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Vertical motion

What have we learned so far?

The resistive force against motion in fluids may be considered to consist of two terms, one term, the viscous term, linear in the velocity v, the other term, the pressure term, quadratic in the velocity v.

The critical velocity distinguishes two regimes.

• In regime I the velocities of the object with respect to the fluid are much smaller than the critical velocity.
  
• In regime II the velocities of the object with respect to the fluid are much larger than the critical velocity.

We have not discussed velocities of the object with respect to the fluid which are of the same order of magnitude as the critical velocity.

The effect of the buoyant force on the dynamical equations for a solid object with density ρ_s submerged in a fluid with density ρ_fluid, boils down to the substitution of the gravitational acceleration g by

g(ρ_s-ρ_fluid)/ρ_s

Objects which are freely falling inside a fluid reach after a while a constant terminal velocity with respect to the fluid.

• In regime I the terminal velocity with respect to the fluid of a solid spherical object with radius r and with density ρ_s is given by
  
  v_term = 4πρ_sr²g(ρ_s-ρ_fluid)/3ρ_sC₁
  
  
• In regime II the terminal velocity with respect to the fluid of a solid spherical object with radius r and with density ρ_s is given by
  
  v_term = √( 4πρ_srg(ρ_s-ρ_fluid)/3ρ_sC₂ )
  
  

Laminar and turbulent flow