Biomedical Engineering Reference
In-Depth Information
Limit IV
Further increases in elasticity lead to the next limit of the minimum velocity
for a perfectly elastic surface film. Here, the mode coupling leads to the case
when the transverse motion is favored at the expense of lateral motion that
is now practically stopped. Treating this case as a departure from the infinite
lateral modulus case, Limit V, expressions for
ω
o
and
α
are given as follows.
ω
o
=
1-
10
64
,
σ
k
3
ρ
-19
/
28
ψ
(26)
and
=
√
2
4
ση
1
/
4
1+
5
3
y
-25
/
121
.
2
k
7
ρ
α
(27)
3
Limit V
At the limit of infinite complex lateral modulus, still restricted to
κ
=0,
ε
∗
=
wherein we deal with the purely elastic limit such that
,thelon-
gitudinal mode should cease. As such, the picture of the surface is one in
which the surface elements are restricted to purely transverse mode, as de-
picted in Fig. 5. Thus, the frequency and damping coefficient are given as
ω
o
=
ε
d
→∞
1-
1
4
k
3
ρ
σ
-1
/
4
ψ
(28)
and
=
√
2
4
ση
1
/
4
1+
1
4
.
2
k
7
ρ
-1
/
α
2
ψ
(29)
3
Here, an earlier work by Reynolds [65] provided an expression for the damp-
ing of a surface film with an infinite lateral modulus, and Eq. 29 is called the
corrected Reynolds equation, for it carries a correction as discussed above.
Limit VI
ε
∗
=
i
ω
∗
κ
→∞
The limit is when
inthecaseofapurelyviscousfilmwith
ε
d
= 0. Putting it differently, the lateral modulus is a complex quantity,
ε
∗
=
(Eq. 18), hence the infinite limit must apply not only to the
purely elastic surface film,
ε
d
+
i
ωκ
= 0, but also to the purely viscous surface film,
ε
d
= 0. The dynamics are then closely related to those of a surface with the
κ
