Hydrodynamic stability of multi-layer Hele-Shaw flows
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Upper bound results on the growth rate of unstable multi-layer Hele-Shaw flows are obtained in this paper. The cases treated are constant viscosity layers and variable viscosity layers. As an application of the bound, we obtain some sufficient conditions for suppressing instability of two-layer flows by introducing an arbitrary number of constant viscosity fluid layers in between. This sufficient condition has very practical relevance because it narrows the choice of internal-layer fluids on the basis of the surface tensions of all interfaces and viscosities of fluids in various layers. The importance of this condition which has been hitherto unknown is also discussed. Other consequences of these upper bounds and sufficient conditions are discussed. The case of internal fluid layers having stable and unstable viscous profiles is also treated for three-layer and four-layer flows. The connection of these variable viscosity results to viscous fingering in complex fluids is also established. Implications of these stability results for these various multi-layer flows are discussed and compared from a practical standpoint. © 2008 IOP Publishing Ltd.
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