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Viscous media

The critical velocity

The velocity of the sphere at which the two terms of the resistive force of the medium are equal

C₁rv_crit = C₂r²v_crit²

is called the critical velocity and is given by

v_crit = C₁/C₂r

There are two regimes of interest, namely

• regime I for which v << v_crit
  
  The only relevant term of the resistive force in this regime is the viscous term. The pressure term may be ignored.
  The terminal velocity is in this regime given by
  
  v_term = mg/C₁r
  
  
• regime II for which v >> v_crit
  
  The only relevant term of the resistive force in this regime is the pressure term. The viscous term may be ignored.
  The terminal velocity is in this regime given by
  
  v_term = √(mg/C₂r²)
  
  

For example for air at normal conditions

C₁ = 3.1×10^-4 kg/ms     and     C₂ = 0.89 kg/m³

Hence, for the critical velocity in air we find

v_crit = C₁/C₂r = (3.6×10^-4 m²/s)/r

For a plastic ball with a radius of 10 cm, we find v_crit=0.0036 m/s. Except when the ball is very near the highest position, its velocities are well over 0.0036 m/s. Moreover, very near the highest position the ball has almost no velocity, hence feels anyhow an ignorable resistive force. Consequently, we are for that case indeed in regime II, as we assumed earlier.

drag coefficient