We report analytical results for the development of interfacial instabilities in rotating Hele-Shaw cells. We execute a mode-coupling approach to the problem and examine the morphological features of the fluid-fluid interface at the onset of nonlinear effects. The impact of normal stresses is accounted for through a modified pressure jump boundary condition. A differential equation describing the early nonlinear evolution of the interface is derived, being conveniently written in terms of three relevant dimensionless parameters: viscosity contrast $A$, surface tension $B$, and gap spacing $b$. We focus our study on the influence of these parameters on finger competition dynamics. It is deduced that the link between finger competition and $A$, $B$, and $b$ can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations. Our results show good agreement with existing experimental and numerical investigations of the problem both in low and high $A<0$ limits. In particular, it is found that the condition of vanishing $A$ suppresses the dynamic competition between fingers, regardless of the value of $B$ and $b$. Moreover, our study enables one to extract analytical information about the problem by exploring the whole range of allowed values for $A$, $B$, and $b$. Specifically, it is verified that pattern morphology is significantly modified when the viscosity contrast $-1⩽A⩽1$ varies: increasingly larger values of $A>0$ $(A<0)$ lead to enhanced competition of outward (inward) fingers. Within this context the role of $B$ and $b$ in determining different finger competition behaviors is also discussed.

• Received 7 April 2004

DOI:https://doi.org/10.1103/PhysRevE.70.066308