Let us consider the three-dimensional problem of the steady flow of a heavy ideal fluid past a surface-piercing obstacle in a rectangular channel of constant depth. The flow is parallel at infinity upstream, with constant velocity c. We discuss an approximate linear problem obtained in the limit of a "flat obstacle". This is a boundary value problem for the Laplace equation in a three-dimensional unbounded domain, with a second order condition on part of the boundary, the Neumann-Kelvin condition. By a Fourier expansion of the potential function, we reduce the three-dimensional problem to a sequence of plane problems for the Fourier coefficients; for every value of the velocity c, these problems can be described in terms of a two parameter elliptic problem in a strip. We discuss the two dimensional problem by a special variational approach, relying on some a priori properties of finite energy solutions; as a result, we prove unique solvability for 〖c≠c〗_(m,k) where c_(m,k) is a known sequence of values depending on the dimensions of the channel and on the limit length of the obstacle. Accordingly, we can prove the existence of a solution of the three-dimensional problem; the related flow has in general a non trivial wave pattern at infinity downstream. We also investigate the regularity of the solution in a neighborhood of the obstacle. The meaning of the singular values c_(m,k) is discussed from the point of view of the nonlinear theory.
|Data di pubblicazione:||2003|
|Titolo:||Solvability of a plane elliptic problem for the flow in a channel with a surface-piercing obstacle|
|Autore/i:||S. GATTI; PIEROTTI D|
|Codice identificativo ISI:||WOS:000185604700007|
|Codice identificativo Scopus:||2-s2.0-0041829450|
|Citazione:||Solvability of a plane elliptic problem for the flow in a channel with a surface-piercing obstacle / S. GATTI; PIEROTTI D. - In: ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN. - ISSN 0232-2064. - STAMPA. - 22(2003), pp. 357-381.|
|Tipologia||Articolo su rivista|
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