Biomedical Engineering Reference
In-Depth Information
�
F
=
g vol
D
ρ
y
ˆ
(6.88)
grav
p
where
g
is the acceleration of gravity,
vol
p
is the volume of the particle,
y
is the verti-
cal unit vector (oriented downwards), and
r
is the difference between the volumic
mass of the particle and that of the liquid. After substitution of (6.87) and (6.88) in
(6.86), one obtains the equation for the particle's velocity
�
� �
dV
p
(6.89)
m
= -
6
πη
r V V
(
-
)
+
g vol
D
ρ
y
ˆ
h
p
f
p
d t
This relation can be decomposed along each coordinate (here we choose a 2D
configuration)
d u
p
m
= -
6
πη
r u
(
-
u
)
h
p
f
d t
(6.90)
d v
p
m
= -
6
πη
r v
+
g vol
D
ρ
h p
p
d t
Using the notations
6
πη
r
h
c
=
1
m
(6.91)
g vol
D
ρ
p
c
=
2
m
this system becomes
d u
p
= -
c u
(
-
u
)
1
p
f
d t
(6.92)
d v
p
= -
c v
+
c
1
p
2
d t
and this system can be solved analytically
-
c t
-
c t
u
=
u
e
+
u
[1
-
e
]
1
1
p
p
,0
f
(6.93)
c
-
c t
2
-
c t
v
=
v
e
+
[1
-
e
]
1
1
p
p
,0
c
1
By definition,
x
and
r
coordinates of the particle at a given time are linked to
the velocity by the relations
d x
p
-
c t
-
c t
=
u
=
u
e
1
+
u
[1
-
e
1
]
p
p
,0
f
d t
(6.94)
d r
c
p
-
c t
2
-
c t
=
v
=
v
e
+
[1
-
e
]
1
1
p
p
,0
d t
c
1
