Biomedical Engineering Reference
In-Depth Information
Figure 3.51
Schematic of the geometry of a droplet constrained between two parallel planes:
(a) case of a convex interface
q
1
<
π
/2,
q
2
>
π
/2, and
q
1
+
q
2
>
π
, (b) case of a concave interface
q
1
<
π
/2,
q
2
>
π
/2, and
q
1
+
q
2
<
π
, and (c) case of a flat interface
q
1
+
q
2
=
π
.
The volume of such a droplet is often useful to know. The calculation is com-
plicated except in the case where the two contact angles are equal (
q
1
=
q
2
=
q
). In
this case, the exact formula has been derived in [8]
ì
3
ü
é
sin 2
(
θ π
-
ù
δ
1
δ
π
)
ï
æ
ö
ï
2
2
2
(
(
)
V
=
2
π
R
-
2
r R
+
2
r
)
-
+
R r r
-
θ
-
+
(3.77)
í
ç
÷
ê
ú
ý
è
ø
2
3 2
2
2
ï
ë
û
ï
î
þ
In the literature, the Nie et al. correlation is sometimes used [24]
π
3
é
2
ù
(
)
(
)
V
=
2
R
-
(
2
R
-
δ
) 4
R
+
δ
(3.78)
ë
û
12
However, this formula does not take into account the contact angle. Discrep-
ancy up to 8% can result, depending on the Young contact angle. The difference
between the exact and approximate expressions is shown in Figure 3.52.
