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Orr-Sommerfeld equation

The Orr-Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow.

The equation takes the form:

{1 \over i \alpha \, Re} \left({d^2 \over d z^2} - \alpha^2\right)^2 \varphi = (U - c)\left({d^2 \over d z^2} - \alpha^2\right) \varphi - U'' \varphi

with the boundary conditions (for no-slip boundaries at z = z1 and z = z2)

\alpha \varphi = {d \varphi \over d z} = 0 at z = z1 and z = z2,

where the base fluid velocity \mathbf{u} = (U(z), 0, 0) and the disturbance is given by \mathbf{u'} = \mathbf{\hat{u}}(z) \exp(i \alpha (x - c t)) (real part understood). Re is the Reynolds number.

The eigenvalue parameter of the problem is c and the eigenvector is \varphi.


For all but the simplest of velocity profiles U, numerical or asymptotic methods are required to calculate solutions. For Pouseille flow, it has been shown that the flow is unstable (i.e. one or more eigenvalues c has a positive imaginary part) for some α when Re > Rec = 5772.22 and the neutrally stable mode at Re = Rec having αc = 1.02056, c = 0.264002. [1]


  • P. G. Drazin, W. H. Reid (1981) 'Hydrodynamic Stability', Cambridge University Press
  1. ^ Orszag S. A. (1971) 'Accurate solution of the Orr-Sommerfeld stability equation' J. Fluid. Mech. 50, 689-703
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Orr-Sommerfeld_equation". A list of authors is available in Wikipedia.
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