New results on the motion of interfaces of multi-layer radial Hele-Shaw flows

We study the dynamical system arising from the linear stability analysis of multi-layer radial Hele-Shaw flows with time-dependent injection rates. An analysis of the system provides conditions under which the disturbances on all interfaces either monotonically decrease or monotonically increase. Additionally, we show that in any multi-layer radial Hele-Shaw flow, if all of the interfaces are circular initially except for one perturbed interface then there exists a time-dependent injection rate such that the circular interfaces remain circular as they propagate and the disturbance on the perturbed interface decays. The motion of the interfaces is also investigated numerically by integrating the dynamical system for the case of constant and time-independent injection rates. This reveals some very interesting dynamics of interfaces in multi-layer flows. It is found that: (i) A disturbance of one interface can be transferred to the other interface(s); (ii) The disturbances on the interfaces can develop either in phase or out of phase from any arbitrary initial disturbance; and (iii) The dynamics of the flow can change dramatically with the addition of more interfaces. With time-dependent injection rate, there are more possibilities. In particular it is shown that an appropriate injection rate can result in a stable configuration, which consists of all perfectly circular interfaces except for one that is perturbed with a monochromatic wave of infinitesimal amplitude. The prescribed injection rate results in the perturbation decaying even if the configuration (i.e., the setup consisting of all but one circular interface and one perturbed circular interface) itself is unstable to disturbances on any one or more of the circular interfaces. However, it may not be without a challenge to realize such a flow in practice.