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218
The ‘dot’ in
.
γ
indicates it is a time derivative of
γ
, where
γ
is the shear
deformation (dx/dz). This is easily demonstrated:
γ
=
dv
dz
=
d dx
dt
⎛
⎝⎜
⎞
⎠⎟
dz
=
d dx
dz
⎛
⎝⎜
⎞
⎠⎟
dt
=
d
γ
dt
In practical terms these equations are somewhat cumbersome. In a typical
viscometer we generally measure the flow-through-time (t) for flowing a
certain volume of liquid (V). For a capillary viscometer (or any tube where
laminar flow prevails), Poiseuille’s equation applies:
Flow rate
U
=
dV
dt
=
π
(
P
1
−
P
2
)
r
4
8
η
l
P
1
−
P
2
: Pressure difference
r
: radius
l
: length of tube
The flow rate (U) is inversely proportional to the flow-through-time. Hence, the
viscosity is directly proportional to the flow-through-time (t):
η
∝
t
Therefore, the relative and specific viscosities (which we need) are easily
found:
Solution (with polymer) viscosity:
η
Solvent viscosity:
η
0
Relative viscosity:
η
r
=
η
η
0
=
t
t
0
Specific viscosity:
η
sp
=
η
−
η
0
η
0
=
η
r
−
1
=
t
t
0
−
1