Abstract
In this paper, we investigate steady two-dimensional free-surface flows of an inviscid and incompressible fluid emerging from a nozzle, falling under gravity and impinging onto a horizontal wall. More precisely, for any given atmosphere pressure \(p_{atm}\) and any appropriate incoming total flux Q, we establish the existence of two-dimensional incompressible impinging jet with gravity. The two free surfaces initiate smoothly at the endpoints of the nozzle and become to be horizontal in downstream. By transforming the free boundary problem into a minimum problem, we establish the properties of the flow region and the free boundaries. Moreover, the asymptotic behavior of the impinging jet in upstream and downstream is also obtained.
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Acknowledgements
Cheng and Xin are supported in part by the Zheng Ge Ru Foundation and Hong Kong RGC Grants: 14300819, 14300917, 14302819. Cheng is also supported partially by National Key R &D Program of China (Grant No. 2022YFA1007700) and by National Natural Science Foundation of China (Grant No. 12001387). Xin is also supported partially by the Key Project of National Natural Science Foundation of China (Grant No. 12131010) and by Guangdong Providence Basic and Applied Basic Research Foundation (Grant No. 2020B1515310002). Du is supported by National Nature Science Foundation of China (Grant No. 11971331, 12125102), and Sichuan Youth Science and Technology Foundation (Grant No. 2021JDTD0024).
Funding
Part of the work was done when the first author was visiting The Institute of Mathematical Sciences, The Chinese University of Hong Kong. He thanks the institute for its hospitality and support
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Communicated by Neil S Trudinger.
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