§ 13. Bernoulli's Equation & Venturi Effect (Grade 10)
Relation between flow speed and static pressure in a variable-cross-section pipe. In steady flow of an ideal incompressible fluid, the sum of static pressure and dynamic pressure head remains constant along any streamline.
Theory Summary & Scientific Mechanism
Steady laminar flow of an ideal incompressible fluid (ρ = const) through a variable-cross-section pipe obeys two conservation laws. First is the continuity equation: volume flow rate Q_V = S₁ v₁ = S₂ v₂ = const is invariant along the tube, forcing the fluid to accelerate wherever the cross-section narrows (S₂ v₁).
Second is Bernoulli’s equation (conservation of mechanical energy per unit volume): along any streamline, the sum of static pressure p, dynamic pressure (1)/(2)ρ v², and hydrostatic head ρ g h remains constant: p₁ + (1)/(2)ρ v₁² = p₂ + (1)/(2)ρ v₂² = const for a horizontal pipe.
This explains the Venturi effect: accelerating a fluid parcel into the constriction requires a forward pressure gradient p₁ - p₂ = (1)/(2)ρ v₁² ≤ft[≤ft((S₁)/(S₂) )² - 1 ] > 0. Consequently, where flow speed is highest, static pressure against the pipe walls (shown by the manometer column) is lowest!
Everyday Analogy
Put your thumb over the end of a garden hose and water shoots out much faster! Remarkably, right where the fluid speeds up in a narrow throat, its side pressure drops, the very principle that helps airplane wings generate lift!
Interactive Question & Derivation
Fluid flows horizontally through a Venturi tube from a wide section of area S₁ into a narrow throat of area S₂ < S₁. How do the flow speed v₂ and manometer liquid column height h₂ above the narrow throat change?
- A: In the narrow throat speed decreases, while pressure and column height h₂ rise
- B: Flow speed in the constriction increases (v₂ = v₁ S₁/S₂), while static pressure and column height h₂ drop
- C: Flow speed increases, but static pressure P remains identical in all cross-sections
Study Hint: In the narrow constriction, flow speed v₂ increases (A₁ v₁ = A₂ v₂). What must happen to static pressure P₂ if P + (1)/(2)ρ v² = const?
Proof & Explanation: By the continuity equation S₁ v₁ = S₂ v₂, fluid speed in the narrow throat increases inversely with cross-sectional area: v₂ = v₁ (D₁/D₂)². By energy conservation (Bernoulli's equation), an increase in dynamic pressure head (1)/(2)ρ v₂² is accompanied by a drop in static pressure: P₂ = P₁ - (1)/(2)ρ(v₂² - v₁²), causing the middle manometer column to drop below the outer columns.
Key Equations & 3D Simulation Parameters
v₂ = v₁ ≤ft((D₁)/(D₂) )^ 2, Δ P = P₁ - P₂ = (ρ v₁²)/(2) ≤ft[≤ft((D₁)/(D₂) )^ 4 - 1 ]
- Inlet flow speed v₁: 0.6 … 2.8 m/s (default: 1.5 m/s)
- Wide pipe diameter D₁: 1.8 … 2.6 m (default: 2.2 m)
- Throat diameter D₂: 0.8 … 1.7 m (default: 1.2 m)
- Fluid density ρ: 700 … 1400 kg/m³ (default: 1000 kg/m³)