Fluid Dynamics

Hydrostatics primer

Pressure in a static fluid increases with depth: P = P₀ + ρgh. A diver at 10 m in seawater (ρ ≈ 1030 kg/m³) experiences gauge pressure ≈ 1.01×10⁵ Pa, roughly one extra atmosphere. Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished — the basis of hydraulic lifts where F₁/A₁ = F₂/A₂. Archimedes' principle gives buoyant force B = ρ_fluidV_displacedg; a body floats when its average density is less than that of the fluid.

Equation of continuity

For an incompressible, steady flow, A₁v₁ = A₂v₂. The product Av is the volumetric flow rate (m³/s). Squeezing a hose nozzle from 1 cm² to 0.25 cm² quadruples the exit speed if the source flow is constant. This conservation law follows from mass conservation and underlies almost every fluid-flow MCQ on the MDCAT.

Bernoulli's equation

For non-viscous, incompressible, steady flow along a streamline, P + ½ρv² + ρgh = constant. Faster-moving fluid has lower pressure — explaining lift on aircraft wings, the curve of a spinning ball, and the operation of a Venturi meter. From a tank with a small hole at depth h, the efflux speed is v = √(2gh) (Torricelli's theorem), the same as a freely falling body. A 5 m deep tank delivers v = √(98) ≈ 9.9 m/s at the bottom.

Viscosity and Stokes' law

Viscosity η is the internal friction between adjacent fluid layers; SI unit is Pa·s. For a sphere of radius r falling through a viscous fluid at speed v, the drag is F = 6πηrv (Stokes' law). At terminal velocity, gravity = drag + buoyancy, giving v_t = (2r²(ρ_s − ρ_f)g)/(9η). A small steel ball thus falls slower in glycerine than in water because η is much larger.

Reynolds number and turbulence

R_e = ρvD/η classifies flow: laminar for R_e < ~2000, turbulent for R_e > ~3000, transitional in between. Bernoulli strictly applies only to laminar flow; for turbulent or viscous flow, energy losses must be added. The FSc Punjab Textbook Chapter 6 introduces Stokes and Reynolds explicitly; HRW Chapter 14 treats Bernoulli rigorously with worked Venturi-meter problems.