Abstract
We analyse the stability of the interface between two immiscible fluids both flowing in the horizontal direction in a thin cell with vertically varying gap width. The dispersion relation for the growth rate of each mode is derived. The stability is approximately determined by the sign of a simple expression, which incorporates the density difference between the fluids and the effect of surface tension in the along- and cross-cell directions. The latter arises from the varying channel width: if the non-wetting fluid is in the thinner part of the channel, the interface is unstable as it will preferentially migrate into the thicker part. The density difference may suppress or complement this effect. The system is always stable to sufficiently large wavenumbers owing to the along-flow component of surface tension.
Highlights
The parallel co-flow of two fluids occurs in many industrial, biological and environmental processes
We investigate how the stability of the interface between the two fluids in a thin cell with vertically varying gap width is controlled by cross-layer buoyancy and capillary effects
The combination of surface tension and the cross-cell variation in thickness introduces a newstabilising process for small wavenumbers in the case that thewetting fluid is in the thinner part of the channel
Summary
The parallel co-flow of two fluids occurs in many industrial, biological and environmental processes. The combination of surface tension and the cross-cell variation in thickness introduces a new (de)stabilising process for small wavenumbers in the case that the (non-)wetting fluid is in the thinner part of the channel This effect may complement or suppress the effect of a density difference between the two fluids on the stability of the interface. The impact of variations in the surface tension associated with variations in the channel width have been explored in detail for the related problem in which an input fluid displaces an ambient fluid in a cell whose width varies in the direction of flow (Homsy 1987; Al-Housseiny, Tsai & Stone 2012; Dias & Miranda 2013; Grenfell-Shaw & Woods 2017) These studies have identified that the effect of cross-cell curvature can complement or suppress the classical Saffman–Taylor instability (Saffman & Taylor 1958).
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