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HYDRODYNAMIC STABILITY OF SOME SPATIALLY PERIODIC FLOWS,
MICHIGAN UNIV ANN ARBOR INST OF SCIENCE AND TECHNOLOGY
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The stability of the incompressible, boundary-free, parallel flow whose velocity profile is the cosine is considered. A two-dimensional cellular flow is also considered. The method of small perturbations is used to linearize the equations of motion about the basic flow. The boundary condition imposed is that the perturbation shall be bounded in space. The time dependence of the perturbation, assumed exponential, is chosen so that stability depends on the imaginary part of a parameter, c, which is considered to be the eigenvalue. Other parameters of interest are the Reynolds number, R, and, for the parallel flow, the wave number of the perturbation, alpha. Formulating the eigenvalue equation as the vanishing of an infinite determinant aids in calculating the neutral curve near the critical Reynolds number. The curve intersects the R axis at Rc. For large values of R it approaches alpha 1 asymptotically. The eigenvalue spectrum consists of an infinity of bands, separated by small gaps, lying along the imaginary axis of the c plane. Author
APPROVED FOR PUBLIC RELEASE