Mechanical Properties of Fluids

Earlier mechanics — Newton's laws, work-energy, rotational motion — dealt with rigid bodies, whose shape and size do not change under force. A fluid (liquid or gas) has no fixed shape and takes the shape of its container. This chapter studies fluids at rest (pressure, Pascal's law, buoyancy), fluids in motion (streamline flow, the equation of continuity, Bernoulli's theorem), and two properties from intermolecular forces inside the fluid: viscosity, which resists relative motion between flowing layers, and surface tension, which makes a liquid's free surface behave like a stretched elastic membrane.

These are not textbook abstractions. Pascal's law runs the hydraulic brake in every car. Bernoulli's theorem explains an aircraft wing's lift. Viscosity decides how a raindrop reaches a safe falling speed instead of hitting the ground at free-fall speed. Surface tension is why small insects walk on a pond without sinking.

Pressure is the normal force exerted by a fluid per unit area: P = F/A. It is a scalar — at any point inside a fluid at rest, pressure is the same in every direction (proved by the force balance on a small wedge-shaped fluid element, which forces P_x = P_y = P_z). This differs from a solid, where a force acts only along a fixed direction.

For a fluid at rest under gravity, pressure increases with depth because each layer supports the weight above it. Balancing forces on a thin horizontal slab gives P = P₀ + ρgh, where P₀ is the pressure at the free surface and h is the depth below it. This is why a dam wall is thicker at the bottom, and why a diver's ears hurt more the deeper she goes.

Pascal's law follows: a change in pressure applied to an enclosed, incompressible fluid is transmitted undiminished to every point of the fluid and the container's walls — the principle behind the hydraulic lift and brake, where a small force on a narrow piston produces a much larger force on a wide piston, since P = F₁/A₁ = F₂/A₂.

Archimedes' principle: a body partly or wholly immersed in a fluid experiences an upward buoyant force equal to the weight of fluid it displaces. A body floats when this force can equal its own weight; it sinks if even fully immersed the displaced fluid weighs less than the body.

When a fluid flows so that every particle passing a given point follows the same path as the particle before it, the flow is streamline (or laminar). No two streamlines can cross — a particle at the crossing point would need two velocities at once. A bundle of streamlines forms a tube of flow, which no fluid crosses. Beyond a certain critical speed, the orderly layers break into eddies and whorls — turbulent flow.

For streamline flow of an incompressible fluid through a pipe of varying cross-section, mass entering any section per second equals mass leaving another section per second; since density is constant, this gives the equation of continuity: A₁v₁ = A₂v₂, i.e. Av is constant along a tube of flow. This is why water speeds up when a hose nozzle narrows, and a river flows faster through a gorge than a plain.

Bernoulli's theorem is the work-energy theorem applied to a flowing, non-viscous, incompressible fluid. Take a fluid element flowing steadily through a pipe of varying cross-section and height, between a section of area A₁, height h₁, speed v₁, pressure P₁, and a second section A₂, h₂, v₂, P₂. In time Δt, the pressure forces do net work (P₁ − P₂)ΔV on the fluid of volume ΔV = A₁v₁Δt = A₂v₂Δt (equation of continuity). This work changes the fluid's kinetic and potential energy: ΔKE = ½Δm(v₂² − v₁²), ΔPE = Δmg(h₂ − h₁), where Δm = ρΔV. The work-energy theorem then gives (P₁ − P₂)ΔV = ½Δm(v₂² − v₁²) + Δmg(h₂ − h₁); dividing by ΔV and rearranging gives Bernoulli's equation: P + ½ρv² + ρgh = constant along a streamline.

Three applications follow. Speed of efflux (Torricelli's law): for a wide tank with a small hole at depth h, Bernoulli's equation gives v = √(2gh), the same as free fall through height h. Venturimeter: a pipe constriction speeds up flow there, which lowers pressure there — measuring the drop gives the flow rate. Dynamic lift: faster air over an aerofoil's curved upper surface means lower pressure above than below, giving a net upward force; the same unequal-speed idea (Magnus effect) curves a spinning ball.

Real fluids offer internal friction between adjacent layers moving at different speeds — this property is viscosity. Consider a fluid flowing over a fixed surface: the layer in contact with it is at rest, and successive layers above move faster, so velocity varies with distance from the surface (a velocity gradient dv/dx). Newton's law of viscosity gives the tangential force needed to maintain this sliding as F = ηA(dv/dx), where η is the coefficient of viscosity and A is the contact area between layers; its SI unit is Pa s. Viscosity of liquids falls as temperature rises (thermal agitation weakens the cohesive forces causing the friction), while viscosity of gases rises with temperature (molecular momentum transfer between layers increases).

Stokes' law gives the viscous drag on a small sphere of radius r moving at speed v through a fluid of viscosity η: F = 6πηrv. A sphere falling through a viscous fluid speeds up until this drag plus the buoyant force exactly balance its weight, after which it falls at a constant terminal velocity: v_t = 2r²(ρ − σ)g/(9η), where ρ and σ are the densities of the sphere and the fluid. This is why a raindrop reaches the ground at a safe, steady speed instead of accelerating all the way down.

Poiseuille's equation gives the volume of viscous fluid flowing per second through a narrow tube of radius r and length l under a pressure difference P: V = πPr⁴/(8ηl). The fourth-power dependence on radius means even a small narrowing of a blood vessel drastically cuts blood flow — the medical reason atherosclerosis is dangerous.

Whether a flow stays streamline or turns turbulent depends on a dimensionless combination of the flow variables called the Reynolds number, Re = ρvd/η, where ρ is fluid density, v is the average flow speed, d is a characteristic length (pipe diameter, for flow through a tube), and η is the coefficient of viscosity. Re compares the inertial forces (which produce eddies) to the viscous forces (which damp them). Flow through a pipe stays streamline for Re below about 2000, is transitional up to about 3000, and is turbulent above that. The flow speed at Re ≈ 2000 is the critical speed.

Reynolds number explains everyday observations: smoke from an incense stick flows smoothly at first and breaks into swirls as its speed increases; blood flow becomes turbulent (a detectable murmur) if a vessel narrows; and aircraft/car designers test scale models in wind tunnels, since matching the Reynolds number reproduces the full-size flow pattern in a small model.

A molecule inside a liquid is pulled equally in all directions by neighbours, so the net cohesive force on it is zero. A molecule at the free surface has liquid molecules only below and around it, not above, so it feels a net inward pull. Bringing a molecule to the surface needs work against this pull, so surface molecules have extra potential energy. The free surface, at constant temperature, tends to minimise this extra energy, i.e. minimise its area — why an isolated drop is spherical (least area for a given volume) and why a stretched liquid film behaves like a taut elastic sheet.

Surface energy is the extra potential energy per unit area of the liquid surface. Surface tension T is the force per unit length acting along the surface, tangential to it, tending to pull it taut: T = F/l. Considering a rectangular film on a wire frame with a sliding wire, and equating the work done stretching it to the rise in surface energy, shows surface energy per unit area equals surface tension numerically: W = T·ΔA. Surface tension (N m⁻¹) falls as temperature rises, becoming zero at the critical temperature.

Where a liquid surface meets a solid, it meets at a definite angle of contact (θ), measured inside the liquid between the tangent to the liquid surface at the contact point and the solid surface. Its value depends on the relative strength of the cohesive force (between molecules of the same liquid) and the adhesive force (between liquid and solid molecules). If adhesion dominates, the liquid wets the solid and spreads, giving an acute angle (θ < 90°) — water on clean glass has θ ≈ 0°, meniscus concave. If cohesion dominates, the liquid beads up, giving an obtuse angle (θ > 90°) — mercury on glass has θ ≈ 140°, meniscus convex.

Angle of contact decides whether a liquid rises or falls in a capillary tube, and can be changed by an impurity: detergent in water lowers both surface tension and the angle of contact, letting soapy water spread into narrow crevices in cloth fibres — why detergents clean better than plain water.

Because a curved liquid surface behaves like a stretched membrane, the pressure just inside it exceeds the pressure just outside — the excess pressure. For a spherical liquid drop of radius r, equating the work done by surface tension in a small expansion dr to the work done by the excess pressure (P·4πr²dr = T·8πr dr) gives the excess pressure inside a drop: P = 2T/r. The same holds for a spherical air bubble trapped inside a liquid, since it too has only one curved liquid surface.

A soap bubble in air has two liquid surfaces (outer and inner, both ≈radius r), so the same derivation doubles the result: the excess pressure inside a soap bubble is P = 4T/r. Smaller drops/bubbles always have larger excess pressure — why, when two soap bubbles of different sizes are joined by a tube, the smaller one shrinks and blows air into the larger, not the reverse.

When a capillary is dipped into water, water rises up the tube; dipped into mercury, mercury is depressed below the outer level. This capillarity follows from surface tension and the meniscus curvature. Inside a narrow tube, the liquid surface curves into roughly a hemispherical meniscus of radius R, related to the tube's radius r and angle of contact θ by r = R cosθ. Because the meniscus is curved, the pressure just below it is less than atmospheric pressure by 2T/R — this deficit pushes liquid up the tube until the weight of the risen column restores the balance.

Equating the excess pressure 2T/R = 2T cosθ/r to the hydrostatic pressure ρgh of the risen column gives h = 2T cosθ/(ρgr). For water in clean glass, θ ≈ 0° so cosθ ≈ 1 and the liquid rises; for mercury, θ > 90° so cosθ is negative, giving a depression, exactly as observed. Rise is inversely proportional to the tube's radius — a finer tube rises more — which is how water reaches the top of tall trees through narrow xylem vessels, and how a wick draws up liquid.