Interactions of solitary waves and compression/expansion waves in core-annular flows

The nonlinear hydrodynamics of an initial step leads to the formation of rarefaction waves and dispersive shock waves in dispersive media. Another hallmark of these media is the soliton, a localized traveling wave whose speed is amplitude dependent. Although compression/expansion waves and solitons have been well-studied individually, there has been no mathematical description of their interaction. In this talk, the interaction of solitons and shock/rarefaction waves for interfacial waves in viscous, miscible core-annular flows are modeled mathematically and explored experimentally. If the interior fluid is continuously injected, a deformable conduit forms whose interfacial dynamics are well-described by a scalar, dispersive nonlinear partial differential equation. The main focus is on interactions of solitons with dispersive shock waves and rarefaction waves. Theory predicts that a soliton can either be transmitted through or trapped by the extended hydrodynamic state. The notion of reciprocity is introduced whereby a soliton interacts with a shock wave in a reciprocal or dual fashion as with the rarefaction. Soliton reciprocity, trapping, and transmission are observed experimentally and are found to agree with the modulation theory and numerical simulations.