IX-8
Absolute Instability in a Single Jet
Anuj Chauhan, University of California at Berkeley
Charles Maldarelli, David Rumschitzki; Levich Institute, City College of New York
Demetrios Papageorgiou, New Jersey Institute of Technology
A single jet is a column of liquid issuing out of an orifice. It is unstable due to capillarity. This instability results in the formation of drops whose size can be predicted very accurately by a linear stability analysis. The exact configuration at breakup, the satellite formation, and the finite time singularity, can only be predicted by a non-linear theory.
The character of the breakup of the jet is primarily controlled by the Weber number, We = where
r and s are the fluid's density and surface tension respectively and R and V are the jet's radius and the uniform axial velocity, respectively. At very high Weber numbers, the disturbances introduced at the orifice tip, either specifically imposed or picked up due to noise, convect with the jet velocity with no dispersion and grow axially with the growth rates of the temporal stability analysis. At lower Weber numbers there is dispersion accompanied by growth, i. e., the temporal results no longer apply and the growth and propagation can be predicted by the spatial stability analysis. On further lowering the jet velocity, the jet enters the regime of absolute instability. In this regime, the instability also grows exponentially in time for all z.If a disturbance in the form of a wave packet is introduced into the jet, the front and the rear end of the wave packet move with a velocity of V±V
c respectively where V is the jet velocity and Vc is the critical jet velocity below which it becomes absolutely unstable. Thus, if |V|>|Vc|, the front and the back end move in the same direction but if |V|<|Vc|, the front and the back end move in opposite directions. Thus in the absolutely unstable regime a disturbance introduced into the jet propagates backwards too. However, the absolutely unstable waves have a positive phase velocity only. this and the time growth are the chief distinguishing features of absolutely instability, which could help observe it in experiments.For experiments at the high Weber numbers, we use a piezoelectric crystal to introduce a disturbance of a specific frequency at the needle tip from which the jet issues. We observe the jet at fixed spatial locations and record the spatial growth of the disturbance with a Kodak fast video camera (6000 frames/second). The pictures are then used to obtain digitized images of the jet interface from which we calculate the wave length and the spatial growth of the disturbance. At high Weber numbers, we see convective growth of the disturbances causing breakup at a fixed spatial location. The measured values of the growth rates and the wave numbers in this regime match the experimental predictions.
At Weber numbers below the critical value, we observe upstream propagation of disturbances and forward traveling waves whose wavelength matches the predicted wavelength of the absolutely unstable waves. Interestingly, this wavelength is significantly shorter than (by about half) typical waves induced by the usual capillary-driven break-up. These short waves do not seem to grown towards breakup over the time scales of our observation. This fact could indicate that their origin is not from an absolute instability since such waves grow linearly in time and should cause jet breakup. However, we show that other explanations for their origin such as the propagation of capillary waves with negative phase velocity do not match expected wavelengths. Thus we think it more likely that these upstream propagating waves derive from the absolute instability, and nonlinear saturation accounts for the fact that their expected linear growth does not break up the jet.