Mechanical Properties of Fluids
This chapter tests pressure and its transmission, surface tension and capillarity, Bernoulli's principle with its applications, and viscosity with terminal velocity. Expect one derivation (Bernoulli or Stokes) and one numerical on capillary rise or terminal speed.
Core rules
- Pressure at depth: P = P₀ + hρg; gauge pressure excludes atmospheric pressure.
- Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished; a hydraulic lift multiplies force by the area ratio A₂/A₁.
- Excess pressure: 2T/R inside a liquid drop, 4T/R inside a soap bubble.
- Capillary rise: h = 2T cosθ/(rρg). Mercury (θ > 90°) therefore shows depression.
- Equation of continuity: A₁v₁ = A₂v₂. Bernoulli: P + ½ρv² + ρgh = constant along a streamline for an ideal, non-viscous, incompressible, steady flow.
- Torricelli: efflux speed v = √(2gh). Venturimeter uses the pressure drop in a constriction.
- Viscous force F = 6πηrv (Stokes). Terminal velocity v(t) = 2r²(ρ − σ)g/(9η), reached when weight equals buoyancy plus viscous drag.
- Reynolds number Re = ρvd/η; below about 2000 the flow is streamline, above 4000 turbulent.
Worked example
Water rises in a tube of radius 0.2 mm. With T = 0.072 N/m, θ = 0°, ρ = 1000 kg/m³: h = (2 × 0.072)/(2×10⁻⁴ × 1000 × 9.8) = 0.144/1.96 = 0.0735 m, that is about 7.4 cm.
Traps
- Substituting the diameter where the formula needs the radius - the commonest lost mark here.
- Forgetting cosθ when the liquid does not wet the tube fully.
- Applying Bernoulli across a viscous or turbulent flow, where it is invalid.