Solving the Orr-Sommerfeld equation with integral equation method in terms of the green function
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In this paper, an alternative method has been presented to solve the Orr-Sommerfeld equation in the linear theory of stability. To begin with, we define a Green function which is expressed in terms of matrix. Subsequently, its reciprocity has been shown. Finally, a linear integral equation equivalent to the original Orr-Sommerfeld equation is derived. The method is applied to the cases with two solid walls and any velocity distribution of the main flow at any Reynolds number.
KeywordsMathematical Modeling Reynolds Number Integral Equation Velocity Distribution Industrial Mathematic
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- 1.Lin, C. C.,The Theory of Hydrodynamic Stability, Cam. Uni. Press, (1955).Google Scholar
- 2.Lakin, W. D., Ng, B. S. and Reid, W. H., Approximation to the eigenvalue relation for the Orr-Sommerfeld problem, Phi. Trans. Roy. Soc. Lon. 289, No. 1360, (1978).Google Scholar
- 3.Li, J. C., Hydrodynamic stability and asymptotic method, Mechanics and Practice, 3, (1981), No. 4.Google Scholar
- 5.Dolph, C. L. and Lewis, D. C.,On the application of finite systems of ordinary differential equations to perturbations of plane Poiseuille flow, Quar. Appl. Math. 16, (1958), 97.Google Scholar
- 6.Gallagher, A. P. and Mercer, A. M.,On the behaviour of small disturbances in plane Couette flow, Jour. Fluid Mech. 13, (1962), 91.Google Scholar
- 7.Grosch, C. E. and Salwen, H.The Stability of steady and time-dependent plane Poiseuille flow, Jour. Fluid Mech. 34, (1968), 177.Google Scholar
- 8.Orszag, S. A.,Accurate solution of the Orr-Sommerfeld stability equation, Jour. Fluid Mech. 50, (1971), 689.Google Scholar