Strongly coupled interaction between a ridge of fluid and an inviscid airflow

Abstract

The behaviour of a steady thin sessile or pendent ridge of fluid on an inclined planar substrate which is strongly coupled to the external pressure gradient arising from an inviscid airflow parallel to the substrate far from the ridge is described. When the substrate is nearly horizontal, a very wide ridge can be supported against gravity by capillary and/or external pressure forces; otherwise, only a narrower (but still wide) ridge can be supported. Classical thin-aerofoil theory is adapted to obtain the governing singular integro-differential equation for the profile of the ridge in each case. Attention is focused mainly on the case of a very wide sessile ridge. The effect of strengthening the airflow is to push a pinned ridge down near to its edges and to pull it up near to its middle. At a critical airflow strength, the upslope contact angle reaches the receding contact angle at which the upslope contact line de-pins, and continuing to increase the airflow strength beyond this critical value results in the de-pinned ridge becoming narrower, thicker, and closer to being symmetric in the limit of a strong airflow. The effect of tilting the substrate is to skew a pinned ridge in the downslope direction. Depending on the values of the advancing and receding contact angles, the ridge may first de-pin at either the upslope or the downslope contact line but, in general, eventually both contact lines de-pin. The special cases in which only one of the contact lines de-pins are also considered. It is also shown that the behaviour of a very wide pendent ridge is qualitatively similar to that of a very wide sessile ridge, while the important qualitative difference between the behaviour of a very wide ridge and a narrower ridge is that, in general, for the latter one or both of the contact lines may never de-pin.