The development of Poiseuille flow

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Abstract
The problem considered is that of two-dimensional viscous flow in a straight channel. The steady Navier-Stokes equations are linearized on the assumption of small disturbance from the fully developed flow, leading to an eigenvalue equation resembling the Orr-Sommerfeld equation. This is solved in the limiting cases of small and large Reynolds numberR, and an approximate method is proposed for moderateR. The main results are (i) the dominant mode of the disturbance velocity (i.e. that which persists longest) is antisymmetrical; (ii) for largeRthere are two sequences of eigenvalues. Both sequences are asymptotically real asR→ ∞. The members of the first sequence areO(1) asR→ ∞ and are complex for all finiteR. The members of the second sequence areO(R−1) and the imaginary part isO(R−N) for allN. It is the eigenvalues of the second sequence which will dominate the flow at largeR.

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