Strongly coupled interaction between a ridge of fluid and an inviscid airflow
KAUST Grant NumberKUK-C1-013-04
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Abstract© 2015 AIP Publishing LLC. 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.
CitationPaterson C, Wilson SK, Duffy BR (2015) Strongly coupled interaction between a ridge of fluid and an inviscid airflow. Phys Fluids 27: 072104. Available: http://dx.doi.org/10.1063/1.4926623.
SponsorsThe first author (C.P.) gratefully acknowledges the financial support of the University of Strathclyde via a Postgraduate Research Scholarship. All three authors gratefully acknowledge valuable discussions about the rain–wind-induced vibrations of cables and related problems with Dr. Ian J. Taylor and Mr. (now Dr.) Andrew C. Robertson (Department of Mechanical Engineering, University of Strathclyde). This work was begun while the second author (S.K.W.) was a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), Mathematical Institute, University of Oxford, United Kingdom, and completed while he was a Visiting Fellow at the Isaac Newton Institute for Mathematical Sciences in Cambridge, United Kingdom as part of the programme on “Mathematical Modelling and Analysis of Complex Fluids and Active Media in Evolving Domains” and a Leverhulme Trust Research Fellow supported by Award No. RF-2013-355 “Small Particles, Big Problems: Understanding the Complex Behaviour of Nanofluids.” This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
JournalPhysics of Fluids