WOLFRAM|DEMONSTRATIONS PROJECT

Terminal Velocity of Falling Particles

​
gravity g
980.
​
particle density
ρ
p
5
fluid density ρ
1.
particle diameter
D
p
0.01
fluid viscocity μ
0.01
This Demonstration calculates the terminal velocity of a spherical solid particle falling in a fluid under the force of gravity.
Three forces act on the particle: the downward force of gravity, an upward force of buoyancy, and a drag force that acts opposite to the direction of motion of the particle. The equation relating these forces to the particle acceleration is:
4
3

3
r
ρ
p
dV
dt
=
4
3

3
r
(
ρ
p
-ρ)g-sign(V)
1
2

2
r
C
d
ρ
2
V
,
where
r
is the radius of the sphere;
ρ
and
ρ
p
are the densities of the fluid and the particle, respectively;
g
is the gravitational constant;
V
is the particle velocity; and
C
d
is the drag coefficient that varies with the Reynolds number,
Re=
D
p
Vρ
μ
, as follows [1]:
C
d
=
24
Re
+
2.6
Re
5.0
1+
1.52
Re
5.0
+
0.411
-7.94
Re
263000
1+
-8.00
Re
263000
+
0.8
Re
461000
,
where
μ
is the viscosity in
g/cms
and
D
p
is the particle diameter; CGS units are used throughout. The equation for
C
d
is valid over the entire range of the available experimental data; use beyond
Re=
6
10
is not reliable. For
Re<0.1
the equation follows the creeping flow result
C
d
=24/Re
. You can calculate the terminal velocity, the Reynolds number, and the drag coefficient over a wide range of the variables
ρ
,
ρ
p
,
D
p
,
μ
, and
g
. The artificial values of gravity included in the calculation can be achieved particularly in space, but also on Earth.