Introduction
Fluid mechanics studies how liquids and gases behave at rest and in motion. Where rigid-body mechanics treats an object as a single unchanging shape, fluids continuously deform, so the subject needs new concepts — pressure, density, viscosity and flow rate — built on top of the mechanics already learned.
1. Fluid statics: pressure, buoyancy, Pascal’s and Archimedes’ principles
Pressure is force per unit area (P = F/A) and, in a static fluid, acts equally in all directions at a given point and increases with depth: P = P₀ + ρgh, where ρ is fluid density. This is why a diver feels increasing pressure with depth and why dams are built thicker at the bottom.
Pascal’s principle: a pressure change applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of its container. This is the operating principle of hydraulic systems (car brakes, hydraulic lifts): a small force on a small piston creates a pressure that, transmitted to a larger piston, produces a much larger force — the trade-off being that the larger piston moves a proportionally smaller distance, so no energy is gained.
Archimedes’ principle: an object submerged in a fluid experiences an upward buoyant force equal to the weight of fluid it displaces. An object floats if its average density is less than the fluid’s, sinks if greater, and is neutrally buoyant if equal — this is why steel ships float (their overall shape displaces enough water) while a solid steel block sinks.
2. Fluid dynamics: continuity, Bernoulli’s equation, viscosity
Equation of continuity: for an incompressible fluid flowing through a pipe of varying cross-section, A₁v₁ = A₂v₂ — the flow rate is constant, so fluid speeds up where the pipe narrows. This is why water jets faster from a narrow nozzle than a wide hose at the same pressure.
Bernoulli’s equation: P + ½ρv² + ρgh = constant along a streamline in an ideal (non-viscous, incompressible) flow. It expresses conservation of energy for a flowing fluid: where speed increases, pressure must decrease (and vice versa), at constant height. This explains lift on an aircraft wing (air moves faster over the curved top surface, so pressure there is lower than beneath the wing, producing a net upward force) and why a shower curtain is pulled inward when water flows past it.
Viscosity is a fluid’s internal resistance to flow, arising from friction between adjacent layers moving at different speeds. Viscous fluids (honey) flow more slowly than low-viscosity fluids (water) under the same pressure difference. The Reynolds number, a dimensionless ratio of inertial to viscous forces, predicts whether flow will be smooth (laminar, low Reynolds number) or chaotic (turbulent, high Reynolds number).
3. Surface tension and capillarity
Surface tension arises because molecules at a liquid’s surface are pulled inward by neighbouring molecules (with no molecules above to balance the pull), so the surface behaves like a stretched elastic membrane, minimising its area. This is why small water drops are nearly spherical and why some insects can walk on water.
Capillarity is the rise or fall of liquid in a narrow tube due to the combined effects of surface tension and the liquid’s adhesion to (or cohesion away from) the tube’s walls. Water rises in a narrow glass tube (adhesion to glass exceeds cohesion within water), while mercury is depressed (cohesion within mercury exceeds its adhesion to glass). Capillarity explains water rising through soil and the fine root structures of plants.
4. Elasticity and basic material response
Elasticity describes how a solid material deforms under applied stress (force per unit area) and whether it returns to its original shape once the stress is removed. Strain is the resulting fractional deformation (change in length divided by original length, for example). Within the elastic limit, stress and strain are proportional — Hooke’s law, σ = Eε, where E is the material’s Young’s modulus, a measure of stiffness. Beyond the elastic limit, the material undergoes permanent (plastic) deformation and, at the breaking point, fractures. Different moduli describe different types of deformation: Young’s modulus for stretching/compression, the shear modulus for twisting, and the bulk modulus for uniform compression (relevant to how fluids themselves respond to pressure, linking this topic back to fluid statics).
Key Takeaways
- Static fluid pressure increases linearly with depth and is transmitted equally in all directions (Pascal), while buoyancy equals the weight of displaced fluid (Archimedes) and determines floating or sinking.
- The continuity equation and Bernoulli’s equation together describe moving fluids: narrower cross-sections mean faster flow, and faster flow at constant height means lower pressure.
- Viscosity is internal fluid friction and, together with the Reynolds number, determines whether flow is smooth (laminar) or chaotic (turbulent).
- Surface tension and capillarity arise from intermolecular forces at a liquid’s surface and explain droplet shape, insect flotation, and the rise of liquids in narrow tubes.