) and the decrease in the
growth rate of the instability in the moderately confined
regime, we turn to a simple (in principle) calculation of
the surface area of a thread when the fluctuation is no
longer axisymmetric but somewhat flattened as depicted
in Figure 1. The simplifying assumption
behind this
calculation is that the principal effect of the complex
hydrodynamic interactions between the thread and the
confining wall is to cause a fluctuation that is nonaxisymmetric. The cross section of the thread is no longer
circular, but rather ellipsoidal at the bulges. We consider the nonaxisymmetry to be a fixed feature of the
instability and then calculate the critical wavenumber
Xc.
We assume that the surface equation of our nonaxisymmetric sinusoidal thread is expressed by the following equation:
where
Equation 6 describes that the cross section at the
maximum perimeter (z = ) is an ellipse with a major
axis of r
' + 
and a minor axis of r
' +
, as depicted
in Figure 1.
Here, we assume that the amplitude growth toward
the wall, , is
and f can be defined as a degree of flattening. Equation
9 is supported by the observation described in the last
section that / is approximately constant during the
growth of the instability. Next, we calculate the surface
area of the nonaxisymmetric sinusoidal thread as a
function of the dimensionless wavenumber, the degree
of flattening, and the relative amplitude.
Utilizing volume conservation of the thread during
nonaxisymmetric growth and assuming that the cross
section at the neck is circular (r' -
= r' -
) (as
found in the simulations), we obtain the following
relations:
Then surface area of the thread per is as follows:25
Equation 12 cannot be integrated analytically. However,
its first and second derivative with respect to at 0
can be obtained analytically. The results are as follows:
Equation 14 is a necessary condition for the distortion
growth.
In Figure 12, we present the surface area of the
sinusoidal thread divided by that of initial thread vs
the amplitude at two different values of f. For the
axisymmetric case of f = 1 (Figure 12A), we
plot the
normalized surface area for three different values of X.
We see that only distortions with X < 1 cause a decrease
in surface area, as discussed previously, i.e., Xc(f=1) =
1. In Figure
12B, making f < 1 captures the nonaxisymmetric growth of the thread, and we see its effect
on surface area is different. We find that the critical
wavenumber is shifted. For f = 0.5, we now have Xc
(f=0.5) = 0.687. We present a plot of the critical
wavenumber (2
R0/c) vs f in Figure 13. The critical
wavenumber decreases with decreasing f. Thus, the
wavelength increases as a result of confinement in a
manner similar to the experiment. It is interesting that
the critical wavenumber becomes zero below a certain
value of f (
0.36). That may explain our experimental
result that the thread placed between a gap whose size
slightly larger than the thread diameter does not show
the distortion growth at all. The dominant wavenumber,
Xm, at which the amplitude grows fastest will also
decrease as the critical wavenumber decreases. Experimentally, we observe Xm rather than Xc.
Intuitively, we can understand the above result on the basis of eqs 4 and 5. The reduction of surface area upon the growth of a fluctuation occurs in an axisymmetric thread because the average radius (denoted by R' in Figure 1) of the thread must decrease in order to conserve volume. In the nonaxisymmetric case, this decrease in the average radius is less, and so the increase in surface area due to the contour length increase becomes dominant. Note that f cannot be predicted from the present analysis for a given degree of confinement and viscosity ratio. Our primary goal in the surface area calculations is to show that forcing the growth to be nonaxisymmetric can cause the wavelength to increase and to reduce the driving force for breakup. We note that other choices for the shape of the nonaxisymmetric thread are possible other than that described. For example, we also considered a shape wherein the inflection point was circular and both the bulges and necks were elliptical. This too produced a similar result with a slightly smaller value of f where the critical wavenumber becomes zero. While our choices of axisymmetric growth is consistent with experiment and simulation, a more rigorous approach would utilize a calculus of variations coupled with knowledge of the initial grow rates in the parallel and perpendicular directions.
