Fluid Kinematics, Bernoulli and Momentum

Fluid dynamics links velocity, pressure and force through three conservation laws: mass (continuity), energy (Bernoulli) and momentum.

Describing flow

• Steady: properties at a point do not change with time. Uniform: they do not change along the flow.
• Streamline: a line tangent to the velocity everywhere. In steady flow, streamlines, pathlines and streaklines coincide.
• Acceleration in a flow field = local (∂V/∂t) + convective (u ∂V/∂x + …).

Continuity

For incompressible flow:

• In a pipe: Q = A₁V₁ = A₂V₂.
• In 2D: ∂u/∂x + ∂v/∂y = 0.

Stream function ψ: u = ∂ψ/∂y, v = −∂ψ/∂x. It satisfies continuity automatically; the flow between two streamlines = ψ₂ − ψ₁.

Velocity potential φ: u = ∂φ/∂x, v = ∂φ/∂y. It exists only for irrotational flow (vorticity ∂v/∂x − ∂u/∂y = 0).

Bernoulli's equation

For steady, incompressible, inviscid flow along a streamline:

p/(ρg) + V²/(2g) + z = constant (pressure head + velocity head + elevation head)

With losses and pumps: p₁/ρg + V₁²/2g + z₁ + h_pump = p₂/ρg + V₂²/2g + z₂ + h_loss.

Applications:

• Venturimeter: Q = C_d A₁A₂ √(2gh)/√(A₁² − A₂²), with h the head difference.
• Pitot tube: measures stagnation pressure; V = √(2gh) (times a coefficient).
• Torricelli: flow from an orifice at depth H: V = √(2gH).

Momentum equation

The net force on a fluid in a control volume = rate of change of momentum:

ΣF = ρQ(V_out − V_in) (vectors)

• Jet on a fixed flat plate normal to it: F = ρAV².
• Jet on a plate moving away at u: F = ρA(V − u)².
• Jet deflected through angle θ by a fixed vane: F_x = ρQV(1 − cos θ).

Worked example

Water flows through a pipe that narrows from 0.2 m to 0.1 m diameter, at 0.05 m³/s. Find the velocities and the pressure drop (same level, no losses).

• V₁ = 0.05/(π × 0.01) = 1.59 m/s; V₂ = 0.05/(π × 0.0025) = 6.37 m/s.
• Δp = ρ(V₂² − V₁²)/2 = 500 × (40.53 − 2.53) = 19.0 kPa.

Practice