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

Paterson, C. and Wilson, S. K. and Duffy, B. R. (2015) Strongly coupled interaction between a ridge of fluid and an inviscid airflow. Physics of Fluids, 27 (7). 072104. ISSN 1070-6631 (

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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 a rising 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 but 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.