The instability of a moving interface in a narrow tapering channel of finite length

We analyse the displacement of one fluid by a second immiscible fluid through a narrow channel of finite length which connects two reservoirs. We assume that the channel width slowly decreases in the direction of flow, and that the fluids have different viscosity and density. We examine the stability of the interface and find that there are Saffman–Taylor and Rayleigh–Taylor type modes, which may dominate in the narrow and wide regions of the channel, respectively. The gradient of the pressure jump across the interface associated with the surface tension acts to stabilise the interface, and for intermediate channel widths, this effect may dominate the destabilisation associated with both the Rayleigh–Taylor and Saffman–Taylor instabilities, provided the rate of change of the channel width with distance along the channel is sufficient. We also note that the effect of the converging channel leads to instability of long-wavelength modes owing to the quasi-static acceleration of the flow through the cell: we consider cases in which this effect only occurs at much lower wavenumbers than the most unstable Saffman–Taylor and Rayleigh–Taylor modes. We show that there is a maximum wavenumber for instability, which varies with position in the channel. By integrating the growth rate of each wavenumber in time as the interface moves across the channel, we predict the mode which grows to the greatest amplitude as the interface traverses the channel.