In studying the stability due to axisymmetric disturbances of the developing flow of an incompressible fluid in the entrance region of a circular tube, a generalized version of the Orr-Sommerfeld equation was derived which takes account of the radial velocity component in the main flow. The new terms in the generalized Orr-Sommerfeld equation are inversely proportional to the Reynolds number. The resulting eigenvalue problem consisting of the disturbance equation and the boundary conditions was solved by a direct numerical integration scheme along with an iteration procedure. Neutral stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of the present results with those from the conventional Orr-Sommerfeld equation in which the effect of the main-flow radial velocity is neglected, shows that inclusion of the radial velocity contributes to a destabilization of the main flow.
The Effects of Main-Flow Radial Velocity on the Stability of Developing Laminar Pipe Flow
Shen, F. C. T., Chen, T. S., and Huang, L. M. (June 1, 1976). "The Effects of Main-Flow Radial Velocity on the Stability of Developing Laminar Pipe Flow." ASME. J. Appl. Mech. June 1976; 43(2): 209–212. https://doi.org/10.1115/1.3423810
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