Pressure at any depth h in a static liquid of density ρ is the sum of atmospheric pressure at the surface and the weight of the liquid column above per unit area. This is because the weight of the fluid above pushes down on everything below it, increasing pressure linearly with depth.

This is why submarines, deep-sea fish, and scuba divers all face engineering and biological challenges — pressure increases relentlessly with depth. Every 10 m of water adds approximately 1 atm of pressure. A diver at 30 m experiences about 4 atm total (1 atm + 3 atm water), causing nitrogen to dissolve in blood — this is why divers must ascend slowly to avoid decompression sickness ("the bends").

Absolute pressure is the total pressure at a point, measured against perfect vacuum. Gauge pressure is the excess above atmospheric pressure. Most everyday instruments measure gauge pressure — a car tyre showing 30 psi means 30 psi above atmospheric. In this problem, the submarine's structural limit is 100 atm absolute. At the water surface, it already faces 1 atm atmospheric, so the water column must provide at most 99 atm gauge pressure, giving depth = 990 m.

Pascal's Law states that pressure applied to an enclosed incompressible fluid is transmitted equally in all directions throughout the fluid and to the walls of the container. This is the principle behind all hydraulic systems:

Any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This is Archimedes' Principle — discovered when Archimedes noticed water spilling as he stepped into his bath and shouted "Eureka!"

Object floats if F_buoyancy ≥ Weight → ρ_object ≤ ρ_fluid. A submarine controls buoyancy by flooding (to dive) or blowing compressed air into (to surface) its ballast tanks — changing average density relative to seawater. At neutral buoyancy, weight = buoyancy and the submarine hovers at constant depth without using propulsion.

Surface tension arises from cohesive forces between liquid molecules. Surface molecules have fewer neighbours than interior molecules, resulting in a net inward pull — making the surface act like a stretched elastic sheet. Surface tension T = force per unit length (N/m).

Viscosity η is the resistance of a fluid to flow — internal friction between fluid layers moving at different speeds. For a sphere of radius r moving at speed v through fluid of viscosity η, the viscous drag force (Stokes' Law) is:

At terminal velocity, drag + buoyancy = weight. Terminal velocity ∝ r² — larger drops fall faster. Viscosity of liquids decreases with temperature (honey flows better when warm) while gas viscosity increases with temperature.

For steady, non-viscous, incompressible fluid flow along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant:

Applications: where cross-section narrows, velocity increases and pressure drops — this drives Venturi meters, sprayers/atomisers, and aircraft wing lift. Torricelli's theorem (speed of water draining from a tank) is a special case: v = √(2gh). Equation of continuity: A₁v₁ = A₂v₂ (volume flow rate conserved for incompressible fluid).