Fluid Flow & the Equation of Continuity

A waterfall crashing onto rocks, smoke curling from a wood fire, clouds drifting: every droplet obeys Newton’s laws, yet predicting each one is hopeless. To make progress we focus on the simplest kind of flow.

Steady and turbulent flow#

Watch the liquid flowing slowly through a glass tube and concentrate on one point A. If every particle arriving at A has the same speed and direction, and every particle that passes A goes on to the same point B, the flow is steady (or streamline). The velocity at each point stays the same over time, though it may differ from point to point.

Push the liquid through faster and the flow can become turbulent: particles passing the same point have different, erratically changing velocities. A fast river or a high waterfall is turbulent.

Streamlines and tubes of flow#

Two streamlines can never cross: a particle at the crossing would have to move in two directions at once. Draw streamlines through every point on the edge of a small area S, and they enclose a tube of flow. Fluid never crosses its walls, even though no physical wall exists. A pipe carrying a slow, steady flow is itself one tube of flow.

The ideal fluid#

To keep the mathematics simple we assume the fluid is

• incompressible: constant, uniform density (very good for liquids);
• nonviscous: no internal friction between layers, so no mechanical energy is lost;
• in irrotational flow: the fluid particles have no net spin (unlike water stirred round in a bucket, or the small whirlpools in a river).

The equation of continuity#

Fluid entering a tube of flow at one cross-section must leave through any other, since none crosses the walls. At section 1 (area A₁, speed v₁), in a time Δt the fluid in a cylinder of length v₁Δt passes through: a volume A₁v₁Δt. Similarly A₂v₂Δt leaves at section 2. For an incompressible fluid these are equal: