Develop Theories for Spikes at Richtmyer-Meshkov Unstable Mixing Interface

Unstable interfacial mixing occurs in many natural phenomena and in many industrial processes. Understanding unstable interfacial mixing is extremely important for both scientific development and industrial needs. Spikes and bubbles are the most important structures at an unstable material interface. Spikes are the portions of the heavy material penetrating into the light material. Developing theories for spikes is much more difficult than developing theories for bubbles which are the portions of the light material penetrating into the heavy material. This is due to the fact that spikes are more unstable than bubbles. As one changes the values of physical quantities, such as the density ratio of the two fluids across the material interface, the behavior of bubbles only changes quantitatively. However, the changes in the dynamic behavior of spikes can be dramatic. For example, for the shock-driven interfacial instability, the growth rates of bubbles always tend to zero for systems with any density ratio, including systems with an infinite density ratio. However, for spikes, the growth rates tend to zero only for systems with a finite density ratio, and tend to an initial-condition-dependent nonzero constant for systems with an infinite density ratio. For these reasons, most of the theoretical studies have been focusing on bubbles. So far, no accurate theory is available in the literature for spikes in systems with arbitrary density ratios of fluids across the material interface and for the entire time domain from early to late times. In this proposal, we plan to undertake the challenging task of developing theories for spikes at an unstable material interface. We will develop theories for spikes valid in a general setting: for systems with arbitrary density ratios, with arbitrary incident shock strengths, and over the entire time domain from early to late times.