Fluid Mechanics

Fluid Mechanics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Density (ρ) = mass/volume, measured in kg/m³. Relative density has no unit — it’s the ratio of a substance’s density to water’s density (1000 kg/m³). Pressure (P) = force/area, unit: N/m² or Pa. In a static liquid, P = ρgh (ρ = density, g = 10 m/s², h = depth). Pressure acts equally in all directions at a given depth.

Upthrust (U) = ρVg — weight of fluid displaced. Archimedes’ principle: upthrust equals the weight of displaced fluid. A body floats when U = weight; sinks when weight > U. Pascal’s law: pressure applied to an enclosed fluid is transmitted equally to every part of the fluid.

Viscosity resists fluid flow; higher viscosity = slower flow. Surface tension (γ = F/L) lets insects walk on water.

JAMB High-Yield: P = ρgh questions appear almost every year. Memorize upthrust = weight of displaced fluid (not total fluid weight). Know when a body floats vs sinks.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Pressure in Fluids

Pressure at a depth h in a fluid of density ρ is given by P = ρgh. This comes from: weight of fluid column = mg = ρVg = ρAhg, so P = weight/area = ρgh. The key trap: pressure depends only on depth, not on the volume or shape of the container. A 10 m deep pool exerts the same pressure as a narrow 10 m deep tube at the same depth. Atmospheric pressure (≈ 1.01 × 10⁵ Pa) adds to liquid pressure, so absolute pressure = P_atm + ρgh.

Upthrust and Archimedes’ Principle

A body immersed in fluid experiences an upward force called upthrust. Mathematically, U = ρ_f V_d g, where ρ_f is the fluid density and V_d is the volume of fluid displaced — not the volume of the body. For a fully submerged body, V_d equals the body’s volume. For a floating body, only part is submerged and V_d is the submerged volume. A body floats when ρ_body < ρ_fluid; it sinks when ρ_body > ρ_fluid.

Pascal’s Law and Applications

Pressure applied to any part of an enclosed fluid is transmitted equally to all parts. In hydraulic presses, F₁/A₁ = F₂/A₂, so a small force applied to a small piston can lift a large load using a larger piston. This is how hydraulic brakes and jacks work.

Fluid Flow: Continuity and Bernoulli

For streamline flow, A₁V₁ = A₂V₂ (continuity equation). Fluid speed increases where the pipe narrows. Bernoulli’s equation relates pressure, speed, and height: P + ½ρv² + ρgh = constant. As fluid speed increases, pressure decreases — this explains lift on an airplane wing. In horizontal flow (h constant), increased velocity means decreased pressure.

JAMB Pattern: Questions on pressure in liquids and upthrust calculations dominate. Numerical problems combining Archimedes’ principle with density comparisons are frequent.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

The Law of Floatation

When a body floats, the upthrust exactly equals the weight of the body: ρ_body V_total g = ρ_fluid V_submerged g, giving V_submerged/V_total = ρ_body/ρ_fluid. For ice floating in water (ρ_ice ≈ 920 kg/m³, ρ_water = 1000 kg/m³), about 92% of the ice is submerged. This principle explains how ships, submarines, and hot-air balloons maintain equilibrium. A hydrometer uses this: a denser liquid displaces less liquid, making it float higher.

Bernoulli’s Principle — Edge Cases

Bernoulli’s equation P + ½ρv² + ρgh = constant assumes steady, streamline, incompressible, non-viscous flow. In real fluids with viscosity, energy is lost as heat, so the equation is approximate. For horizontal flow (Δh = 0), the equation simplifies to P + ½ρv² = constant. In Venturi meters (a pipe that narrows), fluid speeds up in the constriction, pressure drops, and this pressure difference can be measured to find flow rate.

Viscosity and Stokes’ Law

Viscosity arises from intermolecular forces resisting relative motion between fluid layers. Terminal velocity occurs when drag force equals weight minus upthrust: v_t = (mg - U)/6πηr (for a sphere of radius r, viscosity η). This is tested in free-fall through viscous media.

Common Mistakes to Avoid

• Confusing mass and density: density is mass per unit volume, not weight per unit volume.
• Forgetting that pressure = ρgh requires g and h; g ≈ 10 m/s² in JAMB problems.
• Using the body’s volume instead of displaced volume when calculating upthrust.
• Neglecting atmospheric pressure when asked for absolute pressure.
• Misapplying Bernoulli in non-horizontal situations — always include the ρgh term.

Practice Prompts

1. A block of wood (density 600 kg/m³) floats in water. What fraction of its volume is submerged? (Answer: 0.6 or 60%)
2. Water flows through a pipe of radius 0.1 m at speed 2 m/s and enters a narrower pipe of radius 0.05 m. Find the speed in the narrower pipe. (Answer: 8 m/s via A₁V₁ = A₂V₂)

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