Gravity-driven flows of viscous liquids over two-dimensional topographies

Using phase-stepped interferometry, we have measured full two-dimensional maps of the free-surface shape of a thin liquid film of water flowing over an inclined plate with topography. The measurement technique allows us to image automatically the shape of the free surface in a single field of view of about 2.4 by 1.8 mm, with a lateral resolution of 3.1 μm and a height resolution of 0.3 μm. By imaging neighbouring regions and combining them, complete two-dimensional free-surface profiles of gravity-driven liquid films with a thickness ranging between 80 and 120 μm are measured, over step, trench, rectangular and square topographies with depths of 10 and 20 μm, and lateral dimensions of the order of 1 to several mm. The experimental results for both one- and two-dimensional flows are found to be in good agreement with existing models, including a recent two-dimensional Green's function of the linearized problem by Hayes et al. This extends the applicability of simple models to cases with a high value of topography steepness and low-viscosity liquids as in our experiments. A corollary of the agreement with the linear two-dimensional model is that our experimental results behave linearly, a convenient property that allows the free-surface response to complex topographies to be worked out from knowledge of the response to an elementary topography like a square.