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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.

### Additional recommended knowledge

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 $\varphi$.

### Solutions

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]

## References

• 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