Solids and Fluid Dynamics

Pull on a wire and it stretches. How much depends on the force, but also on how thick the wire is and how long it was to begin with — so comparing materials needs quantities that remove those.

Stress is force per unit area, which removes the thickness. Strain is extension divided by original length, which removes the length. Strain is a ratio of two lengths, so it has no units at all.

For small loads, stress and strain are proportional. That is Hooke's law, and the constant of proportionality is the Young modulus — a property of the material itself, not of the particular wire. Steel has a Young modulus about twice that of copper, meaning it takes twice the stress for the same strain.

Beyond the elastic limit the material no longer returns to its original length when the load is removed; it has been permanently deformed. Beyond that comes the yield point, where it stretches rapidly for little extra load, and finally it breaks.

σ

stressPa

ε

strain

E

Young modulusPa

e

extensionm

L

original lengthm

A fluid at rest exerts pressure on everything in it, and that pressure grows with depth because the weight of fluid above increases. It depends on depth, density and gravitational field strength — and on nothing else. The shape of the container is irrelevant, which surprises people the first time they meet it.

At any given depth the pressure acts equally in all directions. That is what makes hydraulics work: pressure applied anywhere in an enclosed fluid is transmitted undiminished throughout it, so a small force on a small piston becomes a large force on a large one.

Upthrust is a consequence of the same fact. The pressure on the bottom of a submerged object is greater than on the top, because the bottom is deeper. The difference multiplied by the area is a net upward force, and Archimedes' principle states that it equals the weight of the fluid displaced.

An object floats when the upthrust equals its weight — which happens when its average density is less than the fluid's.

p

pressurePa

ρ

fluid densitykg m⁻³

h

depthm

V

volume displacedm³

The gradient of the line is ρg. Switch to mercury and it steepens dramatically — the same depth gives 13.6 times the pressure, which is why a mercury barometer is 760 mm tall while a water one would need ten metres.

Honey is viscous; water is not. Viscosity is internal friction between layers of a fluid sliding past each other, and it is what makes a fluid resist an object moving through it.

For a small sphere moving slowly through a fluid, that resistance is given by Stokes' law, and the crucial feature is that the drag is proportional to speed. Faster means more drag.

That proportionality produces terminal velocity. A ball bearing dropped into oil starts with weight and upthrust acting, and no drag. It accelerates. As it speeds up the drag grows, so the resultant force shrinks and the acceleration falls. Eventually drag plus upthrust equals weight, the resultant force is zero, and the speed stops changing.

The ball is still moving quickly at terminal velocity — it has simply stopped accelerating. This is the same physics as a skydiver, and the same equation governs both.

Viscosity falls sharply as a liquid is heated, which is why engine oil is graded for temperature and why honey pours easily when warm.

η

viscosityPa s

r

radius of the spherem

v

speedm s⁻¹

When a fluid flows steadily through a pipe of varying width, the same volume must pass every cross-section each second — nothing is created or destroyed. So where the pipe narrows, the fluid must speed up. That is the equation of continuity, and it is why a thumb over a hose makes the water shoot further.

Bernoulli's principle follows from energy conservation applied to that flow: where a fluid moves faster, its pressure is lower. This is genuinely counter-intuitive — squeezing the pipe raises the speed and lowers the pressure at the narrow point.

The consequences are everywhere. An aerofoil is shaped so air travels faster over the top than underneath, so the pressure above is lower and there is a net upward force — lift. A spinning ball drags air round with it, making the air faster on one side than the other, and it swerves. Two ships steaming close alongside are drawn together, because the water between them moves faster.

Bernoulli assumes steady, non-viscous, incompressible flow. Real fluids are viscous and real flow becomes turbulent above a certain speed, so the principle is a very good approximation rather than an exact law.

A

cross-sectional aream²

v

flow speedm s⁻¹

p

pressurePa