9.2 Fluid Dynamics & Bernoulli

9.2 Fluid Dynamics & Bernoulli

Pressure

Pressure (P) is force per unit area: P = F/A. The SI unit is the pascal (Pa), where 1 Pa = 1 N/m². In a static fluid of density ρ at depth h, pressure increases with depth:

• P = ρgh

Pressure at a point acts equally in all directions — this is why a dam wall is built thicker at the base, and why a diver's ears hurt long before the air supply runs out.

Worked example — hydraulic piston: A 50 N force is applied to a piston of area 0.01 m². What pressure is transmitted through the fluid?

• P = F/A = 50 / 0.01 = 5000 Pa = 5 kPa

Hydraulic systems exploit the fact that pressure is transmitted undiminished: a small force on a small piston can lift a heavy load on a large piston. This is the principle behind aircraft hydraulic landing gear and brakes.

Buoyancy (Archimedes' Principle)

The buoyant force on a body submerged (fully or partly) in a fluid equals the weight of the fluid it displaces:

• F_b = ρ_fluid · g · V_displaced

A body floats when its average density is less than the fluid's. A hot-air balloon rises because heated air is less dense than the surrounding atmosphere. Note that a 400-tonne aircraft does not float on air by buoyancy — it is lift, not buoyancy, that holds it up. Buoyancy needs a displaced volume whose weight matches the load; for a steel aircraft that would require a vacuum-filled hull the size of a stadium. Lift, by contrast, is generated dynamically by the wing.

The Continuity Equation

For an incompressible fluid (essentially all liquids, and air at speeds well below the speed of sound) flowing through a pipe of varying cross-section, the volume flow rate is conserved:

• A₁v₁ = A₂v₂

Where the pipe narrows, the fluid speeds up; where it widens, it slows down. This is why a nozzle constricts a jet of water into a fast, focused stream, and — crucially for aviation — why air accelerates as it streams over the narrowed flow paths above a curved wing.

Worked example — pipe expansion: Water flows at 4 m/s through a pipe of cross-section 2 cm² that widens to 8 cm². Find the new speed.

• A₁v₁ = A₂v₂ ⇒ (2)(4) = (8)(v₂) ⇒ v₂ = 8/8 = 1 m/s

Doubling the area halves the speed; quadrupling it quarters the speed. The same physics governs the airflow above and below an airfoil.

Bernoulli's Principle

For steady, incompressible, frictionless flow along a streamline, the total mechanical energy per unit volume is constant:

• P + ½ρv² + ρgh = constant

The three terms are static pressure (P), dynamic pressure (½ρv²), and hydrostatic pressure (ρgh). At the same height, if speed rises, static pressure must fall — Bernoulli trades one for the other.

Aviation application — how a wing makes lift: An airfoil is shaped so that air travels faster over the curved upper surface than over the flatter lower surface. By Bernoulli, faster air means lower static pressure on top. The pressure difference (P_bottom − P_top) multiplied by the wing area gives the lift force:

• L = (P_bottom − P_top) × A_wing

At takeoff the pilot rotates the nose up to increase the angle of attack, which intensifies the pressure difference; lift grows with the square of speed until it exceeds weight, and the aircraft climbs.

Worked example — lift estimate: Airspeed over the upper wing is 250 m/s, below is 200 m/s. Taking ρ ≈ 1.2 kg/m³, the pressure difference is:

• ΔP = ½ρ(v_top² − v_bottom²) = ½(1.2)(250² − 200²) = 0.6(62 500 − 40 000) = 0.6(22 500) = 13 500 Pa
• Over a 30 m² wing, lift = 13 500 × 30 = 405 000 N (about 41 tonnes of force).

In reality, lift is also explained by Newton's third law: the wing deflects air downward, and the air pushes the wing up. The two explanations are complementary, not contradictory — Bernoulli describes the pressure field, Newton describes the momentum transfer.

Viscosity

Viscosity (η) is a fluid's internal resistance to flow — think of honey versus water. The dynamic viscosity SI unit is the pascal-second (Pa·s), where 1 Pa·s = 1 N·s/m². Typical values at room temperature:

• Air: ≈ 1.8 × 10⁻⁵ Pa·s
• Water (20 °C): ≈ 1.0 × 10⁻³ Pa·s
• Glycerin: ≈ 1.4 Pa·s
• Honey: ≈ 10 Pa·s

Higher viscosity means more energy lost to internal friction, which shows up as skin-friction drag on a moving aircraft. Reducing viscosity-related drag is why aerodynamic surfaces are polished and why laminar-flow wings are designed to keep the boundary layer smooth for as long as possible.

Note the distinction: dynamic viscosity (η, Pa·s) appears in F = ηA(dv/dy); kinematic viscosity (ν = η/ρ, m²/s) is what you use in Reynolds-number calculations. Don't confuse the two on the exam.

Common Traps

• Static vs dynamic pressure: Static pressure is what a tyre gauge reads; dynamic pressure (½ρv²) is the pressure a moving fluid exerts when brought to rest. Bernoulli trades static for dynamic pressure along a streamline — they are not the same quantity, and you cannot add one to itself to get the other.
• Bernoulli is not the whole lift story: Real lift also relies on Newton's third law (the wing deflects air downward; the air pushes the wing up). Examiners may probe whether you confuse the two — they are complementary, not contradictory.
• Continuity needs incompressible flow: It holds well for liquids and low-speed air, but breaks down at compressible (near-sonic) speeds where density changes along the streamline.
• Bernoulli assumes no viscosity: Near the wing surface, viscous effects in the boundary layer matter; Bernoulli describes the flow just outside that layer.