In this paper, we investigate the dynamics of Hele-Shaw flow in a multiply connected domain. We consider multiple fluid domains enclosed by a moving outer interface in a radial Hele-Shaw cell, with several inner interfaces separating these fluid regions. To compute the long-time evolution of interface morphologies, we develop a spectral-accurate boundary integral method and a nonstiff time updating scheme, together with a time adaptive scheme to speed up the computation. Our numerical results reveal that interface instabilities are strongly influenced by the initial configuration of fluid domains and physical parameters such as fluid viscosities. Notably, at early times, the fingering instability observed on the outer interface—triggered by the presence of inner interfaces—differs fundamentally from the classical perturbation-initiated growth in single-interface scenarios. Depending on their configuration, the inner interfaces can either promote or suppress the development of fingering on the outer boundary, offering a potential strategy for morphological control in practical applications. Motivated by the self-similar theory and the shape-control concept introduced in Li et al. [Phys. Rev. Lett. 102, 174501 (2009)], where self-similarity is only realized at large interface sizes under time-dependent flux, we demonstrate that one can preselect a self-similar limiting morphology and promote its formation at significantly smaller scales. This is achieved through a carefully designed multi-interface setup that promotes the desired symmetry while suppressing other unfavorable instabilities.

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• Fingering instability
• Hele-shaw flows
• Instability control
• Saffman-Taylor instability