The Uniqueness of Certain Flows in a Channel With Arbitrary Cross Section
The Uniqueness of Certain Flows in a Channel With Arbitrary Cross Section
Arthur S Peters
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Besides this. It Is n ° n not difficult to verify that the eigenvalues A are either real or pure Imaginary numbers. It Is known that there can be only a finite number of eigenvalues In any bounded domain of the complex A-plane. It Is also well known that the above second order system defines a complete set of elgenfunctlons {^„(y)} such that a twice dlfferentlable function G(y) can be expanded In the Fourier series 00 G(y) = XZ "n^n^y) • n=0 13 It follows from the above remarks that for any fix...ed value of X, ')(l(x, y) has the unique expansion X (x, y) := ^^ = ZZ ^n^^)^n^y) > "1 1 Y 1 • o^*^ "^ n=0 Here, the coefficient ct^fx) is a^{x) =J PQ(y)u^{y)X(x, y)^n^y^^y -1 and as a function of x it must satisfy dx _-, V- = j po^o^xx^y = -/ [|rPo-oXy-Poy^l^n^y -1 = - ^n / Po^o-X V^ • -1 Hence the coefficient a^(x) must satisfy ^ — + -K'^a (x = , 2 n n^ ^ dx from which a^(x) = a^ cos A^x + b^ sin A^(x), \ ^ ^ (3. 15) Ik This shows that if 'X (x, y) Is to be bounded everywhere then If a.
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