Unstable Shock-Driven Interfacial Mixing in Cylindrical Tubes

The study of unstable interfacial mixing has a very long history. The research on unstable interfacial mixing driven by shock waves started as early as 1960. Since then, due to the importance of this instability in natural processes and industrial applications, the research on this instability has been very active among mathematicians, physicists and engineers. The material interface driven by a shock is unstable. Small disturbances at the material interface grow in size to form fingers. To understand and to predict how these fingers grow in time are the central issues for this unstable interfacial mixing problem. The dynamics of finger growth involves both compressibility and nonlinearity. This makes theoretical studies extremely difficult. Therefore, most theoretical studies carried out in the literature are focused on ideal settings with periodic boundary conditions. However, it is highly desirable to study the dynamics of finger growth in a setting of a shock propagating through a cylindrical tube. Unfortunately, the presence of tube walls complicates the theoretical analysis dramatically. For this reason, no theories are available in the literature for shock-driven interfacial instability of compressible fluids in cylindrical tubes. Most of the theoretical results obtained from studying unstable interfacial mixing in periodic settings cannot be extended to the unstable interfacial mixing in cylindrical tubes. For example, the wave number of the perturbation at the material interface is irrelevant for systems with periodic boundary conditions since the results of different wave numbers can be obtained by a simple linear scaling of the wave number. The situation is very different for unstable interfacial mixing in cylindrical tubes, since the tube walls make the system non-periodic and therefore this scaling approach does not work. In fact, fingers in cylindrical tubes can have many new patterns that do not exist in periodic settings. The purpose of this proposal is to carry out the challenging task of developing quantitative theories for fingers at the shock-driven unstable material interface between two compressible fluids of different densities in cylindrical tubes.