Bernoulli's Principle: Streamline Visualization

Streamlines in Bernoulli’s Principle — What the picture means

Big idea.
A streamline is a curve that is always tangent to the local velocity. In steady, laminar, incompressible flow, the streamline picture is stable and very useful: closer streamlines mean higher speed. Bernoulli’s principle then ties speed to pressure along a streamline.

1) When the streamline picture applies

1. 1. Steady flow: conditions at a point don’t change with time, so a tiny dye speck follows a fixed path.
   2. Laminar flow: motion is smooth, little mixing;
   3. Incompressible fluid: ρ is roughly constant (liquids; gases at low speed).

Under these conditions:

1. 1. No-crossing rule: streamlines never intersect.
   2. Wall behavior: no-slip at solid walls (v = 0 at the wall), so nearby streamlines run parallel to the wall.

2) Why spacing shows speed (Continuity)

Consider a “tube of streamlines” with cross-sectional area A. For incompressible steady flow, the volume flow rate is the same everywhere in the tube, so
A v = constant.
If the area narrows (A↓), the speed must rise (v↑). In the simulation, you’ll see streamlines crowd into the constriction and tracer particles accelerate.

3) Bernoulli along a streamline

When viscous losses are small and no pump/turbine adds or removes energy between two points on the same streamline,

P + ½ρv² + ρ g h = constant

At the same height (h₁ = h₂), a faster region (v₂ > v₁) must have lower pressure (P₂ < P₁). This is the pressure drop you’d measure with a manometer across a throat.

Useful relations in a horizontal tube (same h):

• Continuity: v₂ = v₁(A₁/A₂).
• Pressure change: ΔP = P₂ − P₁ = ½ρ(v₁² − v₂²) (negative when v₂ > v₁).

4) How to read the simulation

• Change radii (A₁, A₂) and inlet speed (v₁): watch streamlines crowd in the throat and particles speed up.
• Click in the flow: see the local particle velocity and which streamline it lies on.
• Compare two locations at the same height: faster region ⇒ lower P by Bernoulli; verify with the velocity bar.

Bernoulli must be applied along a single streamline (or between two points in a region where all streamlines share the same total head). If points differ in elevation, include ρ g h.

6) Take-away

• Crowding ⇒ speeding: narrow area → higher v (A v = constant).
• Speeding ⇒ pressure drop (at same height): higher v → lower P (P + ½ρv² + ρ g h = constant).
• Streamlines give a clean, snapshot picture of steady, laminar, incompressible flow—perfect for applying Bernoulli’s principle.