Streamline flow and continuity

Fluid dynamics studies fluids in motion. A tap opened slowly gives a smooth stream; opened fully, the stream loses its smoothness.

Steady flow: at any fixed point the velocity of each fluid particle passing it stays the same in time. Velocities may differ from point to point, but every particle reaching a given point behaves like the one before it.

A streamline is the path of a fluid particle in steady flow: a curve whose tangent at every point gives the direction of the fluid velocity there. The map of streamlines does not change with time.

Two streamlines never cross: at a crossing a particle would have two possible velocities and the flow could not be steady.

A tube of flow is bounded by the same streamlines, so the mass crossing sections P, R and Q in time Δt is the same: ρ_P A_P v_P Δt = ρ_R A_R v_R Δt = ρ_Q A_Q v_Q Δt.

For an incompressible fluid the densities are equal and A_P v_P = A_R v_R = A_Q v_Q, or Av = constant. This is the equation of continuity, a statement of conservation of mass.

Av is the volume flux (flow rate). Where the tube narrows and streamlines crowd together the speed rises; where it widens the speed falls. The fluid speeds up passing from a wide section R to a narrow section Q.

Steady flow needs low speeds. Above a critical speed the flow becomes turbulent, like the foamy whirls of white-water rapids where a fast stream meets rocks.

Laminar flow: velocities at different points may differ in size but point in parallel directions. A jet of air hitting a flat plate held across it gives turbulent flow.