Viscosity and the Properties That Matter

A solid's shear stress depends on how far it has been deformed. A fluid's depends on how fast. Writing that down gives τ = μ·(du/dy): shear stress is proportional to the velocity gradient, and the constant of proportionality is the dynamic viscosity.

The gradient is the quantity that matters, not the velocity. Fluid moving uniformly at 100 m/s has no gradient anywhere and therefore no viscous stress at all. It is the *difference* between adjacent layers that costs energy.

A fluid that obeys this with constant μ is called Newtonian. Water, air and most oils are, across enormous ranges. Paint, blood, toothpaste and cornflour paste are not.

Dynamic viscosity μ (Pa·s) measures the force needed to shear the fluid. Kinematic viscosity ν = μ/ρ (m²/s) measures how quickly momentum diffuses through it. Dividing by density is not a cosmetic change — it reverses the ordering.

The Reynolds number uses ν. So for the purpose that matters most in this subject, air is the more viscous fluid — which is exactly the opposite of everyday intuition.

In a liquid, molecules are packed closely and viscosity arises from cohesive forces between them. Heating shakes those bonds apart, so μ falls — steeply. Engine oil at 0 °C is roughly ten times thicker than at 100 °C, which is the entire reason multigrade oils exist.

In a gas, molecules are far apart and cohesion is negligible. Viscosity comes instead from momentum exchange: molecules wander between layers, carrying their momentum with them. Heating speeds that wandering, so μ *rises*.

Same word, two mechanisms, opposite signs. This is worth remembering as a check: if a problem has gas viscosity falling with temperature, something has gone wrong.

Density ρ

Mass per unit volume. Water ≈ 1000 kg/m³, air ≈ 1.2 kg/m³ at sea level.

Specific weight γ

Weight per unit volume, γ = ρg. Water ≈ 9810 N/m³. It varies with gravity; density does not.

Specific gravity SG

The ratio to water's density. Dimensionless, so it reads the same in any unit system — mercury is 13.6 everywhere.

Surface tension σ

Energy per unit area of surface, ≈ 0.073 N/m for water in air. Decides droplet shape and capillary rise.

Vapour pressure p_v

The pressure at which the liquid boils at the current temperature. Water: 2.3 kPa at 20 °C. Drop below it and cold water boils.

Bulk modulus K

Resistance to compression, ≈ 2.2 GPa for water. Large enough to ignore — except that it sets the water-hammer wave speed √(K/ρ) ≈ 1400 m/s.

A molecule inside the bulk is pulled equally in all directions. One at the surface has no neighbours above it, so the net pull is inward — and the surface behaves like a stretched membrane. A droplet is spherical because that is the minimum surface area for its volume.

Inside a droplet, Δp = 2σ/r. Smaller bubbles therefore hold higher pressure than larger ones, so a small bubble connected to a large one empties into it rather than equalising.

Capillary rise adds a second competition: adhesion to the wall against cohesion within the liquid, encoded in the contact angle θ. Water wets glass and climbs; mercury does not and is depressed. The rise h = 4σcosθ/(ρgd) goes as 1/d, which is why manometer tubes are kept wide and why fine soil draws water metres above the water table.

Boiling is a pressure condition, not a temperature. A liquid boils when local pressure falls to its vapour pressure — usually reached by heating, but lowering the pressure works just as well.

1. 1Flow accelerates through a pump inlet, a venturi throat or past a propeller tip.
2. 2Bernoulli says pressure falls where velocity rises.
3. 3If it falls to the vapour pressure, vapour cavities form in cold liquid.
4. 4Downstream the pressure recovers and the cavities collapse — violently, against metal.
5. 5This is cavitation. It sounds like gravel in the pump and it erodes the impeller.

The design rule is that NPSH available must exceed NPSH required, and it is one of the few places in fluid mechanics where the margin is checked explicitly on every installation.