Stability of the laminar flow in a rectangular duct

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Abstract
The stability of the laminar flow in a rectangular duct of an arbitrary aspect ratio is investigated numerically by expanding the flow fields of both the main flow and the disturbance into series of Legendre polynomials and solving the eigenvalue problem of the resulting matrix equation. The stability of the flow is found to depend upon the aspect ratio of the duct and the mode of the disturbance. The flow is unstable to two of the four possible modes of different parity and stable to the other two. With respect to the most unstable mode, the flow is stable or unstable according as the aspect ratio is below or above a critical value of 3.2 respectively, and the critical Reynolds number decreases monotonically with increasing aspect ratio towards the known value of 5772 for plane Poiseuille flow. The flow field of the disturbance shows the existence of strong vortex layers along the critical layer at which the velocity equals the phase velocity of the disturbance.

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