Scaling properties of coating flows in rectangular channels

We present an experimental and numerical study of the aspect- ratio dependence of two-phase displacement flows in channels of rectangular cross-section. In square and near-square channels, the bulk features of the flow appear to be virtually independent of aspect ratio, $\alpha$, so that the capillary number, $Ca$, is the only governing parameter. Saffman-Taylor fingering, governed by the modified capillary number, $1/B \propto Ca\, \alpha^2$, is observed for $\alpha> 7$. For channels of intermediate aspect ratio, the interfacial flow depends solely on a generalised modified capillary number consistent with both limits, $\widehat{\mathrm{Ca}} = f(Ca, \alpha)$, above a threshold value $\widehat{\mathrm{Ca}}_t$.. This novel scaling has tremendous practical significance since it implies that the bulk features of any two-phase displacement flow can be inferred from those of the square channel for $\widehat{\mathrm{Ca}} > \widehat{\mathrm{Ca}}_t$.