Biomedical Engineering Reference
In-Depth Information
surface results from a resonant mode coupling between the transverse and
longitudinal modes.
We now turn to illustrate the mode coupling by plotting general solutions
to the dispersion equation, first put forth by Hård and Neuman [63], and
shown in Fig. 4. We first examine the limiting behaviors. The figure repre-
sents the temporal damping coefficient
α
plotted against the real capillary
wave frequency
o
. It stands for a specific solution at a specified wave vector
on the water surface at 25
◦
C. The isopleths of constant dilational elasticity
ω
ε
d
are drawn in radial solid curves emanating from point V, which will be ex-
plained shortly. The corresponding isopleths of constant dilational viscosity
κ
are drawn in circular dashed curves, emanating from a solid curve desig-
nated as VI which is the isopleth of
ε
d
= 0. We should dwell on these limits at
this point because this gives the basis for the non-monotonic appearance of
the experimental frequency and damping coefficient with respect to surface
mass density
Γ
and surface pressure
Π
.
Fig. 4
General solution for the dispersion equation on water at 25
◦
C. The damping coeffi-
cient
α
vs. the real capillary wave frequency
ω
o
, for isopleths of constant dynamic dilation
elasticity
(
dashed circular curves
).
The plot was generated for a reference subphase at
k
= 32 431 m
-1
,
ε
d
(
solid radial curves
), and dilational viscosity
κ
σ
d
= 71.97 mN m
-1
,
µ
= 0 mN s m
-1
,
ρ
= 997.0 kg m
-3
,
η
= 0.894 mPa s and
g
= 9.80 ms
-2
.Thelimitscorres-
pond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface
Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit,
V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coeffi-
cient for a Perfectly Viscous Surface Film
