body falling in a fluid

Force balance involving volume and surface forces

VELOCITY AS A FUNCTION OF TIME, vf(t)
A skydiver falling in the atmosphere is subject not only to gravity but also to forces which resist the motion. These resistive forces (pressure and viscous) both act at the surface of a solid moving through a fluid.

Forces Acting on the Skydiver
Gravity Acts on the entire volume of the skydiver
Pressure gradient Acts normal to the surface of the skydiver
Viscous (Frictional) Acts tangent to the surface of the skydiver

It is usual to lump the pressure and viscous forces together and refer to the combined effect as "the drag force", Fd.
For a given body moving through a particular fluid, the drag force,Fd, depends on the velocity of the body relative to the fluid. There is no simple analytic expression for Fd as a function of vf but a useful simple function consistent with experiments is

   

where k is a constant called the drag coefficient. k has units of Ns2/m2.

In reality, k is not actually a constant. For a fuller discussion of this matter, click here.

Thus the forces acting on the skydiver are


   
(directed downward)

   
(directed upward)

The force balance (Newton's second law) taken in the direction of the gravitational field is

    (3.1)

    (3.2)

The third term in equation (3.1), kvf2, is the drag force. It results from surface forces acting on the skydiver and opposes the motion of the diver.

There are two interesting aspects of the equations above:

  1. The skydiver does not fall with a constant acceleration, g, as in a vacuum. This is apparent from equation (3.2).
  2. The acceleration (and therefore the velocity) of the skydiver is not independent of the mass of the body. The shape of the body is taken into account in the drag coefficient, k.

The initial condition is assumed to be


    (3.3)

which means that when the skydiver jumps out of a plane we have assumed that his or her velocity is zero.

The solution to eqn. (3.1) with initial condition (3.3) is


  Derivation DERIVE! (3.4)

Copyright 1998-1999 Rensselaer Polytechnic Institute. All Rights Reserved.