Fluid Dynamics and Bernoulli's Equation

Conservation of Mass (Continuity)

For steady, incompressible flow: $A_1 V_1 = A_2 V_2 = Q$

where Q = volumetric flow rate (m³/s).

Mass flow rate: ṁ = ρAV = ρQ (kg/s)

Bernoulli's Equation

For steady, inviscid, incompressible flow along a streamline:

$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2$

Each term has units of length (head):

• P/γ = pressure head
• V²/2g = velocity head
• z = elevation head

Total head (Bernoulli constant) = P/γ + V²/2g + z = constant along a streamline

Applications of Bernoulli's Equation

Pitot Tube: Measures velocity from stagnation pressure $V = \sqrt{\frac{2(P_{stag} - P_{static})}{\rho}}$

Venturi Meter: Measures flow rate from pressure difference at a constriction $Q = C_d A_2 \sqrt{\frac{2g(h_1 - h_2)}{1 - (A_2/A_1)^2}}$

Free Jet: Velocity of fluid exiting a tank $V = \sqrt{2gh}$

(Torricelli's theorem — same as free-fall velocity from height h)

Reynolds Number

$Re = \frac{\rho V D}{\mu} = \frac{VD}{\nu}$

Dimensionless Numbers Summary

Pipe Flow and Head Losses

The Energy Equation (Modified Bernoulli)

$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L$

where hL = total head loss = major losses + minor losses.

Major Losses (Pipe Friction) — Darcy-Weisbach Equation

$h_f = f \frac{L}{D} \frac{V^2}{2g}$

where f = Darcy friction factor (from Moody diagram or Colebrook equation).

For laminar flow (Re < 2,300): $f = \frac{64}{Re}$

For turbulent flow: Use the Moody diagram with Re and relative roughness ε/D.

Minor Losses (Fittings, Valves, Etc.)

$h_m = K \frac{V^2}{2g}$

Open-Channel Flow

Manning's Equation

$V = \frac{1}{n} R_h^{2/3} S^{1/2} \quad (\text{SI units})$

where:

• n = Manning's roughness coefficient
• Rh = hydraulic radius = A/P (area/wetted perimeter)
• S = slope of the energy grade line

Froude Number

$Fr = \frac{V}{\sqrt{gD_h}}$