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We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. There is a critical mass for blow-up and a rich set of dynamics including families… (More)

- A P Higginbotham, A Guillen, N Jones, T D Donnelly, A J Bernoff
- The Journal of the Acoustical Society of America
- 2011

A popular method for generating micron-sized aerosols is to submerge ultrasonic (ω~MHz) piezoelectric oscillators in a water bath. The submerged oscillator atomizes the fluid, creating droplets with radii proportional to the wavelength of the standing wave at the fluid surface. Classical theory for the Faraday instability predicts a parametric instability… (More)

- A. J. Bernoff, Rachel Kuske, Bernard J. Matkowsky, Vladimir A. Volpert
- SIAM Journal of Applied Mathematics
- 1995

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can… (More)

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