Chemistry Reference
In-Depth Information
Fig. 7.14
Velocity of droplets in a channel as a function of rescaled time, shown for 4 representative
droplets. The traces are
coloured
for distinguishability. In the frame of the rescaled time, the velocity
now remains constant over time
Fig. 7.15
Frequency distribution of the rescaled time between the droplet collisions
crossover to the diffusive regime, we see a corresponding loss of correlation in the
velocity of the droplets. This clearly points to a collision time when the direction of
motion changes. Indeed, the time of 88 s is the mean time for the first collision as we
discussed above. Therefore, it seems that the transition from the ballistic to diffusive
regimes is set by the mean time of first collision, as one would expect. Subsequently
there are oscillations in the velocity auto-correlation at multiples of the mean first
collision time.
We can calculate the diffusivity
D
at long times by doing a linear fit to the MSD,
at long times, as shown by the blue line in Fig.
7.16
. The diffusivity in one-dimension
is given by the 1
/
2
·
slope of the linear f it
and we get a value for the enhanced
m
2
diffusivity as
D
∼
1137
.
7
μ
/
sec. Using the expression
D
=|
v
|
/ (
2
ρ)
from
m
−
1
(5 droplets per
m
2
the bSF theory, we get
D
∼
1167
.
8
μ
/
sfor
ρ
∼
0
.
0018
μ
2.8mm) and
m/s, in good agreement with the diffusivity calculated
from the experiments. The value
|
v
| ∼
4
.
17
μ
|
v
| ∼
4
.
17
μ
m/s, corresponds to the ensemble
