Thesis: Modeling Faraday Excitation of a Viscous Fluid(in pdf)
Advisor: Professor Andrew J. Bernoff
Ciriculum Vitae: Forrest_CV.html
Modeling Droplet Ejection with Faraday Excitation
of a fluid can be formed by the tips of waves on the surface of the fluid
breaking off. This droplet ejection occurs when the surface waves
reach a certain critical amplitude. Hence, droplets can be produced
through any mechanism that causes growing surface waves. One such
mechanism is Faraday Excitation, where the bottom of the container of fluid
is subjected to a forcing oscillation/vibration. I produced a short
animation (in mpg and avi
formats) illustrating a growing droplet ejecting wave caused by the Faraday
Previous research has shown that the size of an ejected droplet is determined by the wavelength of the surface that ejects it.[1-3] There is in fact a universal constant factor that relates the diameter of a droplet to the wavelength of the surface wave that ejected that droplet. Currently, Professor Tom Donnellyís particle sizing research group (HMC Physics) is generating droplets through the Faraday mechanism, and precisely measuring the size of these droplets through MIE scattering techniques. The goal of my research has been to provide the Donnelly research group with predictions for the wavelength of the surface waves generated by Faraday Excitation on any fluid with any set of values for the oscillation amplitude and oscillation frequency of the forcing vibration.
I used linear stability analysis to the Faraday Excitation model as
my main method of analysis of this problem. My analysis is similar
to that used by Kumar and Tuckerman in 1994.
In their work, Kumar and Tuckerman generated neutral stability curves for
a system of two fully viscous fluids subjected to Faraday Excitation.
These stability curves predict the regions of parameter space for which
unstable surface waves, surface
If in the experimental setting, the amplitude of the oscillator is slowly increased until droplets are observed, this exclusively selects the minimum amplitude unstable wave mode. In other words, the waves that are producing droplet ejection must correspond to the point on the neutral stability curve that is a minimum in amplitude.
I generated the neutral stability curves for a generalized fluid over a 25 decade range of forcing frequency. Figure 1 is the plot of the neutral stability curves at low frequency, while Figure 2 is a plot of the neutral stability curves at high frequency. I also produced a short animation (in mpg and avi formats) that shows how these neutral stability curves change qualitatively as the non-dimensional forcing frequency is increased from 10^-10 to 10^5.
Figure 3 is a log log plot of the surface wave number corresponding to the point of minimum oscillation amplitude vs. angular forcing frequency. The relationship depicted in this plot is the final prediction that was provided to the experimental research group. Note that there are two distinct scaling regimes, as two different slopes on exist in the figure. For low forcing frequency, surface tension is the dominant effect, while for high forcing frequency surface tension is dominated by viscosity.
Animation 2: Qualitative depiction of how the neutral
stability curves for the Faraday Excitation system change as the non-dimensional
forcing frequency increases logarithmically from 10^-10 to 10^5 (in mpg
W = w ( n^3 /(s/r)^2),
A = a ( (s/r)/ n^2 ), and
K = k ( n^2 / (s/r) ),
s is the fluid's surface tension,
r is the fluid's density,
n is the fluid's kinematic viscosity,
w is the dimensional angular forcing frequency,
a is the dimensional forcing amplitude, and
k is the dimensional surface wave number.
For higher quality versions of any of these figures, click on the figure.