Continuity Law of Incompressible Flow

fluid-mechanics physics aerodynamics equations
Last updated: 2025-01-29

Continuity Law of Incompressible Flow

The continuity law is perhaps the most intuitive of the conservation principles. It states that what flows into a pipe must flow out—matter is neither created nor destroyed within the system.

Statement of the Continuity Law

For incompressible flow (constant density) confined to a duct or pipe:

Flow is constant throughout the duct, regardless of cross-sectional area changes.

Mathematically:

v₁A₁ = v₂A₂ = constant = U

where:

  • v₁ = particle velocity at location 1 (m/s)
  • A₁ = cross-sectional area at location 1 (m²)
  • v₂ = particle velocity at location 2 (m/s)
  • A₂ = cross-sectional area at location 2 (m²)
  • U = flow (m³/s)

Physical Interpretation

Flow through constricted pipe Figure 3.14: Flow lines in a constricted pipe. Locations 1 and 2 refer to different pipe diameters.

The Constriction Effect

Imagine flow through a pipe that narrows from location 1 (wider) to location 2 (narrower):

If fluid is incompressible:

  • All fluid particles passing location 1 must also pass location 2
  • No leakage through walls
  • No accumulation between locations

Since the pipe narrows:

  • Particles must speed up
  • Maintains same number of particles per unit time
  • Velocity increases proportionally to area decrease

The Velocity-Area Relationship

From the continuity equation:

v₂ = v₁(A₁/A₂)

This reveals the inverse relationship:

  • If area decreases by factor of 2, velocity increases by factor of 2
  • If area decreases by factor of 10, velocity increases by factor of 10
  • Smaller constrictions produce proportionally higher velocities

Application to the Glottis

The glottis represents the most dramatic constriction in the respiratory tract:

Typical Dimensions

Trachea:

  • Diameter: ~2.5 cm
  • Area: ~5 cm²

Glottis (during phonation):

  • Width: ~0.3-0.5 cm
  • Depth: ~1.0-1.5 cm
  • Area: ~0.03-0.08 cm²

Area ratio: Approximately 100:1 (trachea to glottis)

Velocity Increase

For typical phonation with flow U = 0.1 L/s:

In trachea:

v₁ = U/A₁ = (0.1 L/s)/(5 cm²) 
   = (0.0001 m³/s)/(0.0005 m²)
   = 0.2 m/s

At glottis:

v₂ = U/A₂ = (0.1 L/s)/(0.05 cm²)
   = (0.0001 m³/s)/(0.000005 m²)
   = 20 m/s

The air accelerates from 0.2 m/s to 20 m/s—a 100-fold increase—as it passes through the glottis!

Implications for Voice Production

High Velocities at Glottis

The dramatic velocity increase has several consequences:

  1. Bernoulli forces: High velocity creates low pressure (discussed in next section)
  2. Kinetic energy: Fast-moving air carries significant energy
  3. Turbulence: High velocities can trigger turbulent flow
  4. Acoustic generation: Rapid velocity changes create sound

Flow Conservation During Vibration

Even as the glottis opens and closes during vocal fold vibration:

Continuity still applies:

  • Instantaneous flow equals v×A at any point
  • Flow varies with glottal area changes
  • Velocity inversely tracks area

Dynamic effects:

  • As glottis opens: velocity decreases, flow increases
  • As glottis closes: velocity increases, flow decreases
  • Flow oscillation creates acoustic pulses

No Air “Storage”

The continuity law means:

  • Air cannot accumulate in the vocal tract
  • Subglottal pressure drives constant mass flow
  • Vocal tract acts as flow-through system
  • Supraglottal pressure remains near atmospheric

Traffic Analogy Revisited

Consider a two-lane highway merging to one lane:

Ideal traffic flow (incompressible):

  • All cars entering must exit
  • No cars created or destroyed
  • When lanes merge, cars must travel twice as fast

Real traffic:

  • Cars can “compress” (slow down, get closer)
  • Creates traffic jams
  • Flow becomes compressible

Air in speech:

  • Pressures are small enough that air remains essentially incompressible
  • Behaves more like ideal traffic than real traffic
  • Continuity law applies well

Limitations

The continuity law assumes:

Incompressible Flow

Valid when:

  • Pressure changes are small compared to atmospheric
  • True for speech (≤3 kPa vs. 101 kPa atmospheric)
  • Density remains essentially constant

Breaks down when:

  • Pressures approach or exceed atmospheric
  • Shock waves form (supersonic flow)
  • Not relevant to normal phonation

Rigid Walls

Assumed:

  • Duct walls don’t move
  • No volume changes from wall compliance

Reality:

  • Vocal tract walls can expand slightly
  • Effect is generally small for flow calculations
  • More important for acoustic resonance

No Leakage

Assumed:

  • All flow through main duct
  • No side branches

Reality:

  • Nasal port can provide leakage path
  • Must account for velopharyngeal opening when relevant

Summary

The continuity law states that incompressible flow remains constant throughout a duct system. Velocity and area are inversely related: v₁A₁ = v₂A₂. Application to the glottis reveals dramatic velocity increases (100-fold) as air accelerates through this narrow constriction. This velocity increase has profound implications for pressure forces (Bernoulli effect), energy transfer, and acoustic generation during phonation.

Key Takeaways

  • ✅ Continuity law: v₁A₁ = v₂A₂ = U = constant for incompressible flow
  • ✅ Velocity is inversely proportional to area: smaller constrictions produce higher velocities
  • ✅ Glottal constriction accelerates air from ~0.2 m/s to ~20 m/s (100-fold increase)
  • ✅ Flow conservation applies instantaneously, even during dynamic vocal fold vibration

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.
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