Sustained inertial-capillary oscillations and jet formation in displacement flow in a tube

Abstract

We study inertial effects in the displacement of a fluid in a capillary by a more viscous fluid, using a numerical method. A level-set approach is employed to track the meniscus, and a Navier slip boundary condition is imposed in order to alleviate a stress singularity at the moving contact line. Various flow regimes are identified with a Reynolds number and a capillary number as the main parameters. At relatively low Reynolds number, the meniscus forms a steady shape, and the interfacial curvature at the tube centre can change from being concave to convex upon increasing the Reynolds number if the displacing fluid is wetting to the tube surface. For wetting displacing fluids, beyond a critical Reynolds number, a quasi-steady solution is no longer found: instead, the interface undergoes non-dampened periodic oscillations and, at even larger values of the Reynolds number, quasi-periodically, and the interface evolves from simple wavy shapes to complex shapes with multiple wavy units. This oscillating state is observed for sufficiently small contact angle values defined from the displacing fluid (<80°). Beyond a second critical Reynolds number, the displacing fluid forms a jet and the meniscus advances with a nearly constant speed which decreases with Re. This is also observed at large contact angle values. In a developing jet, however, the interface shape remains partially quasi-steady, near the contact line region and the tube centre. The flow behaviour is shown to be robust over a range of other governing parameters, including the capillary number and the slip length. The potential implications of the work on network models of two-phase flow through porous media are discussed.