Biomedical Engineering Reference
In-Depth Information
where
n
and U are the normal and tangent vectors at each point on the interface,
and L is the local mean curvature of the interface (e.g., (34, ch. 5). Equation [3]
is known as the Laplace-Young condition, and eq. [4] represents Marangoni
stresses.
The clefting force is modeled as a point force applied at selected points
along the interface. Since surface tension can be considered to produce a net
force in the normal direction, we represent the inward-directed point forces as
localized reductions of the surface tension:
i
i
HH
=
f
E
()
s
,
[5]
0
where H
0
is the uniform surface tension everywhere on the epithelial surface,
f
is
the magnitude of the local point force density, and E is the delta function localiz-
ing the force at points
s
i
. In the interest of simplicity, we keep the point forces
pointing in the same directions as initially, regardless of the motion of the inter-
face.
Because the forces of one rudiment on another in vivo or in vitro can gener-
ally be neglected if the rudiments are not very close, the outside fluid is modeled
as rectangular, with periodic boundary conditions.
3.2. Nondimensionalization
If we write the internal and external viscosities as N
-
and N
+
, respectively,
and characteristic length and time scales as
L
and
T
, the characteristic surface
tension as H
0
, and the characteristic pressure as N
-
/
T
, we can define three nondi-
mensional parameters:
+
w ,
N
B
[6]
N
H
T
L
,
0
C
w
[7]
N
w
f
K
L
.
[8]
0
We call B the viscosity ratio, C the nondimensional surface tension, and K the
nondimensional clefting force. These three nondimensional parameters are all
that governs the behavior of the model. There are thus only a small number of
numerical experiments needed to explore all the model behavior.
