Biomedical Engineering Reference
In-Depth Information
� �
2
Figure 4.52
Assuming that
g
1
is larger than
g
2
+
g
3
, the resultant of the forces
γ γ
3
projected on
the direction of
g
1
cannot equilibrate
γ
�
1
.
verified by remarking that if the magnitude of a force is larger than the sum of the
magnitudes of the two others, equilibrium cannot be reached (Figure 4.52).
Hence, it can be shown that the condition for nonengulfment is
γ γ γ
<
+
rc
ct
tr
(4.75)
and
γ γ γ
<
+
ct
rc
tr
A more strict approach of engulfment of a liquid droplet can be done by using
energy considerations. Let us examine the case where a spherical droplet or solid
particle is at the interface between oil and water (Figure 4.53). Let the symbols
A
,
W
, and
O,
respectively, stand for the sphere, water, and oil. If we assume that grav-
ity can be neglected because the droplet Bond number is small
2
g
R
D
ρ
γ
Bo
=
<<
1
We are left with a purely capillary problem. The surface energy of the system
is
(4.76)
E E
=
+
E
-
E
AW
AO
WO
The third term on the right hand side of (4.76) corresponds to the exclusion of
the interface SWO. Equation (4.76) yields
(4.77)
E
=
γ
S
+
γ
S
-
γ
S
AW AW
AO AO
WO WO
Figure 4.53
Sketch of the sphere at the interface.
