The problem of the stability of plane Poiseuille flow to small disturbances leads to a characteristic value problem for the Orr-Sommerfeld equation with given boundary conditions. It happens that negative values of the imaginary parts of the characteristic numbers, which indicate instability, are small, at any rate over the region here investigated, and considerable accuracy is required to establish them, while the Reynolds numbers for which they occur are large.

In this paper the fourth-order differential equation is replaced by a difference system of the same order with a truncation error involving the eighth derivative, so that the error is sufficiently small with a reasonably large interval. The resulting system of linear algebraic equations is solved by direct Gaussian elimination, which avoids the difficulties due to rapid exponential growth of error for high Reynolds number which beset the standard integration procedure.

The characteristic value is obtained for a range of Reynolds numbers and wavelengths of the disturbance, and the critical Reynolds number found to be 5780 for wavelength 3.062 times the width of the channel. A detailed discussion of the accuracy of the work is given for the (unstable) case of wavelength $\pi$ and Reynolds number 10 000, and a table of the profile of the disturbance is given for this case.

• Received 15 April 1953

DOI:https://doi.org/10.1103/PhysRev.91.780