What Makes a Fluid a Fluid

Push a solid sideways and it distorts until its internal elastic stress balances the applied force, then stops. The strain is finite and proportional to the stress — Hooke's law, and the whole of the strength of materials subject.

Apply the same shear to water and it never stops. It deforms continuously, for as long as the force is applied, and there is no threshold below which it behaves like a solid.

This is a much sharper definition than "takes the shape of its container". A block of butter takes the shape of its container too, given enough time, and the difference is exactly whether the deformation ever stops.

1. 1Suppose a shear stress existed somewhere in a fluid at rest.
2. 2By the definition, that region would be deforming continuously — that is, moving.
3. 3But the fluid was stated to be at rest. Contradiction.
4. 4Therefore the only stress in a static fluid is normal to any surface considered.

This is why hydrostatic pressure at a point is the same in every direction — Pascal's law — and why the force on a submerged gate is perpendicular to the gate regardless of its angle. The entire statics module rests on this three-line argument.

A *moving* fluid does resist — but it resists the rate of deformation rather than its amount. Drag a plate over a thin fluid layer and the force needed is proportional to how fast you drag it, not how far you have dragged it.

Newton's law of viscosity: τ = μ·(du/dy). The velocity gradient du/dy is the rate of shear strain, and the constant μ is the dynamic viscosity. This one equation is the difference between fluid and solid mechanics written down.

Dynamic viscosity μ

Units Pa·s. Water is about 1 mPa·s at 20 °C, air about 18 μPa·s — water is roughly 55 times more viscous.

Kinematic viscosity ν = μ/ρ

Units m²/s. Because it divides by density, air's kinematic viscosity is about 15 times water's — the opposite ranking to dynamic viscosity, and it is ν that appears in the Reynolds number.

Temperature dependence

Liquid viscosity falls sharply with temperature (molecules escape each other's attraction); gas viscosity *rises* with it (more molecular momentum exchange). The mechanisms are different, which is why the trends oppose.

The no-slip condition: fluid in contact with a solid boundary moves at exactly the boundary's velocity. It is an experimental fact rather than a derivation, and it holds for every ordinary flow.

1. 1At the wall the velocity is zero, and far from it the velocity is the free-stream value.
2. 2So a velocity gradient must exist in between.
3. 3A gradient times viscosity is a shear stress — this is skin friction drag.
4. 4The thin region containing that gradient is the boundary layer.

It is why a fan blade collects dust: the air at its surface is not moving at all. And it is why boundary layer behaviour — growth, transition, separation — determines drag, stall and most of the loss in a pipe.

The syllabus assumes Newtonian throughout, and for water and air that is excellent. Paint is shear-thinning deliberately, so it spreads under a brush and stops running once on the wall. A Bingham plastic technically fails the definition of a fluid below its yield stress — a useful reminder that the definition is a model, not a law of nature.

Density ρ

Mass per volume. For liquids essentially constant, which is what the incompressible assumption means; for gases it depends strongly on pressure and temperature.

Specific weight w = ρg

Weight per volume. This is the quantity that actually appears in hydrostatic pressure, p = w·h.

Bulk modulus K

Resistance to compression. Water's is about 2.2 GPa — high enough to justify treating it as incompressible, and low enough that water hammer is a real and destructive phenomenon.

Surface tension σ

Energy per unit area of a free surface. Dominates at small scales — capillary rise, droplet formation, and why an insect can stand on water.

Vapour pressure

The pressure at which the liquid boils at a given temperature. If local pressure falls below it — behind a pump impeller or a propeller — bubbles form and collapse violently. That is cavitation, and it destroys metal.