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 Ns
2/m
2.
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
The force balance (Newton's second law) taken in the direction of the gravitational field is
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:
- The skydiver does not fall with a constant acceleration, g,
as in a vacuum. This is apparent from equation (3.2).
- 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
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
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1998-1999
Rensselaer Polytechnic Institute. All Rights Reserved.