Fluid Mechanics & Hydraulics — Formula Sheet

Mass density: ρ = m/V    [kg/m³]

Specific weight: γ = ρg    [N/m³];   γ_water = 9810 N/m³

Specific gravity: SG = ρ_fluid/ρ_water = γ_fluid/γ_water   (dimensionless)

⭐ Newton’s law of viscosity: τ = μ (du/dy)    [Pa]

Kinematic viscosity: ν = μ/ρ    [m²/s]

Units: μ in Pa·s; 1 Poise = 0.1 Pa·s;   ν in m²/s; 1 Stoke = 10⁻⁴ m²/s

Temperature effect: Liquids → μ decreases with ↑T;   Gases → μ increases with ↑T

Surface tension — pressure inside droplet: Δp = 4σ/d

Pressure inside soap bubble (2 surfaces): Δp = 8σ/d

Pressure inside liquid jet: Δp = 2σ/d

Capillary rise/depression: h = 4σ cosθ/(ρgd)

Water–glass: θ ≈ 0° → capillary rise;   Mercury–glass: θ ≈ 135° → capillary depression

Bulk modulus: K = –V(dp/dV) = ρ(dp/dρ)    [Pa]

Water: K ≈ 2.1 GPa;   Speed of sound: c = √(K/ρ) ≈ 1450 m/s in water

Hydrostatic pressure at depth h: p = ρgh = γh    [Pa]

Pressure head: h = p/(ρg) = p/γ    [m of fluid]

Pascal’s law: Pressure at a point acts equally in all directions in a static fluid.

p_abs = p_atm + p_gauge

⭐ Total hydrostatic force on a plane surface:

F = ρg Ȳ A = γ Ȳ A    [N]

where Ȳ = vertical depth of centroid; A = surface area

⭐ Centre of pressure depth:

h_cp = Ȳ + I_G/(AȲ)   [h_cp > Ȳ always]

I_G for rectangle (b×d): bd³/12;   Circle (D): πD⁴/64

Curved surface forces:

F_H = force on vertical projection of curved surface

F_V = weight of fluid above curved surface (real or imaginary)

F_resultant = √(F_H² + F_V²)

⭐ Buoyant force (Archimedes): F_B = ρ_f g V_submerged

Floating body: V_sub/V_total = ρ_body/ρ_fluid

Metacentric height: GM = BM – BG;   BM = I_WL/V_s

GM > 0 → stable;   GM = 0 → neutral;   GM < 0 → unstable

U-tube manometer (gauge pressure):

p_A = ρ_m g h_m – ρ_f g h_f

Differential manometer:

p_A – p_B = h_m(ρ_m – ρ_f)g

Mercury–water: Δp = 12.6 × x × ρ_w × g   (x = manometer deflection in m)

⭐ Bernoulli’s equation (ideal flow along streamline):

p₁/(ρg) + V₁²/(2g) + z₁ = p₂/(ρg) + V₂²/(2g) + z₂

or: p + ½ρV² + ρgz = constant    [Pa]

Total head H = p/(ρg) + V²/(2g) + z = constant    [m]

⭐ Continuity equation (incompressible): A₁V₁ = A₂V₂ = Q

For circular pipe: Q = (πD²/4) × V

⭐ Modified Bernoulli (with losses and pump/turbine):

p₁/(ρg) + V₁²/(2g) + z₁ + h_p = p₂/(ρg) + V₂²/(2g) + z₂ + h_t + h_L

Pump power: P = ρgQh_p/η    [W]

Turbine power: P = ρgQh_t × η    [W]

Pitot tube: V = C_v√(2gΔh)   where Δh = stagnation head – static head

Torricelli’s theorem (orifice in tank): V_th = √(2gH)

Siphon maximum crest height (approx): z_c,max ≈ 7–8 m (practical; 10.33 m theoretical)

HGL = p/(ρg) + z;   EGL = HGL + V²/(2g);   EGL – HGL = V²/(2g)

⭐ Venturimeter / Orifice meter actual discharge:

Q = C_d × (A₁A₂/√(A₁² – A₂²)) × √(2gΔh)

Differential head from manometer: Δh = x[(ρ_m/ρ_f) – 1]   (mercury–water: Δh = 12.6x)

C_d: venturimeter 0.96–0.99; orifice meter 0.61–0.65; flow nozzle 0.95–0.99

Orifice in tank: Q = C_d × A₀ × √(2gH)

C_d = C_v × C_c;   C_v ≈ 0.97–0.99;   C_c ≈ 0.61–0.64

C_v from jet trajectory: C_v = x/(2√(yH))

C_d (sharp-edged orifice) ≈ 0.62; C_d (external mouthpiece) ≈ 0.82

⭐ Rectangular notch/weir:

Q = (2/3) C_d L √(2g) H^3/2 = 1.838 L H^3/2   (Francis, C_d = 0.623)

Francis end contraction correction: Q = 1.838 (L – 0.1nH) H^3/2

n = 0 (suppressed); n = 1 (one contraction); n = 2 (both ends free)

⭐ Triangular (V-notch) weir:

Q = (8/15) C_d tan(θ/2) √(2g) H^5/2

90° V-notch (C_d = 0.611): Q = 1.417 H^5/2   (Thompson’s formula)

Broad-crested weir: Q = C_d × 1.705 × L × H^3/2   (C_d ≈ 0.848)

Cipolletti (trapezoidal, side slopes 1:4): Q = 1.859 L H^3/2

⭐ Darcy-Weisbach equation (major friction loss):

h_f = fLV²/(2gD) = 8fLQ²/(π²gD⁵)

Laminar flow: f = 64/Re

Turbulent smooth pipe (Blasius, Re < 10⁵): f = 0.316 Re^–0.25

Swamee-Jain (explicit): f = 0.25/[log(ε/3.7D + 5.74/Re^0.9)]²

Hagen-Poiseuille (laminar only):

Q = πD⁴Δp/(128μL);   h_f = 32μLV/(γD²) = 128μLQ/(πγD⁴)

Velocity profile: u(r) = V_max[1–(r/R)²];   V_max = 2V_avg

Minor losses: h_m = K V²/(2g)

Sharp entry: K = 0.5; Exit (any): K = 1.0; Globe valve (open): K = 4–10; 90° elbow: K ≈ 0.9

⭐ Sudden expansion (Borda-Carnot): h_e = (V₁–V₂)²/(2g)

⭐ Pipes in series: Q same; H_total = Σh_fi

H = Q² × Σ(8f_iL_i/π²gD_i⁵)

⭐ Pipes in parallel: h_f1 = h_f2; Q_total = ΣQ_i

Equivalent pipe (series, same f): L_eq/D_eq⁵ = Σ(L_i/D_i⁵)

Water hammer (Joukowsky): Δp = ρcΔV;   Δh = cΔV/g

Wave speed: c ≈ 900–1400 m/s (steel pipe); ≈ 300–500 m/s (PVC/HDPE)

Critical closure time: T_c = 2L/c; if t_closure < T_c → sudden closure

⭐ Reynolds number: Re = ρVD/μ = VD/ν = 4Q/(πDν)

Flow regimes (pipe): Re < 2000 → Laminar; 2000–4000 → Transitional; Re > 4000 → Turbulent

Critical velocity: V_c,laminar = 2000ν/D;   V_c,turbulent = 4000ν/D

Laminar velocity profile: u(r) = V_max[1–(r/R)²];   V_max = 2V_avg

Wall shear stress: τ_w = 8μV/D = 4τ̄ (average is half wall value)

Turbulent (1/7 power law): u/V_max = (y/R)^1/7;   V_avg/V_max ≈ 0.817

Entry length — Laminar: L_e/D ≈ 0.06 Re;   Turbulent: L_e/D ≈ 4.4 Re^1/6

Hydraulic diameter (non-circular): D_h = 4A/P

Open channel Re: Re = VR/ν (R = hydraulic radius); critical Re ≈ 500–2000

⭐ Manning’s equation: V = (1/n) R^2/3 S^1/2

Q = (1/n) A R^2/3 S^1/2

n_concrete = 0.013;   n_{earthen canal} = 0.025;   n_{natural river} = 0.025–0.050

Chezy’s formula: V = C√(RS);   C = R^1/6/n

Hydraulic radius: R = A/P (NOT pipe radius)

Section formulas:

Rectangle (b, y): A = by; P = b+2y; R = by/(b+2y)

Trapezoid (b, z, y): A = (b+zy)y; P = b+2y√(1+z²); R = A/P

Full circle (D): A = πD²/4; P = πD; R = D/4

⭐ Most economical sections:

Rectangle: b = 2y → R = y/2

Trapezoid (z given): b = 2y(√(1+z²)–z) → R = y/2 (inscribed semi-circle)

Best trapezoid (z free): z = 1/√3 (half-hexagon, 60° sides)

All best sections: R = y/2

⭐ Froude number: Fr = V/√(gD) = V/√(gy) (rectangular)

Fr < 1: subcritical; Fr = 1: critical; Fr > 1: supercritical

Critical depth (rectangular): y_c = (q²/g)^1/3;   q = Q/b

Critical velocity: V_c = √(gy_c)

Critical slope (wide channel): S_c = gn²/y_c^1/3

GVF equation: dy/dx = (S₀–S_f)/(1–Fr²)

M1: y > y_n > y_c (mild slope; backwater behind dam)

M2: y_n > y > y_c (mild; drawdown)

M3: y < y_c < y_n (mild; below sluice gate)

Circular pipe — maximum flow: occurs at y/D ≈ 0.94; Q_max/Q_full ≈ 1.076

Maximum velocity at y/D ≈ 0.81; V_max/V_full ≈ 1.14

⭐ Specific energy: E = y + V²/(2g) = y + q²/(2gy²)   [m]

At critical flow: E_min = (3/2)y_c;   y_c = (2/3)E_min

Velocity head at critical depth: V_c²/(2g) = y_c/2

General critical condition (any section): Q²/g = A³/T

Hump analysis:

E over hump = E₁ – Δz (neglecting friction)

Max hump height without choking: Δz_max = E₁ – E_min = E₁ – (3/2)y_c

If Δz > Δz_max: flow choked; critical flow at hump; upstream depth increases

⭐ Hydraulic jump sequent depth ratio:

y₂/y₁ = ½[√(1 + 8Fr₁²) – 1]

y₁ = upstream (supercritical) depth; y₂ = downstream (subcritical) sequent depth

Fr₁ > 1 always; y₂ > y₁ always

⭐ Energy dissipated in hydraulic jump:

ΔE = (y₂ – y₁)³/(4y₁y₂)   [m]

Power dissipated: P = ρgQΔE   [W]

Height of jump: h_j = y₂ – y₁

Jump length (approximate): L_j ≈ 5–7 × y₂

⭐ Euler’s turbomachinery equation:

H = (u₂V_w2 – u₁V_w1)/g   (pump, no pre-swirl: H = u₂V_w2/g)

H = (u₁V_w1 – u₂V_w2)/g   (turbine)

u = ωr = πDN/60   (peripheral blade velocity)

Pump power & efficiency:

Power to fluid: P_fluid = ρgQH_m

Overall efficiency: η = ρgQH_m/P_input

P_input = ρgQH_m/η

η_manometric = H_m/H_Euler = gH_m/(u₂V_w2)

⭐ Specific speed — Pump: N_s = NQ^1/2/H^3/4

⭐ Specific speed — Turbine: N_s = N√P/H^5/4

Pelton: N_s = 4–30; Francis: N_s = 51–255; Kaplan: N_s = 255–860   (rpm, kW, m)

⭐ Affinity laws (same impeller, speed change):

Q₂/Q₁ = N₂/N₁;   H₂/H₁ = (N₂/N₁)²;   P₂/P₁ = (N₂/N₁)³

Size change (same speed):

Q ∝ D³; H ∝ D²; P ∝ D⁵

Pelton wheel hydraulic efficiency:

η_h = 2(u/V₁)(1 + k cosβ)(1 – u/V₁)

Maximum at u/V₁ = 0.5;   η_h,max = (1 + k cosβ)/2

Jet velocity: V₁ = C_v√(2gH);   C_v ≈ 0.97–0.99

⭐ Cavitation / NPSH:

NPSH_available = H_atm – H_v – H_s – h_f,suction

Condition: NPSH_avail > NPSH_required

H_s,max ≈ 10.33 – H_v – NPSH_req – h_f,s

Thoma’s cavitation number: σ = (H_atm – H_s – H_v)/H

No cavitation if: σ > σ_c (critical Thoma number from tests)

Reciprocating pump discharge:

Single-acting: Q_th = A × L × N/60

Double-acting: Q_th = (2A – A_rod) × L × N/60

Q_act = η_v × Q_th

⭐ Buckingham Pi theorem: Number of Pi terms = n – m

n = total variables; m = fundamental dimensions (usually 3: M, L, T)

F(π₁, π₂, …, π_n–m) = 0

Rules for repeating variables: Choose m variables; span all m dimensions; exclude dependent variable; avoid same-dimension pairs.

⭐ Key dimensionless numbers:

Reynolds: Re = ρVL/μ = VL/ν   (inertia/viscous)

Froude: Fr = V/√(gL)   (inertia/gravity)

Euler: Eu = Δp/(ρV²)   (pressure/inertia)

Weber: We = ρV²L/σ   (inertia/surface tension)

Mach: Ma = V/c   (inertia/elastic)

Strouhal: St = fL/V   (unsteady/convective inertia)

⭐ Froude model law (open channel, same fluid, g_r=1):

V_r = √L_r;   T_r = √L_r;   Q_r = L_r^5/2;   F_r = L_r³;   P_r = L_r^7/2

Reynolds model law (same fluid): V_r = 1/L_r

Dimensions of key quantities (MLT system):

Force: MLT⁻²; Pressure: ML⁻¹T⁻²; Energy: ML²T⁻²; Power: ML²T⁻³

Viscosity μ: ML⁻¹T⁻¹;   Kinematic viscosity ν: L²T⁻¹

Surface tension σ: MT⁻²;   Bulk modulus K: ML⁻¹T⁻²