previous
next
Viscous media
Viscosity
As we have seen
before,
for an object that moves vertically in a fluid,
the resistance is usually
a combination of terms linear and quadratic
in the velocity v
of the object with respect to the fluid:
a = -g + A×v + B×v2
The term A×v
is called the viscous term,
whereas the term
B×v2
is refered to as the pressure term.
For spherical objects of radius
r
and mass m
which are vertically freely moving in the fluid
with velocity v,
the above expression can be written in the form
ma = -mg + C1rv +
C2r2v2
The resistive force of the medium is thus given by
Fres
= C1rv +
C2r2v2
The viscosity coefficient
C1
depends on the stickiness of the fluid (usually a liquid)
and therefor on its temperature,
as we all know that most substances (e.g. Brasilian wax)
are much less sticky at higher temperatures.
The coefficient
C2
depends basically on the density of the medium.
Let us also study the units of
C1
and C2.
The units of force are N
(
Newton) and
1 N = (1 unit of mass)×(1 unit of acelleration)
= (1 kg)×(1 m/s2)
= 1 kg m/s2
The unit of C1
follows from
1 N =
(1 unit of C1)×(1 unit of
length)×(1 unit of velocity)
Hence,
(1 unit of C1) =
(1 N)/((1 unit of length)×(1 unit of velocity)) =
= (1 kg m/s2)/((1 m)×(1 m/s)) =
1 kg /ms
The unit of C2
follows from
1 N =
(1 unit of C2)×(1 unit of
length squared)×(1 unit of velocity squared)
Hence,
(1 unit of C2) =
(1 N)/((1 unit of length squared)×(1 unit
of velocity squared)) =
= (1 kg m/s2)/((1 m2)×(1 m2/s2)) =
1 kg /m3
The latter is indeed the unit of density,
which makes it plausible that
C2
is related to the density of the fluid.
critical velocity