Nonlinear instability of liquid layers.

摘要

The nonlinear instability of two superposed viscous liquid layers in planar and axisymmetric configurations is investigated. In the planar configuration, the light layer fluid is bounded below by a wall and above by a heavy semiinfinite fluid. Gravity drives the instability. In the first axisymmetric configuration, the layer is confined between a cylindrical wall and a core of another fluid. In the second, a thread is suspended in an infinite fluid. Surface tension forces drive the instability in the axisymmetric configurations.; The nonlinear evolution of the fluid-fluid interface is computed for layers of arbitrary thickness when their dynamics are fully coupled to those of the second fluid. Under the assumption of creeping flow, the flow field is represented by an interfacial distribution of Green's functions. A Fredholm integral equation of the second kind for the strength of the distribution is derived and then solved using an iterative technique. The Green's functions produce flow fields which are periodic in the direction parallel to the wall and have zero velocity on the wall.; For small and moderate surface tension, planar layers evolve into a periodic array of viscous plumes which penetrate into the overlying fluid. The morphology of the plumes depends on the surface tension and the ratio of the fluid viscosities. As the viscosity of the layer increases, the plumes change from a well defined drop on top of a narrow stem to a compact column of rising fluid.; The capillary instability of cylindrical interfaces and interfaces in which the core thickness varies in the axial direction are investigated. In both the unbounded and wall bounded configurations, the core evolves into a periodic array of elongated fluid drops connected by thin, almost cylindrical fluid links. The characteristics of the drop-link structure depend on the core thickness, the ratio of the core radius to the wall radius, and the ratio of the fluid viscosities. The factors controlling the relative volumes of the drop and link are discussed. The nonlinear evolutions are compared with the predictions of a variational theory and with those of lubrication theory.