Finite Reynolds number effects in the Bretherton problem

Abstract

In this paper we investigate the effect of fluid inertia in the classical Bretherton problem in which a semi-infinite air finger displaces viscous fluid in a two-dimensional channel. The governing free-surface Navier–Stokes equations are discretized by a finite element method and the system’s behavior is studied for capillary and Reynolds numbers in the ranges 0.05<Ca<5 and 0<Re<280, respectively. It is shown that at a finite Reynolds number, a sequence of closed vortices develops in the recirculating flow region ahead of the bubble tip. The largest of these vortices is located near the bubble tip where it significantly affects the pressure distribution and leads to a noticeable increase in the overall pressure rise across the bubble tip. The underlying physical mechanisms are discussed in detail.