe
Rapid closure (t < t_c): full Joukowsky pressure rise

📝 GATE Tip: Darcy-Weisbach: h_f = fLV²/(2gD). Laminar: f=64/Re. Exit loss K=1.0; entry K=0.5. Series pipes: total head loss additive; parallel: same head loss. Water hammer: ΔP = ρcΔV. L_eq/D_eq^5 = ΣL_i/D_i^5 (series equivalent pipe).

9Drag & Lift Force►

9.1 Drag and Lift Definitions

Drag force (parallel to flow): F_D = C_D × ½ρV²A
Lift force (perpendicular to flow): F_L = C_L × ½ρV²A
A = reference area (frontal area for drag; planform area for lift)

C_D, C_L = drag and lift coefficients (dimensionless; from experiments or theory)

9.2 Types of Drag

Pressure (form) drag: Due to pressure difference between front (high) and back (low, wake); dominant for bluff bodies
Friction (skin) drag: Due to shear stress on surface; dominant for streamlined bodies
Wave drag: Free-surface or compressibility effects (ships, supersonic aircraft)
Induced drag: Due to vortex shedding from tips of finite-span wings

9.3 Drag Coefficients for Standard Bodies

Body Shape	Re Range	C_D	Notes
Sphere	Re < 0.5	24/Re (Stokes)	Creeping flow; F_D = 3πμDV
Sphere	0.5–1000	Intermediate; decreasing	Oseen, Intermediate range
Sphere	10³–3×10⁵	≈ 0.4	Newton's drag regime
Sphere	> 3×10⁵	≈ 0.1–0.2	Drag crisis; turbulent BL (dimples on golf ball!)
Long circular cylinder	10³–10⁵	≈ 1.0–1.2	High form drag; separated flow
Flat plate (normal)	Turbulent	≈ 1.9	Full pressure drag; wake = plate size
Flat plate (parallel)	Laminar/Turbulent	≈ 0.001–0.005	Friction drag only
Aerofoil (thin)	Large Re	0.01–0.05	Streamlined; mostly friction drag

9.4 Stokes' Law (Creeping Flow)

For Re < 1 (sphere in viscous fluid):
F_D = 3πμDV (Stokes drag)
C_D = 24/Re

Terminal velocity of a settling sphere:
W − F_B − F_D = 0
V_t = (ρ_s − ρ_f) × g × D² / (18μ) [Stokes settling velocity]

Applications: sedimentation tanks, centrifuges, air quality (particle settling)

For laminar: F_D = F_pressure + F_friction = πμDV × 3 = πμDV(form) + 2πμDV(friction)

📝 GATE Tip: Stokes: F_D = 3πμDV; C_D = 24/Re. Terminal velocity: V_t = (ρ_s−ρ_f)gD²/(18μ). C_D(sphere) ≈ 0.4 for 10³ < Re < 3×10⁵. Drag crisis at Re ≈ 3×10⁵ — C_D drops sharply (reason for golf ball dimples). F_D = C_D × ½ρV²A is the universal drag formula.

10Boundary Layer Theory►

10.1 Boundary Layer Concept (Prandtl)

Prandtl (1904) introduced the boundary layer concept: for high Re flows over a body, viscous effects are confined to a thin layer near the surface — the boundary layer. Outside this layer, flow is essentially inviscid (potential flow).

Boundary layer thickness δ: distance from wall where velocity = 0.99U (free stream)
Displacement thickness: δ* = ∫₀^δ (1 − u/U) dy [measure of flow deficit]
Momentum thickness: θ = ∫₀^δ (u/U)(1 − u/U) dy [measure of momentum deficit]

Shape factor: H = δ*/θ

10.2 Laminar Boundary Layer (Blasius Solution)

Boundary layer thickness: δ/x = 5.0/√Re_x = 5.0x/√(Ux/ν)
Displacement thickness: δ*/x = 1.72/√Re_x
Momentum thickness: θ/x = 0.664/√Re_x
Local skin friction coefficient: C_f = τ_w / (½ρU²) = 0.664/√Re_x
Average drag coefficient (laminar, flat plate): C_D = 1.328/√Re_L

Transition from laminar to turbulent: Re_x,cr ≈ 5×10⁵ (smooth flat plate)

10.3 Turbulent Boundary Layer

Boundary layer thickness: δ/x = 0.37/Re_x^0.2 (1/5 power law)
Local skin friction: C_f = 0.0576/Re_x^0.2
Average drag (turbulent, flat plate): C_D = 0.074/Re_L^0.2 [Re < 10⁷]
Mixed (laminar + turbulent): C_D = 0.074/Re_L^0.2 − A/Re_L
A = 1742 for Re_cr = 5×10⁵

Law of the wall (turbulent velocity profile):
Viscous sublayer (y⁺ < 5): u⁺ = y⁺
Log-law region (30 < y⁺ < 300): u⁺ = (1/κ) × ln(y⁺) + B ≈ 2.5 ln(y⁺) + 5.0
κ = von Kármán constant = 0.41; u⁺ = u/u*; y⁺ = yu*/ν; u* = √(τ_w/ρ)

10.4 Boundary Layer Separation

Separation occurs when: ∂u/∂y|_{y=0} = 0 (wall shear stress zero) due to adverse pressure gradient
(pressure increases in flow direction → ∂p/∂x > 0)

Causes of separation: bluff bodies, sharp corners, diverging channels, airfoil at high angle of attack
Effects: large wake, increased pressure drag, loss of lift (stall)
Prevention: streamlining, suction, turbulent BL (trips), vortex generators

10.5 von Kármán Momentum Integral Equation

τ_w / (ρU²) = d θ/dx + (2+H) × θ/U × dU/dx
For zero pressure gradient (flat plate): τ_w = ρU² × dθ/dx
This relates wall shear to growth of momentum thickness; basis of approximate BL methods

📝 GATE Tip: Laminar BL: δ = 5x/√Re_x; C_D = 1.328/√Re_L. Turbulent BL: δ = 0.37x/Re_x^0.2; C_D = 0.074/Re_L^0.2. Transition at Re_x ≈ 5×10⁵. Separation: ∂u/∂y = 0 at wall; caused by adverse pressure gradient. Von Kármán constant κ = 0.41.

11Turbulent Flow►

11.1 Nature of Turbulence

Turbulent flow is characterised by chaotic, irregular velocity fluctuations in space and time. It involves mixing, vortex stretching, energy cascade from large eddies to small eddies, and dissipation as heat at the Kolmogorov scale.

Reynolds decomposition: u = ū + u' (mean + fluctuating components)
By definition: u' has zero time average: ū' = 0
Reynolds stresses (turbulent shear): τ_t = −ρ × ū'v' (Reynolds normal/shear stresses)
Total shear: τ_total = τ_viscous + τ_turbulent = μ(du/dy) − ρu'v'

11.2 Velocity Distribution in Turbulent Pipe Flow

Prandtl's mixing length theory: τ_t = ρ l² |du/dy| du/dy
l = mixing length ≈ κy (near wall; κ = 0.41)

Power law velocity profile: u/u_max = (y/R)^(1/n) or (1 − r/R)^(1/n)
n ≈ 7 at Re ≈ 10⁵; n → ∞ as Re → ∞
V_mean / u_max = 2n² / [(n+1)(2n+1)] ≈ 0.817 (for n=7)

Friction velocity: u* = √(τ₀/ρ) (τ₀ = wall shear stress)
Log-law (smooth pipe): u/u* = 5.75 log(y × u*/ν) + 5.5
Log-law (rough pipe): u/u* = 5.75 log(y/ε) + 8.5 (ε = roughness height)

11.3 Turbulent Flow in Pipes

Kinetic energy correction factor α = 1.05–1.10 (turbulent; approximately 1.0 for rough estimates)
Momentum correction factor β = 1.02–1.05 (turbulent)

Moody diagram: f = f(Re, ε/D)
Smooth: f = 0.316/Re^0.25 [Blasius; 4000 < Re < 10⁵]
Fully rough: 1/√f = 2 log(3.7D/ε) + const [f independent of Re; turbulence dominated by roughness]
Transition zone: Colebrook-White equation

📝 GATE Tip: α = 2.0 (laminar) vs α ≈ 1.05 (turbulent). Turbulent velocity profile: u/u_max = (y/R)^(1/7). V_mean/u_max ≈ 0.817 for n=7 (turbulent). τ_total = μ(du/dy) + ρl²(du/dy)². Friction velocity u* = √(τ₀/ρ).

12Dimensional Analysis & Model Analysis►

12.1 Buckingham π Theorem

If a physical phenomenon involves n variables and m fundamental dimensions (M, L, T):
Number of dimensionless π groups = n − m

Procedure:
1. List all variables (n total); identify fundamental dimensions (m)
2. Choose m repeating variables (must span all dimensions; e.g., ρ, V, D)
3. Form (n−m) π groups by multiplying each remaining variable with repeating variables raised to unknown powers
4. Solve powers by dimensional homogeneity; resulting π groups are dimensionless

12.2 Important Dimensionless Numbers

Number	Symbol	Formula	Significance	Model Similarity
Reynolds	Re	ρVL/μ = VL/ν	Inertia / Viscous; pipe flow, BL	Viscous models (pipe, BL)
Froude	Fr	V/√(gL)	Inertia / Gravity; free-surface flow	Open channel, ship, spillway
Euler	Eu	p / (ρV²)	Pressure / Inertia; cavitation	Pressure-dominated flows
Weber	We	ρV²L/σ	Inertia / Surface tension; droplets	Capillary flow, atomisation
Mach	Ma	V/c	Inertia / Compressibility; sound waves	Compressible (aerodynamics)
Strouhal	St	fL/V	Vortex shedding frequency; oscillatory flows	Vortex-induced vibration
Prandtl	Pr	μc_p/k = ν/α	Momentum / Thermal diffusivity	Heat transfer similarity

12.3 Model Analysis and Similarity Laws

Complete similarity: geometric + kinematic + dynamic similarity

Scale ratios (prototype:model = p:m):
L_r = L_p/L_m (length scale ratio)

FROUDE LAW (free-surface models): Fr_p = Fr_m
V_r = √(L_r); T_r = √(L_r); Q_r = L_r^(5/2); F_r = L_r³ (weight/pressure force)

REYNOLDS LAW (viscous flow models): Re_p = Re_m
V_r = ν_r / L_r; Q_r = V_r × L_r² = ν_r × L_r

MACH LAW (compressible flow): Ma_p = Ma_m
V_r = c_r = √(K_r/ρ_r)

Note: Froude and Reynolds similarity cannot both be satisfied simultaneously in the same model (unless the fluid is changed) — this is the model scale problem.

📝 GATE Tip: π groups = n − m. Froude: Q_r = L_r^(5/2); V_r = √L_r. Reynolds similarity: V_r = ν_r/L_r. Key dimensionless numbers: Re (pipe/BL), Fr (open channel/ships), Ma (compressible). Weber for surface tension. These appear in every GATE dimensional analysis question.

13Notches & Weirs►

13.1 Rectangular Notch / Weir

Theoretical flow (Francis formula derivation):
Q_th = (2/3) × L × √(2g) × H^(3/2)

Actual flow: Q = C_d × (2/3) × L × √(2g) × H^(3/2)
C_d ≈ 0.611 for sharp-crested weir; 0.62 typical
H = head above weir crest; L = weir length

Francis formula (with end contractions, n = number of contractions):
Q = C_d × [L − 0.1nH] × (2/3)√(2g) × H^(3/2)
n = 2 (two end contractions); n = 1 (one side contracted); n = 0 (suppressed, no contractions)

Velocity of approach: H_eff = H + V_a²/(2g) [corrected head]
Q = C_d × (2/3)L√(2g) × [H_eff^(3/2) − (V_a²/2g)^(3/2)]

13.2 Triangular (V-Notch) Weir

Q = (8/15) × C_d × tan(θ/2) × √(2g) × H^(5/2)
θ = apex angle of notch; H = head above vertex
C_d ≈ 0.62 for 90° notch

For 90° V-notch: Q = (8/15) × 0.62 × √(2×9.81) × H^(5/2) ≈ 1.417 × H^(5/2)

Advantages over rectangular: more sensitive at low flows (Q ∝ H^2.5 vs H^1.5); self-cleaning

13.3 Trapezoidal Weir (Cipolletti)

Side slopes 1:4 (H:V) compensate for end contractions:
Q = (2/3) × C_d × L × √(2g) × H^(3/2)
Same formula as suppressed rectangular (no end contraction correction needed)
C_d ≈ 0.62

13.4 Broad-Crested and Ogee Weirs

Broad-crested weir (flow becomes critical on crest):
Q = C_d × L × H × √(g × 2H/3) = C_d × L × (2/3)^(3/2) × √g × H^(3/2)
Q ≈ 1.705 × C_d × L × H^(3/2) [C_d ≈ 0.848 for well-rounded; 0.848 theoretical]

Ogee (overflow spillway): profile follows underside of sharp-crested nappe at design head H_d
C_d ≈ 0.74 (at design head); most efficient spillway design

Side weir (diversion): used to divert flow from canal; lateral weir formula is complex; Q_diverted depends on varying head

13.5 Submerged Weir

When tailwater rises above weir crest (submergence factor S = H₂/H₁):
Q_subm / Q_free = [1 − (H₂/H₁)^1.5]^0.385 [Villemonte equation]
H₁ = upstream head; H₂ = downstream head above weir crest
Submergence ratio S < 0.67 → <10% reduction; S > 0.9 → severe reduction

📝 GATE Tip: Rectangle: Q = C_d × (2/3)L√(2g) × H^(3/2). V-notch: Q = (8/15)C_d × tan(θ/2) × √(2g) × H^(5/2). For 90° V-notch: ≈ 1.417H^(5/2). C_d ≈ 0.62 (sharp-crested). Francis end contraction: L_eff = L − 0.1nH. These are the most directly tested formulae.

14Impact of Jets & Turbines►

14.1 Impact of Jets on Stationary Vanes

Flat plate (perpendicular to jet):
F = ρ × a × V² (a = jet area, V = jet velocity)
Power = 0 (plate stationary)

Curved vane (series of vanes; steady state):
F_x = ρaV(V − u)(1 − cosβ) [β = blade exit angle; u = vane velocity]
F_y = ρaV(V − u) sinβ
Work done per second = F_x × u = ρaV(V − u)²(1 − cosβ)

Maximum work (optimum u): u = V/2
Maximum efficiency (curved vane, single jet): η_max = (1 − cosβ)/2
For β = 180°: η_max = 1.0 (theoretically; semicircular vane, no friction)

14.2 Pelton Wheel (Impulse Turbine)

Jet velocity: V₁ = C_v × √(2gH) [H = net head; C_v ≈ 0.96–0.99]
Work done per kg: W = (V₁ − u) × u × (1 + k×cosβ)
k = friction factor on bucket (0.85–0.95); β = bucket angle (165°–170°; ≈ 180° theoretically)

Hydraulic efficiency:
η_h = 2u(V₁ − u)(1 + kcosβ) / V₁²
Maximum η_h when u = V₁/2 (jet:speed ratio = 0.5 at best efficiency):
η_h,max = (1 + kcosβ)/2 ≈ 90–95%

Speed ratio: φ = u/√(2gH) = 0.43–0.47 (best efficiency; theoretical 0.5)
Specific speed: N_s = N√P / H^(5/4) [P in kW; N in rpm; H in m]
N_s for Pelton: 4–70 (low N_s; high head, low flow)

14.3 Francis Turbine (Mixed Flow, Reaction)

Work done (Euler turbine equation):
W = (V_w1 × u₁ − V_w2 × u₂) / g [head equivalent]
V_w = whirl component of absolute velocity; u = peripheral blade velocity

Draft tube: allows turbine to be set above tailwater level; converts velocity head to pressure
η_dt = (V₃² − V₄²)/(2g) / {H_s + (V₃² − V₄²)/(2g)} (draft tube efficiency)
H_s = setting height above tailwater; V₃ = entry velocity to draft tube; V₄ = exit

N_s for Francis: 60–300; medium head (30–500 m); mixed head and flow
Francis turbine: η_max ≈ 90–92%

14.4 Kaplan / Propeller Turbine

Axial flow; adjustable blades (Kaplan) or fixed blades (propeller)
Low head (2–30 m); high flow rates
N_s: 300–900 (high specific speed)
η_max ≈ 91–93% (Kaplan; adjustable blades maintain efficiency over range of loads)

Comparison of turbine types:
Pelton: H > 300 m; N_s = 4–70; impulse; 1–2 jets
Francis: 30–500 m; N_s = 60–300; reaction; radial inflow
Kaplan: H < 50 m; N_s = 300–900; axial flow; adjustable blades

📝 GATE Tip: Pelton max efficiency: u = V₁/2. Work = ρaV(V−u)²(1+cosβ). Euler: W = (Vw1·u1 − Vw2·u2). N_s: Pelton 4–70, Francis 60–300, Kaplan 300–900. Pelton = impulse; Francis/Kaplan = reaction. Speed ratio φ ≈ 0.45 for Pelton at best efficiency.

15Pumps►

15.1 Classification of Pumps

Type	Mechanism	N_s	Application
Centrifugal (radial)	Centrifugal action; impeller; most common	20–80	High head, moderate flow; water supply
Mixed flow	Combines centrifugal + axial	80–160	Medium head and flow
Axial flow (propeller)	Axial lift; low head; high flow	160–400	Irrigation, drainage; very low head
Reciprocating (positive displacement)	Piston in cylinder; fixed volume per stroke	—	Dosing, very high pressure, viscous fluids
Gear / Rotary	Intermeshing gears or rotors; positive displacement	—	Lubrication, hydraulic systems

15.2 Centrifugal Pump — Performance

Euler's pump equation (work done on fluid per kg):
H_m = (V_w2 × u₂ − V_w1 × u₁) / g
V_w = whirl component; u = blade tip speed
For radial entry (V_w1 = 0): H_m = V_w2 × u₂ / g

Net Positive Suction Head (NPSH):
NPSH_a = p_s/(ρg) + V_s²/(2g) − p_v/(ρg) [available at pump inlet]
NPSH_r = required NPSH (from pump curve; depends on pump design)
Cavitation occurs when NPSH_a < NPSH_r

Suction head limit: H_s ≤ p_atm/(ρg) − p_v/(ρg) − NPSH_r − h_fs
Practical limit: H_s ≤ 6–8 m (for water at normal conditions)

15.3 Specific Speed and Affinity Laws

Specific speed: N_s = N × √Q / H^(3/4) [N in rpm, Q in m³/s or litres/min, H in m]
Unit specific speed N_su = N_s / 51.64 [gives dimensionless form]

Affinity (similarity) laws for same pump (varying speed):
Q ∝ N; H ∝ N²; P ∝ N³

For geometrically similar pumps (different sizes, same N):
Q ∝ D³; H ∝ D²; P ∝ D⁵

Combining (same head and fluid): Q ∝ N × D³; H ∝ N²D²; P ∝ ρN³D⁵

15.4 Pumps in Series and Parallel

Pumps in SERIES (same flow, heads add):
Q_total = Q_each; H_total = ΣH_i
Use when: high head required; single pump insufficient in head

Pumps in PARALLEL (same head, flows add):
H_total = H_each; Q_total = ΣQ_i
Use when: high flow required; demand exceeds single pump capacity

Operating point: intersection of pump curve (H vs Q) with system curve (H_system = H_static + kQ²)

15.5 Reciprocating Pump

Single-acting: Q_mean = A × L × N / 60 [A = piston area, L = stroke, N = rpm]
Double-acting: Q_mean = (2A − a) × L × N / 60 [a = piston rod area]

Slip: S = Q_theoretical − Q_actual
Coefficient of discharge: C_d = Q_actual/Q_theoretical = 1 − S/Q_theoretical
Negative slip: actual > theoretical (possible due to inertia and valve timing)

Air vessel: smooths pulsating flow; reduces acceleration head; placed close to pump

📝 GATE Tip: Affinity laws: Q∝N, H∝N², P∝N³. Specific speed N_s = N√Q/H^(3/4). Series → H adds; Parallel → Q adds. NPSH_a < NPSH_r → cavitation. Suction head < 6–8 m for water. These are the most frequently tested pump facts.

16Open Channel Flow►

16.1 Types and Basic Equations

Chezy's formula: V = C × √(R × S)
Manning's formula: V = (1/n) × R^(2/3) × S^(1/2) [SI units; most used]
Q = A × V = (A/n) × R^(2/3) × S^(1/2)

R = hydraulic radius = A/P (A = flow area; P = wetted perimeter)
S = slope of energy gradient (bed slope for uniform flow)
n = Manning's roughness coefficient

Relation: C = R^(1/6)/n (Chezy C in m^0.5/s; R in m)

16.2 Manning's n Values

Channel Type	Manning's n
Smooth concrete	0.011–0.013
Unfinished concrete	0.014–0.017
Brick lined	0.013–0.016
Clean earth channel	0.020–0.025
Grassed earth channel	0.030–0.040
Natural (clean, winding)	0.025–0.040
Natural (weedy, deep pools)	0.050–0.100
Flood plain (cultivated areas)	0.030–0.040

16.3 Most Efficient Channel Section

Best hydraulic section = section with minimum wetted perimeter P for given area A
→ maximum R = A/P → maximum Q for given slope, n and area

Best hydraulic section for each shape:
Rectangle: B = 2y (width = 2 × depth); R = y/2
Trapezoid: θ = 60° (equilateral triangle half); side slope 1:√3 (H:V); R = y/2
Circle: most efficient at d = 0.94D (Q_max); V_max at d = 0.81D
Semi-circle: theoretically the most efficient cross-section of all shapes
Triangle: m = 1 (45° sides); V-shaped

16.4 Specific Energy and Critical Flow

Specific energy: E = y + V²/(2g) = y + Q²/(2gA²)
y = flow depth; V = mean velocity; A = flow area

Critical depth y_c: depth at which specific energy is minimum for given Q
dE/dy = 0 → 1 − Q²T/(gA³) = 0 → Fr = V/√(g×A/T) = 1
T = top water surface width

For RECTANGULAR channel:
y_c = (Q²/(gB²))^(1/3) [B = channel width]
V_c = √(g × y_c) [critical velocity]
E_c = (3/2) × y_c [minimum specific energy = 1.5 × critical depth]
Fr = V/√(gy)

Fr < 1: Subcritical (tranquil; y > y_c; gentle slope; controlled by downstream)
Fr = 1: Critical
Fr > 1: Supercritical (rapid; shooting; y < y_c; steep slope; controlled by upstream)

Fig. 16.1 — (Left) Specific energy diagram: E_min at y_c; two depths for same E (alternate depths). (Right) Hydraulic jump: supercritical → subcritical; sequent depth y₂/y₁ = ½(√(1+8Fr₁²)−1)

16.5 Hydraulic Jump

Sequent (conjugate) depth ratio (rectangular channel):
y₂/y₁ = ½ × [−1 + √(1 + 8Fr₁²)]
Fr₁ = V₁/√(gy₁) > 1 (supercritical inflow)

Energy loss in hydraulic jump:
ΔE = E₁ − E₂ = (y₂ − y₁)³ / (4y₁y₂)

Length of hydraulic jump: L_j ≈ 5 to 7 × y₂
(No theoretical formula; empirical only)

Classification of hydraulic jumps:
Fr₁ = 1–1.7: Undular; weak oscillation
Fr₁ = 1.7–2.5: Weak; small rollers
Fr₁ = 2.5–4.5: Oscillating; jet oscillates bottom to surface
Fr₁ = 4.5–9.0: Steady; best energy dissipation; stable
Fr₁ > 9.0: Strong; rough; choppy; large energy loss

16.6 Gradually Varied Flow (GVF)

GVF equation: dy/dx = (S₀ − S_f) / (1 − Fr²)
S₀ = bed slope; S_f = friction slope = n²V²/R^(4/3) [Manning]; Fr = Froude number

Normal depth y_n: depth at which flow is uniform (S_f = S₀)
Critical depth y_c: depth at which Fr = 1

Profile classification:
M1: y > y_n > y_c (backwater; mild slope; subcritical)
M2: y_n > y > y_c (drawdown; mild slope; subcritical)
M3: y_n > y_c > y (mild slope; supercritical; after sluice gate)
S1: y > y_c > y_n (steep slope; subcritical backwater)
S2: y_c > y > y_n (drawdown; steep slope; supercritical)
S3: y_c > y_n > y (steep slope; supercritical)
C1, C2: critical slope (y_n = y_c); H1, H2, H3: horizontal slope; A1, A2: adverse slope

16.7 Uniform Flow — Non-Circular Sections

Trapezoidal section: A = (B + my)y; P = B + 2y√(1+m²)
R = (B+my)y / (B+2y√(1+m²)); m = side slope (H:V)

Circular section at partial depth d:
θ = 2 × arccos(1 − 2d/D) [half central angle]
A = (D²/8)(θ − sinθ); P = Dθ/2; R = (D/4)(1 − sinθ/θ)
Q_max at d/D = 0.94; V_max at d/D = 0.81

📝 GATE Tip: Manning: V = (1/n)R^(2/3)S^(1/2). Critical: E_c = 1.5y_c; y_c = (Q²/gB²)^(1/3). Fr < 1 subcritical; > 1 supercritical. Hydraulic jump: y₂/y₁ = ½(√(1+8Fr₁²)−1). ΔE = (y₂−y₁)³/(4y₁y₂). Best rectangle: B = 2y. Best trapezoid: m = 1/√3 (60° walls). These are perennial GATE questions.

★Quick Revision – All Formulae, Tables & Mnemonics►

Fluid Properties

τ = μ(du/dy) [Newton's law]; ν = μ/ρ; S_Hg = 13.6; γ_w = 9.81 kN/m³
Droplet: ΔP = 4σ/d; Bubble: 8σ/d; Jet: 2σ/d
Capillary rise: h = 4σcosθ/(ρgd); Liquid μ↓ with T; Gas μ↑ with T

Hydrostatics

p = ρgh; F = ρgȳA; y_cp = ȳ + I_G/(ȳA)
Curved surface: F_H = ρgȳ_proj × A_proj; F_V = weight of fluid above
Fraction submerged = ρ_body/ρ_fluid; BM = I_WL/V_sub; GM = BM − BG
Rectangle pontoon: BM = B²/(12d); Stable if GM > 0

Fluid Kinematics

A₁V₁ = A₂V₂ = Q; u = ∂ψ/∂y, v = −∂ψ/∂x
Steady flow: streamline = pathline = streakline
Irrotational: ∂v/∂x = ∂u/∂y; ∇²φ = 0

Fluid Dynamics

Bernoulli: p/ρg + V²/2g + z = H (constant along streamline)
Torricelli: V = √(2gh); Orifice: Q = C_d × A × √(2gh); C_d = C_v × C_c
Venturimeter: C_d = 0.96–0.99; Orifice meter: C_d ≈ 0.61; C_c ≈ 0.64; C_v ≈ 0.98
ΣF = ρQ(V₂ − V₁) [momentum equation]

Viscous and Pipe Flow

H-P: Q = πD⁴ΔP/(128μL); u_max = 2V_mean; α = 2.0; β = 4/3 [laminar]
Darcy-Weisbach: h_f = fLV²/(2gD); Laminar: f = 64/Re
Entry loss K=0.5; Exit loss K=1.0; Sudden expansion: h_e = (V₁−V₂)²/(2g)
Water hammer: ΔP = ρcΔV; Colebrook-White for turbulent f

Boundary Layer

Laminar BL: δ = 5x/√Re_x; C_D = 1.328/√Re_L [flat plate]
Turbulent BL: δ = 0.37x/Re_x^0.2; C_D = 0.074/Re_L^0.2
Transition at Re_x ≈ 5×10⁵; Stokes drag: F_D = 3πμDV; C_D = 24/Re
Terminal velocity: V_t = (ρ_s−ρ_f)gD²/(18μ)

Dimensional Analysis

π groups = n − m; Re = ρVL/μ; Fr = V/√(gL); Ma = V/c; We = ρV²L/σ
Froude law: Q_r = L_r^(5/2); V_r = √L_r
Reynolds law: V_r = ν_r/L_r; Q_r = ν_r × L_r

Weirs

Rectangle: Q = C_d(2/3)L√(2g)H^(3/2); C_d ≈ 0.62
V-notch: Q = (8/15)C_d·tan(θ/2)·√(2g)·H^(5/2); 90° ≈ 1.417H^(5/2)
Francis end correction: L_eff = L − 0.1nH

Turbines and Pumps

Pelton: max η at u = V₁/2; φ = 0.43–0.47; N_s = 4–70
Francis: N_s = 60–300; Euler: H = (Vw1·u1 − Vw2·u2)/g
Kaplan: N_s = 300–900; Affinity: Q∝N; H∝N²; P∝N³
N_s(pump) = N√Q/H^(3/4); Series → ΣH; Parallel → ΣQ
NPSH_a < NPSH_r → cavitation; H_s < 6–8 m practical limit

Open Channel Flow

Manning: V = (1/n)R^(2/3)S^(1/2); Q = AV
Critical: y_c = (Q²/gB²)^(1/3); E_c = 1.5y_c; Fr = V/√(gy)
Hydraulic jump: y₂/y₁ = ½(√(1+8Fr₁²)−1); ΔE = (y₂−y₁)³/(4y₁y₂)
Best rectangle: B = 2y; Best trapezoid: m = 1/√3
Circular Q_max at d/D = 0.94; V_max at d/D = 0.81

Key Number Summary

Parameter	Value	Context
S_mercury	13.6	Specific gravity; manometry
C_d (orifice)	≈ 0.61	Sharp-edged; 0.96–0.99 for Venturi
C_v	≈ 0.98	Velocity coefficient
C_c	≈ 0.64	Contraction coefficient
Laminar Re	< 2000	Pipe flow transition
Turbulent Re	> 4000	Pipe flow
f (laminar)	64/Re	Darcy friction factor
α (laminar / turbulent)	2.0 / ~1.05	Kinetic energy correction
Exit loss K	1.0	Pipe discharges to reservoir
Entry loss K	0.5	Sharp-edged entry
Re (BL transition)	5 × 10⁵	Flat plate
Pelton speed ratio φ	0.43–0.47	u/√(2gH) at best efficiency
C_d weir (sharp)	0.62	Rectangular and V-notch
E_c / y_c	1.5	Critical specific energy
Best rectangle	B = 2y	Most efficient open channel

Mnemonics

"Droplets Bubble Jets — 4, 8, 2":
ΔP: Droplet = 4σ/d (two surfaces) | Bubble = 8σ/d (soap, two films) | Jet = 2σ/d (one surface)

"CP is always Below Centroid — like CG is the Boss, CP is the Employee below":
y_cp = ȳ + I_G/(ȳA); always y_cp > ȳ (CP deeper than centroid)

"BM = Big Moment / Volume" — BM = I_WL / V_sub
"Stable if GM is Good and Positive (M above G)"

"Bernoulli's Bank Account: pressure + kinetic + potential = constant":
p/ρg + V²/2g + z = H (conservation of energy along streamline)

Orifice coefficients: "Cd = Cv × Cc → 0.61 = 0.98 × 0.64"
(Check: 0.98 × 0.64 = 0.627 ≈ 0.62 ✓)

"SERIES flows through all pipes — Head Adds; PARALLEL splits the flow — Head Same"
Same rule as electrical: series R add; parallel V same

"Pelton's perfect speed is Half": u = V₁/2 for maximum efficiency
Pelton N_s = 4–70 (low); Kaplan N_s = 300–900 (high) — "K is the King of Low Head, High N_s"

"Fr < 1 = Subcritical = Slow = Tall" (deep water, slow)
"Fr > 1 = Supercritical = Shooting = Shallow" (fast, shallow)
"Hydraulic Jump goes from Supercritical to Subcritical — from Fast & Shallow to Slow & Deep"

Best hydraulic section: "Rectangle half the width is depth: B=2y; Trapezoid at 60° walls"

Affinity laws mnemonic: "Q-H-P = 1-2-3 powers of N"
Q ∝ N¹; H ∝ N²; P ∝ N³ — "One Head Powers Two, Three"

Exam-Angle Comparison

Topic	GATE Focus	ESE Focus	SSC JE Focus
Fluid Properties	Newton's law; viscosity types; capillarity formulae; surface tension ΔP	Non-Newtonian fluids; Andrade equation; compressibility; cavitation number	Definitions; units; S_Hg = 13.6; μ increases/decreases with T
Manometry	U-tube calculation; differential manometer p_A−p_B	Inverted U-tube; micro-manometer; inclined manometer amplification	Simple U-tube reading; manometer types; Pascal's law
Hydrostatics	F=ρgȳA; y_cp formula; curved surface F_H and F_V	Full curved surface analysis; dam overturning; lock gate forces	F=ρgȳA; CP below CG; dam pressure triangle
Buoyancy	BM = I_WL/V_sub; GM; experimental GM; fraction submerged	Stability conditions; metacentric height calculation; layered fluids	Archimedes; fraction submerged; floating vs sinking condition
Kinematics	Continuity A₁V₁=A₂V₂; stream function; irrotationality condition	Vorticity; circulation; flow net; potential functions	Continuity concept; streamline definition; laminar vs turbulent
Bernoulli / Flow Measurement	Venturimeter Q formula; C_d values; Pitot tube; momentum on bend	Modified Bernoulli; Euler equation derivation; impact force on plate	Bernoulli equation; Torricelli V=√(2gh); Venturimeter concept
Pipe Flow	Darcy-Weisbach; f=64/Re; minor losses; series/parallel equivalent pipe	Hardy-Cross; water hammer; Moody diagram; Colebrook-White	Darcy-Weisbach formula; f=64/Re for laminar; minor loss names
Boundary Layer	BL thickness formulae; C_D laminar/turbulent flat plate; transition Re	Von Kármán integral; separation conditions; Blasius solution	BL concept; laminar vs turbulent BL; Stokes settling
Dim. Analysis	π theorem (n−m groups); Froude Q_r = L_r^5/2; Re similarity	Full model scale law derivations; combined similarity problems	Dimensionless number names and physical meanings
Weirs	Rectangle and V-notch Q formulae; Francis end correction; C_d values	Broad-crested weir; ogee spillway; submerged weir Villemonte	Rectangular weir Q formula; V-notch 90°; C_d ≈ 0.62
Turbines	Pelton speed ratio u=V/2; Euler equation; N_s ranges; draft tube	Complete turbine design; Francis runner velocity triangles; cavitation	Types of turbines; impulse vs reaction; N_s comparison
Pumps	Affinity laws Q∝N,H∝N²,P∝N³; series vs parallel; NPSH	Pump characteristics; system curve; reciprocating pump; cavitation	Centrifugal pump concept; series vs parallel; affinity laws basic
Open Channel	Manning; critical depth y_c; hydraulic jump sequent depth; E_c=1.5y_c	GVF profiles (M1,M2,S1,S2); best hydraulic section; gradually varied flow	Manning's formula; critical flow concept; hydraulic jump concept

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Hydraulics & Fluid Mechanics – Complete Study Notes

1.1 Definition and Classification

1.2 Newton's Law of Viscosity

1.3 Viscosity — Temperature Dependence

1.4 Compressibility and Bulk Modulus

1.5 Surface Tension and Capillarity

1.6 Cavitation

2.1 Pressure and Pascal's Law

2.2 Types of Manometers

2.3 Manometer Calculation Procedure

2.4 Pressure Prism and Hydrostatic Paradox

3.1 Force on a Plane Submerged Surface

3.2 Force on a Curved Submerged Surface

3.3 Pressure on Dams and Lock Gates

4.1 Archimedes' Principle

4.2 Metacentre and Stability of Floating Bodies

4.3 Metacentric Height

5.1 Types of Flow

5.2 Flow Lines

5.3 Continuity Equation

5.4 Stream Function and Velocity Potential

5.5 Vorticity and Circulation

6.1 Bernoulli's Equation

6.2 Flow Measuring Devices

6.3 Orifice and Vena Contracta

6.4 Momentum Equation (Linear Momentum)

7.1 Hagen-Poiseuille Flow (Laminar Pipe Flow)

7.2 Laminar Flow Between Parallel Plates

7.3 Reynolds Number and Transition

8.1 Darcy-Weisbach Equation

8.2 Minor Losses

8.3 Pipes in Series and Parallel

8.4 Pipe Network Analysis (Hardy-Cross)

8.5 Water Hammer

9.1 Drag and Lift Definitions

9.2 Types of Drag

9.3 Drag Coefficients for Standard Bodies

9.4 Stokes' Law (Creeping Flow)

10.1 Boundary Layer Concept (Prandtl)

10.2 Laminar Boundary Layer (Blasius Solution)

10.3 Turbulent Boundary Layer

10.4 Boundary Layer Separation

10.5 von Kármán Momentum Integral Equation

11.1 Nature of Turbulence

11.2 Velocity Distribution in Turbulent Pipe Flow

11.3 Turbulent Flow in Pipes

12.1 Buckingham π Theorem

12.2 Important Dimensionless Numbers

12.3 Model Analysis and Similarity Laws

13.1 Rectangular Notch / Weir

13.2 Triangular (V-Notch) Weir

13.3 Trapezoidal Weir (Cipolletti)

13.4 Broad-Crested and Ogee Weirs

13.5 Submerged Weir

14.1 Impact of Jets on Stationary Vanes

14.2 Pelton Wheel (Impulse Turbine)

14.3 Francis Turbine (Mixed Flow, Reaction)

14.4 Kaplan / Propeller Turbine

15.1 Classification of Pumps

15.2 Centrifugal Pump — Performance

15.3 Specific Speed and Affinity Laws

15.4 Pumps in Series and Parallel

15.5 Reciprocating Pump

16.1 Types and Basic Equations

16.2 Manning's n Values

16.3 Most Efficient Channel Section

16.4 Specific Energy and Critical Flow

16.5 Hydraulic Jump

16.6 Gradually Varied Flow (GVF)

16.7 Uniform Flow — Non-Circular Sections

Fluid Properties

Hydrostatics

Fluid Kinematics

Fluid Dynamics

Viscous and Pipe Flow

Boundary Layer

Dimensional Analysis

Weirs

Turbines and Pumps