Asymptotic Approximation of the Eigenvalues and the Eigenfunctions for the Orr-Sommerfeld Equation on Infinite Intervals  

作　　者：Victor Nijimbere 

机构地区：[1]School of Mathematics and Statistics, Carleton University, Ottawa, Canada

出　　处：《Advances in Pure Mathematics》2019年第12期967-989,共23页理论数学进展（英文）

摘　　要：Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.

关 键 词：EIGENVALUES EIGENFUNCTIONS Infinite Intervals WKB Methods Long-Wave LIMIT APPROXIMATION Short-Wave LIMIT APPROXIMATION Generalized HYPERGEOMETRIC Functions 

分 类 号：O17[理学—数学]