Conservation of Energy (Bernoulli’s Law)

Bernoulli’s energy conservation law is central to understanding pressure-velocity relationships in airflow. Named after Swiss scientist Daniel Bernoulli (1700-1782), this principle explains seemingly counterintuitive pressure changes in constricted airways.

Bernoulli’s Energy Law

Energy in a fluid stream has two components:

1. Potential energy: Related to pressure
2. Kinetic energy: Related to particle velocity

Bernoulli recognized these can be summed in the form of pressures:

P + ½ρv² = constant

where:

• P = pressure (Pa)
• ρ = fluid density (kg/m³)
• v = particle velocity (m/s)

The term ½ρv² represents kinetic energy expressed as an equivalent pressure.

Bernoulli’s Principle

The law leads directly to Bernoulli’s principle:

If total energy in a fluid stream is constant (no losses), an increase in particle velocity must be accompanied by a decrease in pressure.

This follows from the equation: to keep the left side constant, P must decrease when v increases.

The Counterintuitive Result

Consider flow through a pipe with decreasing cross-sectional area:

Intuition might suggest:

• Constriction should increase pressure
• Like a traffic jam building up

Bernoulli’s principle states:

• Constriction decreases pressure
• Higher velocity means lower pressure

Resolution:

• Traffic (compressible) ≠ fluid flow (incompressible for speech)
• In ideal incompressible flow, particles speed up through constrictions
• This acceleration requires a pressure gradient: high pressure upstream, low pressure in constriction

The Traffic Analogy (Corrected)

For truly incompressible traffic:

Two-lane highway → one-lane:

• Traffic should speed up from 55 mph to 110 mph
• This would indeed reduce “traffic pressure” in the narrow section
• Cars are farther apart at higher speed for same flow rate

In reality, traffic compresses (slows down), violating incompressibility. Air at speech pressures, however, remains essentially incompressible.

Application to Airflow and Lift

Airplane Wings

A wing is a familiar—and often misexplained—example of Bernoulli’s law:

What actually happens:

• Air passing over the upper surface really does move faster than air passing beneath, and the pressure there really is lower
• The faster flow is not caused by the upper path being longer: the popular “equal transit time” story (air parting at the leading edge must reunite at the trailing edge) is a fallacy—air over the top arrives at the trailing edge well before air underneath
• The speed difference arises because the curved, tilted wing deflects the airstream downward; the flow pattern (circulation) that does this is what produces the pressure difference
• Newton’s third law (the wing pushes air down, air pushes the wing up) and Bernoulli’s law (fast flow, low pressure) are two descriptions of the same lift, not two separate contributions

Why it matters here:

• Bernoulli’s law is valid along a streamline whenever energy losses are small, which is what makes it useful for the glottis
• It never explains why the velocity changes; that comes from geometry and continuity (as in the glottal constriction) or from flow deflection (as with the wing)

Glottal Flow

At the glottis during phonation:

Velocity increase:

• Air accelerates from ~0.2 m/s (trachea) to ~20 m/s (glottis)
• Continuity law requires this acceleration

Pressure decrease:

• Pressure in glottis drops below subglottal pressure
• This pressure drop has two components:
  • Bernoulli effect (velocity-related)
  • Viscous losses (friction)

Force on vocal folds:

• Lower intraglottal pressure
• Higher pressure on medial surfaces
• Creates suction effect
• Assists in closing movement

Quantitative Example

Consider flow at two locations:

Location 1 (trachea):

• Pressure: P₁ = 1000 Pa (1.0 kPa subglottal)
• Velocity: v₁ = 0.2 m/s
• Area: A₁ = 5 cm²

Location 2 (glottis):

• Velocity: v₂ = 20 m/s (from continuity)
• Area: A₂ = 0.05 cm²
• Pressure: P₂ = ?

Using Bernoulli’s equation:

P₁ + ½ρv₁² = P₂ + ½ρv₂²

P₂ = P₁ + ½ρ(v₁² - v₂²)
   = 1000 Pa + ½(1.2 kg/m³)(0.04 - 400) m²/s²
   = 1000 Pa - 240 Pa
   = 760 Pa

The pressure drops by 240 Pa (24%) from trachea to glottis due to the Bernoulli effect alone.

Energy Losses

The equation P + ½ρv² = constant assumes no energy losses. In reality:

Sources of Loss

1. Viscous friction: Boundary layers near walls
2. Turbulence: Chaotic flow patterns dissipate energy
3. Flow separation: Vortices at sharp edges
4. Unsteady effects: Acceleration/deceleration losses

Modified Bernoulli Equation

Accounting for losses:

(Energy)₂ = (Energy)₁ - Losses

P₂ + ½ρv₂² = P₁ + ½ρv₁² - Losses

For the vocal tract:

• Losses can be substantial (30-50% of pressure drop)
• Vary with Reynolds number (flow regime)
• Depend on geometry (sharp vs. rounded edges)

Application to Vocal Fold Vibration

Bernoulli forces play a role in the self-sustained oscillation of vocal folds:

During Opening Phase

Glottis widens:

• Flow increases
• Velocity in narrowest region decreases (area increasing)
• Pressure rises (Bernoulli effect diminishes)

Force reversal:

• Lower pressure difference
• Reduces suction on medial surfaces
• Elastic forces can dominate
• Initiates closing movement

During Closing Phase

Glottis narrows:

• Velocity increases (area decreasing)
• Pressure drops (Bernoulli effect intensifies)
• Suction on medial surfaces increases

Combined with momentum:

• Inward-moving tissue has momentum
• Bernoulli forces assist closure
• Leads to complete adduction

Historical Perspective

Early theories (van den Berg, 1957) attributed vocal fold vibration primarily to Bernoulli forces. Modern understanding recognizes:

• Bernoulli forces are present and measurable
• But they’re not the dominant mechanism
• Pressure variations from flow interruption are typically larger
• Tissue properties and geometry also crucial

The Bernoulli effect exists and matters, but it’s been traditionally overstated in voice science literature.

Practical Significance

For Voice Production

Understanding Bernoulli forces helps explain:

• Pressure distributions during vibration
• Why certain geometries favor oscillation
• Energy transfer from flow to tissue
• Onset conditions for phonation

For Clinical Assessment

Bernoulli principles inform:

• Interpretation of pressure measurements
• Analysis of high-speed videography
• Understanding of pathological patterns
• Computational models of phonation

For Voice Training

While vocalists don’t consciously manipulate Bernoulli forces:

• Understanding clarifies mechanism
• Informs imagery and technique
• Helps troubleshoot problems
• Provides scientific foundation

Summary

Bernoulli’s law states that P + ½ρv² = constant for energy-conserving flow. This produces the counterintuitive result that constrictions (which increase velocity) decrease pressure. Applied to the glottis, Bernoulli forces create suction on medial vocal fold surfaces, contributing to (but not dominating) the self-sustained oscillation mechanism. Energy losses in real systems reduce the magnitude of Bernoulli effects, but the principle remains important for understanding vocal fold aerodynamics.

Key Takeaways

• ✅ Bernoulli’s law: P + ½ρv² = constant (no losses)
• ✅ Bernoulli’s principle: increased velocity causes decreased pressure
• ✅ Glottal constriction reduces intraglottal pressure by ~200-300 Pa
• ✅ Bernoulli forces contribute to vocal fold vibration but aren’t the dominant mechanism

Related Topics

• Continuity Law of Incompressible Flow
• Flow Resistance (Glottal Resistance)
• Vocal Fold Vibration (Chapter 4)

Further Reading

1. Titze, I. R. (2000). Principles of voice production (2nd ed.). National Center for Voice and Speech.
2. Van den Berg, J., Zantema, J., & Doornenbal, P. (1957). On the air resistance and Bernoulli effect of the human larynx. Journal of the Acoustical Society of America, 29, 626-631.
3. Alipour, F., & Scherer, R. (2004). Flow separation in a computational oscillating vocal fold model. Journal of the Acoustical Society of America, 116(3), 1710-1719.